Patterns and Interfaces in Dissipative Dynamics (Springer Series in Synergetics)

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Springer Complexity Springer Complexity is a publication program, cutting across all traditional disciplines of sciences as well as engineering, economics, medicine, psychology and computer sciences, which is aimed at researchers, students and practitioners working in the ﬁeld of complex systems. Complex Systems are systems that comprise many interacting parts with the ability to generate a new quality of macroscopic collective behavior through self-organization, e.g., the spontaneous formation of temporal, spatial or functional structures. This recognition, that the collective behavior of the whole system cannot be simply inferred from the understanding of the behavior of the individual components, has led to various new concepts and sophisticated tools of complexity. The main concepts and tools – with sometimes overlapping contents and methodologies – are the theories of selforganization, complex systems, synergetics, dynamical systems, turbulence, catastrophes, instabilities, nonlinearity, stochastic processes, chaos, neural networks, cellular automata, adaptive systems, and genetic algorithms. The topics treated within Springer Complexity are as diverse as lasers or ﬂuids in physics, machine cutting phenomena of workpieces or electric circuits with feedback in engineering, growth of crystals or pattern formation in chemistry, morphogenesis in biology, brain function in neurology, behavior of stock exchange rates in economics, or the formation of public opinion in sociology. All these seemingly quite different kinds of structure formation have a number of important features and underlying structures in common. These deep structural similarities can be exploited to transfer analytical methods and understanding from one ﬁeld to another. The Springer Complexity program therefore seeks to foster cross-fertilization between the disciplines and a dialogue between theoreticians and experimentalists for a deeper understanding of the general structure and behavior of complex systems. The program consists of individual books, books series such as “Springer Series in Synergetics", “Institute of Nonlinear Science", “Physics of Neural Networks", and “Understanding Complex Systems", as well as various journals.

Springer Series in Synergetics Series Editor

Hermann Haken Institut für Theoretische Physik und Synergetik der Universität Stuttgart 70550 Stuttgart, Germany and Center for Complex Systems Florida Atlantic University Boca Raton, FL 33431, USA

Members of the Editorial Board Åke Andersson, Stockholm, Sweden Gerhard Ertl, Berlin, Germany Bernold Fiedler, Berlin, Germany Yoshiki Kuramoto, Sapporo, Japan Jürgen Kurths, Potsdam, Germany Luigi Lugiato, Milan, Italy Jürgen Parisi, Oldenburg, Germany Peter Schuster, Wien, Austria Frank Schweitzer, Zürich, Switzerland Didier Sornette, Nice, France and Zürich, Switzerland Manuel G. Velarde, Madrid, Spain SSSyn – An Interdisciplinary Series on Complex Systems The success of the Springer Series in Synergetics has been made possible by the contributions of outstanding authors who presented their quite often pioneering results to the science community well beyond the borders of a special discipline. Indeed, interdisciplinarity is one of the main features of this series. But interdisciplinarity is not enough: The main goal is the search for common features of self-organizing systems in a great variety of seemingly quite different systems, or, still more precisely speaking, the search for general principles underlying the spontaneous formation of spatial, temporal or functional structures. The topics treated may be as diverse as lasers and ﬂuids in physics, pattern formation in chemistry, morphogenesis in biology, brain functions in neurology or self-organization in a city. As is witnessed by several volumes, great attention is being paid to the pivotal interplay between deterministic and stochastic processes, as well as to the dialogue between theoreticians and experimentalists. All this has contributed to a remarkable cross-fertilization between disciplines and to a deeper understanding of complex systems. The timeliness and potential of such an approach are also mirrored – among other indicators – by numerous interdisciplinary workshops and conferences all over the world.

L.M. Pismen

Patterns and Interfaces in Dissipative Dynamics

With a Foreword by Y. Pomeau

With 163 Figures and 1 Table

ABC

Professor L.M. Pismen Department of Chemical Engineering Technion - Israel Institute of Technology Technion City, Haifa 32000, Israel E-mail: [email protected]

Library of Congress Control Number: 2005939040 ISSN 0172-7389 ISBN-10 3-540-30430-4 Springer Berlin Heidelberg New York ISBN-13 978-3-540-30430-2 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the author and techbooks using a Springer LATEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper

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Foreword

Nature is full of marvels, be they in the inanimate world or in the realm of life. Quite often their beauty and attractive power come from the regularity of some sort of “pattern,” or – more subtly – from some obvious or hidden irregularity (as the French poet Verlaine said, “always favour the uneven”). Obvious natural patterns are found in the stripes of sea-shells or on the fur of mammals, such as our domestic cats. Stones such as malachites or agathes also show ﬁnely crafted layers of matching or contrasting colors. A less obvious case of “pattern” is seen in the opals, whose opalescence reﬂects wonderfully regular arrays of spheres of silica. Civilizations of the Far East highly praised these miraculous examples of beauty in minerals. The way things and living beings are made or have been made began to attract the interest of scientists of the Classical Age, particularly after Newton in the query 31 at the end of his Opticks claimed that the microscopic world, made of atoms, follows the same kind of laws as the Universe at large. The ﬁrst instance of a “regular pattern” produced artiﬁcially was by the spontaneous modulation of the surface of a liquid vibrated vertically. It was ﬁrst believed that this Faraday instability was a kind of linear resonance of the capillary waves at the frequency of the excitation. It was Rayleigh who ﬁrst explained the striking fact that actually the dominant frequency of the surface oscillations is half that of the exciting acceleration, a result that is not obvious. Rayleigh’s theory gave rise to a whole ﬁeld of research, the study of ordinary diﬀerential equations with time periodic coeﬃcients. Once transformed into maps of the circle this class of problems has been a major source of mathematical progress over the years. The next example of an artiﬁcial pattern in experiments was the famous (and ﬁrst very poorly understood) B´enard cells. Henri B´enard, one of the ﬁrst great experimentalists in ﬂuid mechanics, showed that a thin layer of whale oil heated from below set itself spontaneously in motion in a remarkably regular array of hexagonal cells. B´enard himself did explain this phenomenon in a rather confused way and another 50 years passed before Scriven and Sterling showed that the instability is due to the Marangoni eﬀect, that is the temperature dependence of the surface tension. It took another 10 years to see the explanation of the hexagonal pattern by Palm, who proved that hexagons are the generic structure to expect beyond a bifurcation yielding wavy patterns with an arbitrary orientation in a plane. The Rayleigh–B´enard instability due to buoyancy (instead of the

VI

Foreword

Marangoni eﬀect) of a ﬂuid layer heated from below does show rolls, because of a particular, nongeneric symmetry. Over the years an almost inﬁnite number of “pattern-forming” bifurcations in physical systems have been discovered, as well as many kinds of instabilities. The famous Brusselator explained the oscillating chemical reactions in the well-known Belousov–Zhabotinsky system. The Turing instability shows up in driven systems with widely diﬀerent diﬀusion coeﬃcients. Quite often the scientists responsible for the discoveries and/or the theory of patternforming systems saw in a more or less explicit way a relationship between their observations and the structures of living beings. Such a rather close connection surely exists for patterns on sea-shells that can be explained by simple reaction–diﬀusion equations without any recourse to a detailed “program” written in the DNA of the snails. May be less noticed than discoveries in the experimental ﬁeld, there has been a tremendous progress in our rational understanding of what I would call loosely the qualitative behavior of solutions of partial diﬀerential equations and/or ordinary diﬀerential equations. A very crucial idea, ﬁrst developed by Poincar´e and by Andronov, is the one of “robustness.” Loosely speaking, it allows us to describe a dynamical system (= the equation(s) of motion of the system we consider) independently of their detailed form. This is of course a “topological theory” in the deepest sense, as it amounts to consider that two systems are “the same” if one can deform one smoothly into the other. Except at “bifurcation” value this holds true for almost all values of the parameters into the equation(s) of motion: changing a bit a parameter value changes only numerical values typical of the solutions, not their structure. The study of the qualitative properties of dynamical systems has experienced an explosive growth in the late 1970s and early 1980s. This is very nicely presented in the present book. The mathematicians managed later on to prove many results that had only been guessed by physicists in this ﬁeld. Emboldened by the success of this notion of “structural stability,” some scientists, Len Pismen among them, tried to deal in the same way with the pattern-forming systems. This amounted to summarizing the properties of the system at hand into a set of equations as simple as possible, but with all the relevant symmetries. This method traces back its origin in Poincar´e’s normal forms; near a bifurcation point, one can reduce the complexity of the equations by keeping as variables a few time-dependent amplitudes only, with simple polynomial nonlinearities. This reduction of the complexity is, after all, what we do spontaneously when we see a set of rolls in aligned clouds or chaos and turbulence in a cloud painted by Tiepolo or Corot. The question is posed by this reduction: Can we obtain a picture of the phenomena general enough to yield the most obvious features of the solutions we would like to describe, although the mathematics remain simple enough to be amenable to our methods of analysis? Of course, such a reductionist view is at the heart of our physical description of the world. To take an example, the Navier–Stokes equations of a viscous ﬂuid reduce all the complex motion of an

Foreword

VII

almost inﬁnite number of atoms and molecules to the knowledge of the value of the shear viscosity of this ﬂuid and, eventually, to the pressure–density relation. Len Pismen starts from quite general equations, kind of generalized reaction–diﬀusion equations for an arbitrary number of chemical species. He shows masterfully how to reduce them to simpler forms near bifurcation points (the famous Newell–Withehead–Segel amplitude equation, the Kuramoto– Sivashinsky equation, the Ginzburg–Landau equation to name just a few). He then follows this general scheme of reduction, but with very little “hubris.” Reading his text, I am amazed by the conciseness and eﬃciency of his presentation. Only a few lines of text and some very clever mathematical steps bring all the necessary light in very messy questions, in the great tradition of scientists such as Zel’dovich, Kolmogorov, and Landau. He balances very ﬁnely the logical step, the development of a new intuition, with the help of classical analysis at its best. This text has the double interest to introduce the reader to a whole fascinating new ﬁeld, still in a stage of fast development, and to show how to sort out quite complex issues in an eﬃcient way. Besides its obvious pedagogical aspects, this monument of scholarship is also the very ﬁrst one to cover in a fully uniﬁed way an extremely wide range of topics. It reaches quite often the cutting edge of present-day research. For instance, it gives a very thorough review of everything known about the spirals in excitable media, including the instabilities of the center of the spiral that may oscillate, meander, and so on. Although the amplitude equations have been known for a number of years, this is – as far as I can tell – the ﬁrst time all their properties are discussed in a uniﬁed text. As the title shows, part of the book is devoted to fronts and interfaces. This is done in Chap. 2, which is very welcome as it puts in a single framework topics such as the Maxwell–van der Waals theory of liquid–vapor interface, the growth of a stable pattern in an unstable medium (the so-called KPP problem), and other “phase ﬁeld models” that have been lately very thoroughly studied. The amplitude equation section (Chap. 4) is an exposition by a master of the ﬁeld. Everything is covered in a smooth way, including the very tricky question of the motion of defects, where the distortion brought to the phase ﬁeld by the motion is analyzed in depth. The latest progress in pinned fronts, grain boundaries, etc. is well covered. The advantage of a single book is most evident here, because the discussion relies on a rather delicate analysis of bifurcations of trajectories in a 3D space. The ﬁnal chapter is about wave patterns and spirals, with all their instabilities, interactions, etc. This book follows another one by the same author, on vortices in nonlinear ﬁelds, and the two bring a uniﬁed view of an extremely active and interesting ﬁeld of modern research.

Paris, France

Yves Pomeau

Preface

This book involved considerable rewriting and reshuﬄing, despite the fact that I thought when I started it that I knew the material well. As usual, it was diﬃcult to decide what to include and what not. A monograph should not imitate a journal review; it is too easy to send readers to ﬁnd their own way in the thick of the woods. However, the narrative may expand without limit when attempting to be self-contained. The ﬁnal compromise may sometimes be subjective and imperfect. Few appreciated the hard labor of Procrustes, who had to ﬁt all kinds of strangers into his infamous bed. The author ﬁnds himself in a somewhat similar position, with an additional constraint on the use of violent means and an obligation to keep his customers alive and well, while resizing them to common logic and notation. This is not easy even when processing one’s own work of yesteryear. One wishes to give up and write another paper instead, but this becomes impossible after a critical point has been passed and the invested eﬀort has grown too heavy to discard. The result is before the reader. Patterns and interfaces live in space; I could not avoid, nevertheless, to include a long chapter on dynamical systems, which live in time only, but provide both technical means and examples to follow for more robust spatial structures. Ardent students of dynamical systems may ﬁnd it almost sacrilegious to see a variety of bifurcations and endless abyss of chaos crowded into a few dozen pages, still shared with the dynamics of regular patterns (which imitate far simpler systems by forfeiting their imperfections and natural variability). These are, of course, pure essentials that are needed to understand further material. The core of the book lies in various applications of perturbation theory, which attempts to tackle nonlinear spatiotemporal structures that, at ﬁrst sight, are likely to break our modest analytical tools. A modern paradox is that these structures are often easier to simulate numerically than to carve analytically. Simulation is, however, not a substitute for understanding, and one hopes that analytical skills will not fade away with the change of generations.

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Preface

Just imagine a feeble knight-errant, armed only with a paper sword – and, perhaps, a laptop with symbolic computation software: can he dare to overcome a horde of nonlinear beasts? There are, essentially, two tricks known from ancient times: separate them or try to surprise them while they are still waking up. The ﬁrst method is called scale separation and is the principal tool in Chaps. 2 and 3, where we study ﬁrst fronts, and then structures built up by their combinations in reaction–diﬀusion and related equations. The second method, called amplitude expansion, deriving and analyzing universal equations near symmetry-breaking transitions, prevails in Chaps. 4 and 5. Reaction–diﬀusion systems, with their plethora of chemical and biological applications and common but versatile structure, provide the bulk of material of the book. Other common pattern-forming systems rooted in ﬂuid mechanics and nonlinear optics are not considered explicitly, but they converge to the same universal equations of amplitude and phase dynamics. I am indebted to many colleagues and friends of our “nonlinear community”, which has forged during the late decades of 20th century our basic understanding of pattern formation far from equilibrium. The new century, that seems to be interested, in its youth, in the particular more than in the universal, may still ﬁnd this knowledge useful for building up future biomorphic technologies based on bottom-up self-organization rather than top-down manufacturing.

Technion – Israel Institute of Technology Haifa, November 2005

L.M. Pismen

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quest for Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conservative and Dissipative Systems . . . . . . . . . . . . . . . . . . . . . . Closed and Open Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Field Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Historical and Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 3 4 6

Dynamics, Stability and Bifurcations . . . . . . . . . . . . . . . . . . . . . . 1.1 Basic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Reaction–Diﬀusion System . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Vertical Structure and Representative Equations . . . . . . 1.1.4 Spectral Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5 Trajectories and Attractors . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Linear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Instabilities of Stationary States . . . . . . . . . . . . . . . . . . . . 1.2.2 Turing and Hopf Instabilities . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Dispersion Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Instabilities of Periodic Orbits . . . . . . . . . . . . . . . . . . . . . . 1.3 Weakly Nonlinear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Multiscale Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Bifurcation of Stationary States . . . . . . . . . . . . . . . . . . . . 1.3.3 Derivation of Amplitude Equations . . . . . . . . . . . . . . . . . 1.3.4 Hopf Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.5 Degenerate Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Global Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Global Dynamics: An Overview . . . . . . . . . . . . . . . . . . . . . 1.4.2 Systems with Separated Time Scales . . . . . . . . . . . . . . . . 1.4.3 Almost Hamiltonian Dynamics . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Bifurcation Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Deterministic Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 9 9 13 15 17 19 21 21 23 24 25 27 27 29 32 35 38 39 39 42 45 46 51

1

XII

Contents

1.5.1 Chaotic Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Shilnikov Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 Period Doubling Cascade . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.4 Lorenz System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Dynamics of Planforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Interaction of Turing Modes . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 Resonant Planforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.3 Degenerate Wave Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.4 Biscale Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

51 54 57 61 66 66 69 74 77

Fronts and Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 2.1 Planar Fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 2.1.1 Space-Dependent Amplitude Equations . . . . . . . . . . . . . . 83 2.1.2 Propagating Front as a Heteroclinic Trajectory . . . . . . . 85 2.1.3 Computation of the Propagation Speed . . . . . . . . . . . . . . 87 2.1.4 Maxwell Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2.1.5 Unstable Nonuniform States . . . . . . . . . . . . . . . . . . . . . . . 90 2.1.6 Front Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 2.2 Weakly Curved Fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 2.2.1 Aligned Coordinate Frame . . . . . . . . . . . . . . . . . . . . . . . . . 94 2.2.2 Expansion Near Maxwell Construction . . . . . . . . . . . . . . 97 2.2.3 Burgers Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 2.2.4 Ripening under Global Control . . . . . . . . . . . . . . . . . . . . . 100 2.3 Propagation into an Unstable State . . . . . . . . . . . . . . . . . . . . . . . . 103 2.3.1 Continuum of Propagating Solutions . . . . . . . . . . . . . . . . . 103 2.3.2 Asymptotic Theory of the Leading Edge . . . . . . . . . . . . . 105 2.3.3 Stability of the Leading Edge . . . . . . . . . . . . . . . . . . . . . . 106 2.3.4 Pulled and Pushed Fronts . . . . . . . . . . . . . . . . . . . . . . . . . . 108 2.4 Cahn–Hilliard Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 2.4.1 Order Parameter and Energy Functional . . . . . . . . . . . . . 110 2.4.2 Conservative Gradient Dynamics . . . . . . . . . . . . . . . . . . . 111 2.4.3 Inner and Outer Equations . . . . . . . . . . . . . . . . . . . . . . . . . 112 2.4.4 Solvability Condition and Matching . . . . . . . . . . . . . . . . . 114 2.4.5 Propagation and Coarsening in 1D . . . . . . . . . . . . . . . . . . 117 2.4.6 Ostwald Ripening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 2.5 Phase Field Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 2.5.1 Variational Principle for Phase Field Model . . . . . . . . . . 123 2.5.2 Sharp Interface Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 2.5.3 Mobility Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 2.6 Instabilities of Interphase Boundaries . . . . . . . . . . . . . . . . . . . . . . 127 2.6.1 Instabilities due to Coupling to Control Field . . . . . . . . . 127 2.6.2 Mullins–Sekerka Instability . . . . . . . . . . . . . . . . . . . . . . . . . 129 2.6.3 Instability of a Curved Interface . . . . . . . . . . . . . . . . . . . . 132 2.6.4 Directional Solidiﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 2.6.5 Long-Scale Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

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Systems with Separated Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 3.1 Stationary Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 3.1.1 FitzHugh–Nagumo System . . . . . . . . . . . . . . . . . . . . . . . . . 141 3.1.2 Stationary Structures in 1D . . . . . . . . . . . . . . . . . . . . . . . . 143 3.1.3 Solitary Disk and Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . 145 3.1.4 Migration-Enhanced Structures . . . . . . . . . . . . . . . . . . . . . 148 3.1.5 Quasistationary Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 150 3.2 Symmetry Breaking Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 3.2.1 Stability of a Straight-Line Front . . . . . . . . . . . . . . . . . . . 152 3.2.2 Stability of a Solitary Band . . . . . . . . . . . . . . . . . . . . . . . . . 153 3.2.3 Stability of a Striped Pattern . . . . . . . . . . . . . . . . . . . . . . . 154 3.2.4 Stability of a Solitary Disk and Cylinder . . . . . . . . . . . . . 157 3.3 Dynamic Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 3.3.1 Propagating Fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 3.3.2 Traveling Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 3.3.3 Oscillatory Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 3.3.4 Phenomenological Velocity–Curvature Relation . . . . . . . 165 3.3.5 Long-Scale Evolution Equations . . . . . . . . . . . . . . . . . . . . . 170 3.4 Locally Induced Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 3.4.1 Intrinsic Equations of Motion . . . . . . . . . . . . . . . . . . . . . . 172 3.4.2 Steadily Rotating Spiral . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 3.4.3 Propagation into a Quiescent State . . . . . . . . . . . . . . . . . 177 3.4.4 Spiral Band near Ising–Bloch Bifurcation . . . . . . . . . . . . 179 3.5 Advective Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 3.5.1 Inertial Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 3.5.2 Dispersion Relation for Wave Trains . . . . . . . . . . . . . . . . 183 3.5.3 Chaotic Wave Trains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 3.6 Rotating Spiral Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 3.6.1 Advective Limit for a Rotating Spiral . . . . . . . . . . . . . . . . 188 3.6.2 Propagating Finger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 3.6.3 Slender Spiral Band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 3.6.4 Tip Meandering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 3.6.5 Phenomenology of Complex Spiral Motion . . . . . . . . . . . 200 3.6.6 Scroll Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

4

Amplitude Equations for Patterns . . . . . . . . . . . . . . . . . . . . . . . . 209 4.1 Spatially Modulated Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 4.1.1 Ginzburg–Landau Equation . . . . . . . . . . . . . . . . . . . . . . . . 209 4.1.2 Newell–Whitehead–Segel Equation . . . . . . . . . . . . . . . . . . 211 4.1.3 Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 4.1.4 Stationary Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 4.1.5 Stability of Stationary Solutions . . . . . . . . . . . . . . . . . . . . 216 4.2 Phase Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 4.2.1 Universal Form of Phase Equations . . . . . . . . . . . . . . . . . . 218 4.2.2 Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

XIV

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4.3

4.4

4.5

4.6

5

4.2.3 Long-Scale Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 4.2.4 Covariant Phase-Amplitude Equation . . . . . . . . . . . . . . . 223 Defects in Striped Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 4.3.1 Natural Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 4.3.2 Phase Field of a Dislocation . . . . . . . . . . . . . . . . . . . . . . . . 227 4.3.3 Dislocations in NWS Equation . . . . . . . . . . . . . . . . . . . . . 229 4.3.4 Dislocation Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 4.3.5 Disclinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 4.3.6 Domain Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Motion of Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 4.4.1 The Nature of the Driving Force . . . . . . . . . . . . . . . . . . . . 238 4.4.2 Phase Field of a Moving Defect . . . . . . . . . . . . . . . . . . . . . 239 4.4.3 Dissipation Integral and Peach–K¨ ohler Force . . . . . . . . . 241 4.4.4 Matched Asymptotic Expansions . . . . . . . . . . . . . . . . . . . 244 4.4.5 Interaction of Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . 247 4.4.6 Motion in a Supercriticality Ramp . . . . . . . . . . . . . . . . . . 252 Propagation of Pattern and Pinning . . . . . . . . . . . . . . . . . . . . . . . 254 4.5.1 Wavelength Selection in a Propagating Pattern . . . . . . . 254 4.5.2 Self-Induced Pinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 4.5.3 Geometry of Pinned States . . . . . . . . . . . . . . . . . . . . . . . . . 260 4.5.4 Crystallization Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 4.5.5 Pinning of Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Hexagonal Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 4.6.1 Triplet Amplitude Equations . . . . . . . . . . . . . . . . . . . . . . . 271 4.6.2 Skewed Triplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 4.6.3 Penta-Hepta Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 4.6.4 Domain Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 4.6.5 Propagation of the Hexagonal Pattern . . . . . . . . . . . . . . . 283

Amplitude Equations for Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 5.1 Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 5.1.1 Complex Ginzburg–Landau Equation . . . . . . . . . . . . . . . 287 5.1.2 Perturbations of Plane Waves . . . . . . . . . . . . . . . . . . . . . . 289 5.1.3 Phase Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 5.1.4 Propagation of Wave Pattern . . . . . . . . . . . . . . . . . . . . . . . 294 5.1.5 Coupled CGL Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 5.2 One-Dimensional Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 5.2.1 Classiﬁcation of 1D Solutions . . . . . . . . . . . . . . . . . . . . . . . 297 5.2.2 Holes and Wavenumber Kinks . . . . . . . . . . . . . . . . . . . . . . . 299 5.2.3 Modulated Waves and Phase Turbulence . . . . . . . . . . . . . 302 5.2.4 Transition to Defect Chaos . . . . . . . . . . . . . . . . . . . . . . . . . 305 5.2.5 Sources and Sinks in Coupled CGL Equations . . . . . . . . 307 5.3 Spiral Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 5.3.1 Symmetric Spiral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 5.3.2 Asymptotic Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

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XV

5.3.3 Nondissipative Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 5.3.4 Acceleration Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 5.4 Interaction of Spiral Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 5.4.1 Nonradiative Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 5.4.2 Weakly Radiative Vortices . . . . . . . . . . . . . . . . . . . . . . . . . 322 5.4.3 Strong Radiation and Shocks . . . . . . . . . . . . . . . . . . . . . . . 326 5.4.4 Multispiral Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 5.4.5 Period Doubling in Spirals . . . . . . . . . . . . . . . . . . . . . . . . . 333 5.5 Line Vortices and Scroll Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 5.5.1 Curvature-Driven Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 336 5.5.2 Nondissipative Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 5.5.3 Dissipative Nonradiative Line Vortex . . . . . . . . . . . . . . . . 343 5.5.4 Instability of Line Vortices . . . . . . . . . . . . . . . . . . . . . . . . . 344 5.6 Resonant Oscillatory Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 5.6.1 Broken Phase Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 5.6.2 Forcing at Double Frequency . . . . . . . . . . . . . . . . . . . . . . . 348 5.6.3 Ising and Bloch Fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 5.6.4 Higher Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

Introduction

Quest for Complexity Human eye is bored by monotonous landscape. We are depressed by gray skies, unending plain under the wing of an airliner, featureless urban sprawl, geometric canvasses in empty museum halls. Having evolved to meet the challenges of a complex world, we enjoy complexity. Stormy sea entertains us by a spectacle of spatio-temporal complexity. It is imitated by entertainers overwhelming us by the information deluge of blinking images in video clips. We also do enjoy, however, the calm of spatial complexity: a mountain landscape, a forest, a meadow, a mosaic. Complexity is particularly appealing when it arises spontaneously, rather than by design. This is where nature and industry diﬀer most: nature starts from a simple form and builds complexity by evolution and diﬀerentiation, while industry manufactures objects according to a preconceived form. As products of modern technology become increasingly complex, and the size of details goes down to micro- and nanoscale, industrial methods may become biomorphic as well. They would then make use of natural processes of spontaneous structure formation to grow complex entities in the way they are nurtured by nature itself. Our aim is to understand how relatively simple nonlinear systems obeying symmetric equations may generate a dazzling variety of forms. We shall be mostly concerned with distributed systems dependent on spatial coordinates as well as on time. The standard physical setting for our study is an extended (ideally, inﬁnite) homogeneous isotropic system, and the description will invariably be on a macroscopic (mean ﬁeld, or hydrodynamic) level. A much simpler dynamical system arises when the spatial dependence is suppressed. Dynamical systems may still exhibit complex temporal behavior, and serve as an indispensable tool for understanding more complex spatially distributed systems when a few dominant modes are suﬃcient for adequate approximate description of spatio-temporal dynamics.

2

Introduction

Conservative and Dissipative Systems Earliest dynamical models have originated in classical mechanics. These models contain one or more integrals of motion, which are conserved in the course of evolution. Systems of this kind, called conservative, may exhibit persistent motion without relaxing to a stationary equilibrium state. Until the end of 19th century, classical mechanics was preoccupied with integrable systems, which can be separated into an ensemble of oscillators and display periodic or quasiperiodic behavior, depending on whether the oscillation periods are commensurate or not. Henri Poincar´e was the ﬁrst to realize that even such a simple system as two interconnected pendula may already exhibit far more complex aperiodic dynamics. It took, however, another half-a-century or more for the notion of deterministic chaos to establish itself in the scientiﬁc community. Paradoxically, it was quantum mechanics that destroyed the belief in classical determinism. The latter’s ultimate expression was the famous Laplace’s maxim: a “supreme intelligence” capable of solving dynamical equations for all particles in the universe and knowing all initial conditions could predict the future in its entirety. Quantum mechanics showed this to be impossible, introducing chance into physics. Many scientists and philosophers, including the great Einstein, who detested God playing dice, regretted this. The paradox is that throwing dice, a symbol of chance game, follows the laws of classical mechanics. Its determinism is illusory, as precision in ﬁxing the initial conditions needed for predicting the future state rapidly increases with the growing length of the prediction interval. Quantum mechanics with its indeterminism actually smoothes out chaotic classical dynamics, and so prediction of averaged expectations becomes easier in quantum theory. The advent of computers played a crucial role in understanding complex chaotic dynamics, or at least in making the notion of deterministic chaos familiar to the wider scientiﬁc community and the general public. Before the computer age, research was concentrated on a tiny and nonrepresentative minority of mechanical systems described by analytically solvable models, and it took imagination of men of genius to understand the complexity of behavior that is transparent nowadays to an ordinary person observing a trajectory on a computer screen. An extension of conservative dynamical systems is ﬁeld theory studying conservative dynamics of spatially extended systems with inﬁnite number of degrees of freedom. Nonlinear ﬁelds described by nonlinear partial diﬀerential equations possess the richest mathematical structure and are capable of generating inﬁnite variety of spatio-temporal structures, both ordered and chaotic. The ﬁrst and perhaps most widely studied nonlinear ﬁeld theory was classical hydrodynamics of ideal ﬂuids. The phenomenon of turbulence has been ﬁrst studied in ﬂuid-mechanical context, and only later extended to a wider class of distributed systems displaying spatio-temporal chaos. Fascinating analytical theory has been developed to describe the behavior of a

Introduction

3

special class of completely integrable nonlinear partial diﬀerential equations that possess an inﬁnite number of integrals of motion. The most prominent feature of these models is the existence of particle-like localized solutions, called solitons, which are able to preserve their identity in interactions and collisions. Completely integrable ﬁeld equations are as nongeneric as completely integrable dynamical systems. They give, however, the only chance to study nonlinear structures analytically, and the results of this study, besides their mathematical beauty, are useful for the analysis of more realistic systems where the conservation laws are weakly broken, so that integrals of motion do not remain strictly constant but slowly evolve in time. As conservation laws are woven into the canvas of physics, dissipative (nonconservative) models always appear as approximate or coarse-grained models describing the averaged behavior of complex systems. Dissipation may appear in a mechanical system as an additional term taking account of friction, which is, in fact, the result of disordered motion of a great number of microscopic particles lumped by a macroscopic model characterized by few degrees of freedom. Mean ﬁeld models of thermodynamic or chemical origin are always dissipative, as they describe averaged dynamics characterized by a small number of thermodynamic variables. In ﬂuid mechanics, dissipation is brought in by viscosity and diﬀusion, stemming, like friction, from disordered motion on microscopic level. It should be noted that dissipative mean ﬁeld models are also usually derived from a mass, energy, or momentum conservation law, but contain sources or sinks that destroy conservation of the respective integrals. Besides dissipation in the common sense of the word – loss of energy, momentum, etc. – nonconservative systems may include sources due to external pumping, or “antidissipation.” Behavior of such systems, constrained neither by conservation laws nor by the boring laws of thermodynamics, is particularly rich and interesting; most pattern-forming systems we shall study belong to this class. From a formal point of view, conservative systems are nongeneric: a randomly generated set of equations will have no conservation laws. Closed and Open Systems We shall mostly deal here with macroscopic dissipative systems. A system of this kind can either be closed or open, i.e., exchanging mass and energy with the environment. The former case is less interesting, since, by the second law of thermodynamics, a closed physical system has to relax to a state of thermodynamic equilibrium. Since thermodynamic free energy decreases in the course of evolution, any system of equations modeling a closed system should possess an integral F monotonically decreasing with time1 . 1

In some models, existence of a monotonically decreasing integral, called a Lyapounov functional can be established ad hoc even when not supported by thermodynamics

4

Introduction

Although this restricts dynamics in a severe way, behavior of closed systems may still be far from trivial, since they may possess multiple equilibria. Any equilibrium state can be characterized by a certain value of free energy, and so, generically, there will be a single state of absolute equilibrium with the lowest value of F, while the other equilibria will be metastable. Evolution to the state of absolute equilibrium may be, however, hindered by high activation barriers, and so various metastable equilibria may persist indeﬁnitely. An open system is sustained in a nonequilibrium state by external ﬂuxes. Such a system, generally, can be assigned no energy functional; its dynamics is more interesting, since it is not restricted by a priori rules. There is, however, a restriction imposed on the geometry of an open system. While a closed system may be isotropic in three dimensions, the direction of external ﬂuxes sets a preferred axis in an open system; we shall tentatively call this direction “vertical.” The system may retain isotropy in the normal “horizontal” plane, provided neither the properties of the system itself nor the strength of external ﬂuxes depend on spatial coordinates explicitly and the inﬂuence of lateral boundaries can be discarded. A suitable geometric setting is an inﬁnite plane normal to the direction of external ﬂuxes (Fig. 1.1). The system may be truly two-dimensional (Fig. 1.1a) if the process is taking place on a surface and all relevant ﬁeld variables are surface densities. Otherwise, the system would possess some “vertical structure” in the direction of ﬂuxes (Fig. 1.1b) but retain isotropy in horizontal directions. Field Variables The level of detail on which the physical system is described is always a compromise between faithfulness and transparency, between available tools and desired results. The physicist may be satisﬁed by the simplest description that catches the essence of the phenomena, while the engineer strives for quantitative precision, often illusory. Either one sees (or prefers not to see) that the quest is open-ended, and any problem is solvable only in an imperfect way, on a level that shifts with time, or with changing circumstances and aims. The level on which we operate here is one of macroscopic classical theory. The physical system is described by a certain set of mean ﬁeld variables, which characterize the thermodynamic state of the system. The list of variables may include temperature, density or pressure, concentrations of chemical species, magnetization, electric polarization or potential, etc. It might be convenient to organize these variables in an n-dimensional array u, which should completely determine the state of the physical system in question. The mean ﬁeld description is suﬃcient in very large ensembles, under conditions when ﬂuctuations are negligible. In some cases, ﬂuctuations may be reintroduced phenomenologically through random inputs to mean ﬁeld equations. The same set of thermodynamic variables also serves to characterize a nonequilibrium system. All these variables should be then deﬁned locally and are, generally, functions of spatial variables and time, u(x, t). The aim of

Introduction

5

the mean ﬁeld theory is to formulate and solve evolution equations describing the dynamics of state variables. Since nonequilibrium processes usually involve mass transport, it might be necessary to complement the set of thermodynamic variables by the velocity ﬁeld . The set of evolution equations will then include coupled hydrodynamic equations for ﬂow velocity and transport equations for thermodynamic variables. The mean ﬁeld description of a spatially inhomogeneous and temporally evolving system is, in some way, self-contradictory. Averaged state variables and ﬂow velocities can be deﬁned in a macroscopic volume only, which is formally at odds with the notion of an inﬁnitesimal volume used in derivation of continuum equations. The contradiction is resolved by the requirement that all state variables should vary on spatial and temporal scales far exceeding all relevant “microscopic” scales, such as intermolecular distances or free path length and molecular relaxation times. This requirement is, unfortunately, too stringent in many situations, especially near interphase boundaries or at particular values of parameters (in the vicinity of critical points) when the scale of ﬂuctuations reaches macroscopic dimensions. The crossover from microscopic to mean ﬁeld description is one of “internal frontiers” of modern physics, where problems are challenging and answers are few and incomplete. Even as the mean ﬁeld description is oversimpliﬁed from the fundamental point of view, it may be, at the same time, too complex and technical. A more transparent theory may help to elucidate the most important qualitative features of the system under study. For this purpose, Landau has coined the notion of “order parameter.” The order parameter may distinguish between alternative equilibria, which correspond to diﬀerent phases. When the phases coexist, the order parameter changes continuously across the interphase boundary. Thus, we should speak of an order parameter ﬁeld (with some regret for misuse of the word “parameter” that commonly denotes a ﬁxed coeﬃcient rather than a ﬁeld variable). It may be either one of the physical ﬁelds or a ﬁctitious ﬁeld that lumps the eﬀect of all changes accompanying the phase transition. The free energy should be expressed through the order parameter and its derivatives in such a way that its minima correspond to stable phases. The hierarchy of simpliﬁed descriptions ascending to larger scales and more complex phenomena is a common structure of knowledge. Even the order parameter ﬁeld may be subject to further averaging and coarsening on the next hierarchical level if it develops its own elaborate spatio-temporal structure. Composite entities play a role of elementary objects in diverse applications, from dynamics of granular media to motion of galactic clusters. In biology and social sciences, a mean ﬁeld theory may operate with still more complex and coarse-grained variables, such as population densities, while applying equations and techniques not much diﬀerent from those of chemical physics.

6

Introduction

Historical and Bibliographical Notes Study of dynamics and pattern formation far from equilibrium (a major branch of “nonlinear science”) is now a mature discipline that has passed a common sequence of development stages. The following is an incomplete view from the perspective of this book2 : Prehistory: Faraday instability (Faraday, 1831) of the free surface of a vibrated ﬂuid layer was the ﬁrst example of spontaneous pattern formation; it had to wait, however, for more than a century for theoretical explanation. Dawn: study of particular phenomena; lack of connections (1900s–1960s). Spontaneous symmetry breaking was known in ﬂuid mechanics since the experiments of B´enard (1900) and their forceful, though factually incorrect, explanation by Rayleigh (1916). In chemical and biological applications, the awareness of this phenomenon had to wait another half-a-century till the famous work of Turing (1952), far weaker analytically, though philosophically charged. Chemical oscillations were considered taboo because of thermodynamical misconceptions well into the 1960s, while electrochemical oscillations and waves were known since the turn of the century (Ostwald, 1900). Landau (1944) treated turbulence as a ﬁeld theory with a very large number of degrees of freedom, while wide implications of chaotic behavior of simple mechanical systems, known already to Poincar´e, were not understood. A paradigmal model of chaotic ﬂow by Lorenz (1963) remained unnoticed for 15 years. Development: general theory comes of age (late 1960s to early 1980s). Synergetics of Haken (1983) and the self-organization principle of the Brussels school (Nicolis & Prigogine, 1977) oﬀer paradigms of dynamics far from equilibrium. Belousov–Zhabotinsky reaction (Zhabotinskii, 1974) is explored as a spectacular example of chemical oscillations, prompting theoretical studies of chemical waves and diﬀusion fronts (Ortoleva & Ross, 1974; Kuramoto & Tsuzuki, 1976; Fife, 1979, and many others). Nonlinear theory of convective patterns is ﬁrmly established (Busse, 1978; Normand, Pomeau & Velarde, 1977) and tested in precision experiments (Ahlers, 1974). Other input comes from combustion theory (Zeldovich, 1985; Buckmaster & Ludford, 1983; Sivashinsky, 1977) and the theory of solidiﬁcation (Langer, 1980). Simple patternforming models – Brusselator, Kuramoto–Sivashinsky, Swift–Hohenberg equations – are brought forward and explored. Chaos theory is transformed by the discovery of period-doubling sequence and other generic routes of transition to chaos. Summit (mid-1980s to early 1990s): acme. Common techniques are applied to nonequilibrium structures of diﬀerent physical origin. Space-dependent amplitude equations and phase dynamics lead to understanding of genesis and behavior of realistic patterns, constrained by boundaries and blemished by defects. Chemical patterns are studied under controlled conditions (Ouyang & Swinney, 1991), and the theory of spiral waves, prompted by chemical and 2

This list omits references to original work described in detail elsewhere in the book.

Introduction

7

physiological applications, attracts great attention. The NATO Special Program on Chaos, Order and Patterns did much to unify physicists’ approach and diﬀuse the knowledge, and it might not be by accident that its end coincided with the publication of the review by Cross and Hohenberg (1993), which remains to this day the most comprehensive survey of the ﬁeld. Other monographs, emphasizing diﬀerent aspects of nonequilibrium dynamics and pattern formation were published by Manneville (1990), Mikhailov (1991), Nicolis (1995), and Walgraef (1997), while the books by Berg´e et al (1984) and Ott (1993) concentrated on problems of chaos and turbulence. Speciﬁc problems of patterns and fronts in reaction-diﬀusion equations were treated in (very dissimilar) books by Murray (1989) and Fife (1988) – one geared to biological applications and the other to combustion problems. On a diﬀerent front, theory of bifurcations in dynamical systems was formalized in mathematical monographs, with the emphasis on dynamic behavior (Guckenheimer & Holmes,1983; Hale & Kocak, 1991, and many others) or group-theoretical aspects (Golubitsky & Schaeﬀer, 1985). Maturity (mid-1990s to now): a more detailed look at complexity. Attention returns to speciﬁc applications; among them, nonlinear optics and studies of granular media come to the forefront. Forcing and control of patterns, either enhancing or suppressing the complexity of behavior, are studied in detail. As a humble laptop turns into a supercomputer, more fascinating patterns, envy of abstract expressionists, are generated by model equations of increased complexity run by friendly Matlab programs. Whether or not new ideas will emerge as the new century comes of age, this discipline, dealing with ubiquitous problems of order and chaos, is bound to ﬁnd its way into basic curricula and wealth of practical applications.

1 Dynamics, Stability and Bifurcations

1.1 Basic Models 1.1.1 Dynamical Systems A general dynamical system is just a set of ordinary diﬀerential equations (ODEs) that can be written as a system of ﬁrst-order equations resolved in respect to the time derivatives: dui = fi (u). dt

(1.1)

It is convenient to arrange the dependent variables as components ui of the n-dimensional state array u. It is often advantageous to describe the dynamics in geometric terms, and refer to u as a state array, i.e. a vector in the n-dimensional phase space. Respectively, the right-hand sides (r.h.s.) are presented as components fi of a nonlinear vector-function f (u). In chemical and biological models, the dependent variables ui may be concentrations of chemical species or population densities; then (1.1) represents the balance of the respective species in a spatially homogeneous (e.g., well mixed) system, and fi (u) combine the net production rates and exchange with the environment. Similar equations can be written for other variables, e.g., temperature. Stationary states, or ﬁxed points of (1.1), u = us , are zeros of f (u); they are also often called equilibria, although, of course, they do not coincide with thermodynamic equilibria when exchange with the environment is present. Multiple equilibria are possible even in a closed system, where they are, generally, characterized by diﬀerent values of free energy. Therefore, one can deﬁne the state with the lowest energy as absolutely stable, other states corresponding to local minima of free energy as metastable, and states corresponding to saddle points or maxima as unstable. A closed system relaxing to thermodynamic equilibrium should be described by a gradient dynamical

10

1 Dynamics, Stability and Bifurcations

system that possesses a potential V (u), such that fi (u) = −∂V /∂ui . The potential decreases monotonically in the course of evolution, until a minimum is reached: ∂V dui ∂V 2 dV = =− ≤ 0. (1.2) dt ∂ui dt ∂ui i i Open systems, generally, lack this property, so that one cannot distinguish in this case between absolutely stable and metastable states. Particular chemical or biological models are often constructed on the basis of the mass action law , which stipulates that reaction rates are proportional to concentrations of reactants. The functions fi (u) deﬁned in this way are necessarily polynomials, which, of course, may seem unduly restrictive. On the other hand, the dynamic behavior of a polynomial system may not diﬀer qualitatively from that of a more algebraically complicated system. The mass action principle has been used to construct a number of artiﬁcial easily solvable models speciﬁcally designed to illustrate certain features of dynamics. Thus, if one wishes to illustrate a transition between two alternative equilibria, each of them being stable in a certain range of parameters, the simplest form is u˙ = µu − u3 ,

(1.3)

where the dot denotes the time derivative. The stable stationary state is u = 0 √ at µ < 0 and u = µ at µ > 0. This model can be disguised as a chemical scheme including a monomolecular and a trimolecular reaction, known as the Schl¨ ogl model. This disguise would be dismissed by a physicist, who would call u an “order parameter” and use (1.3) to describe a transition between a “disordered” phase with u = 0 and an “ordered” phase with u = 0. For a mathematician, (1.3) is a “normal form” satisfying the condition of inversion symmetry (see Sect. 1.3.2). This symmetry can be broken by adding a constant term. A simple extension of (1.3) to two-dimensional (2D) phase space is the FitzHugh–Nagumo model (FitzHugh, 1961); its rescaled symmetric form cotaining two parameters µ, γ is γ u˙ = u − u3 − v,

v˙ = −v + µu.

(1.4)

This model is far more interesting, as it generates oscillatory behavior as well as multiplicity of stationary states (see Sect. 1.4.2). The most celebrated artiﬁcial “chemical” system is the Brusselator model (Prigogine and Lefever, 1968). It can be constructed as mass conservation equations describing two ﬁctitious chemical reactions 2A + B → 3A,

B → A,

It is further assumed that the species A, but not B, is supplied from and removed to the environment.1 This leads, after appropriate rescaling, to the 1

In the original derivation, two more reactions were added in lieu of the exchange. The interpretation in the text, mathematically indistinguishable from the original model, is also known as Gray–Scott model.

1.1 Basic Models

11

system of equations u˙ = ν − (µ + 1)u + u2 v,

v˙ = µu − u2 v,

(1.5)

where µ, ν are parameters. The real purpose of the construction is to have an “autocatalytic” system, which always has a unique and easily computable stationary state u = ν, v = µ/ν; this state can be made by additional rescaling independent of the parameters of the system. This is, of course, rather unrealistic, but very convenient, as it allows one to forget about the cumbersome task of ﬁnding the equilibria and to concentrate upon more interesting questions of symmetry breaking and dynamics. Continuing this trend, R¨ ossler (1976) designed several three-variable dynamical systems with simple algebraic r.h.s to illustrate various aspects of chaotic dynamics. Various models of population dynamics are also based on the mass action law and have a general form µij uj . (1.6) u˙ i = νi ui + ui j

The common feature of these models is the presence of trivial “extinction” states with vanishing ui . The interactions are competitive when both µij and µji are negative, cooperative when both are positive, and prey–predator when µij and µji are of the opposite sign. Many realistic chemical models have a more complicated form. The source of complexity is in multistage chemical mechanisms, which may have characteristic rates diﬀering by many orders of magnitude. Some chemical stages rapidly reach equilibrium, which enforces certain algebraic relations among the concentrations of reacting species. Using these equilibrium relations in kinetic equations of slower stages leads to dynamic equations with nonpolynomial functions fi (u) even when all original equations are based on the mass action law. This is characteristic to catalytic or enzyme reactions, where the fast stage is adsorption of reactants on the catalyst surface or attachment to an enzyme molecule and the slow stage is the chemical reaction proper. For example, a bimolecular reaction may proceed according to the following mechanism: Ai + E A∗i ,

A∗1 + A∗2 → P + 2E,

where E is a catalytic site, P is a nonadsorbing product, and asterisks denote adsorbed forms of the reactants Ai . Assuming adsorption equilibrium, the kinetic equations for the concentrations ui of the reactants Ai in an open system reduce to the form u˙ i = −

ku1 u2 + νi (1 − ui ). (1 + µ1 u1 + µ2 u2 )2

(1.7)

An interesting feature of this system is its “autocatalytic” character made apparent under conditions when strong adsorption of one of the reactants

12

1 Dynamics, Stability and Bifurcations

reduces the reaction rate by suppressing adsorption of the other reactant; thus, if µ1 u1 far exceeds both µ2 u2 and unity, the nonlinear term in (1.7) is approximately proportional to u2 /u1 , so that ∂f1 /∂u1 > 0. Both multiple stationary states and oscillations are possible under these conditions. In nonisothermal reactions, one of the dependent variables is temperature; a typical temperature dependence of reaction rates is exponential, stemming from the Arrhenius law. If the reaction is exothermic, this dependence has an autocatalytic character, as the reaction rate grows with temperature causing its further increase. The simplest dynamical system describing a monomolecular exothermic reaction in a well-stirred vessel and allowing for exchange of both material and heat with the environment is u˙ = −ev u + µ(1 − u),

γ v˙ = ν ev u − v,

(1.8)

where u is the reactant concentration, v is the reduced temperature, µ, ν, γ are parameters, and an approximate exponential temperature dependence, replacing the Arrhenius law, is used. This system, as well as (1.7), generates multiple stationary states and oscillations. Another (historically earliest) source of dynamical systems is classical mechanics. Dynamical systems of classical mechanics can be presented as a combination of pairs of equations for conjugate variables: coordinates ui and momenta pi : ∂H ∂H dui dpi = =− , . (1.9) dt ∂pi dt ∂ui The Hamiltonian H is conserved in the course of evolution: ∂H dui ∂H dpi dH = + = 0. dt ∂ui dt ∂pi dt i

(1.10)

The general form of a Hamiltonian derivable from the Newton law is p2i + V (u). (1.11) H = 12 i

The minima of the potential V (u) correspond to stable equilibria. Other levels of H correspond to persisting motion along a certain orbit lying on the manifold H = const in the phase space; the dimensionality of this manifold can be further reduced by applying additional conservation laws. The motion can be periodic, quasiperiodic, or chaotic. As we have already noted in Introduction, the latter behavior is actually more typical for a system with a large number of degrees of freedom, and the notion of a classical mechanical system as wellbehaved as a clock had been caused by a bias to exactly solvable models in precomputer days. Hamiltonians of diﬀerent structure may appear in other applications. A well-known example is the Volterra–Lotka system: u˙ = u(1 − v),

v˙ = v(u − µ).

(1.12)

1.1 Basic Models

13

This is a particular form of (1.6) describing the dynamics of a prey–predator system. The Hamiltonian structure is revealed by transforming to the variables x = ln u, y = ln v, which turn out to be conjugate variables of the Hamiltonian H = ex + ey − µx − y. A more general system conserving a certain function H(u) may lack Hamiltonian structure but be presentable in the form dui ∂H = Mij , dt ∂uj j

(1.13)

where Mij = −Mji form an antisymmetric matrix. “Accidental” conservative systems, such as the Volterra–Lotka system where there is no “natural” Hamiltonian structure dictated by the physics of the problem or innate symmetry, are structurally unstable, so that the conservation law is destroyed by any small modiﬁcation of the system. Conservation of energy is also destroyed in a nonideal mechanical system by dissipation and compensating energy input. The Hamiltonian structure remains, however, useful for the study of weakly dissipative systems described by (1.9) with added small nonconservative terms, which may stabilize certain orbits selected from the available continuous family of orbits parametrized by the value of H. This provides a powerful tool for the study of transitions between diﬀerent stationary states and periodic orbits (see Sect. 1.4.3). 1.1.2 Reaction–Diﬀusion System A simple but general distributed model involving transport and nonlinear interactions but no advection is the reaction-diﬀusion system (RDS). It arises in diﬀerent contexts from conservation laws of a general form γ ∂t u = −∇ · j + f (u).

(1.14)

Commonly, the ﬁeld variables u are concentrations of chemical species or population densities; then (1.14) represents the balance of the respective species, with f being the production rate, j the ﬂux, and γ the capacitance factor. Equations of similar structure can be written for other thermodynamic variables, such as temperature and electric potential, stemming, respectively, from conservation of energy and electric charge. Equations (1.14) have to be closed by assigning a dependence of f on the ﬁeld variables and a dependence of j on the ﬁeld variables and their gradients. Generally, there will be a number of interdependent equations for several ﬁeld variables arranged, as in (1.1), as components ui of an n-dimensional state array u. Respectively, the production rates are presented as components fi of a nonlinear vector-function2 f (u; p), which is also dependent on an array of parameters p. 2

Both vectors in phase space, such as u or f , and vectors in real space, such as j, are set in boldface, but there should be no way to mix them up.

14

1 Dynamics, Stability and Bifurcations

The ﬂuxes j i may, generally, depend on gradients of all ﬁeld variables: Dij ∇uj . (1.15) ji = − j

The diﬀusivity matrix Dij is, generally, nondiagonal, and may depend on the local values of the state variables. This dependence can be expressed in a less formal way by separating diﬀusion proper (which originates in random motion of noninteracting particles) from migration under the action of combined molecular interaction forces. If the latter can be expressed through the gradient of a thermodynamic potential U , we obtain: j i = −Di ∇ui − Mi ∇U.

(1.16)

If the dependence U (u) is local , one can express ∇U through ∇ui explicitly, recovering the form (1.15) with Dij = Di δij + Mi ∂U/∂uj . Explicit migration ﬂuxes can also be written for strongly nonequilibrium transport processes, such as chemotaxis, which may not be derivable from a potential. Equation (1.15) or (1.16) excludes any externally imposed ﬂow, although transport processes may involve net mass ﬂux. Presuming the diﬀusivities to be constant, an RDS can be written in the form Dij ∇2 uj + fi (u; p), (1.17) γi ∂t ui = j

where γi are elements of a diagonal capacitance matrix. If all capacitances γi diﬀer from zero, one can always eliminate them by rescaling Dij and fi , so that (1.17), rewritten in the vector notation, becomes ∂t u = D∇2 u + f (u; p).

(1.18)

It might be advantageous, however, to retain the capacitance matrix in order to distinguish among three types of parameters: • elements of the array p inﬂuence stationary states as well as dynamics; • capacitances γi inﬂuence dynamic behavior but not equilibria; • elements of the matrix Dij aﬀect spatially inhomogeneous solutions only. The full form (1.17) is essential in systems with widely separated time scales where some (fast) components are quasistationary. An RDS can describe a truly 2D open system, such as the one shown in Fig. 1.1a, either when both diﬀusion and interactions are taking place on a surface or when the thickness of a bulk layer in the direction of external ﬂuxes is so small that the composition can be assumed uniform in the “vertical” direction. Then the nonlinear functions in (1.14) should account for both internal interactions, e.g., chemical reaction rates, and external (“vertical”) ﬂuxes due to exchange with the environment. We shall suppose here that these functions do not depend on time and spatial variables explicitly. Then

1.1 Basic Models

15

(a)

(b)

Fig. 1.1. An open system isotropic in two dimensions. (a) A truly two-dimensional system. (b) A cut through a system with vertical structure (shown symbolically by varied shading). Arrows indicate the direction of external ﬂuxes

an RDS always has one or more “trivial” homogeneous stationary solutions (HSS) coinciding with ﬁxed points of a respective dynamical system. This solution may become unstable in a certain parametric domain, giving way to spatially inhomogeneous and/or persistently time-dependent states. 1.1.3 Vertical Structure and Representative Equations More generally, a “trivial” state of a distributed nonequilibrium system will be uniform only in the plane normal to the direction of external ﬂuxes, but dependent on the “vertical” coordinate, as in Fig. 1.1. This is necessarily so in the presence of ﬂuid ﬂow when hydrodynamic equations should be solved together with transport equations to determine a vertical structure satisfying appropriate boundary conditions. Other systems with vertical structure include directional solidiﬁcation, propagation of light in nonlinear media, and orientation textures in liquid crystals. The set of variables u describing systems of this kind may include vector as well as scalar ﬁelds, and the applicable equations will typically include diﬀerential nonlinear terms instead of or in addition to algebraic nonlinearities of an RDS. The most general form is an array of nonlinear operators of the type H(u, ∂t u, ∂z u, ∇u, . . .) = 0,

(1.19)

where we distinguish between the derivatives in the “vertical” direction ∂z and the derivatives in “horizontal” directions, expressed by the 2D gradient operator ∇. Vertical structure can often be eliminated when it remains almost constant in the horizontal plane. More precisely, the reduction is possible when

16

1 Dynamics, Stability and Bifurcations

the characteristic horizontal scale is much larger than the characteristic vertical scale, say, ∂z = O(1), ∇ = O(), 1. Another possibility is slow modulation of a structure periodic in the horizontal direction. This involves multiscale expansion of the gradient operator ∇ = ∇0 + ∇1 (Sect. 4.1.1). Equations obtained as a result of elimination of vertical or periodic structure have, as a rule, a relatively simple form, which is determined by symmetries of the underlying system. Equations of identical structure may emerge therefore in diﬀerent physical applications. Such representative equations are often most suitable for study of typical dynamic phenomena in nonlinear systems. A substantial part of the material of this book is based on the study of such equations. The following equations for a single scalar ﬁeld u(x) can be written simply by requiring spatial isotropy as well as a speciﬁed symmetry to transformations of u, and restricting to a certain order in u and ∇; they can also be written by scaling both u and ∇, as indicated in brackets, and retaining the lowest order terms: • Cubic reaction-diﬀusion equation – second order in ∇, third order in u, symmetry to inversion of u: ut = ∇2 u + u(1 − u2 )

[u = O(), ∇ = O()]

(1.20)

• Swift–Hohenberg (SH) equation (Swift and Hohenberg, 1977; Pomeau and Manneville, 1980) – fourth order in ∇, third order in u, symmetry to inversion of u: √ (1.21) ut = −(1 + ∇2 )2 u + u(µ − u2 ) [u = O(), ∇ = O( )] • Burgers equation (Burgers, 1948) – second order in ∇, second order in u, symmetry to translations of u (which permits the dependence on derivatives of u only, rather than u itself): ut = ∇2 u + |∇u|2

[u = O(1), ∇ = O()]

(1.22)

• Kuramoto–Sivashinsky (KS) equation (Kuramoto and Tsuzuki, 1976; Sivashinsky, 1977) – fourth order in ∇, second order in u, symmetry to translations of u: √ (1.23) ut = −∇2 u − (∇2 )2 u + |∇u|2 [u = O(), ∇ = O( )] • Proctor–Sivashinsky equation (Chapman and Proctor, 1980; Gertsberg and Sivashinsky, 1981) – fourth order in ∇, third order in u, symmetry to translations and inversion of u: √ ut = −∇2 u − (∇2 )2 u + ∇ · (∇u|∇u|2 ) [u = O(1), ∇ = O( )] (1.24) All these equations are written in a rescaled – and mostly parameterless – form, with the sign chosen in a way ensuring stability to short-scale perturbations and preventing runaway to inﬁnity. The SH equation (1.21) is perhaps

1.1 Basic Models

17

the simplest pattern-forming system. Its linear part is designed to produce a dispersion relation with a maximum at a ﬁnite wavenumber, normalized by rescaling to unity. The height of the maximum is controlled by the single parameter of the model, µ. The cubic term serves to saturate the instability developing at µ > 0. Similar equations can be written for a complex scalar variable. Equations of this kind often appear as envelopes of periodic structures, and will be studied in detail in Chaps. 4 and 5. The lowest-order form analogous to (1.20) is the complex Ginzburg–Landau (CGL) equation ut = η∇2 u + u(µ − ν|u|2 ).

(1.25)

The coeﬃcients η, µ, ν are generally complex, but the particular cases when they are either all real or all imaginary are important in their own right. In both cases, all coeﬃcients can be reduced to unity by rescaling; the resulting parameterless forms are called, respectively, the real Ginzburg–Landau (RGL) equation (1.26) ut = ∇2 u + u(1 − |u|2 ), and the nonlinear Schr¨ odinger (NLS) equation −iut = ∇2 u + u(1 − |u|2 ).

(1.27)

Equations (1.20), (1.21), and (1.24), as well as (1.26), are gradient systems, and are derivable from an appropriate “energy” functional: ut = −δF/δu, F = L(u, ∇u, . . .) dx. (1.28) The essential property of gradient dynamics is that it ensures dissipative temporal evolution. Indeed, the energy integral must decrease until a minimum is reached: 2 δF dF δF ∂u = dx = − dx ≤ 0. (1.29) dt δu ∂t δu The gradient property may only be approximate and appear in reduced equations even when the original equations describe an open system and lack gradient structure. On the other hand, although RDSs are commonly associated with nonequilibrium systems, they can also be used for the description of equilibrium phase transitions. In particular, the gas–liquid equilibrium in the van der Waals ﬂuid (van der Waals, 1894) is described in the mean ﬁeld approximation by a single reaction-diﬀusion equation with u standing for density and a function f (u) with two stable zeros corresponding to equilibrium densities of two stable phases. 1.1.4 Spectral Decompositions Spatially extended nonequilibrium systems are capable, in principle, of exhibiting extremely complex dynamics. With a great number of degrees of

18

1 Dynamics, Stability and Bifurcations

freedom present, one could expect turbulent behavior of unfathomable complexity to be a norm. This, however, does not always happen. More often than not, the latent complexity of behavior is not realized, and a homogeneous state or an ordered or weakly distorted pattern prevails. This suggests that a dynamical system involving a limited number of variables might be suﬃcient to faithfully describe the dynamic behavior in lieu of the full partial diﬀerential equation (PDE) system, at least approximately and within a certain range of parameters. The choice of such a set of variables is by no means obvious. In the absence of other hints, one can expand the variables in a Fourier series, say, aj ei(kj ·x−ωj t) . (1.30) u(x, t) = j

The amplitude aj of the mode with the wave vector kj and frequency ωj deﬁned by a dispersion relation ω(k) is complex, unless both kj and ωj vanish. When the expansion is used in the original PDE, diﬀerent modes couple through nonlinear terms, and, generally, no ﬁnite number of modes would ever suﬃce, as, given two modes with wave vectors ki , kj , their sum will always be generated. The expansion can be, however, truncated, leaving only a small number of modes and neglecting the rest. The resulting ODE system with polynomial functions fi (a) dependent on the array a of amplitudes aj of the chosen modes may look like “chemical” systems of Sect. 1.1.1 derived from the mass action law. Of course, one can never be sure which, if any, features of the original system are preserved by such a bold action, but, with good luck, it may bring success. In this way, Lorenz (1963) constructed a threevariable dynamical system (see Sect. 1.5.4) by retaining three modes of the Navier–Stokes and convective diﬀusion equations describing Rayleigh–B´enard convection. The resulting system was not actually faithful to the original, but turned instead (after some 15 years of obscurity) into a celebrated paradigm of chaotic dynamics. A more systematic (though empirical) approach to spectral truncation is oﬀered by the Karhunen–Loewe method used, in particular, for compressing images and discerning “coherent structures” in turbulent ﬂow (Sirovich and Everson, 1992). This method employs statistical analysis of measured or computed data to ﬁnd out an optimal combination of modes suﬃcient to reproduce the data in a faithful but economic way. This combination would, of course, change when the underlying system is modiﬁed or driven to a diﬀerent regime. It is far more appealing to be able to discern relevant modes analytically; this may, indeed, be possible under favorable conditions. The key is given by scale separation: some modes may evolve on far longer characteristic times than do the others. Fast modes relax to their quasistationary values dependent on the current values of slow modes, which remain virtually unchanged during this relaxation period. Following this, slow modes evolve on a longer time scale, while fast modes follow them quasistationarily, being bound by

1.1 Basic Models

19

quasiequilibrium algebraic relations – much in the same way as rapidly reacting chemical species reaching a partial equilibrium (Sect. 1.1.1). This is expressed by Haken’s (1983) slaving principle: “fast modes are slaved by slow modes.” Clearly, only slow modes are relevant for long-time evolution. Therefore, the original PDE system, however complex it might be, can be reduced to a set of ODEs for the amplitudes of slow modes. Their general form can be predicted on the basis of scaling and symmetry considerations. Exact derivation connecting ODE parameters with parameters of the underlying PDE is possible when the amplitudes of slow modes are small, so that the state of the system deviates only slightly from a known “trivial” homogeneous stationary state – see Sect. 1.3. Amplitude equations or “normal forms” derived through this approach can reproduce the dynamics of the original underlying system faithfully only in a limited range of parameters and in a limited region of the phase space, since slow modes would, generally, be modiﬁed or cease to be slow when either parameters or values of dependent variables change. Nevertheless, they may reproduce dynamics correctly in a qualitative sense even outside the domain where their derivation can be justiﬁed formally. 1.1.5 Trajectories and Attractors Both (1.1) and (1.17) or (1.19) may be called evolutionary systems. A distributed system, such as an RDS, can be viewed as an inﬁnite-dimensional dynamical system. Qualitative methods of analysis of dynamical systems can be applied to distributed systems as well in many situations when all but few degrees of freedom are essentially suppressed. Particular solutions, or trajectories of a dynamical system u(t), can be obtained by numerical integration starting from certain initial conditions. In standard cases, ﬁnding a numerical solution is an easy task if one uses any of modern software systems, which require no programming. Moreover, more advanced systems are able to choose an optimal integration algorithm suitable for the given problem, so that even stiﬀ equations are well handled. Solving a PDE, such as an RDS, requires, in addition, deﬁning appropriate boundary conditions. This might be a hard decision to make in a situation when one is interested in studying dynamic behavior in an extended region and wishes to reduce the inﬂuence of the boundaries to a minimum. Quite often, periodic boundary conditions are chosen, which allows one to implement most eﬃcient spectral or quasispectral integration routines. Numerical simulation of RDSs in one or two dimensions, using widely available C- or Fortran-based programs with graphical interfaces, is also becoming routine, except in more diﬃcult cases involving internal boundary layers and widely separated scales. Simulation can be, however, no more than a tool for preliminary exploration: a numerical experiment. Even when such an experiment is carried out, one is usually interested in asymptotic behavior at long times, which may be either totally or to a certain degree independent of initial conditions. On a

20

1 Dynamics, Stability and Bifurcations

deeper level, both a researcher trying to understand the global structure of solutions and a designer (in a broader sense) looking for ways to control the system in order to achieve optimal performance would like to explore longtime behavior in a wide parametric region, which requires a systematic study. An asymptotic state approached by an evolutionary system in the limit t → ∞ is called an attractor . An attractor of a dynamical system may be a stationary state, a periodic, quasiperiodic, or chaotic orbit. The deﬁning property of a periodic orbit is u(t) = u(t + T ), where T is the period. A quasiperiodic orbit never returns to the same point of the phase space and covers a 2D or higher-dimensional torus. A chaotic orbit covers a manifold of fractal dimension; trajectories starting from neighboring points lying on a chaotic attractor diverge with time. The last two types are possible when the dimension of the phase space is at least three. An attractor of an RDS, as well as of any other nonlinear spatially distributed system, may be, in addition, a stationary periodic pattern, a wave train, a quasiperiodic pattern, or a disordered (turbulent) state. In any case, the dimension of the attractor is less than the dimension of the dynamical system itself, so that some degrees of freedom are essentially suppressed after a relatively short relaxation period. Out of an inﬁnite number of degrees of freedom of a distributed system, only few may be relevant in most situations (except in the state of developed turbulence). By deﬁnition, an attractor is an asymptotically stable stationary state or orbit. The property of asymptotic stability implies that the system removed from the attractor by a suﬃciently small perturbation returns hereto asymptotically at t → ∞. Any attractor must therefore have a ﬁnite attraction basin. If there are two or more alternative attractors, each of them attracts trajectories starting from initial conditions lying within its basin. Trajectories starting from points lying precisely at a basin boundary cannot reach any attractor and should approach at t → ∞ a saddle point or an unstable saddle orbit lying on the basin boundary. Such trajectories are, of course, themselves unstable, as any inﬁnitesimal perturbation would deﬂect them into one of the bordering attraction basins. Basin boundaries may be as simple or as convoluted as the attractors; thus, a basin boundary of a chaotic attractor would be a fractal object as well. As the parameters of the system change, so do its attractors and basins. Generically, the changes are only quantitative, but at certain values of parameters qualitative changes take place: attractors change their character or disappear, and new attractors emerge. These qualitative changes are called bifurcations. It is advantageous to view bifurcations in geometric terms by considering a p-dimensional parametric space. Qualitative changes of behavior of a dynamical system take place when certain boundaries are crossed. These boundaries are called bifurcation manifolds. The dimension of a boundary of a p-dimensional domain is p − 1, and its codimension is one; thus, a generic bifurcation manifold is a codimension one hypersurface in the parametric space.

1.2 Linear Analysis

21

The vicinity of a bifurcation manifold is of special interest, since the behavior characteristic to the two neighboring parametric domains can be studied there; moreover, if the structure of the attractor(s) in one of the domains is more complicated, it might be more amenable to rational analysis close to bifurcation. One can advance further and look for degenerate, or singular , bifurcations, which require two or more conditions to be satisﬁed simultaneously and occur at bifurcation manifolds of higher codimension. Singular bifurcations are hubs of complexity: a variety of behaviors can be observed in their close vicinity, and diﬀerent kinds of transitions are possible there.

1.2 Linear Analysis 1.2.1 Instabilities of Stationary States A comprehensive study of a distributed system, such as an RDS or a general form (1.19), may start in a parametric domain where the dynamic behavior is most simple. The system should possess an HSS – a trivial solution that retains all symmetries of the equations. Close to thermodynamic equilibrium, when the driving ﬂuxes are weak, this state is expected to be stable and be the only attractor of the system, i.e., the state approached asymptotically starting from any initial condition. When the system is driven farther from thermodynamic equilibrium, its behavior may become far more complicated. The trivial solution may become unstable, and the system may tend asymptotically to attractors of diﬀerent nature: oscillatory, inhomogeneous (patterned), or chaotic (turbulent) states. The ﬁrst step in a search for nontrivial states is therefore to ﬁnd the locus of a primary bifurcation where the trivial state ﬁrst becomes unstable. This transition is indicated by local linear analysis in the vicinity of the HSS. As an example, consider the general RDS (1.18) in the inﬁnite plane and suppose that it has an HSS u = us satisfying f (us ) = 0. Without loss of generality, we can assume us = 0, since the stationary values can be taken as the origins of the respective variables. Thus, we assume that the vector-function f (u) vanishes at the origin. We shall now suppose the deviations from the trivial state to be arbitrarily small, and linearize (1.18) to obtain (1.31) ∂t u = D∇2 + F u, where F is the Jacobi matrix of the vector-function f (u) with the elements Fij = ∂fi /∂uj evaluated at u = 0. Solutions of (1.31) are sought for in the harmonic form u = U eλt+ik·x .

(1.32)

Using this in (1.31), we see that nontrivial solutions are obtained when λ(k 2 ) are eigenvalues of the matrix

22

1 Dynamics, Stability and Bifurcations

L = F − k 2 D.

(1.33)

Because of the isotropy, the eigenvalues depend only on the absolute value k = |k| rather than on the direction of the wave vector k. In a spatially inﬁnite system, each eigenvalue λj (0) of the Jacobi matrix F is blown up into a continuous3 spectral band λ(k). The stationary state at the origin is stable when all spectral bands lie entirely in the left complex half-plane, i.e, for any j, k, (1.34) Re λj (k) < 0. Let the source function f (u) = f (u; µ) depend on some parameter µ. Suppose for simplicity that f (u; µ) vanishes at the origin for all values of µ, and assume that the stability condition (1.34) is satisﬁed when µ is suﬃciently small. Increasing µ sets the spectral bands λj (k; µ) into motion. We shall refer to a band containing the eigenvalue with the largest real part as the leading band and label the eigenvalues from this band as λ0 ; this band will be responsible for the onset of instability when it ﬁrst occurs at some µ = µ0 . At this value of the parameter, the spectrum of the linearized system (1.31) satisﬁes for some k0 the condition of marginal stability Re λ0 (k0 ) = 0,

Re λj (k) < 0

otherwise.

(1.35)

One has to distinguish among four possibilities, depending on whether the eigenvalues from the leading band are real or complex and whether the rightmost edge of this band corresponds to k0 = 0 or k0 = 0. We shall call the mode at the right edge of the leading band the leading mode. The leading mode, which becomes marginally stable at the critical point µ = µ0 , is • • • •

stationary homogeneous when k0 = 0, Im λ0 (0) = 0, oscillatory homogeneous when k0 = 0, Im λ0 (0) = 0, stationary inhomogeneous when k0 = 0, Imλ0 (k0 ) = 0, and oscillatory inhomogeneous when k0 = 0, Im λ0 (k0 ) = 0.

When the parameter µ further increases beyond the point of marginal stability, one can expect that the growth of perturbations in the leading mode would lead (provided it saturates at a certain amplitude) to a new steady state of the same character, i.e., stationary at Im λ0 (k0 ) = 0, oscillatory at Im λ0 (k0 ) = 0, homogeneous at k0 = 0, and inhomogeneous at k0 = 0. The linear analysis, of course, gives but a hint at the existence of this “nontrivial” state in the postcritical region, and further nonlinear analysis is required to assure that this state does actually exist. In the case λ0 = k0 = 0, the determinant of the Jacobi matrix F vanishes. This means that it is impossible to continue the branch of stationary solutions 3

If the region is ﬁnite, each spectral band includes only a discrete (inﬁnite countable) set of eigenvalues. These eigenvalues come closer when the characteristic spatial scale L increases, and the band becomes continuous in the limit L → ∞.

1.2 Linear Analysis

23

in a unique way when the parameters of the problem are changed. Most commonly, this is a fold , or saddle-node bifurcation point where two stationary states, one stable and one unstable, “collide” and disappear (Sect. 1.3.2). For a scalar (one-component) RDS, the leading mode is always stationary and homogeneous. Indeed, it is easy to see that in this case there is only one band of real eigenvalues λ = fu − Dk 2 , where fu = (df /du)u=0 , and its right edge is always at k = 0 inasmuch as D > 0. But already for two coupled equations (n = 2), diﬀerent kinds of leading modes are possible. 1.2.2 Turing and Hopf Instabilities Two principal symmetry-breaking instabilities can be obtained already in a two-component RDS. We shall write a general two-component RDS with a constant diagonal matrix of diﬀusivities in the form ut = D1 ∇2 u + f (u, v), vt = D2 ∇2 v + g(u, v),

(1.36)

where f (u, v), g(u, v) are source functions depending on the state variables u and v. The explicit form of the matrix L in (1.33) is fu − D1 k 2 fv , (1.37) L= gu gv − D2 k 2 where all derivatives fu , etc. are computed at u = v = 0. Two conditions of marginal stability are provided by vanishing trace and determinant of (1.37): tr(L) ≡ fu + gv − (D1 + D2 )k 2 = 0, det(L) ≡ D1 D2 k 4 − (D2 fu + D1 gv )k 2 + ∆ = 0,

(1.38)

where ∆ = fu gv −fv gu is the Jacobian of the source vector-function. The trace condition corresponds to the onset of oscillatory instability (Hopf bifurcation). It is clear that this instability can occur only in the homogeneous mode.4 The determinant, however, may have a minimum at k02 =

D2 fu + D1 gv . 2D1 D2

(1.39)

This minimum occurs at a real k0 only in the presence of positive feedback, or, in chemical terms, when at least one of the species is autocatalytic (fu > 0 or gv > 0). Under these conditions, a stationary inhomogeneous mode is excited, leading to spatial symmetry breaking. This can actually occur, however, only 4

One would have to increase the number of state variables to three to obtain inhomogeneous oscillatory instability, which would lead to an emergence of a wave pattern directly from the trivial state; nevertheless, as we shall see in Chap. 3, stable wave patterns also exist in two-component systems.

24

1 Dynamics, Stability and Bifurcations

if the condition of the Hopf bifurcation has not yet been satisﬁed. We have to presume therefore fu + gv < 0, which means that spatial symmetry breaking would occur when only one of the species is autocatalytic. To be deﬁnite, let fu > 0 and gv < 0. Then the spatial symmetry breaking (Turing bifurcation) preempts the Hopf bifurcation at D1 < D2 , i.e., when the autocatalytic species is less diﬀusive. Thus, for spatial symmetry breaking in a two-component system, one needs a combination of a slowly diﬀusing activator and a rapidly diﬀusing inhibitor. 1.2.3 Dispersion Relations In a general case, the linearization of (1.19) can be written as L1 (∂t , ∂z , ∇)v = 0.

(1.40)

where v = u − u0 is a deviation from some known solution u = u0 . If this solution is stationary and homogeneous, the operator L1 does not contain explicit time and coordinate dependence (except, perhaps, on the “vertical” coordinate z). Then the eigenfunctions verifying L1 ψ = 0 can be presented in the form (1.41) ψ(x, z, t; k) = ei(k·x+λ(k)t) U (z; k). where λ(k) are eigenvalues. The dependence λ(k) is called a dispersion relation. In the absence of vertical structure, the dispersion relation is obtained by direct substitution ∂t → λ, ∇ → ik and ﬁnding the eigenvalues of the matrix L1 (λ, k) grouped in n spectral bands. In the case of a single ﬁeld u, the dispersion relation in an implicit form is given directly by L1 (λ, k) = 0. When z-dependence is present, the dispersion relation is obtained by solving a boundary problem for an ODE in z with appropriate boundary conditions. The overall evolution of an initial perturbation is described by the Fourier integral (1.42) aj (k) eλj (k)t+ik·x U j (z; k) dk, v(x, z, t) = j

where aj (k) is the amplitude of the mode with the wave vector k from the jth spectral band. This cumbersome formula is of little use as it stands, but becomes practical if one is able to identify physically relevant modes, e.g., slow modes in the vicinity of a bifurcation point. After choosing a relevant set of modes, one can further assume that their amplitudes are slowly changing with time and are modulated on an extended spatial scale, i.e., depend on extended temporal and spatial coordinates T, X rather than being constant: aj (X, T ; kj ) eλj (kj )t+ikj ·x U j (z; kj ). (1.43) v(x, z, t; X, T ) = j

Summation over bands and integration are replaced here by summation over the chosen set of modes. Because of a weak spatial and temporal dependence

1.2 Linear Analysis

25

of the amplitudes, this expression incorporates band segments centered on kj , rather than isolated modes. In an isotropic system, the eigenvalues λj and the eigenfunctions U j (z) depend on kj = |kj | only. As a consequence, an entire ring of slow modes appears near a symmetry-breaking bifurcation in two dimensions. This seems to indicate that integration over angular directions of kj should be reintroduced in (1.43). This is, however, not necessary, since, as a result of competition among diﬀerently directed modes caused by nonlinear interactions, only a small number of modes would survive (more on this in Sect. 1.6.1). While linear analysis is the necessary tool for detecting instabilities and identifying slow modes, one can question the utility of linear theories for the study of actual dynamic behavior: indeed, the dynamics is not interesting in stable situations when the system just relaxes back to the basic state, while in unstable situations exponential growth of perturbations promptly makes nonlinear terms essential. Nevertheless, linear theory can sometimes be used beyond the instability limit. One such case is the retreat of an unstable state under an onslaught of an alternative stable solution. It turns out that the speed of the retreat and even the character of the eventually established stable state is determined by the dynamics at the leading edge of the propagating front where linear theory is still useful (see Sect. 2.3). Another case is convective instability in systems of ﬁnite spatial extent. The instability is called convective when the band of unstable eigenvalues does not encompass k = 0. If the instability is observed at a ﬁnite wavelength and frequency, the relevant eigenfunctions close to criticality are waveforms a(T ) ei(k·x−ωt) , where ω = Im λ. Under these conditions, all perturbations are washed out with a characteristic speed equal to the group velocity dω/dk. Therefore, while Re λ remains small, perturbations may not succeed to grow to a ﬁnite amplitude, provided the spatial extent of the system is suﬃciently small. Under these conditions, the unstable state may persist, provided the intensity of external perturbations is not too large, and dynamics of deviations from the unstable state presented by (1.42) or (1.43) can be reasonably well described in the linear approximation. This will not happen when a mode with k = 0 is unstable, and the instability is absolute. 1.2.4 Instabilities of Periodic Orbits Periodic orbits or nonuniform stationary states emerging at a primary bifurcation may in turn lose stability at secondary bifurcation points. Stability of a T -periodic orbit u0 (t) = u0 (t + T ) to inﬁnitesimal perturbations is determined by linearizing the underlying equation in the vicinity of this solution. Consider a general dynamical system (1.1) and look for solutions in the form u(t) = u0 (t) + v(t), where v(t) is a small deviation from the periodic solution u0 (t). The linearized system for v(t) is ∂t v = F (t)v.

(1.44)

26

1 Dynamics, Stability and Bifurcations

The fundamental solution of (1.44) is a matrix A(t), such that v(t) = A(t)v(0) for any initial condition v(0). Stability of the T -periodic solution u0 (t) is determined by eigenvalues Λj of the monodromy matrix A(T ), called Floquet multipliers. The periodic orbit is stable when all Λj are less than unity by their absolute value. Alternatively, solutions of (1.44) can be sought for in the form v(t) = eiλj t w(t), where w(t) is T -periodic and λj = T −1 ln Λj are Floquet exponents, which should all have nonpositive real parts for a stable orbit. Among the Floquet multipliers, there is always one equal to unity, which reﬂects neutral stability to a shift along the periodic orbit. The respective eigenvector is v 0 = u0 (t). One can see, indeed, that the r.h.s. of (1.44) with v = u0 (t) equals the time derivative of f (u0 (t)), and therefore integrating it over the period yields v 0 (T ) − v 0 (0) = f (u0 (T )) − f (u0 (0)) = 0. Additional neutrally stable eigenvectors exist in the presence of conservation laws. Other Floquet multipliers should lie within the unit circle in the complex plane when the periodic orbit is stable. The loss of stability can occur in three distinct ways, depending on whether the value of the multiplier with the largest absolute value at the critical point of marginal stability is 1, −1, or a complex number Λ = eiω . As in the case of stationary states in Sect. 1.2.1, one can expect that the growth of perturbations in the leading mode would lead (provided it saturates at a certain amplitude) to a new orbit of a kind indicated by the instability mode. At Λ = 1 (i.e., λ = 0), this might be another periodic orbit; most commonly, this indicates a saddle-node bifurcation of periodic orbits where two orbits, one stable and one unstable, collide and disappear. A transition at Λ = −1 (i.e., λ = iπ) indicates period doubling. A complex Floquet multiplier on the unit circle signals emergence of a new frequency. The expected new state is a periodic orbit with the period 2nT if ω/π = m/n is rational, or a quasiperiodic orbit covering a torus if ω/π is an irrational number. A quasiperiodic orbit is usually frequency-locked at larger amplitudes, turning into an orbit with a ﬁnite period. For an RDS, the Floquet stability problem reduces to the form (1.44) with F (t) replaced by F (t) − k 2 D after substituting v(x, t) = V (t; k) eik·x . As in Sect. 1.2.1, each Floquet exponent then expands into a spectral band λj (k). Instabilities breaking spatial symmetry, leading to propagating waves, are possible when λj (k) in the leading band is at maximum at k = 0. As a rule, Floquet multipliers can be computed only numerically. In some special cases, stability can be determined analytically using scale separation, as it is done for time-periodic orbits in Sect. 1.4.3 and for space-periodic solutions in Sect. 3.2.3, or taking advantage of a particularly simple form of a periodic solution as in Sect. 4.1.5. Stability of nonstationary and nonuniform states in the vicinity of a primary bifurcation is determined in a systematic way by weakly nonlinear analysis as described in Sects. 1.3 and 1.6.

1.3 Weakly Nonlinear Analysis

27

1.3 Weakly Nonlinear Analysis 1.3.1 Multiscale Expansion In the vicinity of a bifurcation manifold, the original system of equations can be reduced to a simple universal form that is characteristic to the particular type of bifurcation and retains the qualitative features of dynamic behavior of the underlying system in an adjacent parametric domain. The reduced equations are called normal forms in the mathematical literature, while the term preferred by physicists is amplitude equations. Normal forms are ODEs describing dynamics of the amplitudes of slow modes (cf. Sect. 1.1.4), while amplitude equations may include diﬀerential terms operating on an extended spatial scale. The approach to their derivation also diﬀers: mathematicians base it on “near-identity transformation,” while the favorite tools of physicists is multiscale expansion. The advantage of the latter approach lies in a possibility to connect directly the parameters of amplitude equations and the characteristic time of their evolution to parametric regions of the underlying dynamic system and its time scale. A more abstract mathematical approach may be preferable when one wishes to obliterate these details and concentrate on the qualitative features of the dynamics. The formal procedure of multiscale expansion starts with expanding both variables u and parameters p in powers of a dummy small parameter near some known HSS u = u0 (p0 ). All nonlinear terms are expanded in powers of deviations u − u0 ; e.g., the general nonlinear operator (1.19) is expanded as H(u, ∂t u, ∂z u, ∇u) = L1 (∂t , ∂z , ∇)(u − u0 ) + L2 (∂t , ∂z , ∇)(u − u0 )2 + · · · (1.45) No zero-order term appears here, since u0 (p0 ) must verify (1.19). The operator L1 is the linearization of H; slow modes appear in the spectrum of this operator at p close to p0 if p0 lies on a bifurcation manifold of the original system, so that L1 has an eigenfunction with an eigenvalue with vanishing real part, L1 Φ = iω. It is always assumed that all fast modes are decaying, so that at p = p0 there are no eigenvalues with a positive real part. Next, one introduces a hierarchy of time scales tk rescaled by the factor k and expands the time derivative in a series of partial derivatives ∂/∂tk . The hierarchy of equations approximating the dynamics near the reference point in successive orders is obtained in the consecutive orders of the expansion of (1.19). These equations are solved in turn, and normal forms appear as lowestorder nontrivial solvability conditions. Additional restrictions on parametric deviations may be imposed when needed to make the orders of magnitude of diﬀerent terms compatible. Multiple spatial scales may also be introduced to describe weak spatial modulations of slow modes. Practical computation of the coeﬃcients of the amplitude equations is a tedious task, but simple examples can be worked out explicitly. Since the problem is fully formalized, it can be easily handled by symbolic computation programs (Pismen and Rubinstein, 1999a).

28

1 Dynamics, Stability and Bifurcations

Consider a general dynamical system (1.1) written in a vector form du = f (u; p), dt

(1.46)

where the nonlinear vector-function f (u) of the n-dimensional state array u is also dependent on the array of parameters p. Suppose that we know the location of a zero of the function f (u; p) at a certain point p = p0 in the parametric space, i.e., a solution u = u0 satisfying f (u0 ; p0 ) = 0. Near this point, f (u; p) can be expanded in Taylor series in both variables and parameters. Since we do not know beforehand what are relative orders of magnitude of all deviations, it is convenient to start with expanding both u and p in powers of a dummy small parameter : u = u0 + u1 + 2 u2 + · · · ,

p = p0 + p1 + 2 p2 + · · ·

(1.47)

Then we have f (u; p) = f (u0 ; p0 ) + (f u u1 + f p p1 ) (1.48) 2 1 1 + f u u2 + f p p2 + 2 f uu [u1 , u1 ] + f up [u1 , p1 ] + 2 f pp [p1 , p1 ] + · · · , where the arrays of ﬁrst (f u , f p ), second (f uu , f up , f pp ), etc. (e.g., f uuu ) derivatives with respect to variables and parameters are evaluated at the reference point u = u0 , p = p0 . Higher derivatives are deﬁned as arrays of a higher tensor rank: thus, mth derivative acts at m vectors (state arrays) written to the right of it in brackets. We shall also need to expand the time derivative in a series of partial derivatives ∂k ≡ ∂/∂tk : d = ∂0 + ∂1 + 2 ∂2 + · · · . dt

(1.49)

The hierarchy of equations approximating the dynamics near the reference point in successive orders is obtained in the consecutive orders of the expansion of (1.46). The ﬁrst-order equation is ∂0 u1 = fu u1 + fp p1 .

(1.50)

The homogeneous linear equation Lu1 ≡ (fu − ∂0 )u1 = 0

(1.51)

obtained here by setting p1 = 0 governs the stability of the stationary state u = u0 to inﬁnitesimal perturbations. Stability of a ﬁxed point is determined by the location of the leading eigenvalue in the complex plane, i.e., that with the largest real part. It is usually possible to detect a parametric domain suﬃciently close to thermodynamic equilibrium where there is a single stable stationary state, and follow changes of dynamics along a certain path in the parametric space. In this case, all eigenvalues of L will have negative real parts

1.3 Weakly Nonlinear Analysis

29

on the one side of the bifurcation manifold, while the real part of the leading eigenvalue vanishes on the codimension one bifurcation hypersurface. The only two types of codimension one bifurcations that can be located by local (linear) analysis are a monotonic bifurcation where a real leading eigenvalue vanishes and a Hopf bifurcation where the leading pair of complex conjugate eigenvalues is purely imaginary. Additional conditions may deﬁne bifurcation manifolds of higher codimension. On consecutive stages of the expansion, we shall arrive at inhomogeneous equations of the type Lu + Ψ = 0, (1.52) where the operator L has a zero eigenvalue. By Fredholm alternative, an inhomogeneous equation is solvable if the inhomogeneity Ψ does not project on the corresponding eigenfunction Φ. This solvability condition is expressed as Φ† , Ψ , where . . . is an appropriate scalar product and Φ† is the eigenfunction with zero eigenvalue of the adjoint operator L† = fu † + ∂0 ,

(1.53)

where fu † is the transpose of the real matrix fu . For dynamical systems, we have Φ = eiωt U , where U is the eigenvector of L with zero eigenvalue. The scalar product of two vectors, say, U † and U , is deﬁned simply as † U † , U = U · U , where the overline denotes the complex conjugate. Since the eigenvectors are deﬁned up to a constant, one can impose a normalization † condition, e.g., U · U = 1. 1.3.2 Bifurcation of Stationary States A single real mode with unbroken symmetry corresponds to k = 0, ω = Im λ = 0 and describes homogeneous time-independent deviations from a known homogeneous stationary state. The equation for the amplitude a of this mode truncated at the order n should have the form. a˙ =

n

µj aj .

(1.54)

j=0

A bifurcation at zero eigenvalue is also a bifurcation of equilibria of the dynamical system. The number of distinct equilibria equals to the order n of the polynomial on the r.h.s. In the absence of symmetries, one has to ﬁx n − 1 parameters to arrive at the nth-order form. In this and other dynamic equations of this section, the coeﬃcient at the highest-order term is dependent only on the position of the reference point p0 on the bifurcation manifold, while all other coeﬃcients depend on parametric deviations. Diﬀerent parametric deviations, e.g., in the directions parallel and normal to the bifurcation manifold, may be scaled diﬀerently to balance the orders of magnitude of diﬀerent terms. One of these coeﬃcients, most conveniently, µn−1 , can be eliminated

30

1 Dynamics, Stability and Bifurcations

by shifting the amplitude by a suitable constant, a → a − c. If a = O(), the slow time variable in (1.54) is extended by the factor n−1 . By choosing the time scale, the highest coeﬃcient, µn , can be rescaled to ±1. With n odd, µn = −1 ensures that the amplitude always remain ﬁnite. Equation (1.54) has a gradient structure with the potential a˙ = −

∂V , ∂a

V =

n

j −1 µj aj+1 .

(1.55)

j=0

This structure also appears when the original equations do not possess it; thus it may characterize the dynamics in the vicinity of the bifurcation point only, rather than globally. Setting in (1.54) n = 2, µ1 = 0 yields the lowest-order quadratic equation describing dynamics near a point in the parametric space of the original system where two equilibria, one stable and one unstable, bifurcate. The dependence of the ampitude on the parameter µ0 (shown in Fig. 1.2a for µ2 = −1) has the form of a fold; hence, the name fold bifurcation. If all other eigenvalues of the original system have negative real parts, the unstable state is a saddle; the stable state is a node in the simplest case when the original system is a two-variable dynamical system. The transition is therefore also called a saddle-node (sn) bifurcation. Two ﬁxed points a = ± −µ0 /µ2 exist when the parameters µ0 , µ2 have opposite signs, and none when the signs are the same. In the latter case, the absolute value of the amplitude grows indeﬁnitely with time. Physically, it indicates escape of the original system from the region in its phase space where dynamics is slow to another attractor, which is not described by the reduced one-mode system. This attractor cannot be aﬀected by the change of the parameters µ0 , µ2 and keeps attracting some trajectories repelled by the unstable ﬁxed point, but other trajectories are attracted, following the bifurcation, to the newly created stable ﬁxed point. The same equation with n = 2, µ0 = 0, µ1 = 0 describes a transcritical bifurcation where stability is interchanged between the two ﬁxed points a = 0 and a = −µ1 /µ2 (Fig. 1.2b). Forms of this kind naturally appear in systems with a persistent trivial state, e.g., an extinction state in equations of population dynamics. Inasmuch as negative values of ui are nonphysical and are never reached by trajectories starting in the positive domain, also even-order polynomials are suitable in this case for the description of global dynamics. Equation (1.54) with n = 3 presents parametric unfolding of dynamics in the vicinity of a cusp singularity. Generically, it can be attained by ﬁxing an additional parameter of the underlying system, i.e., at a codimension two manifold in the parametric space. The dynamics is restricted to small amplitudes, provided µ3 < 0. Assuming this to be true, this parameter can be rescaled to −1, while µ2 can be set to zero by an amplitude shift. The resulting cubic equation, a˙ = µ0 + µ1 a − a3 ,

(1.56)

1.3 Weakly Nonlinear Analysis (a)

31

(b) a

a

Μ0

Μ1

(c)

(d)

Μ0

a

Μ1

Μ1

(e)

(f) a

a

Μ1

Μ1

Fig. 1.2. Dependence of the real amplitude on a bifurcation parameter for elementary bifurcations. (a) Fold (saddle-node). (b) Transcritical. (c) Cusp (no symmetry); the region of multiple stationary states in the parametric plane µ0 , µ1 is shown in (d). (e) Supercritical pitchfork, Hopf or Turing bifurcations. (f) Subcritical pitchfork, Hopf or Turing bifurcations described by a symmetric ﬁfth-order normal form. Solid lines denote stable states, and dashed lines denote unstable states

has at µ0 = 0 a bifurcation diagram shown in Fig. 1.2c. Three equilibria – two stable and one unstable – exist in the cusped region µ1 > 0, |µ0 | < 2(µ1 /3)3/2 (Fig. 1.2d). Outside this region, there is a unique stable equilibrium. As the coeﬃcient µ1 increases, a fold bifurcation takes place, creating a pair of stationary states, one stable and one unstable. One branch of solutions is not aﬀected by the bifurcation, and solutions belonging to this branch remain stable. With µ1 decreasing, the system residing initially on the lower branch

32

1 Dynamics, Stability and Bifurcations

of stable solutions jumps to the upper branch; this is a ﬁrst-order transition involving a ﬁnite change of the amplitude. A reverse transition to the lower branch of stable solutions at µ1 exceeding the critical value can take place only as a result of a ﬁnite perturbation, which would bring the system within the attraction basin of the alternative stable state. In systems with inversion symmetry, µ0 must vanish; then the bifurcation diagram is symmetric, as in Fig. 1.2e. In this case, a second-order transition, called a pitchfork bifurcation, takes place: the trivial state a = 0 persists through the bifurcation point but becomes unstable, and the two symmetric stationary states that exist at µ1 > 0 have inﬁnitesimal amplitudes when they emerge at the bifurcation point. Continuing to higher orders, one can obtain equations with a larger number of ﬁxed points, which present unfoldings of higher order singularities attained on manifolds of higher codimension. A symmetric bifurcation diagram obtained at µ3 > 0, µ5 = −1, and even-order coeﬃcients vanishing is shown in Fig. 1.2f. This diagram combines a subcritical bifurcation of a pair of unstable nontrivial solutions at µ1 = 0 and a fold bifurcation at µ1 = −µ23 /4. The nontrivial stable stationary states appear in this case via a ﬁrst-order transition. Singularities of fourth and ﬁfth order have ﬂamboyant names: “butterﬂy” and “swallowtail.” General nonsymmetric forms already generate a variety of bifurcations. The procedure of the analysis of parametric dependence of multiple stationary states, though rather straightforward, has been a subject of a huge body of mathematical literature. A catchy term catastrophe theory was coined (Thom, 1975) to describe the sequence of bifurcations of equilibria. The philosophical balloon growing from this theory was ridiculed by Arnold (1984). Numerous studies of bifurcation of equilibria classiﬁed in minute detail a variety of cuts that can be made through bifurcation manifolds (containing diﬀerently inclined folds, isolas, etc.), as well as their modiﬁcation due to innate or accidental symmetries of the underlying system. 1.3.3 Derivation of Amplitude Equations Formal derivation of amplitude equations is carried out following the multiscale expansion procedure outlined in Sect. 1.3.1. Setting in (1.50) u1 = const, i.e., ∂0 u1 = 0, we ﬁnd the shift of the stationary solution due to a small variation of parameters u1 = −fu −1 fp p1 .

(1.57)

The rectangular array fu −1 fp is recognized as the parametric sensitivity matrix. Continuous dependence on parameters can be used to construct a branch of equilibria as long as the matrix fu has an inverse. This branch terminates at a bifurcation manifold where the matrix fu has zero eigenvalue and, consequently, its determinant vanishes.

1.3 Weakly Nonlinear Analysis

33

Suppose now that p0 is a point on this bifurcation manifold. Then one can neither construct a stationary solution at values of parameters close to this point nor characterize stability of the equilibrium in the linear approximation. The dynamics in the vicinity of the bifurcation manifold is governed by a nonlinear amplitude equation to be obtained in higher orders of the expansion. Generically, the zero eigenvalue is nondegenerate. Let U be the corresponding eigenvector satisfying fu U = 0. Then u1 = a(t1 , t2 , . . .)U

(1.58)

is the solution of the homogeneous linear equation (1.51) that remains stationary on the fast time scale t0 . The amplitude a is so far indeterminate and can depend on slower time scales. The inhomogeneous equation (1.50) has solutions constant on the rapid time scale, provided its inhomogeneity does not project on the eigenvector U . This condition is (1) (1.59) µ0 ≡ U † · fp p1 = 0, where U † is the eigenvector of the transposed matrix satisfying U † fu = 0 normalized by U † · U = 1. Equation (1.59) deﬁnes the tangent hyperplane to the bifurcation manifold at the point p = p0 . Before writing up the second-order equation, we require that the secondorder deviation u2 remain constant on the rapid time scale (otherwise it might outgrow u1 at long times). The dependence upon slower time scales must be expressed exclusively through the time dependence of the amplitude a. Using (1.58), we write the second-order equation as fu u2 = ∂1 a U − fp p2 − afup [U , p1 ] − 12 a2 fuu [U , U ].

(1.60)

The solvability condition of this equation is (2)

(1)

(0)

∂1 a = µ0 + µ1 a + µ2 a2 ,

(1.61)

where the slow time derivative ∂1 is indexed according to (1.49). The parameters of (1.61) are µ0 = U † · fp p2 + 12 U † · fpp [p1 , p1 ], (2)

µ1 = U † · fup [U , p1 ], (1)

µ2 = 12 U † · fuu [U , U ]. (0)

(1.62)

The superscripts correspond to the scaling of respective parametric deviations from the bifurcation point. The parameters with the superscript zero are characteristic to the particular point on the bifurcation manifold. Generically, these parameters diﬀer from zero but may vanish on a manifold of higher codimension that corresponds to a degenerate bifurcation. In a generic case, one can consider only parametric deviations transverse (1) to the bifurcation manifold and set p1 = 0 to satisfy (1.59); then µ1 = 0, and the generic equation for slow dynamics near the bifurcation manifold becomes

34

1 Dynamics, Stability and Bifurcations (2)

(0)

∂1 a = µ0 + µ2 a2 .

(1.63)

This equation can also be obtained by shifting the amplitude in (1.61). Its solutions have been already discussed in the preceding subsection. Nongeneric situations may arise either because of intrinsic symmetries of the system or “accidentally” on a manifold of higher codimension. It may happen that the matrix fp , as well as higher derivatives with respect to parameters, vanish identically. This would be the case when u0 is a “trivial” solution that remains constant at all values of parameters. Then (1.59) is satisﬁed identically, and (1.61) reduces to (1)

(0)

∂1 a = µ1 a + µ2 a2 .

(1.64)

This equation, describing a transcritical bifurcation, has two solutions on both sides of the bifurcation manifold, but the two solutions interchange stability when this manifold is crossed. (0) If µ2 = 0, the expansion should be continued to the next order. The (0) coeﬃcient µ2 may vanish identically because of the symmetry of the original problem to inversion of u. Otherwise, it can be equal to zero at certain values of the parameters of the problem. Generally, the two conditions, vanishing (0) of both the determinant of fu and µ2 , are satisﬁed simultaneously on a codimension two manifold in the parametric space that corresponds to a cusp singularity. In order to continue the expansion, we restrict parametric deviations in such a way that the dependence on t1 be suppressed. Deviations transverse to the bifurcation manifold have to be restricted by the second-order condition (2) µ0 = 0, which is stronger than (1.59). First-order parametric deviations p1 parallel to the bifurcation manifold, which are still allowed by (1.59), should (1) now be restricted by the condition µ1 = 0. If the array p contains two para(1) (1) meters only, the conditions µ1 = 0 and µ0 = 0 imply, in a nondegenerate case, that ﬁrst-order parametric deviations should vanish identically. When more parameters are available, parametric deviations satisfying both these conditions are superﬂuous, since they correspond just to gliding into a closer vicinity of another point on the codimension two bifurcation manifold in a higher-dimensional parametric space. Further on, we shall therefore set p1 to zero identically. The dynamics unfolding on a still slower time scale t2 should be determined from the third-order equation fu u3 = ∂2 a U − fp p3 − afup [U , p2 ] − afuu [U , u2 ] − 16 a3 fuuu [U , U , U ]. (1.65) The second-order function u2 has to be found by solving Eq. (1.60), now reduced to the form fu u2 = −fp p2 − 12 a2 fuu [U , U ].

(1.66)

1.3 Weakly Nonlinear Analysis

35

Only the solution of the inhomogeneous equation, which does not project on the eigenvector U , is relevant. It can be expressed as a quadratic form in a containing two suitable vectors: u2 = U 22 + a2 U 20 .

(1.67)

The solvability condition of (1.65) is obtained then in the form (3)

(2)

(0)

∂2 a = µ0 + µ1 a + µ3 a3 ,

(1.68)

where µ0 = U † · fp p3 , (3)

µ1 = U † · fup [U , p2 ] + U † · fuu [U , U 22 ], (2)

µ3 = 16 U † · fuuu [U , U , U ] + U † · fuu [U , U 20 ]. (0)

(1.69)

This equation presents parametric unfolding of dynamics in the vicinity of a cusp singularity. The dynamics is conﬁned to the vicinity of u0 , provided (0) µ3 < 0. Assuming this to be true, (1.68) reduces to (1.56) by rescaling. (0) The condition µ3 = 0 deﬁnes a singular bifurcation manifold of codimension three. Again, it is possible to ﬁx parametric deviations to suppress the dynamics on the scale t2 , and obtain in the next order a quatric equation that represents the unfolding of the butterﬂy singularity. The procedure can be continued further if a suﬃcient number of free parameters are available, arriving in the nth order at the general form (1.54) with the parameters µj dependent on parametric deviations of O(n−j ). 1.3.4 Hopf Bifurcation The symmetry of the slow mode determines the structure of the normal form in a crucial way. At the Hopf bifurcation point where the frequency ω0 = Im(λ) diﬀers from zero, the amplitude is complex and even-order terms are forbidden by symmetry to phase shifts. The lowest-order normal form involving a single complex amplitude a is then a˙ = a(µ1 + µ3 |a|2 ),

(1.70)

where the coeﬃcients µj are complex. The dynamics is restricted to small amplitudes when the real part of µ3 is negative. Then the Hopf bifurcation is supercritical , and the bifurcation diagram has the form shown in Fig. 1.2e. The stationary state a = 0 is stable at Re(µ1 ) < 0, while at Re(µ1 ) > 0 it is unstable and the trajectories are attracted to the periodic orbit with the amplitude a = ρ eiΩT , where ρ2 = |a|2 = −Re(µ1 )/Re(µ3 ) and Ω = Im(µ1 )+Im(µ3 )ρ2 is an O(2 ) frequency shift. For Re(µ3 ) > 0, the bifurcation is subcritical . An unstable periodic orbit exists then at Re(µ1 ) < 0, i.e., under conditions when the trivial solution is stable. This orbit deﬁnes the boundary of the attraction basin of the trivial state.

36

1 Dynamics, Stability and Bifurcations

The coeﬃcients µj can be obtained, as in the preceding subsection, by applying the multiscale expansion procedure. At the Hopf bifurcation point, the parametric dependence of the stationary solution u = u0 (p) remains smooth; a linear correction can be obtained from the stationary equation (1.57). In order to simplify derivations, one can eliminate this trivial parametric depen = u−u0 (p). The resulting dynamic dence by transforming to a new variable u 0 = 0. system has the same form as (1.46) but has a persistent trivial solution u This way to eliminate parametric dependence is, however, not convenient for practical computations, since the stationary solution hardly ever can be computed explicitly. It is more convenient to present the dynamical system in the form du = f (u; p). (1.71) γ dt Then one can keep the entire set of parameters p ﬁxed, but vary only dynamic parameters entering the capacitance matrix γ, which do not aﬀect the ﬁxed points; this would usually be suﬃcient to locate a Hopf bifurcation point. At a Hopf bifurcation point, the Jacobi matrix J = γ −1 0 fu has a pair of imaginary eigenvalues λ = ±iω0 . In the vicinity of this point, we set γ = γ 0 + γ 1 + · · ·. The ﬁrst-order equation Lu1 ≡ (fu − γ 0 ∂0 )u1 = 0

(1.72)

has a nontrivial oscillatory solution u1 = a(t1 , t2 , . . .)Φ(t0 ) + c.c.;

Φ(t0 ) = eiω0 t0 U

(1.73)

with an arbitrary complex amplitude a(t1 , t2 , . . .) changing on an extended time scale; U is the eigenvector of J with the eigenvalue iω0 : J U ≡ γ −1 0 fu U = iω0 U .

(1.74)

The vector Φ(t0 ) and its complex conjugate Φ(t0 ) are two eigenfunctions of the linear operator L deﬁned by (1.72) with zero eigenvalue. The operator L acts here in the space of 2π/ω0 -periodic complex-valued vector-functions with the scalar product deﬁned as ω0 2π/ω0 u, v = u(t) · v(t) dt. (1.75) 2π 0 The eigenfunctions of the adjoint operator L† = J † + vecγ0 ∂0 are Φ† (t0 ) = eiω0 t0 U † and its complex conjugate; U † is the eigenvector of J † with the † eigenvalue iω0 , and a suitable normalization condition is U γ0 U = 1. The second-order equation is Lu2 = (γ 0 ∂1 + γ 1 ∂0 )u1 − 12 fuu [u1 , u1 ].

(1.76)

The inhomogeneity of this equation contains both the principal harmonic eiω0 t0 contributed by the linear terms and terms with zero and double frequency stemming from the quadratic term. The scalar products of the latter

1.3 Weakly Nonlinear Analysis

37

with the eigenfunction Φ† (t0 ) and its complex conjugate vanish, and the solvability condition of (1.76) is obtained in the form: (1)

†

(j)

where µ1 = −iω0 U · γ j U .

∂1 a = µ1 a,

(1.77)

This equation has a nontrivial stationary solution only if the real part of (1) µ1 vanishes. This condition deﬁnes a hyperplane in the parametric space of the capacitance matrix tangential to the Hopf bifurcation manifold. The shift along this manifold can be safely eliminated as in the preceding section by setting γ1 = 0. Then (1.77) reduces to ∂1 a = 0, so that the amplitude may evolve only on a still slower scale t2 . The second-order function u2 has to be found by solving (1.76), now reduced to the form Lu2 = − 12 (a2 e2iω0 t0 fuu [U , U ] + |a|2 fuu [U , U ] + c.c).

(1.78)

The solution of this equation is u2 = −|a|2 V 0 − 12 a2 e2iω0 t0 V 2 + c.c. V 0 = fu −1 fuu [U , U ],

V 2 = (fu − 2iω0 )−1 fuu [U , U ].

(1.79)

The third-order equation is Lu3 = (γ 0 ∂2 + γ 2 ∂0 )u1 − fuu [u1 , u2 ] − 16 fuuu [u1 , u1 , u1 ].

(1.80)

The amplitude equation is obtained as the solvability condition of this equation. Only the part of the inhomogeinity containing the principal harmonic contributes to the solvability condition, which takes the form coinciding with (1.70): (2) (0) (1.81) ∂2 a = µ1 a + µ3 |a|2 a, (0)

†

†

†

µ3 = 12 U · fuuu [U , U , U ] − U · fuu [U , V 0 ] − 12 U · fuu [U , V 2 ]. (1.82) The expansion can be continued to higher orders, after readjusting the scaling of parametric deviations, near a parametric point where the real part (0) of µ3 vanishes. This leads to a quintic equation on a slower O(−4 ) time scale describing dynamics in the vicinity of a subcritical Hopf bifurcation: (4)

(2)

(0)

∂4 a = µ1 a + µ3 |a|2 a + µ5 |a|4 a.

(1.83)

Although, generally, Im(µ3 ) = 0, this term can be cancelled by adding the respective O(2 ) frequency to the basic frequency ω0 at the Hopf bifurcation. Provided Re(µ5 ) < 0, the bifurcation diagram has the form shown in Fig. 1.2f. Since (1.83) retains the symmetry to phase shifts, all transitions depend only on real parts of the coeﬃcients µj . The only diﬀerence with the description of this diagram in the preceding section is that the saddle-node bifurcation creates a pair of periodic orbits (one stable and one unstable) rather than two symmetric pairs of stationary states.

38

1 Dynamics, Stability and Bifurcations

1.3.5 Degenerate Bifurcations More than one mode is necessary for the description of slow dynamics near a manifold in the parametric space where the underlying linearized system has more than one eigenvalue with vanishing real part. The degeneracy may be either “accidental” (attained at certain values of the parameters of the underlying system) or due to symmetry. Two real modes with unbroken symmetry (k = 0, ω = 0) interact in the vicinity of a singular bifurcation at double zero eigenvalue. In the absence of special symmetries, the bifurcation manifold is of codimension two, and the eigenvalue is degenerate geometrically as well as algebraically. Geometric degeneracy means that there is only one eigenvector U satisfying L0 U = 0. The orthogonal basis spanned by this vector and the rest of eigenvectors of L0 should be then complemented by an additional vector V satisfying L0 V = U . The matrix representation of L0 is reduced 01 in this basis to the normal Jordan form where the cell corresponds 00 to the double zero eigenvalue. This singular bifurcation is known as Takens– Bogdanov (TB) bifurcation. It is particularly interesting since in its vicinity one can study analytically global bifurcations that otherwise can be detected only numerically (see Sect. 1.4.1). If a1 , a2 are, respectively, the amplitudes of the modes with the eigenvectors U , V , a straightforward expression for a normal form with the speciﬁed linear part would be a˙ 1 = a2 + f1 (a1 , a2 ),

a˙ 2 = f2 (a1 , a2 ),

(1.84)

where the polynomials fi (a1 , a2 ) account for nonlinear interactions as well as small parametric deviations from the codimension two bifurcation manifold. This expression can be obtained by projecting the original equations on the eigenvectors U , V , expanding in Taylor series and truncating at a certain order; the formal orders of magnitudes of various terms would not, however, match. Equation (1.84) can be further simpliﬁed by choosing new variables x = a1 , y = a2 + f1 (a1 , a2 ). Taking note that the time variable in (1.84) should be slow, we observe that the two amplitudes x, y should be of diﬀerent orders of magnitude, so that y x. One can therefore neglect all terms nonlinear in y. Then (1.84) is rewritten as x˙ = y,

y˙ = f (x) + y g(x),

(1.85)

where f (x), g(x) are, respectively, nth and mth order polynomials. If the amplitude x is scaled as x = O() and the slow time variable as T = O(−(n−1)/2 ), the ﬁrst term on the r.h.s. of the second equation balances its left-hand side (l.h.s.), and the ratio of the second to the ﬁrst term is δ = O(m−(n−1)/2 ). If m > 12 (n − 1), (1.85) is a perturbed Hamiltonian system with weakly dissipative corrections, which is particularly convenient for analysis (see Sect. 1.4.3).

1.4 Global Bifurcations

39

Other codimension two degenerate bifurcations may involve “accidental” coincidence of monotonic and Hopf or of two Hopf bifurcations with diﬀerent frequencies. The amplitude equations are constructed in a straightforward way as combinations of the amplitude equations for the respective elementary bifurcations with added interaction terms allowed by symmetry. For example, the pair of amplitude equations for the Hopf-monotonic degeneracy truncated at the third order is a˙ 1 = µ0 + µ1 a1 − a31 + µ2 |a2 |2 + µ3 a1 |a2 |2 , a˙ 2 = a2 λ0 + λ1 a1 + λ2 a21 − λ3 |a2 |2 ,

(1.86)

where the amplitude a1 and parameters µi are real, while the amplitude a2 and parameters λi are complex. This equation system, as well as a similar one for a nonresonant double Hopf degeneracy, is not interesting by itself, since, although the phase space dimension is formally, respectively, three and four, the phase of the complex amplitude falls out, and dynamics is essentially two-dimensional. Chaotic behavior can be, however, generated under the inﬂuence of weak phase-dependent perturbations. Near the Hopf-monotonic degeneracy, chaos is likely to arise when a homoclinic orbit exists (see Sect. 1.5.2). Near the double Hopf degeneracy, phase dependence can be caused by resonance between the two oscillatory modes. If the ratio of the frequencies is a fraction with a denominator n, the phase-dependent resonant term is of O(n ) when ai = O(). At n ≤ 3, the resonance is strong and enters the lowest order amplitude equation; otherwise, it acts as a perturbation. Since, because of nonlinear frequency shift, resonances may appear in “Arnold tongues” beyond the bifurcation point, transition to chaos turns out to be a generic phenomenon under these conditions (Ruelle and Takens, 1971).

1.4 Global Bifurcations 1.4.1 Global Dynamics: An Overview There are only two kinds of generic (codimension one) bifurcations in dynamical systems that can be detected by local linear analysis: a saddle-node bifurcation at zero eigenvalue that creates a pair of equilibria – a saddle and a node – and a Hopf bifurcation at imaginary eigenvalue creating a periodic orbit. Other kinds of transitions, which cannot be detected by local analysis alone, are, however, necessary to reconstruct a global generic picture of qualitative changes of behavior including bifurcation of stationary states and periodic orbits. Global analysis of a dynamical system aims at deﬁning all kinds of bifurcations aﬀecting either the number of attractors or their basin boundaries. While the local analysis in the preceding section started from a stable stationary state and concentrated on ﬁnding other attractors in its vicinity, both

40

1 Dynamics, Stability and Bifurcations

stable and unstable states are important for global analysis. A ﬁxed point in n-dimensional phase space has, generically, m eigenvalues with positive real parts and n − m eigenvalues with negative real parts. The subspaces spanned by the respective eigenvectors form an m-dimensional unstable manifold and an (n − m)-dimensional stable manifold . Trajectories of the dynamical system approach the ﬁxed point along its stable manifold and escape from its vicinity along its unstable manifold. For a stable ﬁxed point, m = 0, while at a bifurcation point, when this state becomes marginally unstable, there is a center manifold spanned by eigenvectors with vanishing real part. The dimension of the center manifold is one for a bifurcation at zero eigenvalue and two for a Hopf bifurcation. This dimension may be higher for degenerate bifurcations. A periodic orbit in n-dimensional phase space has, generically, an m-dimensional unstable manifold (m = 0 for a stable orbit) and an (n − m − 1)-dimensional stable manifold. The remaining direction is tangential to the trajectory. The system is neutrally stable to tangential displacements, i.e., phase shifts of the periodic motion. Complete global analysis is possible for two-variable systems, which have two kinds of attractors only: ﬁxed points or periodic orbits. The ﬁxed points may be of three kinds only: nodes (either stable or unstable), which have two real eigenvalues of the same sign, saddles, which have two real eigenvalues with opposite signs, and foci (either stable or unstable), which have two complex conjugate eigenvalues. The three types of global codimension one bifurcations in a two-variable dynamical system are saddle-loop(sl), or homoclinic bifurcation; saddle-node inﬁnite period (sniper , or Andronov) bifurcation; and saddle-node bifurcation of periodic orbits (snp). A saddle-loop bifurcation occurs when a trajectory going out of a saddle point along its unstable manifold returns to the same point along its stable manifold. Such a trajectory, called homoclinic, does not exist generically but (a)

(b)

Fig. 1.3. Parametric evolution of a periodic orbit through a saddle-loop (a) and sniper (b) bifurcation

1.4 Global Bifurcations

41

Fig. 1.4. A sketch of a bifurcation diagram for a two-variable dynamical system (top) and phase portraits in diﬀerent parametric domains (bottom) (Reichert et al., 2001, reproduced with permission. Copyright by the American Institute of Physics). Loci of local bifurcations – supercritical (h) and subcritical (h ) Hopf and saddlenode (sn) – are shown by solid lines, but the sn locus is shown by a dash-dotted line when both bifurcating states are unstable; loci of global bifurcations – saddle-loop (sl), saddle-node of periodic orbit (snp), and sniper – are shown by dashed lines. The codimension two points are cusp C, degenerate Hopf bifurcation DH, Takens– Bogdanov point TB, saddle-node loop SNL, and neutral saddle loop NSL. In the phase portraits, asymptotically stable periodic orbits are indicated by solid lines and unstable ones by dashed lines.

42

1 Dynamics, Stability and Bifurcations

may be found by adjusting a single parameter of the system. A homoclinic trajectory can be viewed as a periodic orbit with inﬁnite period. Starting from a supercritical Hopf bifurcation in a system with at least two ﬁxed points (one of which is a saddle) and changing a suitable adjustable parameter, one can follow the parametric evolution of the emerging periodic orbit and observe that both the amplitude and period increase, until the orbit hits the saddle point and turns into a homoclinic (Fig. 1.3a). Beyond the bifurcation, all trajectories are deﬂected to a faraway attractor. An unstable periodic orbit emerging at a subcritical Hopf bifurcation may also undergo an saddle-loop bifurcation, which now aﬀects, however, only basin boundaries, rather than the attractors themselves. An unstable orbit forms the boundary of the attraction basin of the ﬁxed point it surrounds. Following the saddle-loop bifurcation, this basin opens up, and the ﬁxed point may now attract trajectories starting from far removed points in the phase plane. This picture can be read from the same Fig. 1.3a with inverted direction of ﬂow. A sequence going through formation of either stable or unstable homoclinic orbit can be traced by changing a dynamic parameter that does not aﬀect the ﬁxed point, e.g., a parameter entering the matrix γ in (1.71). A sniper bifurcation occurs when a pair of stationary states – a node and a saddle – emerges on a periodic orbit. The orbits in the phase plane at the bifurcation point are shown in Fig. 1.3b. Like in the case of a saddle-loop bifurcation, the period diverges at the bifurcation point; the sequence going through this bifurcation should be traced, however, by changing one of the parameters aﬀecting the stationary states, since an emerging pair, rather than a “preexisting” saddle, is involved here. The third kind of a global bifurcation, snp, is a birth or annihilation of a pair of periodic orbits, one stable and one unstable. A practical example (Reichert et al., 2001) of a bifurcation diagram for a two-variable dynamical system showing all kinds of local and global codimension one bifurcations lines, as well as degenerate bifurcations at their junctions, is shown in Fig. 1.4. This diagram is schematic; some parametric domains terminate in cusps at codimension two junctions, and are too small to be detected in typical experiments. The codimension two points, as well as behavior in diﬀerent parametric domains, will be discussed in more detail in Sect. 1.4.3. Bifurcation diagrams of this kind can also appear, of course, in dynamical systems with more than two variables. When the phase space dimension is at least three, chaotic dynamics is possible (more on this in Sect. 1.5), and the variety of behavior becomes inexhaustible, but in many practical systems these possibilities are never realized, since some degrees of freedom remain essentially frozen. 1.4.2 Systems with Separated Time Scales Global behavior can be easily understood when the characteristic time scales associated with the two variables are widely separated. As an example, con-

1.4 Global Bifurcations (b)

0.4

0.4

0.2

0.2

0

v

v

(a)

0

0.2

0.2

0.4

0.4

1.5 1 0.5 0 u

0.5

1

1.5 1 0.5 0 u

1.5

0.5

1

1.5

0.5

1

1.5

(d)

0.4

0.4

0.2

0.2

0

v

v

(c)

0

0.2

0.2

0.4

0.4

1.5 1 0.5 0 u

43

0.5

1

1.5

1.5 1 0.5 0 u

Fig. 1.5. Null-isoclines of the system (1.87), (1.88). (a) Oscillatory system; gray line shows the periodic orbit. (b) Excitable system; gray line shows a trajectory starting from the perturbed stationary state. (c) Bistable excitable system. (d) Bistable system

sider the two-variable FitzHugh–Nagumo model (1.4) with a fast activator u and a slow inhibitor v, which we rewrite here in a rescaled asymmetric form5 γ u˙ = u − u3 − v, v˙ = −v − ν + µu,

(1.87) (1.88)

where γ 1 is the ratio of the characteristic time scales. In the limit γ → 0, the dynamics can be read directly from the plot of null-isoclines, i.e., loci of zeros of the r.h.s. of the system (Fig. 1.5). The null-isocline of (1.87) is S-shaped, while that of (1.87) is a straight line with a position and incline ﬁxed by the parameters µ, ν. The ﬁxed points are intersections of the null-isoclines; three solutions (no more than two of which can be stable) exist within the cusped region in Fig. 1.6. The parametric expression of its boundary is µ = 1 − 3u2 , ν = −2u3 . The fast variable u is slaved to v and is attracted to either of the stable (descending) branches of the S-shaped curve. Once it has attained a certain branch, it slips alongside following slow evolution of the inhibitor until reaching a ﬁxed point. If, however, there is no ﬁxed point on this branch, a jump 5

This is a space-independent version of the RDS that will be studied in detail in Chap. 3.

44

1 Dynamics, Stability and Bifurcations 0.6 0.4

1

Ν

0.2 0

O

2

-0.2 1

-0.4 -0.6 -0.5 -0.25

0

0.25

0.5

0.75

1

Μ

Fig. 1.6. Bifurcation diagram of the system (1.87), (1.88). The solid line is the locus of saddle-node bifurcation. The dashed lines mark transitions of the ﬁxed point from a stable to an unstable branch and serve as the boundaries of the regions of relaxation oscillations O, a single stable stationary state 1(±), and bistability 2

on the alternative stable branch takes place after the turning point of the S-shaped curve is reached. If there is no ﬁxed point on this branch as well, as in Fig. 1.5a, the system slips again to another turning point and jumps back to the alternative branch. This is the regime of relaxation oscillations characteristic to oscillatory systems with separated time scales. As the parameters µ and ν change, transition to oscillatory behavior is observed when a stable ﬁxed point moves through an extremum of the S-shaped curve to the unstable (ascending) branch. √ For the ﬁxed points on the two branches, this happens at ν = ±( 23 −µ))/ 3 (dashed lines in Fig. 1.6). This is not a Hopf bifurcation, which could have been reached by varying the dynamic parameter γ and would generate a small-amplitude (either stable or unstable) periodic orbit in the vicinity of the bifurcation point. Neither is it a saddle-loop bifurcation, since it is a stable, rather than an unstable, ﬁxed point, which “collides” with the periodic orbit. The transition is, in fact, highly degenerate, and can be resolved by varying the ratio of the two small parameters: γ and the deviation of the ﬁxed point from the extremum6 . The periodic orbit always has a large amplitude; it cannot coexist with a stationary state and, as long as it exists, is insensitive to changes of µ and ν. Within the stable region but not far from the locus of oscillatory transition, the system is excitable. A typical disposition of null-isoclines is shown in Fig. 1.5b. The ﬁxed point lying close to the turning point is stable, but a small ﬁnite perturbation displacing the phase point below the minimum of the S6

This analysis may resolve a transition from small-amplitude to large-amplitude (relaxation) oscillations taking place within a narrow parametric range – the so called canard phenomenon.

1.4 Global Bifurcations

45

shaped curve initiates transition to the alternative branch. The system returns to the stationary state only after a very long excursion, following a circumferential route along both branches, as seen in the ﬁgure. In the case shown in Fig. 1.5c, both stationary states are excitable, and the system switches between them following a weak perturbation. A much stronger perturbation is needed to initiate a switch when the null-isoclines are arranged as in Fig. 1.5d. 1.4.3 Almost Hamiltonian Dynamics Global bifurcations can be studied analytically with the help of the amplitude equations (normal forms) obtained as an unfolding of the TB bifurcation (Sect. 1.3.5). We have seen that these equations can take the form of perturbed Hamiltonian systems with weakly dissipative corrections. We shall rewrite (1.85) as x˙ = y, y˙ = f (x) + δ y g(x), (1.89) where f (y), g(y) are, respectively, nth and mth order polynomials and δ is a small parameter. At δ = 0, the system is conservative. The conserved quantity is “energy” V (y) = − f (x) dx. (1.90) E = 12 y 2 + V (x), The minima of the potential V (x) correspond to stable stationary states and the maxima to saddle points. The equation of motion can be obtained by varying the action integral 1 2

S= (1.91) 2 y − V (x) dt. Equation (1.89) at δ = 0 can be easily resolved by eliminating time to obtain trajectories in the phase plane spanned by x and its time derivative (or “momentum” y): y 2 = 2[E − V (x)]. (1.92) Trajectories at diﬀerent levels of E may be qualitatively diﬀerent (see Fig. 1.7). At low energies, each trajectory surrounds one of stable stationary states only; at energies exceeding those of both stationary states, the level E = const would consist of two disconnected parts. At high energies, each trajectory surrounds both stable stationary states. The boundary between the two types corresponds to the energy of the intermediate unstable state. This is a homoclinic trajectory passing through the unstable state. At δ = 0, the energy changes with time: ∂E ∂E x˙ + y˙ = δ y 2 g(x). E˙ = ∂x ∂y

(1.93)

46

1 Dynamics, Stability and Bifurcations 1

0.5

y

0

-0.5

-1 -1.5

-1

-0.5

0 x

0.5

1

1.5

Fig. 1.7. Trajectories of a Hamiltonian system with two stable ﬁxed points

When δ is small, the energy changes slowly, so that it remains nearly constant during an oscillation period T along a particular trajectory. We can then compute the rate of change of energy by averaging over the period T : xmax t+T xmax dx ˙ ≡ ∆E = δ y 2 g(x) dt = δ y(x)g(x) dx E

T T t y(x) xmin xmin (1.94) The integration limits in the last integrals are values of x at turning points where y vanishes, i.e., the suitable roots of E = V (x). For a speciﬁc system, the computed “dissipation rate” ∆E/T can be expressed as a function of energy7 , and may vanish on a certain trajectory. We can observe that even arbitrarily weak dissipation brings about a qualitative change of dynamic behavior. A conservative system can move along any available trajectory, depending on its energy. On the contrary, a weakly dissipative system evolves to a certain energy level where the average change of energy vanishes. A trajectory becomes stationary when the dissipation integral vanishes: xmax √ xmax y(x)g(x) dx = 2 g(x) E − V (x) dx = 0. (1.95) xmin

xmin

1.4.4 Bifurcation Diagrams In order to construct a global bifurcation diagram, we have to focus attention on the values of parameters when special (nongeneric) trajectories, such 7

Dissipation, of course, can be negative only in a system sustained far from equilibrium. Then it is sometimes called “antidissipation”.

1.4 Global Bifurcations

47

as homoclinics, become stationary. Strictly speaking, the above derivation, assuming that the energy change during a single oscillation period is small, fails for an inﬁnite-period homoclinic trajectory. Nevertheless, the stationarity condition for a homoclinic trajectory correctly indicates the location of the saddle-loop bifurcation in a weakly dissipative system of the type (1.89). As an example, consider (1.89) with a cubic f (x) and quadratic g(x): f (x) = µ0 + x − x3 ,

g(x) = µ1 + µ2 x − x2 .

(1.96)

0 The roots of f (x) = 0 are two stable ﬁxed points x± s and the unstable one xs ; they can be found explicitly using a convenient trigonometric expression for the roots of a cubic polynomial: √ √ x0s = (2/ 3) sin ψ, x± s = −(2/ 3) sin(ψ ∓ π/3), √ ψ = 13 arcsin( 32 3µ0 ). (1.97)

Local analysis gives immediately the location of a saddle-node bifurcation µ0 = 2/33/2 . The condition for Hopf bifurcation on the branches of ﬁxed ± points y = 0, x = x± s (µ0 ) is g(xs ) = 0. In order to determine whether this bifurcation is supercritical or subcritical, one can obtain an analytic approximation of the dissipation integral for small-amplitude orbits with E slightly above V (x± s ) taking into account that these orbits are almost harmonic. The homoclinic orbit corresponds to the energy level E = V (x0s ). It is shaped as the ﬁgure eight, with its two loops surrounding each of stable ﬁxed points. The stationarity condition (1.95) evaluated separately for each loop deﬁnes sl± bifurcation on the branch of periodic orbits surrounding the respective point, while the condition evaluated for the entire ﬁgure-eight deﬁnes sl0 bifurcation on the branch of “large” orbits surrounding all three ﬁxed points. Figure 1.8a shows a 2D section of the bifurcation diagram in the 3D parametric space for a typical value µ0 = 0.1 in the domain where three ﬁxed points exist. The loci of Hopf and saddle-loop bifurcations in the parametric plane µ1 , µ2 are straight lines, and each ﬁxed point is stable to the left of the respective Hopf locus. The snp loci are not shown in this ﬁgure, but one involving “large” orbits is seen in Fig. 1.8b, which presents a blow-up near the intersection of the three sl lines. One can observe that the loci of Hopf and sl± bifurcations intersect (as they also do in Fig. 1.4), so that at positive and weakly negative values of µ1 the saddle-loop bifurcation occurs while the ﬁxed point is still stable. This suggests that the Hopf bifurcation might be subcritical at these values of the parameters and the sl bifurcation takes place on the branch of unstable periodic orbits. This is not precisely so, since the intersection of these loci has no particular signiﬁcance, while the dynamically important codimension two points in the parametric space are those where the third-order coeﬃcient in the expansion at Hopf bifurcation vanishes (DH) and where the homoclinic orbit is neutrally stable (NSL). These points do not generically coincide and should be connected by an snp branch. The snp loci (not discernable on the

48

1 Dynamics, Stability and Bifurcations (a) 2

1.5

000

001

1

Μ2

0.5

0

001

000

-0.5 -1

-1.5

000 -1

-0.5

001

0

1

0.5

1.5

Μ1

(b) 0.75

0.5

000

002

001

0.25

101

010 012 011 002 111 102 101

0 Μ2

000 -0.25 100

-0.5 -0.75

00 1

011

002

000 NSL 0.6

001 0.7

0.8

0.9

1

1.1

Μ1

(c)

Μ2

0.04

NSL

000

100

0.02 snp 0 -0.02

200

h

h’

DH

100 -0.3

sl -0.2

000 -0.1 Μ1

0

0.1

Fig. 1.8. (a) Loci of Hopf (solid lines) and saddle-loop(sl) (dashed lines) bifurcations in the parametric plane µ1 , µ2 at µ0 = 0.1. (b) Blow-up near the intersection of the three sl lines, showing also an snp locus (gray line). (c) Blow-up of the loci of Hopf (h or h ), sl, and snp bifurcations for the orbits surrounding the lower ﬁxed point. See the text for explanations

1.4 Global Bifurcations

49

scale of Fig. 1.8a) can be computed by requiring both the dissipation integral (1.95) and its derivative with respect to E to vanish. The blow-up of bifurcation loci for the lower state is shown separately in Fig. 1.8c where, to get a clearer picture, the diﬀerence ∆µ2 between the values of µ2 for snp or sl and Hopf bifurcations is plotted against µ1 , so that the locus of Hopf bifurcation is mapped onto the axis ∆µ2 = 0. The phase portraits corresponding to diﬀerent domains are denoted by a code including either one or three numbers equal to the number of periodic orbits enclosing a ﬁxed point. In the case when there are three ﬁxed points, the numbers refer, in this order, to the lower and upper states and all three ﬁxed points together. The ﬁrst two digits are written with an overbar when the respective ﬁxed point is unstable. This code allows us to reconstruct the changes in phase portraits quite easily.8 The code digit changes by one at the Hopf and sl bifurcations, and by two at snp. The snp line approaches the other two tangentially, forming cusps at the points DH and NSL. The ﬁxed point is stable above the Hopf line (the axis ∆µ2 = 0). An unstable periodic orbit surrounds this point in the region 100, conﬁning its attraction basin to its close vicinity. As µ2 further increases, this orbit opens up at the saddle-loop bifurcation locus to the right of the point NSL bordering the region 000. This, however, cannot happen to the left of the point NSL where the homoclinic orbit is stable. What really happens at this line segment is bifurcation of a stable periodic orbit surrounding both the ﬁxed point and the unstable orbit that serves as the boundary of their attraction basins. Both attractors coexist with the stable upper state in the region 200. At the snp locus, the stable and unstable orbits annihilate, and in the region 000 the ﬁxed point attracts trajectories originating beyond the homoclinic loop. In the region ¯100 there is a stable periodic orbit surrounding the unstable ﬁxed point. To the left of the point DH, this orbit is formed as a result of a supercritical Hopf bifurcation, while to the right of this point it is just continuing from the region 200, being unaﬀected by the subcritical Hopf bifurcation annihilating the unstable orbit. The stable orbit opens up at the saddle-loop bifurcation, and in the region ¯000 all trajectories are deﬂected to a faraway attractor. The dynamics near the higher ﬁxed point is similar, the only diﬀerence being that the ﬁxed point is stable below the Hopf line. Both ﬁxed points are attractors in the region 000, while in the regions ¯100 and 0¯10 either higher or lower ﬁxed point is replaced as attractor by a surrounding stable orbit. The boundary between the attraction basins, formed by a highly convoluted stable manifold of the saddle point, cannot be well resolved in the weak dissipation approximation. Since all trajectories spiral down along the orbits of the corresponding conservative system, slowly approaching either an attractive orbit or a ﬁxed point, the attraction basins are neatly rolled around each other, so 8

For the phase portraits in Fig. 1.4, the codes are 0 (1), ¯ 1 (2), ¯ 000 (3), ¯ 100 (4), 000 (5), 2 (6), ¯ 0¯ 01 (7), ¯ 011 (8), ¯ 001 (9), ¯ 002 (10), ¯ 200 (11), 100 (12).

50

1 Dynamics, Stability and Bifurcations

1 0.5 y

0 -0.5 -1 -1.5 -1.5

-1

-0.5

0 x

0.5

1

1.5

Fig. 1.9. An example of trajectories attracted to two alternative stable ﬁxed points

that either higher or lower attractor might be reached starting from nearby points lying outside the homoclinic ﬁgure eight (Fig. 1.9). Next, the “large” orbits surrounding both ﬁxed points have to be brought into picture. A stable “large” orbit exists to the right of the line formed by the segment of the line sl0 to the left of the point NSL (where the homoclinic orbit is stable) and the respective snp locus. This orbit opens on the sl0 line, while on the snp line it annihilates with an unstable orbit. The latter lies within the stable one but still outside the homoclinic loop and opens up on the unstable segment of the sl0 locus. To the left of the sl0 –snp line, all trajectories are attracted to the higher state in the region ¯ 000 and to the lower state in the region 0¯00; in the regions 100 and 010, outlying trajectories behave in the same way, but the trajectories lying within an unstable orbit spiral down to the nearby ﬁxed point. In the region 000, both ﬁxed points are attractors, as described above. When the snp line is crossed, the dynamics remains qualitatively the same within the emerging “large” unstable orbit, which now forms the basin boundary of the “old” attractors; outside this orbit, the trajectories are attracted to the “large” stable orbit. We observe that three attractors coexist in the regions 002, ¯102, and 0¯ 12, and two in the regions ¯002 and 0¯02. Following the breakdown of the unstable orbit at the unstable segment of the sl0 line, the attraction basins of the “large” orbit and the surviving stable ﬁxed point become intertwined in the regions ¯ 001 and 0¯ 01, being separated by a convoluted stable manifold of the saddle point. The “large” orbit becomes the sole attractor only in the region ¯ 0¯ 01. If the parameter µ0 increases above the critical value µc = 2/33/2 , the phase portraits 000, ¯ 000, 0¯ 00 transform to 0, the portraits ¯100, 0¯10, ¯001, 0¯01, ¯ ¯ ¯ ¯ ¯ ¯ 001 to 1, and 002, 002, 011, 1¯01 to 2. Some of these transitions are seen in Fig. 1.10, presenting the parametric plane µ1 , µ0 at µ2 = 0, as well as in

1.5 Deterministic Chaos

51

0.5 0

2

1

0.4 000 002

Μ0

0.3 0.2

001 101 012 011 111

010 000

0.1

0.2

0.4

0.6

0.8 Μ1

1

001

1.2

1.4

Fig. 1.10. Loci of Hopf (solid lines), saddle-loop (dashed lines), and snp (gray line) bifurcations in the parametric plane µ1 , µ0 at µ2 = 0

Fig. 1.4. The saddle-node bifurcation is, however, impossible in the regions 100 and 010, and therefore they have to terminate in a cusp at a TB point on the sn locus. The sniper bifurcation line cannot appear in this diagram, since at δ 1 the stable ﬁxed points are foci rather than nodes, unless in a close vicinity of the saddle-node locus. The latter’s intersection with the saddle-loop loci is not well resolved in the weakly dissipative approximation and depends on the ratio of two small parameters: δ and λ = µ0 − µc . Using this ratio as an additional parameter, one can resolve, e.g., the point in Fig. 1.4, where the 0¯ 01 region terminates on the sn locus, into a sniper bifurcation line.

1.5 Deterministic Chaos 1.5.1 Chaotic Attractors Chaos is a fascinating phenomenon that prompted scientists and philosophers in the last third of the 20th century to rethink fundamental questions of determinism, predictability, and randomness (see Introduction). It had always been a commonplace knowledge that complex phenomena, inﬂuenced by a great number of diﬀerent factors, are unpredictable (at least, from a practical point of view) and are amenable to statistical methods only. Statistics gives, however, little consolation when one has to live through a single decisive event, as gamblers, market traders, and weather forecasters know too well. Predicting the future has been a preoccupation of people of diﬀerent walks of life, from clairvoyants to investment bankers, using no less diverse methods. It was therefore a kind of cultural shock (at least for those who

52

1 Dynamics, Stability and Bifurcations (a)

(b)

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

Fig. 1.11. Triangular (a) and Bernoulli shift (b)maps. Thin lines show successive iterations

were able to comprehend it) that systems as simple as a three-variable ODE or even a single-variable discrete mapping could be capable of unpredictable – chaotic – behavior. As we have brieﬂy noticed in Sect. 1.1.5, an attractor of a chaotic system covers a manifold of fractal dimension , and trajectories starting from neighboring points lying on the attractor diverge with time. This is the source of unpredictability: if initial conditions are measured or assigned with a ﬁnite precision, the uncertainty of the values of the dynamic variables grows with time in such a way that after a ﬁnite time interval, the system may be found in a vicinity of any point within the attractor. This would happen if the phase space is stretched locally in a certain direction in the course of evolution. The stretched volume can remain in a ﬁnite volume of the phase space if it folds upon itself. This is known as baker’s transformation: it is, indeed, exactly what the baker does with a piece of dough. In its simplest form, it is expressed by either of the one-dimensional (1D) maps in Fig. 1.11. In both of them, the unit interval is stretched twice over and the second part is mapped back on the same interval by either bending or cutting (i.e., either inverting or preserving its direction). If the position of a point on the unit interval is expressed as an inﬁnite binary string, the leading digit 0 or 1 would mean, respectively, that it is located in either the ﬁrst or the second half of the interval. Following the transformation, the second digit of the original string will classify the position; thus, the mapping in Fig. 1.11b is equivalent to removing the ﬁrst digit and shifting the remaining sequence to the left (Bernoulli shift). It is clear that however close the two points be originally, they will eventually be separated and sent to the two alternative half-intervals: if the diﬀerence in the original positions

1.5 Deterministic Chaos

(a)

(b)

53

(d)

(e)

(c) (f)

Fig. 1.12. Smale horseshoe. See the text for explanations

was in nth digit, this will happen after nth iteration. The transformation in Fig. 1.11a will separate nearby points and eﬀectively “mix” the 1D phase space in the same way. In both cases, the total length of the interval, i.e., the 1D phase volume, is conserved. When the system is dissipative, as most systems we are interested in are, the phase volume decreases with time. It may, however, stretch in a certain direction while shrinking in the others. Shrinking of the phase volume combined with local stretching leads to the formation of an attractor of fractal dimension . This mechanism is explained qualitatively by the Smale horseshoe map. The unit square (Fig. 1.12a) is stretched vertically and shrunk horizontally in such ratios that its total area decreases. It is folded and mapped back in such a way that the bended area remains outside the square (Fig. 1.12b). The part of the original square that is mapped back onto it following the transformations is restricted to two horizontal stripes (Fig. 1.12c), and its image is the two vertical stripes shown in Fig. 1.12b. The area that remains invariant following

54

1 Dynamics, Stability and Bifurcations

the transformation, i.e., is mapped upon itself, is conﬁned to the four squares shown in Fig. 1.12d. Next iteration restricts the invariant area to the 16 small squares shown in Fig. 1.12e. Upon successive iterations, the invariant area shrinks further, all but fading from view in Fig. 1.12f, and turns eventually into a fractal object of zero 2D volume, parts of which, nevertheless, can be found in far-removed locations within the original unit square. The structure of the attractor, though topologically complex, is well deﬁned, but predicting the location of the image of a certain point after a large number of iterations would require ever-increasing precision in deﬁning the initial data. 1.5.2 Shilnikov Bifurcation Motion in the vicinity of a homoclinic orbit is particularly sensitive to perturbations when it slows down as it passes close to a saddle point. In a three-variable nonlinear dynamical system, a saddle point may have either a 1D stable and a 2D unstable manifold or the other way around. A particularly interesting case is when two eigenvalues are complex conjugate; then the ﬁxed point is called a saddle-focus. Shilnikov (1965) showed that a horseshoe map and, consequently, a chaotic attractor can be generated by a threevariable nonlinear dynamical system when it has a homoclinic passing through a saddle-focus. If the complex pair of eigenvalues µ = γ ± iω has a positive real part γ, while the real eigenvalue λ is negative, a trajectory starting from the vicinity of the ﬁxed point spirals out, as in Fig. 1.13a, but, lacking other attractors, eventually returns along the stable direction. If, on the opposite, γ > 0, while λ < 0, the same picture can be read in the opposite sense: the trajectory ejected along the stable direction returns and spirals in toward the ﬁxed point. (a)

(b)

(c)

Fig. 1.13. (a) A typical trajectory close to the homoclinic. (b) Mapping of the shaded strip upon the top of the cylinder by ﬂow with γ > 0, λ < 0, γ < |λ| and a typical trajectory in the vicinity of the ﬁxed point. (c) Mapping of the shaded sector upon the rim of the cylinder by ﬂow with γ < 0, λ > 0, λ < |γ|

1.5 Deterministic Chaos (a)

55

(b)

Fig. 1.14. Construction of a Smale horseshoe near the homoclinic. (a) Mapping of the shaded strip into the “Shilnikov snake.” (b) Mapping of the “snake” back upon the strip by global ﬂow

Close to the ﬁxed point, the dynamical system can be linearized. Suppose that the coordinate system spanning the 3D phase space has been transformed in such a way that the ﬁxed point is placed at the origin, the eigenvector corresponding to the real eigenvalue is directed along the z-axis, and the 2D manifold spanned by the eigenvectors corresponding to the complex conjugate pair spans the plane z = 0, which we parametrize by the polar coordinates r, φ. Then, the trajectory of the linearized system can be presented as z = z0 eλ(t−t0 ) ,

r = r0 eγt ,

φ = φ0 + ωt,

(1.98)

where t0 , z0 , r0 , φ0 are integration constants. A typical trajectory close to the homoclinic looks as in Fig. 1.13b. Take a small cylinder 0 < ζ0 < z < z0 , 0 < r < r0 and consider the map of the line r = r0 , φ = 0, ζ0 < z < z0 upon the top of the cylinder z = z0 , r < r0 generated by ﬂow with γ > 0, λ < 0. Then, t0 in (1.98) is recognized as the traversal time of a trajectory starting at the point r = r0 , ζ = z0 e−λt0 . Eliminating t0 , we see that the image in question is a logarithmic spiral: r = r0

z0 ζ

γ/λ ,

φ=

ω z0 ln , λ ζ

(1.99)

or, in a parametric form, r = r0 exp

γ φ , ω

0<φ<

ω z0 ln . λ ζ0

(1.100)

56

1 Dynamics, Stability and Bifurcations

The length of the spiral, as estimated by the total span of the angular variable, grows without limit as ζ0 → 0. The mapping of a narrow section of the rim of the cylinder (the shaded area in Fig. 1.13b) upon its top is a fattened spiral (“Shilnikov snake”) seen in the same ﬁgure. The mapping reduces the total area when γ < |λ|. Take now a small segment of the rim close to the entrance of the homoclinic trajectory into the vicinity of the ﬁxed point. The “snake” image of this segment (Fig. 1.14a) is mapped back upon the area encircling the ﬁxed point by global ﬂow, as seen in Fig. 1.14b. Inasmuch as the global ﬂow is nonsingular (e.g., similar to that in Fig. 1.13a), it rotates and extends or shrinks the image but does not aﬀect its topology. The transformation by the global ﬂow can be assumed to be linear, as long as the image is small. The points of the segment mapped back upon it by the combined transformation due to the ﬂow near the ﬁxed point and global ﬂow lie within several strips seen in Fig. 1.14b. This invariant area is further cut, shrunk, and dispersed upon successive iterations, turning eventually into a fractal object of zero 2D volume. This is qualitatively similar to the Smale horseshoe mapping discussed in the preceding subsection and should result likewise in the formation of a fractal chaotic attractor. The essential diﬀerence is that the number of strips increases without limit as the lower edge of the segment comes closer to the entrance point of the homoclinic trajectory. Thus, an inﬁnite number of horseshoes are formed in the course of an inﬁnite series of bifurcations as the homoclinic orbit is approached. In the opposite case γ < 0, λ > 0, one can construct in a similar way the map of the line z = z0 , φ = 0, ρ0 < ρ < r0 upon the rim of the cylinder generated by ﬂow going down toward the ﬁxed point and spiraling out: λ/γ r0 ω r0 z = z0 (1.101) , φ = ln , ρ γ ρ or, in a parametric form, z = z0 exp

λ φ , ω

0<φ<

ω r0 ln . γ ρ0

(1.102)

The mapping is a helix, which again stretches without limit at ρ0 → 0. The mapping of the shaded sector in Fig. 1.13c upon the rim is a fattened helix, and the total area shrinks when λ < |γ|. The Smale horseshoe is constructed in this case in the same way as before. The Shilnikov bifurcation can be found in the unfolding (1.86) of the degenerate Hopf-monotonic bifurcation, but simpler ad hoc dynamical systems can reproduce it as well. A simple dissipative system that generates global ﬂow returning trajectories into the vicinity of a saddle-focus was devised by R¨ossler (1976): x˙ = −y − z,

y˙ = x + µy, z˙ = ν + (x − λ)z. √ Two ﬁxed points exist at |λ| > λc = 2 µν:

(1.103)

1.5 Deterministic Chaos (a)

(b) 5

y y

57

0

2

-5

0 -2 -4

15 6 z

z 10

4 2

5

0

0 -5

-2.5 0 x

0 2.5

x

5

5

10

Fig. 1.15. Attractors of the R¨ ossler system. (a) A period two orbit at µ = ν = 0.2, λ = 2.9. (b) Chaotic attractor at µ = ν = 0.2, λ = 5 ± ± x± s = −µys = µzs =

1 2

λ±

λ2 − 4µν

(1.104)

Under certain conditions, e.g., in the particular case µ = ν, both ﬁxed points are unstable. The “upper” one (marked by the plus sign) has a 1D unstable and a 2D stable manifold and the “lower” one, the other way around. At suﬃciently small µ, both are saddle-foci, and the real eigenvalue is larger by its absolute value than the real part of the complex pair. Thus, a nearly homoclinic orbit spiraling out of the lower ﬁxed point and returning to it along its 1D stable manifold satisﬁes the conditions required to form Smale horseshoes. At |λ| λc , we have zs− 1, and the unstable manifold of the lower ﬁxed point is close to the plane z = 0 where the ﬂow is almost linear, so that the linearized equations used in the above proof also remain valid outside the immediate vicinity of a homoclinic. The R¨ossler model is not realistic; its trajectories escape to inﬁnity from a larger fraction of the phase space but may be trapped in the region adjacent to the lower ﬁxed point. A periodic orbit found in this region at appropriate values of parameters undergoes a series of period doubling bifurcations. An example of a period two orbit is shown in Fig. 1.15a. The bifurcation sequence results in an apparently chaotic orbit seen in Fig. 1.15b. 1.5.3 Period Doubling Cascade Straightforward integration of ODEs is ill-suited for studies of chaotic dynamics, since proper characterization of chaotic attractors and distinguishing them from periodic orbits with long periods or long transients requires exceedingly

58

1 Dynamics, Stability and Bifurcations (a)

(b)

2

y

10

0

-2 -4

8

-6 6

4 z

2 4

0 2

-2 0 x

2 4

2

4

6

8

10

Fig. 1.16. (a) Construction of a Poincar´e map. (b) The Poincar´e map of the R¨ ossler system at µ = ν = 0.2, λ = 5. The sign of the coordinates is inverted, and the added segment is shown by the dashed line

long integration times. A better way is to convert an ODE into a Poincar´e map, which presents the relation between successive intersection points (in the same direction) of the trajectories with a given surface, called Poincar´e section. The surface should be transversal everywhere to the ﬂow generated by the ODE. When its position is properly chosen, a period n orbit will produce exactly n intersection points in a given direction (see Fig. 1.16a). The dimension of a Poincar´e map is one less than that of the corresponding dynamical system. In dissipative systems, stretching in a certain direction is, however, compensated by strong shrinking of the phase space in other directions, so that the phase volume tends to zero, as we have seen in the construction of the Smale horseshoe in the two preceding subsections. As a result, the Poincar´e map thins out, e.g., becomes almost 1D for a 3D continuous dynamical system. An example is shown in Fig. 1.16b, presenting the Poincar´e map generated by the chaotic R¨ossler attractor with the plane x = 0 used as the Poincar´e section. It is very close to a 1D map and, when supplemented by a segment connecting it to the origin, looks very much similar to a simple logistic map f (x) = µ x(1 − x). (1.105) xj+1 = f (xj ), At the maximum value of the parameter µ = 4, the logistic map acts in a way similar to the triangular map in Sect. 1.5.1 and satisﬁes the conditions for the formation of a Smale horseshoe. As the parameter µ increases, the route to chaos goes through a period doubling cascade (May, 1976). The ﬁrst stages of the cascade are illustrated in Fig. 1.17. The logistic map has, besides the unstable trivial equilibrium, a single ﬁxed point x = x0 , which is stable

1.5 Deterministic Chaos

59

(a) f x

f x 0.7 0.6

0.6 0.5

0.4

0.4 0.3 0.2

0.2

0.1 0.2

0.4

0.6

0.8

1

x

0.2

0.4

0.6

0.8

1

x

0.2

0.4

0.6

0.8

1

x

(b) f1 x

f1 x 0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.2

0.4

0.6

0.8

1

x

(c)

(d)

f2 x

0.8 0.7 0.6

0.6 0.5 0.4

0.4 0.3 0.2

0.2

0.1

0.2

0.4

0.6

0.8

1

x

0.2

0.4

0.6

0.8

1

Fig. 1.17. (a) The logistic map before (µ = 2.8) and after (µ = 3.1) the period doubling bifurcation; gray lines show an iteration sequence. (b) The composite map f1 (x) before (µ = 3.1) and after (µ = 3.5) the repeated period doubling. (c) The composite map f2 (x) at µ = 3.5. (d) The rescaled central segment of f2 (x) at the third period doubling, compared to f (x) (gray line)

60

1 Dynamics, Stability and Bifurcations (a)

x 1 0.8 0.6 0.4 0.2

3.2

(b)

3.4

3.6

3.8

4

Μ

3.84

3.86

3.88

3.9

Μ

x

0.8 0.6 0.4 0.2

3.82

Fig. 1.18. Points on the attractor at diﬀerent values of µ

as long as the slope f (x) is less than one by its absolute value. At µ = 3, a Hopf bifurcation takes place: above it, f (x0 ) < −1, so that the ﬁxed point is unstable, and the system approaches a period two sequence, alternating between two values x = x± 1 upon each iteration (Fig. 1.17a). Both points are the stable ﬁxed points of the composite map f1 (x) = f (f (x)), while x0 is an unstable ﬁxed point of this map with f1 (x0 ) > 1 (Fig. 1.17b). As µ further grows, |f2 (x± 1 )| increases, and a new period doubling takes place at µ ≈ 3.4495. A period four orbit emerges at larger values of µ, where the four alternating values of x are stable ﬁxed points of the composite map f2 (x) = f1 (f1 (x)) = f 4 (x) (Fig. 1.17c). This sequence of bifurcations continues ad inﬁnitum. One of the stable n ﬁxed points of the composite map fn (x) = f 2 (x) approaches the maximum of the logistic map at xm = 1/2 as n increases, and the form of the composite

1.5 Deterministic Chaos

61

map in the vicinity of this point becomes indistinguishable from the rescaled original logistic map. This similarity is apparent already in the map f2 (x), as seen in Fig. 1.17d , where the rescaled central segment of f2 (x) at the period eight bifurcation point µ ≈ 3.5441 is compared to f (x). On the basis of this self-similarity, Feigenbaum (1978) applied the renormalization group to prove universality of the period doubling sequence. Asymptotically at n → ∞, the period 2n bifurcation point approaches the accumulation point µ∞ = 3.5699456 . . . as µ∞ − µn ∼ δ −n , with the exponent δ = 4.6692016 . . .. The chaotic attractor emerges at µ = µ∞ . The attractor undergoes an inﬁnite number of bifurcations beyond µ∞ . Figure 1.18a shows the points on the attractor at diﬀerent values of µ up to the upper limit µ = 4. This picture still gives a little idea of the true complexity of behavior and can be enlarged to reveal ﬁner structures at smaller scales. The sequence contains an inﬁnite number of periodic “windows” and repeated cascades. A prominent feature is a rather wide period three window seen in the blow-up in Fig. 1.18b. The universality of the period doubling sequence explains its wide applicability to a variety of dynamical systems, which was conﬁrmed in numerous experiments and simulations. As long as the map is continuous and has a maximum, its exact form is irrelevant, since only the vicinity of the maximum, which can always be approximated by a parabola, is relevant at late stages of the period doubling cascade. This is true, e.g., for the Poincar´e map of the R¨ossler model seen in Fig. 1.16b. The period doubling route to chaos, though frequently encountered, is in no way unique. A complicated system containing a number of parameters can enter the chaotic region in diﬀerent ways along diﬀerent parametric paths, e.g., proceeding ﬁrst into the period three region. A still wider variety of routes to chaos is possible for discontinuous maps and ODEs that can be approximated by maps of this kind (see the next subsection). 1.5.4 Lorenz System Historically, the ﬁrst example of chaos in a dissipative system was given by Lorenz (1963). The Lorenz system was originally derived as a qualitative model of thermal convection and included the amplitudes of just three modes in the Fourier expansion of equations of thermal convection: x˙ = σ(y − x),

y˙ = ρ x − y − x z,

z˙ = x y − β z.

(1.106)

At ρ = 1, the system undergoes a pitchfork bifurcation: the trivial equilibrium becomes unstable, and two stable symmetric nontrivial equilibria, xs = ys = ± β(ρ − 1), zs = ρ − 1, emerge. This models the situation when a quiescent convection cell becomes unstable; two nontrivial states correspond to two opposite directions of circulation, which are totally equivalent but exclusive. The trivial equilibrium at ρ > 1 is a saddle with a 1D unstable and 2D stable manifold. The nontrivial equilibria undergo a Hopf bifurcation at

62

1 Dynamics, Stability and Bifurcations

σ = 1 + β,

ρ=

σ(3 + σ + β) . σ−1−β

(1.107)

Beyond the Hopf bifurcation locus, the nontrivial ﬁxed points are saddle-foci. We shall further ﬁx the commonly adopted values β = 8/3, σ = 10 and follow the parametric evolution at increasing ρ. One expects dynamic behavior to be interesting in the parametric region where all equilibria are unstable. In fact, the dynamics becomes nontrivial even at lower values of ρ. A homoclinic orbit exists at ρ ≈ 13.9265. Below this value, trajectories going out of the trivial equilibrium along the unstable direction approach the closest nontrivial equilibrium. Above this point, the trajectories turn around and converge to the nontrivial equilibrium of the opposite sign, as shown in Fig. 1.19a. At larger values of ρ, trajectories starting from some initial conditions may wander around for a long time in an apparently chaotic manner and do not approach any equilibrium point for a considerably long time (see Fig. 1.19b), while starting from other initial conditions, trajectories start spiralling to an equilibrium point immediately. Above the Hopf bifurcation at ρ ≈ 24.7368, trajectories coming to the vicinity of nontrivial equilibria spiral out. The dynamics is chaotic, as trajectories switch erratically between circling either positive or negative stationary point. Being a meteorologist, Lorenz put forward the results of his numerical study of chaos in this apparently simple system as a proof of unpredictability of weather. His colleagues were not impressed, and the work (a)

b 20 y

10

10

y

0

0

-10 -20

-10 20

30 15

z 20

z 10

10

5 0 -10

0 -5

-10

0 x

0

5 10

x

10

Fig. 1.19. Trajectories of the Lorenz system at β = 8/3, σ = 10, and ρ = 14 (a) and ρ = 24 (b)

1.5 Deterministic Chaos (a)

63

b x

8

7.5

6 4

5

2 f x

2.5 0

100

200

300

400

500

n

-2

-2.5

-4

-5

-6

-7.5

-5

-2.5

0 x

2.5

5

7.5

-7.5

Fig. 1.20. (a) The Poincar´e map of the Lorenz system in the chaotic region (β = 8/3, σ = 10, ρ = 28). Gray line shows the approximate map (1.108). (b) A chaotic sequence generated by the map (1.108)

remained obscure for 15 years until studies of chaotic behavior came into fashion. Long-time chaotic behavior should be studied with the help of a Poincar´e map. An appropriate Poincar´e section is the plane z = zs = ρ − 1. As in the case of the R¨ossler model, the map turns out to be almost 1D. It is, however, very much diﬀerent from both logistic and R¨ ossler map, as it has a discontinuity corresponding to the point where the orbit switches between the two parts of the attractor encircling the two alternative saddle-foci (Sparrow, 1982). The map shown in Fig. 1.20a has been made more compact by measuring the positions relative to the nearest equilibrium point; in this way, the segment |x| < |xs | where upwardly moving trajectories never cross the Poincar´e section is excluded. The map is well approximated by the function f (x) = x − [(γ n − xn )1/n − γ]sign x,

(1.108)

where, for the parameter values in Fig. 1.20, n = 4 and γ = 4.726. All computations can be carried out very eﬃciently with the help of the approximate map. One can compute very long sequences in this way, as the one in Fig. 1.20b, and study the statistics of the output. One can see, e.g., that the number of consecutive points in either part of the attractor, dependent on the proximity of the “injection point” to xs , forms a random sequence, and build up statistics of this sequence. The disadvantage of this approach is, of course, that one has to build up an approximate map anew for each combination of the parameters by integrating the original equations. Essential features of the dynamics of the Lorenz system can be demonstrated, however, with the help of a simple model map, much in

64

1 Dynamics, Stability and Bifurcations (a) 0.5 0.4 0.3 Γ

0

0.2 1

0.1

2T 2 1

0

2 Μ

4

3

(b) 0.8

x

0.6 0.4 0.2 0 1 1.5

2T 2

2 2.5 Μ

0 3

3.5

Fig. 1.21. (a) The bifurcation diagram of the map (1.109). (b) Evolution of the attractor along the line γ = 0.08. The number of ﬁxed points on each half-interval is indicated in each parametric region

the same way as the dynamics of the R¨ossler model is imitated by the logistic map. A suitable map deﬁned on the unit interval is the following discontinuous map antisymmetric relative to the point x = 1/2: at x < 12 γ − µx2 (1.109) f (x) = 2 1 − γ − µ(1 − x) at x > 12 . The number of ﬁxed points in either half-interval is shown in Fig. 1.21a. On the boundary between the regions 0 and 2, γ = 14 µ−1 , µ > 1, a saddle-node bifurcation occurs, while on the other boundary, γ = 14 (2−µ), one of the ﬁxed points touches the discontinuity. The parametric evolution of the attractor (for trajectories originating just above x = 1/2) at γ = 0.08 and changing values of µ is shown in Fig. 1.21b. This particular line γ = const passes through all four regions. Proceeding

1.5 Deterministic Chaos f x 1

65

f x 1

escape

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.2

0.4

0.6

0.8

1

x

0.2

0.4

(a)

0.6

0.8

1

x

100

n

(b) x

x

0.8

0.9 0.8

0.6 0.7

0.4

0.6 0.5

0.2 0.4 20

40

60

80

100

n

20

40

60

80

Fig. 1.22. Positions after successive iterations of the map (1.109) in the case of long chaotic transients (a) and intermittency (b)

along this line, the trajectories are consecutively attracted to the nearest ﬁxed point (1), to a chaotic attractor of limited extent (2T), to the ﬁxed point in the opposite half-interval (2), and to a chaotic attractor covering almost the entire unit interval (0). In region 1, a single ﬁxed point found in each half-interval is always stable, and the trajectories converge to the nearest ﬁxed point. When the second pair of ﬁxed points appears (region 2), it is unstable. Under these conditions, convergence to the stable ﬁxed point lying in another half-interval is possible, similar to the behavior of the Lorenz system following the homoclinic connection. A trajectory may be trapped in the region between the two unstable ﬁxed points for a considerable time before escaping to the attraction region of one of the stable ﬁxed points, as shown in Fig. 1.22a. The trapping of trajectories originating suﬃciently close to the point x = 1/2 becomes permanent when the edge of the left segment of the map at f ( 12 − 0) goes below the unstable ﬁxed point on the right segment, which happens at γ < 32 − 14 µ − 2µ−1 , 2 < µ < 3. In this region, denoted as 2T in Fig. 1.21, a chaotic attractor coexists with stable ﬁxed points attracting

66

1 Dynamics, Stability and Bifurcations

trajectories originating farther from the point x = 1/2; the unstable ﬁxed point marks the boundary between the two attraction basins. Finally, chaotic dynamics prevails in region 0, though narrow periodic windows may exist in this region. Near the border of the chaotic region, just after the saddle-node bifurcation, some trajectories may be trapped for a long time in the “shadow” of a former ﬁxed point where dynamics slows down, even though the stable ﬁxed point has already disappeared. The positions after successive iterations of the map are shown in Fig. 1.21b. This is known as the intermittency route to chaos (Manneville and Pomeau, 1979).

1.6 Dynamics of Planforms 1.6.1 Interaction of Turing Modes Space-independent amplitude equations describing regular patterns eﬀectively turn underlying PDEs into dynamical systems for amplitudes of the constituent modes. The amplitude of a mode with k = 0, ω = 0 excited at a Turing bifurcation point, like the amplitude of a Hopf mode, is complex. In one dimension, the amplitude equation describing dynamics in the vicinity of the bifurcation point cannot contain quadratic terms and should have the same form as (1.70), with a substantial diﬀerence that its coeﬃcients must be real. The inhomogeneous solution bifurcates supercritically, provided µ3 < 0; this coeﬃcient can then be set by rescaling to −1. The bifurcation diagram for the real amplitude has the same form as in Fig. 1.2e: the attractor is the trivial state a = 0 at µ1 < 0 and a harmonic stationary pattern with the √ amplitude |a| = µ1 at µ1 > 0. The phase of a is arbitrary, and its shift leads just to shifting the pattern along the wave vector k. In the case when µ3 is positive but small, the expansion can be continued to the ﬁfth order, yielding the bifurcation diagram seen in Fig. 1.2f. Symmetry breaking bifurcations in more than one dimension are degenerate due to spatial symmetries. In an isotropic system, an arbitrary number of diﬀerently directed modes with k = 0 can be excited beyond the bifurcation point. A combination of these modes can give a variety of distinct planforms. Competition among the modes that determines the pattern selection is governed by nonlinear interactions. The product of the amplitudes aj , ak , etc. (where the overline denotes the complex conjugate) may appear in the equation for the amplitude ai if the respective wave vectors add up to zero, ki + kj + kk + . . . = 0. This condition ensures that the modes in question are in resonance. Otherwise, the product of these modes rapidly oscillates and is averaged out when the amplitude equation is derived using the multiscale expansion procedure. Interaction coeﬃcients between diﬀerent modes can be computed similarly to the computation in Sects. 1.3.3 and 1.3.4. This formalism automatically

1.6 Dynamics of Planforms

67

chooses in each order of the expansion combinations of modes satisfying the resonances conditions. We start with a general RDS in the form (1.18). For simplicity, we shall ﬁx, as in Sect. 1.3.4, the parameters p aﬀecting homogeneous equilibria but vary only diﬀusivities entering the matrix D. This would usually suﬃce to locate a Turing bifurcation point where the Jacobi matrix J = fu − D0 k 2 has a zero eigenvalue for a certain value of the wavenumber k. We assume that this eigenvalue is nondegenerate and that all other bands of eigenvalues λ(k) lie entirely in the complex half-plane Re(λ) < 0, and present the diﬀusivity matrix as D = D 0 + D 1 + · · ·. The ﬁrst-order equation u1 ≡ (fu + D0 ∇2 )u1 = 0 has a nontrivial spatially harmonic solution

u1 = U ai (t1 , t2 , . . .) eiki ·x + c.c. .

(1.110)

(1.111)

i

Here ai (t1 , t2 , . . .) is an arbitrary complex amplitude of the ith mode dependent on slower time scales tn , n > 0; all wave vectors ki have the same magnitude, |ki | = k, and U is the eigenvector of the matrix J = fu − D0 k 2 with the zero eigenvalue. The eigenvector of the transposed matrix J † is U † , and the normalization condition is U † · U = 1. The second-order equation is (fu + D 0 ∇2 )u2 = ∂1 u1 + k 2 D 1 u1 − 12 fuu [u1 , u1 ].

(1.112)

The ﬁrst two (linear) terms on the r.h.s. contain the principal harmonics e±iki ·x , while the last (quadratic) term contains composite harmonics e±i(kj ±kk )·x . The solvability conditions of this equation are evaluated by taking the scalar product with the eigenvector U † , multiplying by e∓iki ·x and taking the spatial average. In one dimension, where a single symmetry-breaking mode exists, the contribution of the quadratic term vanishes, and the solv(1) ability condition just requires µ1 = U † · D 1 U to vanish, thereby freezing evolution on the time scale t1 . Restricting for the time being to one dimension, we set D 1 = 0 and compute the second-order function u2 by solving (1.112), which is now reduced to the form

(fu + D 0 ∇2 )u2 = − |a|2 + 12 a2 e2ikx + c.c. fuu [U , U ]. (1.113) The solution of this equation is

u2 = − |a|2 fu −1 + 12 a2 e2ikx (fu − 2D 0 k 2 )−1 + c.c. fuu [U , U ]. (1.114) The third-order equation is (fu +D 0 ∇2 )u3 = (∂2 −D 2 ∇2 )u1 −fuu [u1 , u2 ]− 16 fuuu [u1 , u1 , u1 ]. (1.115)

68

1 Dynamics, Stability and Bifurcations

Using here (1.111), (1.114) and collecting the terms in the inhomogeinity containing the principal harmonic eikx , we compute the solvability condition, which has the same form as (1.70) but contains real coeﬃcients: (2)

(0)

∂2 a = µ1 a + µ3 |a|2 a,

µ1 = U † · D 2 U , (2)

(1.116) µ3 = 12 U † · fuuu [U , U , U ]

− U † · fuu U , fu −1 + 12 (fu − 2D 0 k 2 )−1 fuu [U , U ] . (0)

In higher dimensions, the solvability condition of the second-order equation is trivial only if the system is invariant to the sign change u → −u and, consequently, the quadratic terms vanish identically. In this case, u2 = 0, and the amplitude equations obtained in the third order are νij |aj |2 . (1.117) ∂2 ai = µai − νii |ai |2 ai − 2ai i=j

Here the interaction coeﬃcients between all modes −νij are identical and are (0) equal to the ﬁrst term of the expression for µ3 in (1.116), which should be negative for a supercritical bifurcation. The inversion symmetry never appears in realistic RDSs, while in other models that contain nonlinear diﬀerential terms and, possibly, vertical structure, the interaction coeﬃcients are, generally, distinct. In an isotropic system, the coeﬃcient νij depends only on the angle α between the two modes. A two-mode pattern is preferred if the mutual suppression of the waves directed at a certain angle is weaker than self-suppression. A single-mode striped pattern is preferred a priori, since, if the interactions do not depend on the angle at all, mutual suppression is stronger due to the combinatorial factor 2 in (1.117). If the interaction weakens with growing α, the minimum may be reached, by symmetry, at α = π/2. Then a square pattern comprising both modes will be selected, provided the minimum is suﬃciently deep. In an unlikely case when a deep minimum is reached at a smaller angle9 , a combination of more than two modes will be preferable. It may also happen that the solvability condition of the second-order equation is satisﬁed in the absence of inversion symmetry, either “accidentally” at a certain codimension two manifold in the parametric space or due to some less obvious innate symmetry. The latter cancellation mechanism works in the classical problem of Rayleigh–B´enard convection in a ﬂuid layer heated from below where the source of symmetry is conservation of energy and momentum by nonlinear terms in the commonly used Boussinesq approximation (Gershuni and Zhukhovitsky, 1976). In this case, u2 does not vanish, and solving the second-order equation is required for derivation of the amplitude equation. 9

This may happen when a weakly damped mode with a diﬀerent wavelength is present.

1.6 Dynamics of Planforms

69

1.6.2 Resonant Planforms In a general asymmetric case, the quadratic nonlinearity in (1.113) contains a harmonic with the basic wavenumber k if the excited modes include a triplet with the wave vectors directed at the mutual angle 2π/3. Then, retaining ﬁrst-order deviations in (1.113), and collecting resonant terms, we write the solvability condition of this equation as (1)

(0)

∂2 aj = µ1 aj + µ2 aj−1 aj+1 ,

(1.118)

where µ2 = −U † · fuu [U j , U j ] and aj±1 are amplitudes of the modes with the wave vectors kj±1 turned by ±2π/3 relative to the wave vector of the jth mode. Presenting the complex amplitudes in the polar form aj = ρj exp(iθj ) and separating the real and imaginary parts reduces three complex equations (1.118) to four real equations, since the phases appear only in the combination Θ= θj : (0)

(1)

(0)

ρ˙ j = µ1 ρj + µ2 ρj−1 ρj+1 cos Θ, (0) Θ˙ = −µ2 sin Θ

3 ρj−1 ρj+1 j=1

ρj

,

(1.119) (1.120)

where the time derivative is denoted by the dot and the amplitudes aj are numbered modulo three. The remaining two phase degrees of freedom are translational Goldstone modes, since shifting phases in an arbitrary way while keeping their sum constant is equivalent to shifting the entire pattern in the plane. The total phase Θ is, however, dynamically important, since it determines the action of the quadratic term in (1.119). According to (1.120), Θ (0) relaxes to zero when µ2 > 0, thereby rendering the quadratic term desta(0) bilizing. If, on the opposite, µ2 < 0, the stable equilibrium is Θ = π, and the term is destabilizing again! This is why the hexagonal pattern comprising three resonant modes always bifurcates subcritically, and why the hexagonal planform is ubiquitous in 2D pattern-forming systems (Palm, 1960). Since quadratic terms are always destabilizing, cubic terms are necessary to obtain a stable ﬁnite-amplitude solution. In order to obtain a cubic ampli(0) tude equation with balanced orders of magnitudes, the coeﬃcient µ2 should be assumed small. Generically, this requires restricting the system parameters to the vicinity of a codimension two manifold in the parametric space. Deviations of diﬀusivities should be again reduced to O(2 ), and the evolution on the O(−1 ) time scale frozen, as in the preceding subsection. Solving the second-order equation (1.113) then gives the function√u2 containing composite harmonics e±i(kj ±kj±1 )·x with the wavenumbers 3k and 2k, as well as a constant term and terms proportional to e±ikj ·x V , where V is a vector normal to U , i.e., satisfying U † · V = 0. All these terms are fed into the

70

1 Dynamics, Stability and Bifurcations

third-order equation, which yields as its solvability condition the system of amplitude equations. We shall write this system in a rescaled form

(1.121) a˙ j = aj µ − |aj |2 − ν |aj−1 |2 + |aj+1 |2 + aj−1 aj+1 . The three amplitudes aj are again numbered modulo three; the coeﬃcient at the quadratic term and the cubic self-interaction coeﬃcient have been normalized, respectively, to unity and −1 by choosing the time and amplitude scales. If the former was negative in the unscaled equations, the sign can be reversed by shifting all phases by π/3. Self-interaction is assumed to be suppressing; otherwise, expansion to higher orders would be needed. The polar form of (1.121) is

(1.122) ρ˙ j = ρj µ − ρ2j − ν ρ2j−1 + ρ2j+1 + ρj−1 ρj+1 cos Θ, Θ˙ = − sin Θ

3 ρj−1 ρj+1 j=1

ρj

.

(1.123)

The stationary amplitude of the hexagonal pattern can be computed after setting Θ = 0, ρj = idem = ρ by solving the equation µ + ρ − (1 + 2ν)ρ2 = 0. The solution is ρ=

1+

(1.124)

1 + 4µ(1 + 2ν) . 2(1 + 2ν)

(1.125)

It exists at µ ≥ −[4(1 + 2ν)]−1 (encroaching on the subcritical region µ < 0) and is stable at µ < (ν + 2)/(ν − 1)2 . A single-mode (striped) pattern ρ1 = √ µ, ρ2 = ρ3 = 0 is also a solution of (1.119) in the supercritical region µ > 0 and is stable at ν > 1, µ > (ν − 1)−2 . The dependence of both solutions on µ is shown in Fig. 1.23. Both patterns have been observed in experiments with pattern-forming chemical reactions (Fig. 1.24). Equation (1.121) has a gradient structure a˙ j = −∂V /∂aj

(1.126)

with the potential V =

3 j=1

−µ|aj |2 + 12 |aj |4 + ν |aj |2 |ak |2 − a1 a2 a3 − a1 a2 a3 .

(1.127)

j=k

This representation is most transparent, since every term can be associated with a triangle or rhombus formed by the respective wave vectors. It can be easily extended to higher dimensions, as we shall see below. One can use (1.127) to compare the energy of diﬀerent solutions: the trivial one (all aj = 0), a single-mode (striped) pattern, and the hexagonal pattern |aj | = idem = 0. Stability regions of these solutions overlap, as shown in Fig. 1.23.

1.6 Dynamics of Planforms

Ρ 3

71

Ρ 0.25

2.5 0.1 2 -0.05

0.05

Μ

striped

1.5 1 hexagonal 0.5 1

2

3

4

5

Μ

Fig. 1.23. Nontrivial solutions of (1.122): dependence of the real amplitude of the striped and hexagonal patterns on the bifurcation parameter µ for ν = 2. Stable solutions are shown by thick lines and unstable solutions are shown by thin lines. The inset shows a blow-up near the origin

The general interaction potential also valid in higher dimensions is V = −µ

N

|aj |2 −

ai aj ak +

νijkl ai aj ak al + · · ·

(1.128)

j=1

= −µ

N

ρ2j −

ρi ρj ρk cos Θijk +

νijkl ρi ρj ρk ρl cos Θijkl + · · · ,

j=1

where the summation is carried out over all closed polygons formed by the wave vectors of extant modes, and Θijk = θi +θj +θk , Θijkl are sums of phases around the respective triangles or rhombi. This expression may also include resonant quartic terms combining the wave vectors forming nonplanar rhombi. The dynamic equations for aj follow from the gradient principle (1.126). All cubic terms in (1.128) are based on identical equilateral triangles, which may diﬀer only by orientation; they are therefore universal and all come with the same coeﬃcient that can be taken as unity. The cubic terms are phasedependent, and the potential is at minimum when the sum of the phases of modes forming an equilateral triangle is zero (modulo 2π). When phases adjust in this way, the triplet interactions are destabilizing; this is why symmetrybreaking transitions leading to the formation of stationary patterns are always subcritical and of the ﬁrst order. The emerging pattern is stabilized by quartic terms, which depend on angles between interacting modes and are speciﬁc for particular systems. The quartic terms are phase-dependent only when the rhombus formed by the respective wave vectors is nonplanar.

72

1 Dynamics, Stability and Bifurcations

Fig. 1.24. Striped and hexagonal patterns in the Belousov–Zhabotinsky reaction (Ouyang and Swinney, 1991, reproduced with permission). In the hexagonal patterns in the upper row, the total phase Θ diﬀers by π. The distorted pattern in the lower right picture represents a state intermediate between hexagons and stripes

In three dimensions, triplet interactions favor a combination of modes forming one of regular polyhedra with triangular faces – tetrahedron, octahedron, or icosahedron (Alexander and McTague, 1978). The former two correspond to the same body-centered cubic crystalline lattice, and the latter correspond to a quasicrystalline structure. One could expect that these structures are likely to have the lowest energy close to the bifurcation point, especially in the subcritical region, in the same way as the hexagonal structure is preferable in two dimensions. Since the pattern boosted by the resonant terms can be stabilized only by nonuniversal higher-order interactions, other structures can be actually chosen by the evolving system, and both experiment and numerical simulations show that the face-centered cubic structure is usually favored (Groh and Mulder, 1999). The special character of the icosahedral combination of modes is apparently responsible for the formation of quasicrystals with pentagonal symmetry (Shechtman et al., 1984), which was once seen as a sensational discovery contradicting the dogma of classical crystallography. Quasicrystalline structures in D dimensions can be, generally, formed by a combination of D + 2 or more modes. In two dimensions, a most likely

1.6 Dynamics of Planforms

73

40

20

0

-20

-40 -40

-20

0

20

40

Fig. 1.25. Two resonant triangles and the respective dodecagonal quasicrystalline pattern

(but still exotic) possibility is a “dodecagonal” pattern formed by a combination of two resonant triangles turned by π/6 relative to each other (Malomed et al., 1989). A piece of such a six-mode pattern, comprising almost symmetric dodecagonal and pentagonal “ﬂowers” but never repeating itself, is shown in Fig. 1.25. Fixing two phase relations to satisfy resonance conditions in the two resonant triangles still leaves four phase degrees of freedom. Thus, in addition to the two translational modes, there are two Goldstone “phasons” modifying the pattern in a nontrivial way. Quasicrystalline structure may be lost when some constituent modes are suppressed by higher-order interactions. Various transitions among possible patterns (one-mode striped, two-mode squared, three-mode hexagonal, and six-mode dodecagonal) in the six-mode system of amplitude equations, dependent on the angular dependence of stabilizing quartic interactions, were studied by Echebarria and Riecke (2001). Resonant interactions enhancing a triplet of modes forming an equilateral triangle may also become possible in the presence of inversion symmetry due to interaction with a homogeneous mode (Price, 1994; Hilali et al., 1995). If a homogeneous mode with the amplitude a0 is included, (1.121) is replaced by an enlarged equation system

74

1 Dynamics, Stability and Bifurcations

a˙ j = aj µ − |aj |2 − ν0 a20 − ν |aj−1 |2 + |aj+1 |2 + a0 aj−1 aj+1 ,   3 |aj |2  + σ(a1 a2 a3 + a1 a2 a3 ). (1.129) a˙ 0 = a0 µ0 − β0 a20 − β j=1

This system has no gradient structure, unless β = ν0 , σ = 1, which would be assured automatically only when the underlying system has a gradient structure. The polar form of (1.129) is

(1.130) ρ˙ j = ρj µ − ρ2j − ν0 a20 − ν ρ2j−1 + ρ2j+1 + a0 ρj−1 ρj+1 cos Θ,   3 ρ2j  + 2σρ1 ρ2 ρ3 cos Θ, (1.131) a˙ 0 = a0 µ0 − β0 a20 − β j=1

Θ˙ = −a0 sin Θ

3 j=1

ρj−1 ρj+1 . ρj

(1.132)

The equation for the sum of phases Θ = θj is very similar to (1.120) but now Θ may relax to either 0 or π, depending on whether a0 is positive or negative. In view of the symmetry to simultaneous inversion of the sign of a0 and the phase shift by π, both possibilities are totally equivalent. Thus, the system can relax to two alternative stationary states, which correspond to “up” and “down” hexagonal patterns with either maxima or minima forming a triangular grid, as in the two upper panels of Fig. 1.24. An interesting possibility arises at σ < 0 when both states, a0 > 0, Θ = 0, and a0 > 0, Θ = π, may be unstable, and the attractor is a heteroclinic orbit passing through these states. Chaotic behavior is likely to be observed when this orbit is weakly perturbed. 1.6.3 Degenerate Wave Modes In wave patterns emerging when modes with both k and ω diﬀering from zero are excited beyond the bifurcation point, the degeneracy is possible already in one dimension. The two symmetric forms correspond to waves propagating in the opposite directions. If a± are the complex amplitudes of the two waveforms, ei(kx±ωt) , two symmetric amplitude equations are derived similarly to Sects. 1.3.4 and 1.6.1. As in the case of the Hopf bifurcation, the solvability condition of the second-order equation is satisﬁed automatically. Evolution equations on an O(−2 ) time scale T are obtained as solvability conditions of the third-order equation. They are similar to the form (1.117) derived for coupled Turing modes in the presence of inversion symmetry but contain complex coeﬃcients: (1.133) a˙ ± = a± µ − ν+ |a± |2 − ν− |a∓ |2 .

1.6 Dynamics of Planforms

75

These equations, like (1.70), do not have gradient structure. The nontrivial stationary solutions that exist at Re(µ) > 0 have the form a± = ρ± eiΩT . As in the case of a single equation, the imaginary parts of all coeﬃcients do not aﬀect the real amplitudes ρ± and inﬂuence only the phase shift. If Re(ν+ ) < Re(ν− ), the stable solution is a single propagating wave, coinciding with the respective solution of (1.70). Otherwise, the stable solution is a± = idem = [Re(µ)/Re(ν+ + ν− )]1/2 , which corresponds to a standing wave. One can expect the dynamic equations to have a similar structure also in two dimensions, since triplet interactions of waveforms with (ω = 0) are forbidden. Indeed, the resonance condition now requires, besides vanishing sum of kj , that the sum of ωj also vanish for the waves forming a resonant triangle. Since all ωj are equal, this is evidently impossible, and the lowestorder nonlinear terms appearing in the amplitude equation are cubic. These terms can be, however, resonant when they involve interaction of pairs of ± waves propagating in the opposite directions. Let a± 1 , a2 be the amplitudes of the four waves with identical ω and k propagating in the opposite directions along, respectively, the x-axis and the y-axis. Then, the amplitude equation of a+ 1 is

+2 + + 2 − 2 − 2 + − + γa− a˙ + 1 = a1 µ − ν+ |a1 | − ν− |a1 | − β |a2 | + |a2 | 1 a2 a2 . (1.134) Other three equations are symmetrical. One can check that both wave vectors and frequencies add up to zero along the rectangle formed by all four modes that corresponds to the last term. This term is resonant, being dependent on the combination of phases Θ = θ1+ + θ1− − θ2+ − θ2− (Pismen, 1986; Hoyle, 1994). Only standing waves are enhanced by the resonant interactions that impede their decay into single propagating waves. Further on, we concentrate therefore on standing wave solutions with equal real amplitudes of the counterpropagating waves ρj = |a± j |. After transforming to the polar form, the realpart of (1.134) takes the form ρ˙ 1 = ρ1 µr − νr ρ21 − 2βr − Re γ e−iΘ ρ22 , ρ˙ 2 = ρ2 µr − νr ρ22 − 2βr − Re γ eiΘ ρ21 , (1.135) where ν = ν+ + ν− and the complex coeﬃcients are presented as ν = νr + iνi , etc. These equations can be rearranged by transforming to new variables p = ρ21 − ρ22 , q = ρ21 + ρ22 : q˙ = 2µr q − νr (q 2 + p2 ) − (2βr − γr cos Θ) (q 2 − p2 ), p˙ = 2µr p − 2νr pq − γi sin Θ(q 2 − p2 ).

(1.136)

The imaginary part of the polar form of (1.134) gives the equation of Θ: Θ˙ = −2γr q sin Θ − 2(νi − 2βi − γi cos Θ)p.

(1.137)

76

1 Dynamics, Stability and Bifurcations (a)

(b)

Γi 2

0.8 q p

0.6 0.4

1.5

Π

0.2 0

1

-0.2

0.5

-0.4

0.5

1

1.5

2

Γr

-0.6 t

Fig. 1.26. (a) Stability regions of the synphase square pattern (hatched ) and a single standing wave (shaded ) in the complex plane γ. (b) Oscillations at γr = 0.4, γi = 2. Other parameters: µ = ν = β = 1

Two easily computable stationary solutions of (1.136), (1.137) are synphase and antiphase locked states, which exist in the supercritical region µr > 0: p = 0,

cos Θ = ±1,

q=

2µr . νr + 2βr ∓ γr

(1.138)

Another solution is q = µr /νr , p = ±q, which corresponds to a single standing wave; the phase Θ is indeﬁnite in this case, since the resonance is switched oﬀ. To be deﬁnite, we further assume γr > 0, since changing the sign of this coeﬃcient just interchanges both solutions. The real part of other interaction coeﬃcients should be positive in a standard case when cubic interactions are stabilizing. Moreover, we restrict to conditions when one of the two standing waves would be suppressed in the absence of resonance and require νr < 2βr . The square standing wave pattern is then stabilized in the synphase conﬁguration, provided γr > βr −νr /2. The stability condition for a single-mode pattern is |γ| < 2βr . The bifurcations at both stability limits are of the Hopf type. A typical conﬁguration of stability regions in the complex plane γ is shown in Fig. 1.26a. In the overlap area, where both single-wave and synphase patterns are stable, both bifurcations should be subcritical; the unstable periodic orbit emerging at both points lies on a surface separating the attraction basins of both states. On the opposite, if the Hopf bifurcation takes place in a parametric domain where the alternative state is unstable, it should be supercritical. A stable periodic orbit emerging under these conditions corresponds to a “twinkling” square wave pattern with the amplitudes of both waves oscillating on the long time scale of (1.134), as shown in Fig. 1.26b. For γr = 0, the attractor is a heteroclinic orbit connecting the ﬁxed points q = 1, p = ±1, Θ = ±π/2.

1.6 Dynamics of Planforms

77

1.6.4 Biscale Resonances Other resonant structures may appear “accidentally” at degenerate bifurcations of a higher codimension when planforms with diﬀerent wavenumbers k are excited simultaneously. This can be achieved in a most natural way in two-layer systems where the wavelength of the excited pattern depends on the thickness of each layer, as in convection (Proctor and Jones, 1988), or diﬀerent diﬀusivities, as in a pattern-forming chemical system (Yang et al., 2002). More possibilities arise in nonlinear optics where spatial symmetry breaking may occur on diﬀerent wavelengths at rather close values of a control parameter (Pampaloni et al., 1997; Pismen and Rubinstein, 1999b). The resulting coupled amplitude equations can generate a variety of composite planforms. Various “superposition” patterns with nonresonant wave vectors of diﬀerent magnitude may be quite picturesque (Yang et al., 2002), but the amplitude dynamics can be nontrivial only when the excited modes are in resonance. A resonant triangle may include either three Turing modes or one Turing and two wave modes with ω = 0. In one dimension, the strongest resonance, appearing already in the second order of the bifurcation expansion, exists when the two wavelengths are in the 2:1 ratio, with the wave modes, if they are present, being the shorter ones. Taking into account the necessity to ﬁt the wavenumbers at two degenerate bifurcations to ensure resonance, the codimension of the bifurcation manifold rises to three. In two dimensions, no rigid ﬁtting of wavenumbers is required, since the resonant modes can form an isosceles triangle, as shown in Fig. 1.27a. When both excited modes are of Turing type, proliferation of resonances on equilateral triangles could make, however, the picture quite √ messy. An exceptional case, also requiring exact ﬁt of wavenumbers, is 3:1 resonance. Two reso√ nant triplets with the wavenumber ratio 3 form three additional resonant triangles when the wave vectors of the two triplets are turned at π/6 relative to each other (Fig. 1.27b). Two relations connecting the phases of each triplet are ﬁxed by resonance conditions around the two equilateral triangles. The three isosceles triangles give three more relations, only two of which are independent, leaving two translational degrees of freedom. Depending on the sign of the interaction coeﬃcients, the phases around each resonant triangle should sum up to either 0 or π (modulo 2π). For each case, there are several combinations of phases satisfying these conditions, and there is a possibility that, due to mutual frustration, runaway to inﬁnity may be prevented even without suppression by higher-order interactions. Superposition of all resonant modes yields a superlattice pattern, such as the one shown in Fig. 1.27c. Resonant interaction coeﬃcients can be computed starting from a general RDS in the same manner as described in Sect. 1.6.1. Let U j be eigenvectors of the two excited modes satisfying (fu − D0 kj2 )U j = 0, j = 0, 1. A resonant isosceles all-Turing triangle containing two modes with the wavenumber k1 and amplitudes a± and a mode with the wavenumber k0 and amplitude a0 contributes to the second-order equation of a+ and a− the terms, respectively,

78

1 Dynamics, Stability and Bifurcations (c) (a)

k

20

k 10

k0 (b)

0

-10

-20 -20

-10

0

10

20

Fig. 1.27. (a) A resonant isosceles triangle. (b) Resonant triangles formed by two √ triplets of modes with the scale ratio 3. (c) The respective hexagonal superlattice

ν1 a0 a− and ν1 a0 a+ with the coeﬃcient ν1 = −2U †1 · fuu [U 1 , U 0 ] and to the equation of a0 the term ν0 a+ a− with the coeﬃcient ν0 = −2U †0 · fuu [U 1 , U 1 ]. Generally, all interaction coeﬃcients are distinct, and, unlike generic Turing bifurcation, the system does not have gradient structure, which would exist only in a special case ν1 = ν0 . For a triangle formed by one Turing and two wave modes, the interaction coeﬃcient ν1 entering the equations of wave modes is complex, while the interaction coeﬃcient entering the equation of Turing modes ν0 = −2U †0 · fuu [U 1 , U 1 ] is real. Because of the absence of gradient structure, persistent nonstationary behavior is possible even in the case of all-Turing degeneracy but is most pronounced when wave–Turing resonance is realized. As an example, we shall consider the simplest wave–Turing resonant structure including two wave modes a± eik± ·x−ωt with |k± | = idem, and a Turing mode a0 eik0 ·x , with the wave vectors satisfying the resonance condition k+ − k− + k0 = 0, as shown in Fig. 1.27a. Under these conditions, the amplitudes may remain ﬁnite even when higher-order damping interactions are switched oﬀ (Pismen and Rubinstein, 1999b). The general form of the lowest order amplitude equations is

1.6 Dynamics of Planforms

79

a˙ 0 = µ0 a0 + ν0 a− a+ , a˙ + = µ1 a+ + ν1 a− a0 , a˙ − = µ1 a− + ν1 a+ a0 .

(1.139)

where µ0 < 0 and ν0 are real, while µ1 and ν1 are complex. Since the imaginary part of µ1 can be absorbed in frequency, this parameter will also be viewed as real, while ν1 will be presented in a polar form ν1 = ν eiα . Using the polar representation of the complex amplitudes, a0 = ρ0 eiθ0 , a± = ρ± eiθ± , (1.139) can be reduced to a system of four real equations including a single phase combination Θ = θ0 + θ+ − θ− . Three out of ﬁve real parameters of this system can be eliminated by rescaling the amplitudes and time. To be deﬁnite, we assume ν0 > 0; changing the sign of this parameter can be compensated by turning the composite phase Θ by π. Clearly, the amplitude of the Turing mode can be ﬁnite only in the subcritical region, µ0 < 0. Using √ |µ0 |−1 as a time scale, |µ0 |/ν as a scale of ρ0 , and |µ0 |/ νν0 as a scale of ρ± , we transform (1.139) to a polar form containing just two parameters µ = µ1 /|µ0 | and α: ρ˙ ± = µρ± + ρ∓ ρ0 cos(Θ ∓ α), (1.140) ρ˙ 0 = −ρ0 + ρ+ ρ− cos Θ, ρ ρ ρ ρ− + − + Θ˙ = − sin Θ − ρ0 sin(Θ + α) + sin(Θ − α) . (1.141) ρ0 ρ− ρ+ The stationary values of the amplitudes ρ0 , ρ± obtained by resolving (1.140) are 1/2 |µ| µ , ρ = − . ρ0 = ± 1/2 cos(Θ ± α) cos Θ [cos(Θ − α) cos(Θ + α)] (1.142) The composite phase Θ should be such that the above expressions be real. Plugging (1.142) in (1.141) brings the equation deﬁning stationary values of Θ to the form sin(Θ − α) sin(Θ + α) + tan Θ + |µ| = 0. (1.143) | cos(Θ + α)| | cos(Θ − α)| A simple solution of this equation ensuring a positive value of ρ0 is Θ = 0, and the condition required to keep ρ± in (1.142) real is either µ < 0, α < π/2, or µ > 0, α > π/2. This deﬁnes a symmetric solution with equal amplitudes of the wave modes: µ (1.144) Θ = 0, ρ0 = ρ2± = . cos α The symmetric solution merges at µ = 0 with the trivial solution ρ0 = ρ± = 0. The eigenvalues of (1.140), (1.141) linearized near this solution are

1 1 (1.145) −1 ± 1 − 8µ , 4µ − 1 ± 1 − 16µ2 tan2 α . 2 2

80

1 Dynamics, Stability and Bifurcations

The ﬁrst pair of eigenvalues is always negative at µ > 0, while the second pair deﬁnes the stability conditions π 3π <α< . 2 4

1 1 cos2 α < µ < , 2 4

(1.146)

A supercritical secondary Hopf bifurcation takes place on the branch of symmetric solutions at µ = 1/4, while a pitchfork bifurcation to a pair of asymmetric solutions occurs at 2µ = cos2 α. The latter is also supercritical at µ < 1/4. For cos(Θ ± α) having the same sign, (1.143) reduces to tan Θ −

|µ| sin 2Θ = 0. cos(Θ − α) cos(Θ + α)

(1.147)

A nontrivial root of this equation leads to an asymmetric solution that exists at 2µ < cos2 α: (a)

(b) Μ 1

Μ 0.25

0.8

0.2 0.15

Ρ

0.6

S

0.1

0.4

0.05

0.2

A

Ρ0 Α Π 0.05 0.1 0.15 0.2 0.25

Α Π 0.55 0.6 0.65 0.7 0.75

(c)

(d) 1.6 1.4 1.2

1

Ρ

Ρ

1.2

0.8

1 0.8 0.6

0.6 t

t

Fig. 1.28. Dynamics near the wave–Turing resonance. (a) Stability regions of the symmetric (S) and asymmetric (A) solutions. Solid line denotes the pitchfork and dashed lines denote the Hopf bifurcations. (b) Dependence of amplitudes on µ for α = 0.6π. (c) Oscillations of ρ± (solid lines) and ρ0 (dashed line) at α = 0.65π and µ = 0.27. (d) The same for µ = 0.28, showing loss of symmetry between the two wave modes

1.6 Dynamics of Planforms

µ (1 − 2µ) sin2 α 2 , ρ0 = √ , cos Θ = 1 − 2µ 2 sin α

−1/2 µ(1 − 2µ) . ρ± = cos α ± cos2 α − 2µ sin α

81

(1.148)

The two asymmetric solutions are transformed one to the other by switching the sign of Θ and interchanging the amplitudes of the wave modes. At still lower values of µ, the asymmetric solutions undergo a supercritical Hopf bifurcation. The bifurcation locus in the plane (α, µ) is given implicitly by the relation − (1 − 4µ)(1 − 5µ + 8µ2 ) sin4 α + 3(4 − 7µ + 4µ2 )(cos2 α − 2µ)2 + [3 − 8µ(2 − µ)(1 − 2µ)] (cos2 α − 2µ) sin2 α = 0. (1.149) The resulting bifurcation diagram in the plane µ, α is presented in Fig. 1.28a and an example of solution branches is shown in Fig. 1.28b. Above the Hopf bifurcation at µ = 1/4, the amplitudes ρ± oscillate out of phase (Fig. 1.28c), and the symmetry between the two wave modes is lost as µ is further increased (Fig. 1.28d). The amplitude of oscillations grows without limit as µ is further increased; runaway occurs as well when α draws closer to π. Conﬁnement to the small-amplitude region by quadratic interactions is possible only when the Turing mode is subcritical and the wave mode is supercritical but not too strongly. When runaway to large amplitudes is observed, the actual pattern may be stabilized by nonresonant higher-order interactions.

2 Fronts and Interfaces

2.1 Planar Fronts 2.1.1 Space-Dependent Amplitude Equations If the amplitude is allowed to vary in space, a diﬀusional term is added to the respective amplitude equation. For a bifurcation at zero eigenvalue in a general RDS, spatial dependence can be incorporated in the formal expansion of Sect. 1.3.2 by scaling the spatial derivatives as ∇ = O((n−1)/2 ) when the expansion proceeds to the nth order. Starting from a general RDS (1.18), one obtains then the term D∇2 a with the amplitude diﬀusivity D = U † · DU added to the solvability condition in the respective order, yielding a single (scalar ) diﬀusion equation with a polynomial nonlinearity of degree n. We shall write this equation in a general form ∂t u = D∇2 u + f (u),

(2.1)

where the function f (u) has two or more zeroes that correspond to either stable or unstable homogeneous stationary states (HSS). This function may depend on a number of parameters modifying its shape. In particular, a cubic equation obtained at the cusp singularity (n = 3) can be seen as the simplest model describing coexistence and competition of alternative stationary states. The √ diﬀusivity D can be removed by rescaling the coordinates x → lx, where l = Dτ is the characteristic scale of spatial inhomogeneities in the system u) for and τ is a characteristic time scale, which can be chosen as τ −1 = f ( some nominal u = u . We shall further recognize l as the characteristic width of a front separating alternative states. Equation (2.1) can also model, of course, an actual physical system when a single variable or “order parameter” is suﬃcient for its description. It belongs to a wide class of equations of gradient dynamics (Sect. 1.1.3) and is derivable from the “energy” functional F = L dx, L = V (u) + 12 D|∇u|2 . (2.2) ut = −δF/δu,

84

2 Fronts and Interfaces (a)

(b) fu

Vu

u

u

Fig. 2.1. A function f (u) with three zeros (a) and the respective double-well potential (b)

In thermodynamic models the functional F is identiﬁed with free energy decreasing in the course of evolution to equilibrium. The ﬁrst term in the La grangian L is potential energy V (u) = − f (u)du; the second term is distortion energy taking into account the “costs” of the gradients of the order parameter. The coeﬃcient D is called in this context rigidity rather than diffusivity. In nonequilibrium systems, the energy functional lacks direct physical interpretation, but can be deﬁned formally as stated above. It is remarkable that gradient structure emerges near a bifurcation point even when it is not present in the underlying system, such as a general RDS (1.18), which cannot be assigned an energy functional unless the functions fi = −∂V /∂ui are derivable from a potential V (u). In this and next sections, we shall consider fronts between two HSS which are both stable to inﬁnitesimal perturbations and correspond to two alternative minima of the potential V (u). A typical suitable function f (u) sketched in Fig. 2.1a has three zeros, two of which, u = u± s (with ∂f /∂u < 0), correspond to stable HSS and the middle one, u = u0s (with ∂f /∂u > 0) to an unstable state. The respective double-well potential is shown in Fig. 2.1b. The position and depth of the minima can be regulated by shifting some parameter of the model. Suppose that two stable HSS with diﬀerent energy are approached in spatially removed parts of an extended system. The boundary between them, also called a front or a kink , is a narrow region of a characteristic width l where the dependent variable changes between the two alternative values u = u± s . The state with a higher energy is metastable, and the overall energy of the system will decrease when the boundary between the alternative states propagates in such a way that the area occupied by this state decreases. Since both HSS are attractors, small perturbations of u± s tend to decay. This serves as a localization mechanism, which ensures that the front would retain its stationary form as it propagates. We shall see that the front separating two stable HSS can be characterized by a unique propagation speed. The situation is diﬀerent when the retreating state is unstable; this case will be considered in Sect. 2.3.

2.1 Planar Fronts

85

2.1.2 Propagating Front as a Heteroclinic Trajectory Consider the front propagation in one dimension or, equivalently, propagation of a straight-line front in 2D or a planar front in 3D along the normal direction z. Assuming that the propagation is stationary, (2.1) can be rewritten in the comoving frame propagating with the front velocity c, as yet unknown. By convention, c is positive when the down-state advances. The steadily propagating solution depends on a single coordinate x = z − ct, and (2.1) reduces to an ordinary diﬀerential equation cu (x) + Du (x) + f (u) = 0,

(2.3)

subject to the boundary conditions u = u± s at x → ±∞.

(2.4)

Here u = u± s (with ∂f /∂u < 0) are two stable HSS, to be called up and down state. This problem is overdetermined, so that the solution exists only at a single value of c; thus, the propagation speed is obtained as a solution of a nonlinear eigenvalue problem. This is made transparent when (2.3) is viewed as a damped mechanical system, where the coordinate x is playing the role of time. In this interpretation, diﬀusivity is replaced by mass; −f (u) is a force with the potential −V (u), and c is the friction coeﬃcient. Since the sign of the potential is interchanged, stable equilibria of the reaction-diﬀusion equation are mapped onto unstable mechanical equilibria and vice versa; the energy integral of a one-dimensional reaction-diﬀusion system corresponds to the action integral of a mechanical system. The front solution corresponds to a heteroclinic trajectory connecting the equilibria u = u± s . In mechanical terms, this would be a trajectory of a particle starting from the higher maximum of the potential −V (u), dissipating a part of its energy through friction, and stopping precisely at the lower maximum. This can be made possible only by adjusting the friction coeﬃcient in a unique way. A more precise proof is based on a “counting” argument. Linearize (2.3) in the vicinity of either stable solution to obtain the equation with constant coeﬃcients (2.5) cu (x) + Du (x) − αu = 0. The coeﬃcients α = −f (u± s ) are negative, in view of the presumption that both equilibria are stable. The solutions of (2.5) are combinations of exponents u ∝ exp(λ± x), with the eigenvalues

c (2.6) λ± = −1 ± 1 + 4αD/c2 . 2D For α > 0, both pairs of eigenvalues λ± are real and of diﬀerent sign, so that both HSS are saddles when viewed as equilibria of (2.3). There is a

86

2 Fronts and Interfaces

p

(b)

p

(a)

u

u

Fig. 2.2. Generic trajectories in the phase plane u, p = u (x) (a) and a nongeneric set of trajectories containing a heteroclinic orbit (b)

single trajectory leaving or entering the saddle point. The front solution is a heteroclinic trajectory connecting both equilibria. Such a trajectory does not exist generically (Fig. 2.2), but can be only obtained by adjusting a single parameter of the equation. The adjustable parameter is the propagation speed c which is unknown a priori. Since the motion slows down near the equilibrium, the heteroclinic trajectory takes inﬁnite time to traverse; accordingly, the kink solution exists, strictly speaking, only on an inﬁnite line. The equilibrium is, however, approached exponentially, and therefore, for all practical purposes, the deviation from a HSS becomes negligible at distances of several basic length units. The sign of c can be determined by multiplying (2.3) by u (x) and integrating over the inﬁnite axis: ∞ ∞ − [u (x)]2 dx = − f (u)u (x)dx = V (u+ (2.7) c s ) − V (us ). −∞

−∞

Since the integral on the l.h.s. is positive, the sign of c coincides with the sign of the diﬀerence between the energies of the two competing states. The state with the lower energy advances, and the energy of the entire system is lowered due to increasing area occupied by the more favorable state. The potential diﬀerence equals to the derivative of the energy with respect to the front position ζ(t) = ζ0 + ct, which can be interpreted as a “thermodynamic force” acting on the fronts. The speed can therefore be presented as the ratio of the thermodynamic force and the dissipation integral : ∂F dζ = I −1 , dt ∂ζ where p(u) = u (x).

I=

∞

−∞

[u (x)]2 dx =

u+ s

u− s

p(u)du,

(2.8)

2.1 Planar Fronts

87

2.1.3 Computation of the Propagation Speed As a rule, the exact value of the propagation speed can be only computed numerically. Equation (2.3) is solved most easily in phase plane variables, using u as an independent, and p = u (x) as a dependent variable. Then it is rewritten as (2.9) cp + Dpp (u) + f (u) = 0. Since p vanishes when a HSS is approached, this equation has to be integrated − with the boundary conditions p(u+ s ) = p(us ) = 0. In this formulation, it is obvious that the ﬁrst-order ODE with two boundary conditions is overdetermined. One could argue that a simple transformation to the form (2.9) might have replaced lengthy arguments from the preceding subsection; this transformation is, however, possible only when the unknown function u(x) is monotonic. The propagation speed can now be found by integrating (2.9) starting − from either u+ s or us and ﬁnding the value of c by shooting. This nonautonomous equation is, however, singular at u = u± s , since the coeﬃcient at the derivative vanishes there. The numerical solution cannot start precisely at the equilibrium point. Instead, the initial point should be slightly shifted along the unstable manifold of the saddle point of (2.9). Taking as a starting point u = u− s + δ with δ 1 and choosing the sign of p to be positive, we set the initial condition − (2.10) p(u− s + δ) = λ+ (us )δ, − where the eigenvalue λ+ (u− s ) is deﬁned by (2.6) with α = −f (us ). Using some chosen starting value of c, we continue the integration until either u reaches the maximum value u+ s or p vanishes. In the former case, the guessed value of c is too small and in the latter, too large (see Fig. 2.3). An eﬃcient numerical procedure may start with the upper and lower a priori bounds of c and converge to the true value by iteratively subdividing the interval, until is found, p(u+ s ) drops below a set small target value. Once the function p(u) the proﬁle u(x) is computed (in an implicit form) by quadrature: x = p−1 du. The propagation speed can be computed analytically when f (u) is a cubic: − 0 f (u) = −(u − u+ s )(u − us )(u − us ),

(2.11)

where u0s is the unstable middle root. The solution satisfying both boundary conditions is sought for in the form − p(u) = b(u+ s − u)(u − us ).

(2.12)

When this ansatz is substituted in (2.9), the equation factors out as − 2 0 2 − + (u − u+ s )(u − us )[(2b D − 1)u + bc + us − b D(us + us )] = 0.

(2.13)

The bracketed linear form has just enough adjustable coeﬃcients to make it vanish. This yields

88

2 Fronts and Interfaces

p

(a) 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1

0.5

(b)

c D 1.2 1 0.8 0.6 0.4 0.2 0 u

0.5 0.1

0.2

0.3

Μ

Fig. 2.3. (a) Curves p(u) with an underestimated (upper curve) overestimated (lower curve) and exact (middle curve) values of propagation speed (b) The dependence of the propagation speed on the parameter µ for the cubic function f (u) = u − u3 − µ

√ b = ±1/ 2D,

c=

√

+ 2D u0s − 12 (u− s + us ) .

(2.14)

The speed is assumed to be positive when the lower state advances. The two signs of b correspond to the two alternative orientations of the front. For the cubic in the form f (u) = u − u3 − µ, the speed can be written explicitly using convenient trigonometric expressions for the roots: √ √ u± u0s = (2/ 3) sin ψ, s = −(2/ 3) sin(ψ ∓ π/3), ψ=

1 3

arcsin

√ 3 2 3µ .

(2.15)

Using (2.15) in (2.14) yields c=

√

6D sin

1 3

arcsin

√ 3 2

3µ .

(2.16)

The dependence of the propagation speed on the parameter µ is shown in Fig. 2.3b. Note that the curve becomes vertical at the bifurcation point µ = 2/33/2 wherethe roots u0s and u+ s merge, terminating there at the maximum value cm = 3D/2. The curve continues antisymmetrically into the region µ < 0. The good luck does not extend to higher order polynomials. Take f (u) of order 2n − 1; the order should be odd to prevent runaway to inﬁnity in the dynamical system u (t) = f (u). Then p(u) must be of order n to match the orders of f (u) and p(u)p (u). After the boundary conditions are satisﬁed, n − 1 free parameters are left in the polynomial p(u). Those should be used, together with c, to cancel 2(n − 1) terms of the polynomial remaining after factoring. The numbers match only for n = 2, i.e. for a cubic f (u).

2.1 Planar Fronts

89

2.1.4 Maxwell Construction In the special case when both energies V (u± s ) are equal, called Maxwell construction, the propagation speed is zero, so that both states coexist at equilibrium. This may be achieved by adjusting some parameter µ at a certain level µ = µ0 , e.g. µ = 0 for the cubic form. The resulting nonuniform stationary state carriesinterfacial energy, which, according to (2.2), consists of the potential energy V (u) dx and the distortion energy 2 1 1 u (x) dx = 2 D p(u)du. (2.17) 2D Equation (2.3) with vanishing propagation speed, Du (x) + f (u) = 0

(2.18)

is rewritten in phase plane variables as Dpp (u) + f (u) = 0.

(2.19)

Denoting T = 12 Dp2 , this can be expressed as T (u) = V (u). The value of V (u± s ) can be taken as zero, and T evidently vanishes for any HSS; hence, integrating the latter relation proves that the distortion energy equals the potential energy at any point. This is formally equivalent to the virial theorem of classical mechanics (where distortion energy is replaced by kinetic energy). Thus, the surface energy per unit area, or surface tension 1 of the stationary front can be computed as

∞

2

u (x) dx = D

σ=D −∞

u+ s

u− s

p(u)du =

u+ s

u− s

2DV (u)du.

(2.20)

Note that surface tension is proportional to the dissipation integral (2.8), σ = DI. The stationary front proﬁle can be computed in quadratures. Integrating (2.19) yields 2 V (u) − V (u− (2.21) p(u) = ± s ), D and the front proﬁle is expressed in an implicit form u D du x(u) = . (2.22) 2 u0s V (u) − V (u− ) s

The origin x = 0 may be arbitrary, but, to be deﬁnite, we have placed it at the location of the unstable intermediate HSS u0s . 1

This term stresses the analogy to phase transitions in equilibrium systems – see Sect. 2.4.1.

90

2 Fronts and Interfaces

A useful analytical form is the kink solution of the symmetric cubic equation with f (u) derived from the quartic potential 2 V (u) = 14 1 − u2 . (2.23) This lowest order symmetric form appears as a result of expansion near a pitchfork bifurcation, or a second-order phase transition. The universal parameterless form obtained after rescaling is the cubic reaction-diﬀusion equation (1.20). The inhomogeneous stationary solution of this equation dependent on a single coordinate x normal to the front is elementary: √ √ u(x) = tanh x/ 2 . (2.24) p(u) = 1 − u2 / 2, 2.1.5 Unstable Nonuniform States In addition to the heteroclinic (front) solution, (2.3) with a correctly chosen speed has a continuous family of periodic solutions. The respective trajectories in the phase plane (u, p) shown in Fig. 2.4a lie within the heteroclinic loop and surround the unstable equilibrium u = u0s . For an incorrectly chosen speed, (2.3) has a homoclinic trajectory approaching at x → ±∞ the metastable HSS, interpreted as a propagating pulse, and a continuous family of periodic solutions (Fig. 2.2a), corresponding to wave trains of varying wavelength. A family of pulses propagating with diﬀerent speeds is shown in Fig. 2.9b. All these solutions are dynamically unstable when viewed as solutions of the evolution equation (2.1). Stationary periodic patterns (c = 0) can be computed in quadratures as in the preceding subsection, though expressions are rather more complicated. The solution of (2.19) should be written as (2.25) p(u) = ± 2/D V (u) − V (umin ) (a)

(b)

p

p

u

u

Fig. 2.4. (a) Phase plane trajectories p(u) corresponding to unstable wave trains. (b) A family of homoclinic trajectories corresponding to unstable pulses propagating with diﬀerent speeds. Only half-trajectories are shown. The limiting heteroclinic trajectory is shown in both ﬁgures

2.1 Planar Fronts

x(u) =

D 2

u

umin

du V (u) − V (umin )

91

(2.26)

where umin and umax are the minimum and maximum values of u satisfying the condition V (umin ) = V (umax ). The last formula gives in an implicit form the proﬁle u(x) for a half-period of the solution. The admissible values of V lie between the potential of the unstable state V (u0s ) and the potential of the metastable state, i.e. the larger of V (u0+ ) and V (u0− ). The lower bound of this interval corresponds to a homoclinic orbit (a critical nucleus) when the two minima are unequal and to a heteroclinic orbit (a static front) when V (u0+ ) = V (u0− ). The total wavelength is umax √ du L = 2D . (2.27) V (u) − V (umin ) umin One can see that it diverges when V (u) is extremal, i.e. for umin = us± . For a cubic f (u), the proﬁles u(x) can be expressed in elliptic functions; in particular, for the symmetric f (u) = u − u3 and D normalized to unity, the solution is u(x) = r sn(αx|k), (2.28) where 0 < r < 1, α2 = 1 − r2 /2. The modulus of the elliptic Jacobi function sn(x|k) is k = r2 /(2 − r2 ) and the spatial period of the pattern (2.28) is L = 2K(k)/α, where K(k) is the complete elliptic integral of the second kind. All periodic patterns have ﬁnite mean energy density. Instability of stationary periodic patterns is proven easily, since the linearization of (2.18) is a Sturm–Liouville problem. Presenting, as usually, small perturbations from a stationary solution as u = u1 (x)eλt , we arrive at the eigenvalue problem (2.29) Du1 (x) + f (u0 (x))u1 = λu1 . The left-hand side is the linearization of (2.18) near the stationary solution, which we denote now as u0 (x). It is annulled by the derivative u1 = u0 (x), as proven by diﬀerentiating (2.18). Thus, u0 (x) is the eigenfunction of (2.29) with zero eigenvalue. If u0 (x) is a monotonic stationary front, this eigenfunction, corresponding to translations of the front, is sign-deﬁnite, and therefore zero is the largest eigenvalue, and the solution is stable to all perturbations modifying its shape. For all other solutions, the translational eigenfunction is sign-changing. According to the general theorem on Sturm–Liouville systems, there must exist a sign-deﬁnite eigenfunction with a larger eigenvalue, and therefore all nonmonotonic solutions are unstable. 2.1.6 Front Interactions Consider two fronts of opposite polarity, to be called a kink and an antikink, centered at x = ±L/2 (Fig. 2.5). One can see in this ﬁgure that each front

92

2 Fronts and Interfaces

is only weakly perturbed by the exponential tail of its counterpart when the fronts are removed at a distance of several characteristic diﬀusional lengths. In the absence of interaction, each front would be described by the respective stationary solutions, u0 (x + L/2) and u0 (L/2 − x). As the separation L decreases, the energy F of the kink/antikink pair, equal to 2I at L → ∞, decreases as well. The derivative ∂F/∂L can be interpreted as a “force” bringing the two fronts together. In nonequilibrium thermodynamics this is known as Peach–K¨ ohler force. The drift velocity can be computed in a most transparent way starting from the expression for dissipation rate (1.29). The computation is possible when the drift is slow, so that the proﬁle of u always remains quasistationary and can be expressed by a certain function depending on the current value of L as a parameter and independent of the history of motion. Under these conditions, (1.29) can be rewritten as ∂F dL =− ∂L dt

δF δu

2 dx.

(2.30)

An approximate solution describing a quasistationary kink/antikink pair can be obtained by matching the inner solution near the front and the outer solution in the region between the two fronts where u is close to u+ s (see Fig. 2.5). The inner solution is the standard front proﬁle u(i) = u0 (L/2 ± x), , where the deviation u veriﬁes the while the outer solution is u(o) = u+ s −u linearized equation u = 0, (2.31) u (x) − α with α = −f (u± s ) > 0. A symmetric solution is √ u = A cosh( αx),

(2.32)

where the constant A should be chosen to match the outer asymptotics of the inner solution. For the kink centered at x = −L/2, the latter has the form (a)

(b)

u

u

1

1.5 1 0.5 0.99

0.5 10

5

1

1

1.5

Ζ

0.98 5

0.5

0.5

10

Ζ

0.97 0.96 0.95 0.94

Fig. 2.5. (a) The proﬁle of interacting fronts. (b) The blow-up of the matching solution (2.32)

2.1 Planar Fronts

lim u(o)

x→∞

√ = u+ s − a exp[− α(x + L/2)].

93

(2.33)

For example, for a cubic, a = α = 2, as can be seen by computing the asymptotics of (2.24) at x → ∞. Thus, we have − u(o) = u+ s − 2ae

√

αL/2

√ cosh( αx).

(2.34)

By symmetry, this also matches the antikink centered at x = L/2. Combining the inner and outer solutions matched at some position x = √ ±X, such that exp[ α(X − L/2)] 1, we compute the energy F =2

L/2−X −∞

= 2I −

∞

[u0 (x)]2 dx +

X

{[ u (x)]2 + 2V [u(o) (x)]}dx X √ √ √ a2 αe− α(2x+L) dx + 2 a2 αe− αL cosh(2 αX)dx

L/2−X √ √ = 2I − a2 αe− αL .

0

0

(2.35)

The transformation of the outer integral takes into account that the potential 1 can be presented near u = u+ u2 . The result is independent of s as V = 2 α the matching position √ X, provided the above condition is satisﬁed, so that √ cosh(2 αX) ≈ 12 exp(2 αX). As expected, ∂F/∂L > 0, and the dependence on separation is exponential as long as separation is large compared with the eﬀective front thickness. Diﬀerentiating the ﬁnal expression in (2.35) with respect to L gives the thermodynamic force on the l.h.s. of (2.30). This force can be compensated by shifting the parameters to lower the energy of the state in the segment between the two fronts. Since the total energy gain is proportional to the length of the region occupied by the preferred state, the stationarity condition + is ∂F/∂L = ∆V = V (u− s ) − V (us ). For example, for a simple parametric shift f (u) → f (u) + µ with µ 1, the energy gain for the upper state is − ∆V = µ(u+ s − us ). Using (2.35), we see that the domain with the width L is stationary at √ a2 α −√αL e . (2.36) µ=µ = + us − u− s The equilibrium is unstable, but can be maintained if the parameter µ is adjusted dynamically, for example by maintaining the ratio of domain lengths of the alternative states (see Sect. 2.2.4). Returning to the dynamical problem, it remains to evaluate the r.h.s. of (2.30). According to the gradient dynamics principle (2.2), the variation δF/δu equals in the evolving system to the time derivative ut ≈ u0 (x)dL/dt with an inverted sign. Thus, (2.30) can be rewritten as ∞ √ dL ∂F =− [u0 (x)]2 dx = − 12 I −1 a2 αe− αL . (2.37) dt ∂L −∞

94

2 Fronts and Interfaces

The dissipation integral is replaced here by twice the standard value for two noninteracting fronts, since the correction due to interactions is exponentially small. This formula can be extended to an arbitrary array of interacting fronts centered at x = ζj . Since interactions decrease exponentially with separation, only nearest-neighbor interactions are relevant. The generalization of (2.37) − for a symmetric case α = f (u± s ) = f (us ) is √ √ dζj = 12 I −1 a2 α e− α(ζj+1 −ζj ) − e− α(ζj −ζj−1 ) . dt

(2.38)

In an asymmetric case when α± = f (u± s ) diﬀer, one should replace in this formula α by α+ in the ﬁrst term and by α− in the second term if the front at the jth position is a kink, and the other way around if it is an antikink. The system (2.38) has a family of stationary solutions ζj = jL, j = 0, ±1, ±2, . . . corresponding to an inﬁnite array of equidistant fronts (or to a ﬁnite array on a ring), i.e. to a stationary periodic pattern. We have already argued in the preceding subsection that such a pattern is unstable. The instability is easily proven by linearizing (2.38) in the vicinity of this solution. Equation (2.38) does not describe the last stages of interaction at short distances and the annihilation of the colliding kink and antikink, but it is clear that the duration of these stages is much shorter than the time of slow approach of fronts originally separated at distances far exceeding the characteristic diﬀusional length.

2.2 Weakly Curved Fronts 2.2.1 Aligned Coordinate Frame The front solution is neutrally stable to translations along the x-axis. This Goldstone mode is weakly perturbed when the translation is weakly nonuniform, so that the front becomes curvilinear but the curvature radius still far exceeds the characteristic front thickness. Propagation of a weakly curved front should be studied using a coordinate frame aligned with its deformed shape. First, we have to deﬁne the nominal front position, replacing a diﬀuse transitional region by a planar curve C. The most natural choice is to ﬁx it as the locus of the unstable zero of the function f (u). The curve chosen in this way can be approximated locally by a circular segment, and (2.3) rewritten in polar coordinates r, φ. The local coordinates transverse and parallel to the front can be expressed as x = r − R and y = Rφ, and since the curvature radius R far exceeds the characteristic diﬀusional scale, x R in the transitional region where u diﬀers signiﬁcantly from both alternative HSS. Then the 2D Laplacian can be approximated as ∇2 = r−1 ∂r (r∂r ) + r−2 ∂φ2 = ∂x2 + κ∂x + ∂y2 + O(κ2 ),

(2.39)

2.2 Weakly Curved Fronts

95

where κ = R−1 1 is the local curvature. Generally, any planar curve C can be deﬁned at any moment of time t in a parametric form X(s; t), where s is the arc length. The tangent to C at any point is the derivative X s = l. Since l, by construction, is a unit vector, the vector ls must be orthogonal to l. By deﬁnition, the direction of ls is the normal n, and its absolute value is the curvature κ. Alternatively, the curve can be parametrized by any coordinate y; the inﬁnitesimal arc length ds is deﬁned as ds = g dy, where g = |X (y)| is the metric factor. Then the unit tangent vector is computed as l = X (s) = g −1 X (y). The coordinate frame aligned at any point of the curve with the tangent and normal vectors is set up in the following way. By convention, the x-axis has its origin on the curve and is directed from the region occupied by the down state to the region occupied by the up state, so that the boundary conditions (2.4) are approached asymptotically far from the front location. The vector n is directed, by deﬁnition, toward the local center of curvature removed at the distance |κ|−1 . In view of these direction conventions, the curvature κ should be considered positive when the normal is directed toward the down state, and negative otherwise. The coordinate lines x = const are obtained by shifting the curve along the normal by a constant increment, as shown in Fig. 2.6. Evidently, this shift causes the length to increase on convex, and to decrease on concave side of the curve. The length element is computed as ds2 = dx2 + (1 + κx)2 dy 2 .

(2.40)

Reading the metric tensor from this expression, the Laplacian is expressed as

∇2 = (1 + κx)−1 ∂x (1 + κx)∂x + ∂y (1 + κx)−1 ∂y . (2.41)

us

x

n n

us –

Fig. 2.6. Construction of the aligned coordinate frame. The coordinate lines are shown in gray. Arrows show the local directions of the normal n and the x-axis. Note of a singularity developing on the concave side

96

2 Fronts and Interfaces

The aligned system is well deﬁned only suﬃciently close to the curve, due to a singularity developing on the concave side at a distance of O(κ−1 ). If the curvature radius far exceeds the diﬀusional range l, the aligned system remains regular in the region where u changes between the alternative equilibria u = u± s . If the thickness of this transitional region is l = O(1), the curvature correction can be treated perturbatively when κ 1. We set κ = O(), expressing the order of magnitude through a book-keeping small parameter . The state variable(s) should change slowly in the tangential direction. The appropriate scaling compatible with the scaling of curvature can be estimated by computing the curvature of a slightly bended curve deﬁned in the Cartesian coordinates X, Y by the equation X − ζ(Y ) = 0. Assuming ζ (Y ) 1, the components of the normal vector are, in the leading order, nX = 1, nY = in the leading order, as κ = −ζ (Y ), and the curvature is computed, again √ ζ (Y ). Thus, one should assume ∂Y = O( ) to obtain κ = O(). Adopting the same scaling for the derivatives with respect to the transverse coordinate in the aligned frame, we rewrite the Laplacian (2.41) as (2.42) ∇2 = ∂x2 + (κ∂x + ∂y2 ) − 2 x κy ∂y + 2κ∂y2 + κ2 ∂x + O(3 ). Up to the ﬁrst order, the general derivation is equivalent to the simpliﬁed version using the local polar coordinates, but in the second order there is an additional term due to transverse variation of curvature. In 3D, one can deﬁne in a similar way a coordinate system aligned with a surface X(y i ; t) parametrized by coordinates y i , i = 1, 2. The surface metric tensor is deﬁned as γij = (∂i X)(∂j X), where ∂i = ∂/∂y i . As in 2D, the surface frame is extended into a neighboring layer by translating the surface along the normal. The length element in the aligned frame is expressed as (2.43) ds2 = dx2 + γij + xκij + x2 κki κkj dy i dy j , where κij are elements of the curvature tensor; the indices are raised and lowered with the help of the surface metric tensor γij and summation over repeated lower and upper indices is presumed. An orthogonal frame can be constructed on the surface by choosing coordinate axes directed locally along the principal directions of the curvature tensor. Then the Laplacian is expressed as 1 + κ2 x 1 + κ1 x 1 √ 2 ∂1 + ∂ 2 ∂2 , (2.44) ∇ = √ ∂x g ∂x + ∂ 1 g 1 + κ1 x 1 + κ2 x √ where κi are the two principal curvatures and g = (1 + κ1 x)(1 +√κ2 x). The approximate expression employing the scaling κi = O(), ∂i = O( ) is ∇2 = ∂x2 + (κ∂x + ∇2⊥ ) + O(2 ),

(2.45)

where κ = κ1 + κ2 is twice the mean curvature and ∇2⊥ = ∂12 + ∂22 is the transverse Laplacian. The second-order term in this expansion is computed as

2.2 Weakly Curved Fronts

97

−2 x (κ21 + κ22 )∂x + ∂1 (κ1 − κ2 )∂1 + ∂2 (κ2 − κ1 )∂2 + 2(κ1 ∂12 + κ2 ∂22 ) . The general covariant form of the transverse part of the bracketed expression is 2∂i κij ∂j − ∂ i κ∂i . 2.2.2 Expansion Near Maxwell Construction Near the Maxwell construction µ = µ0 , a curvature correction may become comparable to a slow speed of a straight-line front. Both the deviation µ − µ0 and the propagation speed must be of O() to match the orders of magnitude. The diﬀerence between the energies of both stable uniform states should be − small, V (u+ s )−V (us ) = V1 . Thus, we set f (u) = f0 (u)+f1 (u). Transforming the time derivative in the comoving frame as ∂t → ∂t − c∂x and expanding the state variable as u = u0 (x) + u1 (x, y) + . . . yields, in the zero order Du0 (x) + f0 (u0 (x)) = 0.

(2.46)

This equation is veriﬁed by a stationary front solution u0 (x) satisfying the asymptotic condition (2.4). The ﬁrst-order equation is Du1 (x) + f0 (u0 (x))u1 + Ψ (x, y) = 0,

(2.47)

which contains the inhomogeneity Ψ (x, y) = (c + Dκ)u0 (x) + f1 (u0 (x)).

(2.48)

The term containing curvature κ of a planar curve or mean curvature of an interface in 3D comes from the expansion of the Laplacian (2.42) or (2.45). The propagation speed is determined by the solvability condition of (2.47). The inhomogeneity Ψ (x, y) should be orthogonal to the Goldstone eigenmode u0 (x) that corresponds to the translational symmetry of a stationary front: ∞ u0 (x)Ψ (x, y)dx = 0. (2.49) −∞

This yields the eikonal equation c = c0 − Dκ,

(2.50)

where c0 = V1 /I is the speed of the straight-line front (cf. (2.7) applied in the vicinity of Maxwell construction). The dissipation integral I deﬁned by (2.8) is proportional to the interfacial energy per unit length (surface tension) σ = DI, as shown in Sect. 2.1.4. At the Maxwell construction (f1 = V1 = 0), the motion is driven exclusively by curvature: c = −Dκ. The front propagates along the curvature vector in such a way that a convex region occupied by one of the alternative equilibria shrinks. The stationary condition c0 = Dκ deﬁnes the radius of a critical

98

2 Fronts and Interfaces

nucleus of a preferred state; this stationary state is, evidently, unstable, as the droplet with the curvature larger than critical shrinks, while beyond the critical radius it grows indeﬁnitely. A more explicit form of the stationary − condition, V (u+ s ) − V (us ) = σκ, is linked to the Gibbs–Thomson law relating the equilibrium pressure or temperature with the radius of a droplet (see also Sect. 2.4.3). The speed of the curvature-driven motion can be derived directly from the variational principle (2.2). We observe that both the standard proﬁle of the order parameter across the front u0 (x) and the distortion energy per unit length of a weakly curved front are, in the leading order, independent of curvature. Therefore, the energy functional is proportional to the length (in 2D) or area (in 3D) of the front. In 3D it is computed as √ g dy1 dy2 . (2.51) F = DI dS = DI Since energy is totally determined in this approximation by geometry of the interface, its variation with respect to the order parameter can be expressed in purely geometrical terms. Consider a Cartesian frame (X, Y ) where the X-axis is aligned with the normal at a chosen point and Y spans the tangent plane at the same point. An adjacent segment of the interface can be deﬁned by the equation Φ(X, Y ) ≡ X − ζ(Y ) = 0. (2.52) In this parametrization, the metric factor is g = |∇Φ|2 = 1 + |∇⊥ ζ|2 and the normal vector is n = g −1/2 ∇Φ. We take note that only displacements normal to the front are signiﬁcant, while parallel displacements lead only to reparametrization. The variation can be therefore computed as √ δF = δ g d2 Y = g −1/2 ∇⊥ ζ · δ∇⊥ ζ d2 Y (2.53) = ∇⊥ n δζ d2 Y = κ δX d2 Y . One can add here a shape-independent gain δF = −V1 δX due to advancement of a more favorable state, which is proportional to a weak deviation from the Maxwell construction. The relaxation equation (2.2) can be rewritten therefore, introducing also a mobility coeﬃcient Γ , as c=

δF dX = −Γ = Γ (V1 − DIκ). dt δX

(2.54)

This coincides with (2.50) if one sets Γ = I −1 . 2.2.3 Burgers Equation For a weakly distorted planar front deﬁned by (2.52), the eikonal equation (2.50) can be rewritten explicitly in the Cartesian frame (X, Y ) propagating

2.2 Weakly Curved Fronts

99

with a constant speed c0 . In accordance to the scaling of the transverse coordinates in the preceding subsection, we presume that the transverse gradients compared with those in the propagadenoted by a 2D operator ∇⊥ are small √ tion direction, and scale ∇⊥ = O( ). The extended time variable is scaled accordingly as ∂t = O(). The projection of the normal displacement on the X-axis is expressed as −1/2 = c0 1 − 12 |∇⊥ ζ|2 + O(2 ), c0 1 + |∇⊥ ζ|2 and the curvature is computed as κ = −∇2⊥ ζ + O(2 ). Using this in (2.50) transformed to the moving frame yields in the leading order the Burgers equation (2.55) ∂t ζ = D∇2⊥ ζ − 12 c0 |∇⊥ ζ|2 . The diﬀusional term acts in such a way that convex segments retreat and concave ones advance, so that the front tends to relax to a straight line or planar shape. The nonlinear term is purely advective and reﬂects the tendency of the interface to propagate normal to itself. This term is responsible for asymmetry due to the propagation direction, since segments with a larger slope lag behind, the front tends to be smoother on the onward and steeper on the back side as seen in Fig. 2.7. At the Maxwell construction (c0 = 0), (2.55) degenerates into a linear diﬀusion equation. Otherwise, it can be linearized with the help of the Hopf– Cole transformation (2.56) ζ = −(2D/c0 ) ln χ, transforming it to a linear diﬀusion equation ∂t χ = D∇2⊥ χ.

(2.57)

The “hydrodynamic” form of the Burgers equation obtained by applying the gradient operator is ∂t v + v · ∇v = D∇2 v,

(2.58)

where v = c−1 0 ∇ζ is interpreted as “velocity”, and the ⊥ sign is omitted. This form bears superﬁcial resemblance to the Navier–Stokes equation, but lacks

Fig. 2.7. Successive positions of a front propagating upward with a constant normal velocity. Dots mark successive positions of chosen points. Note a singularity developing on the concave segment

100

2 Fronts and Interfaces

pressure and is not coupled to the continuity equation. One can say that the Burgers equation describes a “supercompressible” ﬂuid with pressure independent of density (and therefore irrelevant). The Burgers equation has been originally introduced (Burgers, 1948) in this form as a model replacement of the Navier–Stokes equation. It is, however, ill suited for studies of hydrodynamic phenomena, especially turbulence, as the velocity ﬁeld deﬁned in this way is potential , while the heart of hydrodynamic turbulence is vorticity. In the “inviscid” limit D = 0, which is approached far from the Maxwell construction when the front propagates with a constant speed and the eﬀect of curvature is negligible, (2.58) degenerates into a ﬁrst-order PDE that generates moving discontinuities – “shocks” forming boundaries between front segments with diﬀerent slopes.√A small diﬀusional term with D = O() resolves discontinuities on an O( ) scale. The characteristic equation of the ﬁrst-order PDE is dv/dt = v, which means that segments with a higher slope propagate faster and thus conquer and obliterate segments with a lower slope. To compute the propagation speed of a shock, to be denoted as q, we rewrite the degenerate form of (2.58) in a 2D frame comoving with the shock as q dv/dz =

1 2

d|v|2 /dz,

(2.59)

where the z axis is directed normally to the shock. Integrating this across the shock and taking into account that the component of v parallel to the shock must be continuous, while the normal component may jump between the values vz = v± , yields q = 12 (v+ − v− ), i.e., the shock propagates with the speed equal to the arithmetical mean of the slopes on the two sides. The dynamics of shocks is in fact a simple geometrical consequence of normal propagation of the interface with a constant speed masked by a chain of mathematical transformations of the eikonal equation (2.50). The Burgers equation is one of the most extensively studied universal equations. A question of interest is the law of decay of originally imposed inhomogeneities (see e.g. Bec et al., 2000). Another well-studied topic is dynamics in the presence of noise described by the Burgers equation with an added random term, commonly called Kardar–Parisi–Zhang (KPZ) equation (Kardar et al., 1986). 2.2.4 Ripening under Global Control Dependence of the front propagation speed on curvature plays a primary role in long-scale evolution of patterns. Consider a pattern consisting of twodimensional “droplets” of a minority phase (say, u = u− s ) immersed in a continuum of an alternative majority phase. Since the front interactions decay exponentially with separation, their inﬂuence on the dynamics of fronts that bound droplets separated at a distance much larger than the characteristic diﬀusional length should be negligible compared to the inﬂuence of curvature.

2.2 Weakly Curved Fronts

101

Thus, well-separated droplets will relax to the circular shape and evolve according to (2.50) that gives preference to growth of larger droplets. For any value of c0 > 0 (i.e. under conditions favoring growth of the minority phase), there is a critical radius Rc = D/c0 , such that the droplets with larger radii R > Rc grow, while droplets with R < Rc shrink and eventually disappear. If c0 remains ﬁxed, the initial minority phase eventually conquers the entire plane. This can however, be, prevented if there is an additional global variable, which is dynamically adjusted to maintain the total area occupied by the minority phase constant. This variable should act to reduce the value of c0 in the course of evolution in order to compensate the decrease of average curvature. As a result, the critical radius grows and, as it overtakes the radius of growing droplets, the latter start to shrink and eventually disappear. The ﬁnal result of evolution minimizing the total energy of the system under the constant area constraint would be a single circular droplet. This ripening process (similar to Ostwald ripening in two-phase systems, see Sect. 2.4.6) may, however, take exceedingly long time at its later stages, when the radii of the majority of surviving droplets become very large compared to the thickness of the interfacial layer. It is interesting therefore to compute the distribution of droplet sizes at late stages of ripening when the system has been evolving suﬃciently long to obliterate the initial size distribution, but the number of droplets still remains suﬃciently large to justify statistical approach. The asymptotic analysis follows the classical work by Lifshitz and Slyozov (1958) who have considered ripening in 3D with a growth rate diﬀerent from that used here. We shall return to this case in Sect. 2.4.6; see also a generalized analysis by Giron et al. (1998). Let f (R) be the number density of droplets with radius R; this function is not normalized, since the total num ber f (R)dR decreases in time as smaller droplets collapse. The evolution equation for f (R) is ∂(cf ) ∂f + = 0, (2.60) ∂t ∂R where the growth rate c(R) is expressed, using (2.50) and denoting c0 = D/Rc (t), as 1 1 c(R) = D − . (2.61) Rc R An asymptotic distribution, if it exists, should admit some quasistationary representation. Quasistationarity may be preserved relative to the timedependent critical radius. A suitable variable, replacing R, is therefore the dimensionless radius ρ = R/Rc . The growth of the respective dimensionless area is proportional to D 1 d(ρ2 ) ρ2 dRc = 2 (ρ − 1) − . 2 dt Rc Rc dt

(2.62)

This expression is further simpliﬁed by using Rc as a measure of time. A convenient transformed time variable is τ = ln(Rc /R0 )2 , where R0 is any

102

2 Fronts and Interfaces

suitable length scale. Denoting q(t) =

1 d(Rc2 ) , 2D dt

(2.63)

we rewrite (2.62) as ρ−1 1 d(ρ2 ) = − ρ2 . 2 dτ q(τ )

(2.64)

It turns out that there is a unique asymptotic value of q at τ → ∞ compatible with the area conservation condition. The r.h.s. of (2.64) is negative at both small and large values of ρ, but becomes positive in an intermediate range when q < 1/4 (see Fig. 2.8a). If such a range existed, the area covered by droplets with respective radii would grow with time without limit, which is forbidden, as it contradicts the area conservation condition. Conversely, if q were negative everywhere, the area occupied by all droplets would keep shrinking, which is again incompatible with the area conservation. The only remaining possibility is that the asymptotic value is q∞ = 1/4; then the r.h.s. of (2.64) reaches the zero maximum at ρ = 2. This deﬁnes the maximum dimensionless radius in the asymptotic distribution. Indeed, if droplets with a larger radius are present in the initial distribution, their radius, notwithstanding their continuous growth, draws closer to the critical radius, until they are swept to the vicinity of the point ρ = 2. The function q(τ ) should approach its limiting value q∞ = 1/4 from above in such a way that a small gap decreasing with time remain between the maximum of the r.h.s. of (2.64) and the ρ axis. The droplets from the tail of the distribution ﬁlter slowly through this gap in the course of evolution. The time-dependent correction to the asymptotic value aﬀects the form of the distribution near the point ρ = 2, but this does not aﬀect the rest of the distribution, since, as we shall see, the contribution of the vicinity of this point to the total area is negligible. (a)

vΡ 2

(b)

1

2

3

4

2

14

4

13

6

Ρ

1.25 1 0.75

ln

1.5

15

0 40 80 1.9 1.95 Ρ

0.5 0.25 0.5

1

1.5

2

Ρ

Fig. 2.8. (a) Plot of the r.h.s. of (2.64); the values of q are indicated at the respective curves. (b) The normalized distribution function. Inset: the tail near ρ = 2 drawn on a logarithmic scale

2.3 Propagation into an Unstable State

103

The asymptotic size distribution can be obtained now by solving (2.60), which is rewritten in the transformed variables as ∂(cf ) ∂f + = 0, ∂τ ∂ρ

2 c(ρ) = − (2 − ρ)2 . ρ

(2.65)

The general solution of (2.65) is f (ρ, τ ) = −

χ(τ + ψ) , c(ρ)

ψ(ρ) = −

dρ , c(ρ)

(2.66)

where χ(. . .) is an arbitrary function that has to be obtained from the normalization condition Rmax

R2 f (R)dR = S0 ,

π

(2.67)

0

or, in dimensionless variables, 2 S0 eτ ρ2 f (ρ)dρ = y ≡ . πR02 0

(2.68)

It is clear from the latter expression that the only choice making the l.h.s. independent of τ is χ(τ ) = Ae−τ with an appropriate constant A. Computing the integrals yields f (ρ, τ ) = 12 Ae−τ −1/(2−ρ) ρ(2 − ρ)−5/2 , A = 2y

2

ρ3 (2 − ρ)−5/2 e−1/(2−ρ) dρ = 1.159y.

(2.69) (2.70)

0

The distribution becomes independent of τ when it is normalized to unity, f (ρ, τ ) = Be−τ ϕ(ρ), where ϕ(ρ)dρ = 1 and B is a normalization constant. The normalized distribution is shown in Fig. 2.8b. To translate the result back to physical variables, √ one has to keep in mind that Rc increases asymptotically proportionally to t and e−τ ∝ t−1 . Convergence to the asymptotic distribution may be, however, impaired for initial distributions with a very long tail (Niethammer and Pego, 1999).

2.3 Propagation into an Unstable State 2.3.1 Continuum of Propagating Solutions The problem of front propagation into an unstable state arises naturally in population dynamics. A stationary state where competitively advantageous specie is absent is formally unstable to inﬁnitesimal perturbations but will be nevertheless preserved at any location until this specie is introduced there. This will typically happen as the habitat boundary propagates outward, i.e.,

104

2 Fronts and Interfaces

via a propagating front connecting the stable and unstable states. The diﬃculty of this problem, which has motivated the pioneering work of Kolmogorov et al. (1937), lies in the nonuniqueness of a solution describing a steadily propagating front. The propagation speed therefore cannot be computed in a straightforward way by solving an ODE. It is selected dynamically, and the full nonstationary problem should be considered to understand the selection mechanism. If the receding state is unstable, the coeﬃcient α in the linearized equation (2.5) √ is positive. Then both eigenvalues (2.6) are real negative at c > cmin = 1 2 αD; below this value they are complex with a negative real part. This means that at c > cmin the unstable state, viewed as an equilibrium point of (2.3), is a stable node, and the unstable manifold of the advancing stable HSS connects generically to this point (see Fig. 2.9). The generic connection persists also at smaller positive velocities, but the approach to the equilibrium is oscillatory. Thus, there is a continuum of one-dimensional front solutions, and restricting to monotonic solutions gives only a lower bound c = cmin . The question is, which speed is actually chosen by the dynamically evolving system. Moreover, it is not evident a priori that transition from an unstable to a stable state should be carried by a propagating front retaining its quasistationary shape. Any perturbation of an unstable state is destined to grow; therefore, dynamics of transition from an unstable to a stable state should strongly depend on initial conditions. The kind of an initial condition that is likely to generate a propagating front is a perturbation localized in a narrow region. This kind of a perturbation is typical for many physical problems. When a perturbation of this kind is introduced into a system that had been rapidly quenched into an unstable state, the transition into a stable state is eﬀected locally, and a sharp front propagating into the unstable state is formed. In problems of population dynamics (which historically were the ﬁrst to prompt the interest in front propagation dynamics), a localized perturbation corresponds to a lo(a) u’x 0.12

(b) u’x

ccmin

0.3

ccmin

0.1 0.08

0.2

ccmin

0.06

0.1

0.04 0.02 0.2

0.4

0.6

0.8

1

u

0.2 0.1

u 0.2

0.4

0.6

0.8

1

Fig. 2.9. Trajectories in the phase plane connecting a stable and an unstable equilibrium

2.3 Propagation into an Unstable State

105

cally introduced successful mutant population that displaces an old genom. In other systems, the stable state may nucleate at the boundary or at a local inhomogeneity. In the absence of other perturbations, the global transition to the stable state is eﬀected through front propagation. 2.3.2 Asymptotic Theory of the Leading Edge The method developed for the reaction-diﬀusion equation by Kolmogorov et al. (1937), and later generalized to pattern-forming systems (Dee and Langer, 1983, Ben-Jacob et al., 1985; see Sect. 4.5.1), is based on the asymptotic linear analysis of the leading edge of the propagating front. The front is called “pulled” when its propagation speed is determined by growth of perturbations at its leading edge. Consider the growth of a perturbation of an unstable state u = u0s . While the perturbation remains small, the linear theory applies, and the solution can be presented as a Fourier integral ∞ u (k)ei[kx−ω(k)]t dk. (2.71) u(x, t) − u0s = −∞

The dispersion relation ω(k) follows from a linearized evolution equation. If the latter has a general form ut = L(∂x2 )u + α(u − u0s ), the dispersion relation obtained in the vicinity of u = u0s is

ω = i α + L(−k 2 ) .

(2.72)

(2.73)

In particular, for the linearized reaction–diﬀusion equation, we have L = −Dk 2 . To determine the front speed c, we evaluate (2.71) asymptotically for large x, t, such that ξ = x − ct = O(1). This solution should describe the leading edge of the front (where the linear theory still applies), advancing with the speed c, as yet undetermined. The integral (2.71) now takes the form ∞ u (k)ei[kξ+h(k)t] dk, (2.74) u(ξ, t) − u0s = −∞

with h(k) = ck − ω(k). If the original perturbation is suﬃciently localized, all Fourier modes are present, and u (k) can be considered to be a smooth O(1) function. The integral (2.74) can be evaluated using the method of steepest descent in the complex k-plane. We require h(k) to be maximal, h (k) = 0, and the imaginary part of h(k) to vanish, so that the solution expressed by (2.74) neither grow nor decay in time. This yields ci (k) = 0, c= cr (k),

c = ωi /ki .

(2.75)

106

2 Fronts and Interfaces

where c = ω (k) is the group velocity, and the subscripts r and i denote the real and imaginary parts respectively. Equation (2.75) is satisﬁed at a certain point k = k in the complex k-plane. The approximation implies that the Fourier integral is dominated at late stages of evolution by this particular value of k, so that the leading edge of the front can be presented as u − u0s ∼ ei[k−ω(k)t] = e−ki (x−ct) ei[kr x−ωr (k)]t .

(2.76)

The imaginary part of k determines the rate of decay in the direction of propagation. According to (2.75), the selected propagation speed coincides with the real part of the group velocity, which determines the speed of propagation of perturbations. In a frame moving with a speed faster than c, the solution would appear to be decaying in time; thus, one can say that propagation plays a stabilizing role, in a sense that one can outrun a growing perturbation by moving suﬃciently fast. If cr were larger than the speed c with which the front moves, perturbations would run ahead of the front and cause the proﬁle to be unstable. The selected speed is such that the perturbation neither grows nor decays, i.e., it satisﬁes the condition of marginal stability (Dee and Langer, 1983). 2.3.3 Stability of the Leading Edge In order to gain more physical understanding of the stability of the leading edge of the propagating front and of the way the leading edge of the propagating front evolves to a marginally stable proﬁle, we generalize (2.76), following van Saarloos (1988, 2003), to u = eiθ(x,t) , where k = ∂θ/∂x is allowed now to change slowly in space and time. The evolution equation of θ, rewritten in a frame propagating with the velocity c, has a general form θt = ck − ω(k) + M(∂x , ∂x2 , . . .)k,

(2.77)

where M is a nonlinear diﬀerential operator derivable from L. One can easily see that the imaginary part of this equation is indeed veriﬁed by θi = const with k = k when the propagation speed is deﬁned by (2.75). To check the stability of this solution, we linearize (2.77) to obtain the equation for a perturbation θ: k) − cr ( k)]θx + M∂x ( k)θxx + . . . . θt = [c(

(2.78)

Omitted terms contain higher derivatives with respect to x. The proﬁle is stable if perturbations bounded at x → ∞, that have the form θ ∼ eikx with k, only the ﬁrst term in (2.78) is relevant, ki > 0, decay with time. For small and therefore the necessary condition for stability is Re{i[c( k) − c( k)] k} < 0 for all k 1, ki > 0.

(2.79)

2.3 Propagation into an Unstable State

107

Separating the real and imaginary parts gives the necessary stability conditions k), ci ( k) = 0. (2.80) c> cr ( The ﬁrst condition coincides with the intuitive statement in the preceding subsection that the velocity c of a stable front must exceed the propagation speed of small perturbations that is determined by the real part of the group velocity c. Vanishing imaginary part of the group velocity is the condition, that has appeared already in (2.75), insuring stability to changes of a local wave number of the emerging pattern. A nonlinear analysis of (2.77) is required to describe the evolution to the marginally stable proﬁle. This analysis has also been carried out by van Saarloos (1988) in the long-scale approximation, assuming a slow spatial variation of k. Suppose that the amplitude of the propagating solution is monotonically decreasing, so that ϑ = θi is a monotonically increasing function of x. It is advantageous then to write the evolution equation of k taking ϑ, instead of x, as an independent variable changing in the direction of propagation. The transformation to the new variable is carried out using the relations (∂x )t = ki (∂ϑ )t ,

(∂t )x = (∂t )ϑ + (ϑt )x (∂ϑ )t ,

where (∂x )t denotes the diﬀerentiation with respect to x at t = const, etc., and the imaginary part of the right-hand side of (2.77) should be substituted for (ϑt )x . Then diﬀerentiating (2.77) with respect to ϑ yields kt = [c(k) − cr (k)] ki kϑ + (higher order terms).

(2.81)

The bracketed combination, that we denote now as C(k), is the same as in (2.78), except that it is now computed at a variable rather than constant value of k. When small terms, containing higher order derivatives with respect to ϑ and powers of kϑ , are omitted, we are left with what appears to be a standard ﬁrst-order quasilinear PDE, but is generally a system of equations for ki , kr that is not even hyperbolic. The standard case is, however, recovered when the stability condition ci = 0 holds. Using this condition to ﬁx the relation between kr and ki , C, which is now real, can be written as a function of ki only. Then it follows from (2.81) that k is conserved along a characteristic ϑt = −ki C(ki ). Since ki > 0, and C must be positive for a stable proﬁle, the characteristics are directed toward decreasing ϑ. This supports our notion that the leading edge plays the premier role in shaping the emerging pattern: the causal propagation along the characteristics is directed toward the stable state. When C (ki ) < 0, steeper parts of the proﬁle propagate back slower, and therefore dominate the leading edge of the front. Under these conditions, the front steepens in the course of evolution, and approaches the slope ki corresponding either to a minimum of C(ki ) or to the marginal value C(ki ) = 0.

108

2 Fronts and Interfaces

2.3.4 Pulled and Pushed Fronts The leading edge analysis allows us to predict with a high precision both the propagation velocity and the wavelength of the emerging pattern. Some caveats are, however, in order. The above analysis has all common drawbacks of both linear and long-scale theories. The front stable according to the leading edge criteria can be destabilized by some “invasion” mode involving rapid changes of the slope, or modiﬁed by nonlinear interactions to give way to a more rapidly propagating front, as shown below. The dispersion relation for the reaction–diﬀusion equation is ω(k) = i(α − Dk 2 ), so that c = −2iDk. The second condition (2.75) yields simply kr = 0; thus, as expected, no oscillations arise here, and both ω and k = iq are purely imaginary. Using c = ω/k= α/q + Dq and c = 2Dq, we see that C(q) = α/q − Dq vanishes at q = α/D, which coincides with the minimum on √ the dependence c(q). Thus, the marginally stable propagation speed c = 2 αD turns to be in this case also the slowest speed satisfying the stability condition (2.80). This simple result of the leading edge theory becomes, under certain circumstances, rather unlikely. Consider equation (2.3) with the standard cubic nonlinearity f (u) = u−u3 −µ. There is an exact solution determining the propagation speed or the front separating two stable states (Sect. 2.1.3). As one of the competing states approaches the bifurcation point µ = 2/33/2 where it becomes marginally stable, the propagation speed grows (see Fig. 2.2b), reaching the maximum value c = 3D/2 at the bifurcation point itself. On the other hand, the propagation speed into the same marginally stable state can be computed, as above, by approaching it along the branch of unstable states. Along this √ branch, the propagation speed predicted by the leading edge theory, c = 12 αD decreases with the decreasing derivative α = f (u0s ) and vanishes at the marginally stable point. The clue to this apparent contradiction is that the special propagating solution obtained in Sect. 2.1.3 is also applicable to a front separating a stable and an unstable state. Such a front, determined by the full nonlinear solution, is called “pushed”, as it is “pushed ahead” by an advancing stable state rather than “pulled along” by perturbations growing at the leading edge. If, say, the state u = u+ s is approached at x → −∞, this solution is expressed as √ 0 − u (x) = −(u − u0s )(u+ c = D/2(u+ s − u)/ 2D, s + us − 2us ), (2.82) 0 where u± s , us are deﬁned by (2.15). The asymptotic value of the slope at the leading edge following from (2.82) is 0 d ln(u − u0s ) p(u0s ) u+ s − us √ q = , (2.83) = = dξ u − u0s 2 ξ→∞

2.3 Propagation into an Unstable State

109

which increases when u0s is shifted toward the marginally stable point µ = −2/33/2 ≈ −0.3849, contrary to a decrease of q = f (u0s ) predicted by the leading edge theory. At µ > µ ≈ −0.2078, the slope q exceeds q (Fig. 2.10a). The system chooses the propagating solution with the steepest slope, which now, unlike the generic situation in the linear leading edge theory, is also moving faster (Fig. 2.10b). The plot of the propagation speed deﬁned by (2.82) against µ is tangential at µ = µ to the plot of the generic propagation speed c = 2Dq. The transition between the generic pulled front and the pushed front determined by the special nonlinear solution (2.82) occurs at µ = µ (Ben Jacob et al., 1985). Beyond this value, the pushed front, which is still faster but has a lower slope than the pulled one, has no physical signiﬁcance. The reason why competition with nongeneric pushed fronts has not surfaced in the mathematically rigorous theories of Kolmogorov et al. (1937) and Aronson and Weinberger (1978) is that the special nonlinear solution (2.82), apparently, cannot be attained as a result of evolution starting from localized initial condition with u ≥ u0s . In problems of population dynamics, which gave motivation to the early theories, states with u < u0s = 0 are nonphysical, and, in any case, solutions of (2.5) with α > 0 starting from strictly nonnegative u initial conditions remain nonnegative at all times. On the other hand, if deviations from u = u0s of any sign are allowed, evolution to the pushed front can be understood in the following way (van Saarloos, 1989). Consider a propagating front of the state u = u− s , which is metastable at µ, slightly above the bifurcation value µ = −2/33/2 , into the unstable state u = u0s , being trailed by a transition front from u− s to , as shown in Fig. 2.11a. Suﬃciently close to the the globally stable state u+ s bifurcation point, (actually, at |µ| > 0.3730 – see Fig. 2.10b), the trailing front moves faster, and catches on. As a result, the intermediate metastable state is exterminated, and u becomes a monotonically decreasing function of ξ, while the front evolves to the non generic proﬁle (2.82). A more complex mechanism of relaxation to a pushed front is required at 0.2078 < |µ| < 0.3730 (a)

q D 1.2

(b)

c D 2

1 1.5

0.8 0.6

1

0.4 0.5

0.2 0.1

0.2

0.3

Μ

0.1

0.2

0.3

Μ

Fig. 2.10. Asymptotic slope (a) and velocity (b) of a pulled (solid curve) and pushed (gray curves) fronts. The upper and lower branches of the gray curve correspond to the advance of the stable state, respectively, to the unstable and metastable states

2 Fronts and Interfaces

u

110

1 0.75 0.5 0.25 0 -0.25 -0.5 Ξ

Fig. 2.11. A nonlinear front trailing a pulled front (solid curve) and a pushed front formed after the nonlinear front has caught up (gray curve)

when a pulled front propagates into the unstable state faster than the globally stable state displaces the metastable one. This indicates an important role of perturbations bringing the system locally below the level u = u0s . While in the ﬁnal state u = u0s is nonnegative, transient states oscillating around u = u0s are necessary for evolution to the rapidly propagating pushed front (Powell et al., 1991).

2.4 Cahn–Hilliard Equation 2.4.1 Order Parameter and Energy Functional Nonlinear models possessing two or more stable HSS (and, hence, interfaces separating these states) appear even in closed systems relaxing toward thermodynamic equilibrium. Indeed, the ﬁrst model of this kind has been proposed more than a century ago by van der Vaals (1894) for description of a diffuse liquid–gas interface. Therefore, fronts separating alternative HSS in nonequilibrium systems are similar in many respects to interphase boundaries in equilibrium systems, and many familiar terms, such as surface tension, critical nucleus and Maxwell construction (deﬁning a stationary front) remain useful also in nonequilibrium setting, provided the dynamics is of the gradient type. The simplest expression for the free energy of a system possessing two equilibrium phases has the form (2.2), where u is now interpreted as an order parameter and diﬀusivity D is replaced by the rigidity coeﬃcient K. This model free energy functional, called Landau energy, can be derived starting from an expression for free energy in density functional approximation 1 U (r)[u(x + r) − u(x)] dx. (2.84) F = V (u(x)) dx + 2 u(x) dx r>d

The two terms in this expression give, respectively, the free energy of a homogeneous state and the distortion energy due to changes of density in space;

2.4 Cahn–Hilliard Equation

111

U (r) < 0 is an isotropic attractive pair interaction kernel with a short-range cutoﬀ d. If the order parameter changes only slightly over distances comparable with the characteristic interaction length, one can expand u(x + r) = u(x) + r · ∇u(x) + 12 rr : ∇∇u(x) + . . .

(2.85)

Using this in (2.84) we see that the contribution of the linear term to the nonlocal integral vanishes when the system is isotropic and, as a consequence, the interaction term is spherically symmetrical, and the lowest order contribution is due to the quadratic term 2 1 1 − 2 K u(x)∇ u(x) dx = 2 K |∇u(x)|2 dx, (2.86) where K=−

2π 3

∞

U (r) r4 dr > 0.

(2.87)

d

The interaction has to fall oﬀ suﬃciently fast to ensure convergence of this integral, as well as of higher order integrals in the expansion. This, unfortunately, is not true for interaction kernels with power decay, including the commonly used Lennard–Jones potential U (r) ∝ r−6 producing divergence in the next nonvanishing (fourth) order. Nevertheless, local Landau functionals are widely used because of a great gain in simplicity of computations, compared to the nonlocal expression (2.84). 2.4.2 Conservative Gradient Dynamics In physical systems relaxing toward equilibrium the dynamics is usually constrained by some conservation laws. For example, in the simplest case of a one-component ﬂuid, the “order parameter” is just density, and conservation of mass should be taken care of when dynamic equations are formulated. Evidently, the variable evolving according to (2.2) is not conserved. To ensure conservative dynamics, the gradient principle should be formulated in a different way, based on a conservation law for the order parameter u: ∂t u = −∇ · j.

(2.88)

The ﬂux j is assumed to be proportional to the gradient of the variation of the free energy functional: j = −Γ (u)∇

δF ≡ −Γ (u)∇µ, δu

(2.89)

where µ = δF/δu is chemical potential . The mobility coeﬃcient Γ > 0 is, generally, dependent on the order parameter u. As before, the free energy

112

2 Fronts and Interfaces

always decreases until a minimum is reached. This is proven by using (2.88), (2.89) in a general free energy functional and integrating by parts: δF 2 δF δF dx ≤ 0. ∇ · Γ (u)∇ dx = − Γ (u) ∇ δu δu δu (2.90) Equations (2.88), (2.89) with F deﬁned by (2.2) combine to the generalized Cahn–Hilliard equation (Cahn and Hilliard, 1959): dF = dt

δF ∂u dx = δu ∂t

∂t u = ∇ · (Γ (u)∇µ),

(2.91)

K∇2 u + f (u) + µ = 0.

(2.92)

A more commonly used version with Γ = const rescaled to unity is

∂t u = −∇2 K∇2 u + f (u) . (2.93) When V (u) = − f (u)du is a double-well potential and µ is ﬁxed in such a way that f (u) + µ = 0 has two stable roots (HSS), spinodal decomposition takes place, and the system separates into two phases corresponding to the alternative HSS. Stationary solutions of the Cahn–Hilliard equation describing equilibrium interphase boundaries coincide with the stationary solutions of its nonconservative counterpart (2.1). When the order parameter is conserved, a single planar interface cannot propagate. Therefore, the Maxwell construction should be reached automatically when the dynamics is described by the Cahn– Hilliard equation. The chemical potential can be viewed as an adjustable constant ensuring the existence of a stationary front solution. The value of µ required to maintain the equilibrium is obtained by multiplying (2.92) by ∇u and integrating across the front. The diﬀerential term vanishes upon integration, while the integral of the algebraic part yields the Maxwell condition µs =

− V (u+ s ) − V (us ) . + − us − us

(2.94)

± This, together with µs = −f (u± s ), deﬁnes the three unknowns µs , us .

2.4.3 Inner and Outer Equations √ The characteristic width of a diﬀuse interphase boundary is l = K. Across this thin layer, the order parameter switches between the two alternative values corresponding to the same constant value of chemical potential. The latter is constant everywhere when the two phases are at equilibrium, while under nonequilibrium conditions it varies on a longer scale L that may be determined by the geometry of the system. Problems of this kind, containing a small parameter = l/L, should be solved by matching expansions in in

2.4 Cahn–Hilliard Equation

113

regions characterized by two widely separated scales: the inner region localized at the interface, and the outer region spreading out to the bulk of both alternative phases. This approach is suitable for tracking slow motion of the interphase boundary at times far exceeding the characteristic relaxation time to a stationary proﬁle of the order parameter in the interfacial layer (Pego, 1989; Karma and Rappel, 1998). In the outer region, we extend the coordinates by the factor −1 . Accordingly, the gradient operator is transformed to ∇; the matching time scale should be extended by −2 . Then the distortion term in (2.92) becomes of O(2 ), and can be neglected. Marking the conserved order parameter u and the chemical potential µ in the two bulk regions corresponding to the two alternative HSS by the superscripts ±, we write the outer equations as

µ± = −f (u± ). ∂t u± = ∇ · Γ (u± )∇µ± , (2.95) Combining the two equations, we see that the evolution in the outer region obeys the linear diﬀusion equation

∂t u± = ∇ · D(u± )∇u± . (2.96) The eﬀective diﬀusivities D± = −Γ (u± )f (u± ) are positive, since f (u± ) < 0 on a branch of stable equilibria; hence, the diﬀusion equation is well posed. In the inner region, we shall use, as in Sect. 2.2.2, the aligned comoving frame. The characteristic scale along the axis x normal to the nominal front position is the short scale l, while the coordinates y parallel to the interface are scaled by L. If the curvature radius is of O(L), the curvature is written as κ when measured on the short inner scale. Inasmuch as the interfacial layer is assumed to be locally at equilibrium, the interface is expected to move under the inﬂuence of long-scale changes of chemical potential. Therefore, the propagation speed should be measurable on a long scale, so that the “Peclet number” based on the long length scale L and a typical value of the diﬀusivity, Pe = cL/D should be at most of O(1). The chemical potential within the front region should diﬀer from the Maxwell construction by O(), and is expressed as µ = µs + µ1 . Using this in (2.91), (2.92), and expanding also the order parameter, u = u0 + u1 + . . ., we obtain the inner equations in successive orders of . The zero-order equations are µ0 (x) = 0,

u0 (x) + f (u0 (x)) + µ0 = 0.

(2.97)

The solution is a stationary front that exists at µ0 = µs = const, where µs is the equilibrium chemical potential deﬁned by (2.94). The proﬁle of the stationary front is computed exactly as in Sect. 2.1.4, and is most conveniently expressed in phase plane variables u = u0 (x), p(u) = u0 (x) as a function p(u). Formally, one could still add to µ0 a linear term, but this would be incompatible with matching conditions when propagation is slow.

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In the ﬁrst order, (2.91) reduces to d [Γ (u0 )µ1 (x) + cu0 (x)] = 0. dx

(2.98)

The solutions of the inner and outer equations should be matched at a distance from the front that is large on the inner but small on the outer scale; the result must be independent of a precise matching position within this range. The matching conditions are lim µ1 (x) = n · ∇µ± ≡ −j ± /Γ (u± ),

x→±∞

(2.99)

where n is the normal to the interface (directed, by convention, in the same way as the x-axis), and the ﬂuxes j ± on both sides of the interface are computed as the inner limit of the outer solution. Integrating (2.98) and using (2.99) yields c(u+ − u− ) = −Γ (u+ )n · ∇µ+ + Γ (u− )n · ∇µ− ≡ j + − j − .

(2.100)

The r.h.s. of (2.100) is the diﬀerence of the ﬂuxes on the two sides; thus, this condition deﬁnes the speed of the local interface displacement required to ensure the mass conservation. Since the variable part of µ is restricted in the interfacial layer to O(), the order parameter in the matching regions ± may deviate from the equilibrium values u± s (µs ) satisfying µs = f (us ) by no more than O(). Using again a linearization of f (u), the mass conservation condition (2.100) can be rewritten as c=−

Ds+ n · ∇u+ − Ds− n · ∇u− . − u+ s − us

(2.101)

Since f (u) is, generally, asymmetric and mobilities depend on u, the eﬀective ± diﬀusivities Ds± = −Γ (u± s )f (us ) are, generally, distinct on the two sides of the interface. 2.4.4 Solvability Condition and Matching It remains to determine the relation between the ﬁrst-order correction to the interfacial chemical potential µ± and the ﬂuxes j ± . The respective values of the order parameter u± = u± , which can be used as interfacial boundary conditions for the outer equation (2.95), are deﬁned by the near-equilibrium linearized relation ± ± (2.102) µ± − µs = −f (u± s )(u − us ). Integrating (2.98) twice yields µ1 (x) = µ1 − 0

x

j1 + cu0 (x) dx, Γ (u0 (x))

(2.103)

2.4 Cahn–Hilliard Equation

115

where µ1 = µ1 (0) and j1 are integration constants. The latter equals to the ﬂux through the interface, which is determined by the matching conditions (2.99) yielding either of the equivalent expressions j1 = −cu± − Γ (u± )n · ∇µ± = j ± − cu± =

+ − j − u+ s − j us . + us − u− s

(2.104)

The ﬁrst-order solution (2.103) is used in the ﬁrst-order expansion of (2.92): (2.105) K [u1 (x) + κu0 (x)] + f (u0 )u1 + µ1 = 0. This equation has the same form as (2.47), and is subject to the solvability condition (2.49). Multiplying this inhomogeneous equation by the Goldstone eigenmode u0 (x) and integrating yields − µ1 (u+ s − us ) = j1 J0 + cJ1 − κK I.

(2.106)

Here I is the dissipation integral deﬁned by (2.8), and the integrals Jk are computed as Jk =

∞

−∞ u+ s

=

u0s

u+ u s uk0 ( x) u k d x= d u du x)) u)Γ ( u) 0 Γ (u0 ( u0s p( u− s u0s k u k (u+ ) u ( u − u− s −u s ) d u− d u ≡ Jk+ − Jk− . (2.107) − p( u)Γ ( u) p( u )Γ ( u ) us

u0 (x)dx

x

The lower integration limit for u corresponding to the nominal origin x = 0 may be arbitrary, but, to be deﬁnite, we have chosen it to coincide with the unstable intermediate HSS u0s . If Γ = const and u0 (x) is symmetric (which follows from the symmetry of the potential V (u)), the integral J0 vanishes, and the constant j1 falls out. The dependence of the excess chemical potential on curvature in (2.106) is immediately recognized as the Gibbs–Thomson relation, while the other two terms give dynamic corrections due to the ﬂux through the interface. The limit of the inner solution (2.103) at x → ±∞ is expressed after using (2.104) and rearranging the integral as lim µ1 (x) = µ1 + cJ0± − j ± K± + x n · ∇µ± ,

x→±∞

(2.108)

where we have separated the converging integrals J0± , deﬁned by (2.107), and u± s 1 1 1 1 1 − − dx = du. Γ (u0 (x)) Γ (u± Γ (u± s ) s ) x0 u0s p(u) Γ (u) (2.109) One can check by diﬀerentiating the integrals Jk± , K± with respect to the variable limit that the limiting value of µ(x) is indeed independent of the value of the order parameter at the origin. K± =

±∞

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On the other hand, the chemical potential in the outer regions close to the interface, i.e., at some distance along the normal n, which is small when measured on the outer scale, is presented by expanding the outer solution in Taylor series as (2.110) µ± = µ± + x n · ∇µ± . The last terms in (2.108) and (2.110) match, and the remaining constant terms yield two matching conditions for computing µ± : µ± − µs = µ1 + cJ0± − j ± K± ,

(2.111)

or, using (2.100), (2.104), and (2.106), − − + J1 + u+ s J0 − us J0 µ − µs = c + j± − u+ s − us

J0 − K± + us − u− s

κKI . − u− s (2.112) This expression yields local relations between the values of chemical potential and ﬂuxes on both sides of the front, which can serve as the boundary conditions for outer equations. This completes the reduction of the Cahn–Hilliard equation to the classical sharp interface Stefan problem, which is fully determined by two linear diﬀusion equations (2.96) with the interfacial velocity deﬁned by (2.101) and the boundary conditions (2.112) at the moving interface corresponding to equilibrium values of the order parameter shifted by O(). If the problem is deﬁned in a bounded region, it should be complemented by boundary conditions at external boundaries. In the presence of external ﬂuxes, the chemical potential µ and the respective values of u may substantially deviate, respectively, from µs and u± s far from the interface. This, however, is still compatible with the adopted scaling, as long as the gradients remain weak. In a symmetric case, ±

− u+ s = −us = us ,

J0+ = J0− ,

−

u+ s

J0 = 0,

(2.112) simpliﬁes to

µ± − µs = (2us )−1 c J1 + 2us J0± − κKI − j ± K± .

(2.113)

If, in addition, Γ = const, the integrals K± vanish, and µ± = idem = µ. Then (2.113) can be recast as a mobility relation, or eikonal equation, similar to (2.50): c = c0 −

KI κ, J1 − 2us J0±

c0 = β (µ − µs ) ,

β=

2us , (2.114) J1 − 2us J0±

where c0 is the propagation speed of a ﬂat interface proportional to the deviation from the Maxwell construction µ − µs , and β is the eﬀective mobility. For the standard case of a cubic f (u) = u(1 − u2 ), we have µs = ± ± 0, u± s = ±1, f (us ) = −2. With Γ = D/2 = const, the integrals K

2.4 Cahn–Hilliard Equation

117

vanish, and the other integrals are evaluated to (2.24), using, in accordance √ √ p(u) = (1 − u)2 / 2K. This yields I = 23 2/K, J0± = 2KΓ −1 ln 2, J1 = √ 2 2KΓ −1 (1 − ln 2), and (2.114) reduces to c = c0 −

Dκ , 6

D ± D ± u ∓1 . c0 = − √ µ − µs = − √ 2 2K 2K

(2.115)

where u± are the values of the order parameter on the two sides of the interface. Although (2.50) and (2.114) or (2.115) have the same form, the substantial diﬀerence between the conservative and nonconservative systems is that in the latter case the propagation speed is additionally restricted by the conservation condition (2.101), which, unlike (2.50), involves the gradient of the order parameter. The conservation condition is commonly viewed as determining the velocity of the interface, while (2.114) or more generally (2.112) as the condition for deviation of the order parameter from its standard equilibrium value at a stationary planar interface. In particular, when extant gradients are weak, the propagation speed may be far less than either term on r.h.s. of (2.114). Then the kinetic contribution to the shift of the order parameter can be neglected and (2.115) reduces to the Gibbs–Thomson relation deﬁning the equilibrium of a curved interface. 2.4.5 Propagation and Coarsening in 1D Following a rapid quench into an unstable state, nuclei of alternative states are formed, being locally triggered by weak inhomogeneities or ﬂuctuations. This is followed by slow evolution, as phase domains coarsen to reduce the interfacial energy. In the case when formation of nuclei is suppressed, spinodal decomposition may be eﬀected by a propagating front invading an unstable homogeneous state. The speed of the front is selected, as in Sect. 2.3.2, at its leading edge, and can be computed with the help of the dispersion relation obtained by linearization in the vicinity of the unstable state. We shall work with the rescaled 1D form of the Cahn–Hilliard equation ∂t u = −∂x2 [uxx + f (u)] . The dispersion relation analogous to (2.73) is ω(k) = i αk 2 − k 4 ,

(2.116)

(2.117)

where α = √f (u0 ) > 0 when the homogeneous state u =2u0 is unstable, e.g., at |u0 | < 1/ 3 for the standard function f (u) = u(1 − u ). The group velocity is (2.118) c = ω (k) = 2ik(α − 2k 2 ).

Using here k = kr + ki and separating the real and imaginary part of c, ω, we ﬁnd that the condition ci = 0 in (2.75) has a nontrivial solution

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2 Fronts and Interfaces

Fig. 2.12. Space–time plot of simulations results for a pulled front propagating into the unstable state u = 0.4 (van Saarloos, 2003, reproduced with permission. Copyright by Elsevier Science). The lines denote the proﬁles of u at successive time intervals, and are shifted upward relative to each other. Note the coarsening of the pattern behind the propagating front

2kr2 = α + 6ki2 .

(2.119)

The indices r and i denote here, as in Sect. 2.3, the real and imaginary parts. Using this relation in the condition of equality of the phase and group velocities in (2.75), we ﬁnd that the only positive solution deﬁning the value of ki (i.e., the slope of the marginally stable proﬁle) is 1 ki2 = (2.120) α − 3 + 9 − 6α + 52 α2 . 6 The propagation speed is now evaluated as c= cr = −2ki (α + ki2 − 3kr2 ) 1/2 1 5 2 5 2 . (2.121) α − 3 + 9 − 6α + 2 α = 6 + α − 4 9 − 6α + 2 α 9 Because of mass conservation, the proﬁle behind the invading front cannot be monotonic, and the phase separation in the front region can only occur through the formation of domains of the two phases. The modulated pattern formed behind the propagating front further coarsens, as seen in Fig. 2.12. While this process can only be simulated numerically, its later stages, when the domains size far exceeds the characteristic diﬀusional range which deﬁnes the width of the interface between the two alternative phases, can be described

2.4 Cahn–Hilliard Equation

Ζn1 Ln1

Ζn Ln

119

Ζn1 Ln1

Fig. 2.13. A scheme of a sequence of phase domains in 1D

through a rational expansion in the scale ratio and presented in terms of front interactions as in Sect. 2.1.6 (Kawasaki and Ohta, 1982). It is reasonable to assume that, due to slow-down of the fronts and increasing domain sizes, the Peclet number Pe = cL/D based on the propagation speed c and a typical length L of phase domains becomes small at late stages of coarsening. Then a quasistationary approximation applies also to the “outer” regions and, since mobilities Γ ± can be considered constant in each phase domain, (2.95) is replaced by the stationary diﬀusion equation µxx = 0. The immediate corollary is that the ﬂuxes j ± = −Γ ± µx are constant within each phase domain bounded by fronts with opposite polarity (a kink and an antikink). Consider a sequence of domains with kink positions ζn and lengths Ln numbered as in Fig. 2.13. We shall restrict to a symmetric case; an asymmetric case is treated in a similar way, but the algebra is more involved because of the necessity to distinguish between µ± , as well as taking into account diﬀerent asymptotics of approach to the alternative states. The driving force of the coarsening process stems from the dependence of the energy of the fronts on their separation computed in Sect. 2.5. The equilibrium chemical potential µs at a stationary front should be replaced therefore by a shifted value ensuring stationarity of the front in the presence of interactions. Taking into account nearest-neighbor interactions only is suﬃcient in view of exponential decay of interactions. One can therefore use (2.36) applied to interactions with two neighboring fronts. If the front at the nth position is a kink, the potential shift is √

√ a2 α −√αLn+1 − αLn−1 − e µ n = + e . (2.122) us − u− s For an antikink, this formula should be taken with the opposite sign. The most straightforward way to the description of coarsening in a system of interacting fronts is to use the material balance and mobility relations derived in the two preceding subsections. The ﬂuxes within each domain are jn = −

Γ± µn+1 − µn . Ln

(2.123)

The chemical potentials at the fronts and propagation velocities are deﬁned by (2.100) and (2.114):

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2 Fronts and Interfaces

cn =

− jn+ − jn−1 = β (µn − µ n ) . 2us

(2.124)

The signs in these expressions pertain to kinks; since, following our common sign convention, all formulas in the preceding subsections were derived for kinks with the lower state on the left, the signs of velocities should be inverted for antikinks with the opposite orientation. The system (2.123), (2.124) comprises 3N equations deﬁning ﬂuxes within phase domains, chemical potentials at the fronts and propagation velocities. The system is closed on a ring, and has to be supplemented by boundary conditions for chemical potentials or ﬂuxes in a bounded domain. As a simple example, consider a single “bump” with the length L (a paired kink and antikink, as in Fig. 2.5) subject to external ﬂuxes, j+ from the left and j− to the right. Denoting the chemical potentials at the kink on the left and the antikink on the right as µ± , their displacement rates as ζ˙± = ±c± , and the ﬂux between them as j, the system (2.123), (2.124) reduces to ﬁve equations j=−

Γ µ+ − µ− , L

j − j± ζ˙± = = ∓β µ± − µ , 2us

(2.125)

where µ (L) is given by (2.36). The solution leads to the dynamic equations for the domain length L and the position of the domain center ζ: j+ − j− L˙ = ζ˙− − ζ˙+ = , 2us

βL(j+ + j− ) ζ˙− + ζ˙+ =− . ζ˙ = 2 2(βus L + Γ )

(2.126)

The ﬁrst relation is trivial, and can be obtained simply by adding up the material balance conditions for the two fronts. The second relation, which describes the migration of the “bump” toward the source of material, requires solving (2.125) and depends on the mobilities as well as on the domain size. The potential shift µ falls out of (2.125), and may come into the action only when coarsening is possible, which requires at least three fronts. For a sequence of two kinks ﬂanking an antikink on an inﬁnite line with the no-ﬂux conditions at x → ±∞, (2.100) and (2.114) give the system of eight equations n (Ln ) deﬁned for jn , µn , cn (numbered as in Fig. 2.13), with j0 = j3 = 0 and µ by (2.122) where L0 and L3 should be set to ∞. The result is 2 − Γ µ1 ) 1 − Γ µ2 ) βΓ (2βus L2 µ βΓ (2βus L1 µ , c3 = − , 4β 2 u2s L1 L2 − Γ 2 4β 2 u2s L1 L2 − Γ 2 1 (Γ + 2βus L1 )] βΓ [ µ2 (Γ + 2βus L2 ) − µ c2 = . (2.127) 4β 2 u2s L1 L2 − Γ 2

c1 =

where the fronts and domains are numbered as in Fig. 2.13. In a symmetric 1 = µ 2 = µ , this reduces to case, L1 = L2 = L, µ c1 = −c3 = βΓ µ /(2βus L + Γ ),

c2 = 0,

(2.128)

2.4 Cahn–Hilliard Equation

121

so that the two kinks move toward the antikink until they annihilate. When the symmetry is broken, the antikink moves in the direction of the closest kink, so that eventually the domain with a larger value of µ disappears. Of course, the theory breaks down just before the annihilation event when the domain size becomes small, but this ﬁnal stage is very brief. 2.4.6 Ostwald Ripening In the absence of external ﬂuxes, the system would evolve to a state close to equilibrium after a period of O(L2 /D), where L is a characteristic size of domains corresponding to the alternative coexisting phases. At the same time, the interfaces would evolve locally to minimize the surface energy. If the volume ratio of the two alternative phases strongly deviates from unity, the resulting intermediate state would consist of well-separated spherical droplets of the minority phase immersed in a continuum of the majority phase – a situation we have already encountered in Sect. 2.2.4. Assuming local equilibrium governed by the Gibbs–Thomson relation, the residual diﬀerence in local chemical potentials will only be caused by a scatter of droplet radii R, and the upper estimate of the interfacial propagation √ velocity deﬁned by (2.100) or (2.101) will be c ∼ lRD/L, where l = K is the characteristic width of the interface and L is a typical distance between droplets. This estimate is by O() lower that the a priori estimate c ∼ D/L. This, ﬁrst of all, justiﬁes neglecting the kinetic contribution to (2.112) or (2.114), thereby validating the above assumption of local equilibrium. Moreover, the characteristic time scale, estimated before as D/c2 ∼ L2 /D, now increases by the factor (R/l)2 1. Therefore, the time derivative in the outer equation (2.95) or (2.96) can be neglected and the diﬀusion equation replaced by the Laplace equation (2.129) ∇2 µ = 0. This is called the Hele–Shaw limit, by analogy with viscous ﬁngering phenomena where the motion of an interface is determined by the pressure ﬁeld obeying the Laplace equation. The equilibrium deviation of the order parameter at the interface u is determined by the Gibbs–Thomson relation for the equilibrium chemical potential at the interface, µ = µs + bl/R, (2.130) √ 2 where b is a numerical constant equal to 3 2 in the standard case of the cubic f (u). If µc ≡ µs + bl/Rc is a prevailing value of chemical potential far from the interface, one can easily compute the ﬂux toward a spherical droplet at the origin by solving (2.129) in spherical symmetric geometry with the boundary conditions µ(R) = µ, µ(∞) = µc : µ + (µc − µ)(1 − R/r) at r ≥ R µ(r) = (2.131) µ at r ≤ R.

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2 Fronts and Interfaces

Using (2.130), (2.131) in the interfacial balance condition (2.100) yields the growth rate of the droplet R blΓ dR = 2 + − 1 . (2.132) c= dt Rc R (us − u− s ) Thus, as in Sect. 2.2.4, larger droplets with radii R > Rc grow, while smaller ones with R < Rc shrink and eventually disappear; the bulk chemical potential µc slowly changing in the course of evolution plays here the role of a global control variable. This is the classical process of Ostwald ripening. The asymptotic distribution of droplet sizes at late stages of the ripening process has been computed in the original work by Lifshitz and Slyozov (1959) for this particular case, which diﬀers from that considered in Sect. 2.2.4 by space dimension and the dependence of the growth rate on the droplet size. The quasistationary asymptotic size distribution is computed along the same lines as in Sect. 2.2.4. The variable characterizing the asymptotic distribution, similar to (2.62), is the dimensionless radius ρ = R/Rc . The growth of the respective dimensionless volume is proportional to blΓ 1 d(ρ3 ) ρ3 dRc = 3 (ρ − 1) − . 3 dt Rc Rc dt

(2.133)

This expression is further transformed by introducing a new time variable τ = ln(Rc /R0 )3 (with an arbitrary R0 setting the origin) and denoting − 3 u+ s − us d(Rc ) . 3blΓ dt

(2.134)

1 d(ρ3 ) ρ−1 = − ρ3 . 3 dτ q(τ )

(2.135)

q(t) = Then (2.133) is rewritten as

As in Sect. 2.2.4, there is a unique asymptotic value of q at τ → ∞ compatible with the volume conservation condition. The r.h.s. of (2.135) is negative at both small and large values of ρ, and reaches the zero maximum at ρ = 3/2 when q = 4/27. The latter must be the asymptotic value of q at τ → ∞, while the location of the maximum on the ρ axis deﬁnes the maximum dimensionless radius in the asymptotic distribution. The function q(τ ) approaches the limiting value from above in such a way that a small gap decreasing with time remain between the maximum of the r.h.s. of (2.135) and the ρ axis as in Fig. 2.8a. The asymptotic size distribution can be obtained as in Sect. 2.2.4 by solving (2.60) with c(R) given by (2.132) after transforming this equation to the new variables ρ, τ .

2.5 Phase Field Model

123

2.5 Phase Field Model 2.5.1 Variational Principle for Phase Field Model Phase ﬁeld models have been developed primarily for description of pattern formation phenomena in solidiﬁcation. They include two variables: a nonconserved order parameter (phase ﬁeld ) that is used to distinguish between the solid and ﬂuid phases, and a conserved control ﬁeld that determines relative thermodynamic stability of the two phases and may be physically identiﬁed with enthalpy. Unlike the models of gas–liquid interface, phase ﬁeld models of solidiﬁcation are usually viewed as a computational device enabling to avoid discontinuities and solve PDEs instead of tracking a sharp interphase boundary, and are not expected to describe actual structure of a diﬀuse interfacial layer. The only requirement is to choose model parameters in a thermodynamically and kinetically consistent way, so that the sharp interface limit of the theory would faithfully describe the motion of the interphase boundary. The free energy functional for an isotropic system is written in the form (Langer, 1986; Karma and Rappel, 1998)

(2.136) F= V (u) + 12 λv 2 + 12 K|∇u|2 dx, where u is a phase ﬁeld and v = W + H(u) is a control ﬁeld (which may be identiﬁed with temperature or concentration) related to the conserved “enthalpy” W ; V (u) and H(u) are potentials of a suitable form, K is the rigidity coeﬃcient deﬁned as in Sect. 2.4.1, and λ is a coupling constant. As in Sect. 2.1.1, V (u) is a double-well potential with two minima corresponding to the two alternative phases; the standard choice is the quartic (2.23). The biasing potential H(u) should be odd; its form is not very important, and may be a matter of convenience. A linear function, H(u) = bu, appears to be a natural choice, but using a higher order odd function u (1 − u2 )k du (2.137) H=b 0

gives an advantage of keeping the potential minima ﬁxed at u = ±1. With k = 1, the control variable, while remaining within the limits |v| < (bλ)−1 , just shifts the unstable equilibrium of the phase ﬁeld; a model with k = 2 has also been used in computations, even though it introduces a spurious fourth root. The coeﬃcient b can be related in the thermal context to the ratio of the solidiﬁcation heat-to-heat capacity. The variational equations for the nonconserved and conserved ﬁelds are δF ∂u = −χ , ∂t δu

∂W D δF = ∇2 , ∂t λ δW

(2.138)

where we have introduced kinetic coeﬃcients χ and Γ = D/λ, presumed to be constant. These equations coincide, up to coeﬃcients, with (2.2) and (2.88),

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2 Fronts and Interfaces

(2.89). As in Sects. 2.1.1, 2.4.2, the free energy decreases monotonically in the course of evolution. Taking the variations of (2.136) explicitly yields the equations for the phase and control ﬁelds: χ−1 ∂t u = K∇2 u + f (u) + λvg(u), ∂t v = D∇2 v − 12 g(u)∂t u,

(2.139) (2.140)

where f (u) = −V (u), g(u) = −H (u). One should note the way the two ﬁelds are coupled. The control ﬁeld skews the eﬀective potential of the phase ﬁeld V (u) = V (u) + λvH(u) in (2.139), thereby making one of the minima lower and setting the interphase boundary into motion. At the same time, a moving boundary layer where u switches between the two alternative values contributes a source in the equation of the control ﬁeld, imitating the heat release or density change at the interface. An advantage of the phase ﬁeld model, compared to the Cahn–Hilliard model of Sect. 2.4 is that it disentangles the phase ﬁeld, now just a nonconserved tag, from conserved variables (density or enthalpy). As a result, equilibrium is enforced only on the interphase boundary, while the bulk may remain in a nonequilibrium (supersaturated or undercooled) state. At the same time, conservation laws are enforced, with due account of density change or heat release accompanying the phase transition, and the system evolves in a thermodynamically consistent way toward a minimum of free energy. 2.5.2 Sharp Interface Limit The sharp interface limit can be obtained much in the same way as in Sect. 2.4.3, using a wide √ separation between the characteristic width of the interphase boundary l = K applicable in the inner region and the characteristic length scale of the control ﬁeld L = l/ applicable in the outer region (Karma and Rappel, 1998). If no extrinsic scale is imposed, the latter can be estimated as L = D/c0 , where c0 is a characteristic velocity of the interface. The matching time scale T = D/c20 . Since the last term in (2.140) vanishes far from the interface, the outer equations are just diﬀusion equations for the control ﬁelds on either side of the interface, denoted as v ± , and are made parameterless by the chosen scaling: ∂t v ± = ∇2 v ± .

(2.141)

In the inner region, we shall use again the aligned comoving frame and scale the derivatives along and √ across the axis x normal to the nominal front position, respectively, by l = K and L = l/. The curvature is then of O(L−1 ), and is written as κ = K −1/2 κ1 when measured on the short inner scale, and the velocity is expressed as c = (D/L)c1 = DK −1/2 c1 . Retaining the terms up to the ﬁrst order, we write the inner equations as

2.5 Phase Field Model

u (x) + (τ c1 + κ1 )u (x) + f (u) + λvg(u) = 0,

v (x) + (c1 + κ1 )v (x) + 12 c1 g(u)u (x) = 0,

125

(2.142) (2.143)

where τ = D/χK is a dimensionless capacitance factor, further assumed to be of O(1). The solution of the inner equations is sought for as an expansion in : u = u0 + u1 + · · · ,

v = v0 + v1 + · · · .

(2.144)

The zero-order equations are u0 (x) + f (u0 ) + λv0 g(u0 ) = 0,

v0 (x) = 0.

(2.145)

The solution is again a stationary front that exists only at v0 = 0. This compels the zero-order control ﬁeld at the interface to be at the level corresponding to the Maxwell construction. In the case of a liquid–solid interface, this just tells that the interface should be at the melting point. The ﬁrst-order equation of v reduces to the form

d v1 (x) − 12 c1 H(u0 ) = 0, dx

(2.146)

which can be integrated to v1 (x) = v 1 + j1 x +

1 2 c1

x

H(u0 (x1 ))dx1 ,

(2.147)

0

where v 1 = v1 (0), j1 are integration constants. This solution has to be matched asymptotically at x → ±∞ with the solution of the outer equation (2.141): (2.148) lim v1 (x) = j1 + 12 c1 H ± = −1 n · ∇v ± , x→±∞

±

H(u± s )

and the derivative n · ∇ along the normal n (directed, by where H = convention, from the solid to the liquid) is computed at the matching interface. Combining the matching conditions on either side and using (2.147) yields the condition of interfacial balance, which is more transparent when rewritten in a dimensional form + − 1 = D(n · ∇v + − n · ∇v − ). (2.149) 2c H − H The ﬁrst-order equation of u is again of the form (2.47) with the inhomogeneity (2.150) Ψ (x) = (τ c1 + κ1 )u0 (x) + λv1 g(u0 (x)). The solvability condition (2.49) requires the inhomogeneity to be orthogonal to the translational Goldstone eigenmode u0 (x). The contribution due to the linear term in (2.147) vanishes, as both u0 (x) and g(u0 (x)) are even in x. Collecting the contributions due to other terms, the solvability condition is computed as

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2 Fronts and Interfaces

λ H + − H − v 1 = I(τ c1 + κ1 ) + 12 c1 J , where I is deﬁned by (2.8) and ∞ J = u0 (x) dx −∞

(2.151)

x

H(u0 (x1 )) dx1 .

(2.152)

0

2.5.3 Mobility Relation The values of the control ﬁeld to be used as interfacial boundary conditions for the outer equation should be of O() to match its value in the inner region. They are denoted as vs± = v ± and obtained by matching the inner limit of the outer solution (2.153) v ± = v ± + x n · ∇v ± with the outer limit of the inner solution (2.147) presented using (2.148) as lim v1 (x) = v 1 + 12 c1 K± + x n · ∇v ± ,

x→±∞

where ±

K =

±∞

[H ± − H(u0 (x))] dx.

(2.154)

(2.155)

0

This yields

v ± = v 1 + 12 c1 K± .

(2.156)

±

The limits v are identical when both H(u) and u0 (x) are antisymmetric, and can be written explicitly using (2.151) and omitting ± sign as Iτ + 12 J κ1 I 1 , (2.157) v = c1 2 K + + + − + λ (H − H ) λ (H − H − ) or, reverting to a dimensional form √ √ Iτ + 12 J c K 1 κ KI vs = 2 K + λ (H + − H − ) + λ (H + − H − ) . D

(2.158)

The integrals appearing in the above expressions are computed most easily by using u0 as the integration variable instead of x; for the standard u0 (x) given by (2.24), the transformation uses the expression for the derivative √ u2 )/ 2. For H(u) deﬁned by (2.137) with k = 1, this p(u) = u (x) = (1 −√ yields K±√= K = b/ √2. Other integrals computed with the same functions are I = 23 2, J = 13 b 2. This completes the reduction to the Stefan problem determined by two linear diﬀusion equations (2.141) with the common value v ± = vs on the interface deﬁned by (2.158) and the interfacial velocity ﬁxed by (2.149). The interfacial condition (2.158) can be again rearranged into a mobility relation similar to (2.50) or (2.115), which, in turn, reduces to a Gibbs– Thomson relation when the kinetic contribution to the shift of the value of the conserved ﬁeld at the interface can be neglected. This happens, for example, at b 1 when the coeﬃcient at c in (2.158) becomes very large, while the coeﬃcient at κ remains of O(1).

2.6 Instabilities of Interphase Boundaries

127

2.6 Instabilities of Interphase Boundaries 2.6.1 Instabilities due to Coupling to Control Field It is quite common for nonequilibrium chemically reacting systems that characteristic spatial and temporal scales associated with diﬀerent reactants and other dynamic variables, like temperature or potential, are widely separated due to great diﬀerences in respective reaction and diﬀusion rates. This scale separation has been proﬁtably exploited by analytical theory in Sects. 2.4.3, 2.5.2 where a sharp interface limit has been derived using rational approximation techniques based on the method of matched asymptotic expansions. From the point of view of numerics, the role of scale separation is ambivalent. On the one hand, it makes computations very diﬃcult, requiring high resolution in narrow mobile front regions. On the other hand, the need in a ﬁne grid can be eliminated if the short-scale ﬁeld is eliminated and replaced by equations of motion of sharp fronts dependent on the local front geometry and on the long-range ﬁelds. The equation of motion of a weakly curved front – the eikonal equation (2.50) or (2.54) – can be coupled to an additional long-scale “control” ﬁeld that determines the speed of a straight-line front c0 or the energy gain V1 . Assuming a linear dependence on the control variable v, the equation of front motion can be then written in the form c = c0 (v − lκ),

(2.159)

where l is the capillary length and the scale of v is chosen in such a way that a planar interface is stationary at v = 0. Equations of motion of this kind remain valid as long as the characteristic length of the control variable far exceeds the front thickness and, consequently, the control ﬁeld remains almost constant across the front. Under certain conditions, coupling to the control ﬁeld causes the front to lose stability and become corrugated (Mullins and Sekerka, 1963; Langer, 1980). This instability may eventually lead to the formation of stationary, moving, or dynamic structures composed of domains with prevailing alternative states of the short-scale ﬁeld separated by sharp fronts and immersed in the ﬁeld of the control variable changing on a longer scale commensurable with the domain size. Consider a weakly distorted planar front in 3D; the front position is deﬁned as a function ζ(y, t) of transverse coordinates y and time. The normal velocity c is approximated in the leading order by the derivative ∂ζ/∂t if the displacement ζ is small, or, more precisely, |∇⊥ ζ(y)| 1, where ∇⊥ is the gradient operator in the transverse plane. Under the same conditions, the interfacial curvature is expressed as κ = −∇2⊥ ζ. Suppose that the control ﬁeld v obeys a linear equation and is fast as well as long-scale, so that it adjusts instantaneously to front displacements. Then the value of the control variable

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2 Fronts and Interfaces

at the front can be expressed using an appropriate Green’s function G(r) as a functional of the front position: ∞ G(|r|)ζ(y − r)dr. (2.160) v(y) = 0

The eikonal equation (2.159) can be written now using the approximate expression for the interfacial curvature in a closed form ∞ ∂ζ 2 = l∇ ζ + G(|r|)ζ(y + r)dr. (2.161) c−1 ⊥ 0 ∂t 0 This integro-diﬀerential equation can be further reduced to a local form if the wavelength of interfacial deformations far exceeds the characteristic scale of the control ﬁeld. The deformation can then be expanded as ζ(y + r) = ζ(y) + r · ∇⊥ ζ(y) + rr : ∇⊥ ∇⊥ ζ(y) + . . .

(2.162)

Due to the symmetry of the Green’s function, the only terms in the expansion contributing to the integral in (2.161) are those proportional to r = |r|. Retaining the terms up to the fourth order, we have 2 2 2 c−1 0 ζt = β0 ζ + (l + β2 )∇⊥ ζ + β4 (∇⊥ ) ζ, ∞ βj = rj+1 G(r)dr.

(2.163) (2.164)

0

The expansion is valid only if G(r) decays suﬃciently fast, so that the integrals (2.164) converge. The dispersion relation for perturbations with the transverse wavenumber k generated by (2.163) is 2 4 c−1 0 λ = β0 − (l + β2 )k + β4 k .

(2.165)

Since (2.163) has been derived by long-scale expansion, this dispersion relation is valid for k 1 and is duly balanced only when β0 and l + β2 are small, |β0 | |l + β2 | |β4 | = O(1). The ﬁrst term on the r.h.s. of (2.163) vanishes when the front is neutrally stable to parallel displacements. This is a usual situation in a homogeneous inﬁnitely extended system, but in a ﬁnite system the front can be either stabilized or destabilized by a gradient of the control ﬁeld or due to the inﬂuence of boundary conditions. A practically important example is instability of diffusionally limited deposition due to the increase of the deposition ﬂux on surface segments drawn closer to the source of deposited particles. This instability may be balanced on shorter scales by surface tension, but is usually not saturated and leads to ﬁngering, formation of dendrites, etc. A more important symmetry-breaking instability independent of extrinsic factors is caused by reversal of the sign of the eﬀective surface tension at β2 < −l. If β0 = 0, this instability ﬁrst arises at long wavelengths and is

2.6 Instabilities of Interphase Boundaries

129

stabilized at shorter scales by the last term, provided β4 < 0. The growth rate reaches maximum at some k = 0, which remains small only close to the point of marginal instability, β2 = −l. This instability, leading to the structure formation on a moving interface, will be studied in more detail further in this section. Dynamic instabilities, leading to interfacial oscillations and waves, are possible when the response of the control ﬁeld to surface displacements is delayed. 2.6.2 Mullins–Sekerka Instability When the interface propagation involves a conserved ﬁeld, the velocity is primarily governed by a conservation condition at the interface, while (2.159) determines the deviation of the order parameter from its standard equilibrium value at a stationary planar interface. We take as a starting point the sharp interface limit of the phase ﬁeld model, and consider ﬁrst a planar interface propagating along the x-axis with a constant speed c. The conserved variable v obeys the 1D diﬀusion equation written in the comoving frame as cv (x) + Dv (x) = 0.

(2.166)

The boundary conditions are c = −D[v (+0) − βv (−0)],

v(0) = c/c0 ,

v(∞) = v∞ .

(2.167)

The conditions at the interface placed at x = 0 are given by (2.149) and (2.159), but the former is generalized by allowing the diﬀusivities to be diﬀerent on the two sides of the interface; β is the ratio of diﬀusivities. This problem can be viewed as a model of solidiﬁcation where v = Tm − T is interpreted as deviation2 of temperature T from its equilibrium value at the interface (melting temperature) Tm . In this context, the asymptotic value of v in the liquid bulk far from the interface v∞ is called undercooling. An alternative is a “chemical” interpretation, where v is the concentration of a contaminant inﬂuencing the melting temperature. In this case, one can expect a concentration discontinuity across the interface, which can be formally eliminated by choosing diﬀerent scales of v in the two phases (more on this in Sect. 2.6.4). The solution satisfying (2.166), (2.167) is at x ≤ 0 c/c0 (2.168) v(x) = v∞ + (c/c0 − v∞ )e−(c/D)x at x ≥ 0, where the propagation speed and undercooling are constrained by the relation c = c0 (v∞ − 1). In a practically important case c c0 , i.e., negligible kinetic undercooling, the interfacial temperature remains at the equilibrium melting 2

The choice of the sign follows our common convention placing a higher HSS at x > 0.

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2 Fronts and Interfaces

Fig. 2.14. Schematic illustration of Mullins–Sekerka instability, showing levels v = const for a planar interface (left) and a perturbed interface shifted to the point ζ (right). Adapted from Langer (1980)

point, v(0) = 0. Then the solution exists at an arbitrary speed, but at the ﬁxed value of undercooling v∞ = 1. This, of course, remains true only as long as propagation is suﬃciently slow. The property of Galilean invariance of a propagating planar interface is justiﬁed physically, since faster propagation generates larger temperature gradients suﬃcient to dissipate increased heat of solidiﬁcation at a higher rate, so that the interface might remain at melting temperature. It can be seen, however, as an indication of a likely transverse instability. Indeed, the interface would become highly corrugated if its diﬀerent segments were allowed to propagate with diﬀerent velocities; this tendency is counteracted by surface tension but enhanced by heat transport, since advanced convex segments dissipate solidiﬁcation heat more eﬃciently (see a schematic illustration in Fig. 2.14). We shall study the stability of a planar front to symmetry-breaking perturbations assuming that the motion is slow, so that the interface remains at equilibrium and (2.159) can be replaced by the Gibbs–Thomson relation v = lκ. Both this relation and the interfacial balance condition (2.149), in spite of their innocuous form, are nonlinear, and have to be linearized assuming the perturbations to be small. As in the preceding subsection, the perturbed front position is deﬁned as a function ζ(y, t) of transverse coordinates y and time. The perturbation of the normal velocity c = ∂t ζ and curvature κ = −∇2⊥ ζ are approximated in the same way as before; the approximation holds as long as the amplitude of the perturbation is much smaller than its wavelength. The control variable is expanded in the vicinity of the unperturbed position as v = vs + v, where vs (x) is the basic solution (2.168) with some ﬁxed c, written now as 0 at x ≤ 0 (2.169) v(x) = 1 − e−(c/D)x at x ≥ 0, and v(x, y, t) is a perturbation. Then the Gibbs–Thomson relation is reduced to the form v (0, y, t) + vs (0)ζ] = 0. (2.170) l∇2⊥ ζ + [

2.6 Instabilities of Interphase Boundaries

131

Under the same assumption of slow propagation, the perturbation ﬁeld can be assumed to be quasistationary in the comoving frame, and described by the stationary equation D± (∇2⊥ v + vxx ) + c vx = 0,

(2.171)

where D+ = D, D− = βD. This equation has to be solved both in the liquid (x > 0) and solid (x > 0) domains, subject to the boundary condition (2.170) at x = 0, as well as the asymptotic conditions v → 0 at x → ±∞, and the resulting ﬂuxes D± vx (±0, y) on the liquid and solid sides used in the perturbed interfacial balance condition ζt = D [ vx (+0) − β vx (−0) + vs (+0)ζ]

(2.172)

to obtain the variable part of the normal velocity ζt . We are looking for a solution in the spectral form ζ(y, t) = χ(k)eλt+ik·y ,

v(x, y, t) = ψ ± (x; k)eλt+ik·y ,

(2.173)

where ψ ± denote, respectively, the solutions in the liquid and solid domains. Using (2.173) in (2.170), (2.171) we compute ψ + (x) = (c/D + l k 2 )χ e−q

+

x

,

ψ − (x) = l k 2 χ eq

−

where q ± is the only positive root of D± (q 2 − k 2 ) ∓ cq = 0:   ± 2 2kD c  1+ q± = ± 1 . 2D± c

x

,

(2.174)

(2.175)

Using this in (2.172) yields the dispersion relation λ(k 2 ) that determines stability of the basic solution: λ = −lk 2 (D+ q + + D− q − ) + cq + − c2 /D.

(2.176)

As expected, λ vanishes at k = 0, which reﬂects the translational symmetry of the front. Stability to long-scale perturbations is determined therefore by the sign of the derivative dλ/d(k 2 ) at k = 0, which is positive at D/c > l, i.e. when the diﬀusion length L = D/c exceeds the capillary length. This is always true in practical situations satisfying the above assumption of slow propagation; moreover, this is a necessary condition for the validity of the sharp interface limit in Sect. 2.5.2. The dispersion relation (2.176) can be approximated then as

(2.177) λ ≈ kc 1 − (1 + β)lLk 2 . The “optimal” wavelength of the fastest growing perturbation is of the same order of magnitude as the geometrical mean of the diﬀusional and capillary lengths, and gradually decreases with growing propagation speed. This trend

132

2 Fronts and Interfaces

is, however, reversed at higher velocities when the approximation (2.176) breaks down, so that the “optimal” wavelength diverges as the stability limit L = l is approached. If the quasistationarity assumption is abandoned, q ± become dependent on λ and the dispersion relation becomes implicit. Additional nonstationary eﬀects appear when kinetic undercooling cannot be neglected. Generally, oscillatory instability becomes possible under these conditions. We shall not consider this case here, as rather nontypical for solidiﬁcation fronts, but similar nonstationary eﬀects will be further discussed in Sect. 3.3.1. 2.6.3 Instability of a Curved Interface The above analysis can be generalized to the case of a curved interface representing, for example, a growing spherical nucleus of solid phase. Under these conditions, the problem is, strictly speaking, always nonstationary, since the radius keeps growing as the interface propagates. Nevertheless, the perturbation ﬁeld v remains quasistationary as long as propagation is slow, and can be obtained by solving the diﬀusion equation in spherical coordinates x, θ, φ. The solution is expressed as a combination of spherical harmonics: − at r ≤ R ψnm (x/R)n Pnm (cos θ)eimφ (2.178) v(x, θ, φ) = + ψnm (R/x)n+1 Pnm (cos θ)eimφ at r ≥ R, where R is the instantaneous radius of the solid sphere and Pnm are Legendre polynomials. The components of the radial perturbation of the interface are χnm Pnm (cos θ)eimφ , (2.179) ζ(θ, φ) = x(θ, φ) − R = n,m

and the approximate expression for the curvature of a weakly perturbed sphere with κ = 2(R + ζ)−1 − ∇2⊥ ζ = 2R−1 + R−2 [n(n + 1) − 2]χnm Pnm (cos θ)eimφ . n,m

(2.180) Note that the curvature correction vanishes at n = 1, since this mode corresponds to translation of the sphere without change of form. The mean curvature is independent of the azimuthal index m, and therefore the azimuthal dependence can be further omitted. The spherically symmetric basic solution is expressed in the quasistationary approximation as 2l/R at x ≤ R (2.181) vs (x) = v∞ + x−1 (2l − Rv∞ ) at x ≥ R. The asymptotic undercooling is not forced any more to be equal to unity, and the propagation velocity is computed as

2.6 Instabilities of Interphase Boundaries

133

Fig. 2.15. Multiexposure photograph of a growing dendrite (Langer, 1980; reproduced with permission. Copyright by the American Physical Society)

D cs = R

2l v∞ − R

2lD = 2 R

R −1 . Rc

(2.182)

The velocity vanishes at R = Rc = 2l/v∞ , which deﬁnes the radius of a critical nucleus; further on, it is convenient to eliminate v∞ in favor of Rc . Using (2.180), (2.181) in the Gibbs–Thomson relation v(R + ζ) = lκ expanded to the ﬁrst order yields ψn+ = l R−2 [n(n + 1) − 2R/Rc )] χn ,

ψn− = l R−2 (n+2)(n−1)χn . (2.183)

The dispersion relation is now obtained from the interfacial balance condition (2.172):

λ = DlR−3 −n(n + 1)2 + 2(n + 1)R/Rc − 2βn − 4(R/Rc − 1) = (n − 1)DlR−3 −[4 + (3 + 2β)n + (1 + β)n2 ] + 2R/Rc . (2.184) As expected, λ vanishes at n = 1. Instability in the symmetric mode n = 0, leading to the collapse of a small nucleus, is observed at R/Rc < 2. The most dangerous symmetry breaking instability occurs in the dipole mode n = 2 at R/Rc > 7 + β; thus, the nucleus loses its spherical shape when it is still not much larger than the critical nucleus. Extending the analysis to more irregular shapes, we can conjecture that the propagating interface tends to destabilize whenever the local curvature radius exceeds Rc . This causes the growing solid to acquire peculiar dendrite forms3 , as seen in Fig. 2.15. The dendrite tip propagates, apparently retaining 3

Actual dendrite forms observed experimentally cannot be reproduced without taking into account anisotropy of the crystalline solid interfaces.

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2 Fronts and Interfaces

its shape, while new protuberances, eventually developing into similar propagating tips, develop further along the stem. The entire structure may keep growing and developing tertiary, etc., branches, as long as it is not foiled by collisions of branches and retarded transport. An interesting question is the existence of a self-preserving form that might model the growing tip. This problem is relevant also for other applications where the interface propagation is controlled by a ﬁeld obeying the Laplace equation, and has been ﬁrst studied in the context of ﬁngering of the interface of a less viscous ﬂuid propagating into a more viscous one. An inﬁnite family of solutions parametrized by the propagation speed has been found by Ivantsov (1947) for the case of vanishing surface tension; see also an elegant solution method given by Kurtze (1987). The radius of the tip is determined by the velocity, but the selection mechanism for both as yet remains unclear. A nonvanishing surface tension destroys the self-similarity of the problem and renders the analytical theory hopeless. It is commonly believed that surface tension selects a unique solution out of Ivantsov’s inﬁnite family, but it is possible that the selection mechanism is dynamic, as in propagation into an unstable state Sect. 2.3, and can operate also in the absence of surface tension, as suggested in a controversial paper by Mineev-Weinstein (1998). 2.6.4 Directional Solidiﬁcation The experimental setup used for puriﬁcation of solids by remelting includes a planar front moving in an imposed temperature gradient (Fig. 2.16a). Two control ﬁelds are involved here: temperature and contaminant concentration. In the simplest formulation of this problem, both the melt and the solid are assumed to be good heat conductors, so that the heat of solidiﬁcation can be neglected and temperature can be assumed to be a linear function of the coordinate, T = Θx. This explicit coordinate dependence breaks the translational invariance of the system. The emergence of spontaneous surface corrugations in this system can be understood in the following way (Wollkind and Segel, 1970). Suppose a protuberance has formed on a solid surface, bringing it into the region of higher temperatures. At a ﬁxed contaminant concentration, this protuberance would melt, but, due to enhanced contaminant transport, the local concentration would decrease and the melting temperature rise, so that the protuberance may keep growing, counteracting the surface tension. Some typical cellular patterns – a regular one close to the point of marginal instability and a more complicated “doublet” structure – are shown in Fig. 2.16b. The contaminant concentration in the liquid phase is described by the diﬀusion equation ∂v = D∇2 v. (2.185) ∂t Diﬀusion in the solid is neglected. The boundary conditions depend on the distribution coeﬃcient K < 1 equal to the ratio of the equilibrium contam-

2.6 Instabilities of Interphase Boundaries (a)

135

(b)

Fig. 2.16. Directional solidiﬁcation. (a) a schematic of experimental setup (adapted from Langer, 1980). (b) experimental cellular patterns (Jamgotchian et al., 1993; reproduced with permission. Copyright by the American Physical Society)

inant concentrations in the solid and in the liquid. The ﬂux at the interface determined by the material balance is −Dn · ∇v(ζ) = c(1 − K)v,

(2.186)

where n is the normal to the interface and c is the normal velocity. The rejected material diﬀuses back into the melt. The asymptotic contaminant concentration far from the interface v(∞) will be taken as unity. The problem is closed by the equation for the equilibrium melting temperature, which, taking into account the postulated linear dependence of temperature on the coordinate as well as the Gibbs–Thomson relation between the melting tem0 − Tm )/Θ = σκ, just relates the front perature and interfacial curvature (Tm position to the local curvature κ and the contaminant concentration in the solid at the interface: ζ = α[1 − Kv(ζ)] − σκ, (2.187) where the constant α = Θ−1 dTm /dv > 0 characterizes a linear dependence of melting temperature on concentration. The stationary concentration ﬁeld v0 (x) ahead of a planar interface steadily propagating with the speed c0 veriﬁes the equation in the comoving frame where the interface position is ﬁxed at ζ = 0: Dv0 (x) + c0 v0 (x) = 0,

−Dv0 (0) = c0 (1 − K)v0 (0). (2.188) The contaminant concentration at the interface is taken as the scale of v. The solution is (2.189) v0 (x) = K + (1 − K)e−c0 x/D . v0 (∞) = K,

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2 Fronts and Interfaces

We shall look for a perturbed solution in a spectral form ζ(y, t) = χ(k, λ)eλt+ik·y ,

v(x, y, t) = v0 (x) + ψ(x; k, λ)eλt+ik·y . (2.190)

The dispersion relation λ(k 2 ) determines stability to inﬁnitesimal perturbations. Using (2.190) in (2.185) yields Dψ (x) + c0 ψ (x) − (λ + Dk 2 )ψ = 0.

(2.191)

The boundary conditions for (2.191) are set at x = ζ; therefore when (2.186) and (2.187) are expanded and linearized, one should use the value of the zeroorder function and its derivative at this location, v0 (ζ) = v0 (0)ζ, v0 (ζ) = v0 (0)ζ. Since the zero-order solution is independent of the transverse coordinates, deviations of the normal from the x-axis do not aﬀect the linearized boundary conditions, so that one can replace n · ∇ ≈ ∂x and express the normal velocity as c ≈ c0 + ∂ζ/∂t. Then (2.186) yields −D[ψ (0) + v0 (0)χ] = (1 − K)[c0 ψ(0) + λv0 (0)χ + c0 v0 (0)χ].

(2.192)

The condition (2.187) reduces to χ = −αK[ψ(0) + v0 (0)χ] − σk 2 χ.

(2.193)

The dispersion relation is obtained (in an implicit form) by solving (2.191) with the boundary conditions (2.192), (2.193), and the asymptotic condition ψ(∞) = 0. A convenient dimensionless expression is obtained using D/c0 as the length scale and D/c20 as the time scale and introducing dimensionless parameters W 1 0.8

0 0.05

0.6 0.4

0.1 0.2 0.3 0.4

0.2

0.5

1

1.5

2

k2

Fig. 2.17. The marginal instability locus at β = 1/2. The numbers indicate the values of K

2.6 Instabilities of Interphase Boundaries

W =

D , αc0 (1 − K)

W c20 σ c0 σ . = D2 (1 − K)αD

β=

137

(2.194)

The result is λ = (1 − W − βk ) K − 2

1 2

+

1 4

+λ+

k2

− K.

(2.195)

The dependence of the critical value of W at the point of marginal instability (λ = 0) on the wavenumber at diﬀerent values of K is shown in Fig. 2.17. The instability sets on at ﬁnite K at suﬃciently small values of W below Wc = 1. 2.6.5 Long-Scale Instability At K 1, the instability occurs at long wavelengths and W close to unity, which implies αc0 /D = O(2 ). The marginal instability locus can be approximated in this parametric range as W = 1 − βk 2 − (1 + k 2 /K)−1 .

(2.196)

The maximum is reached at k 2 = O() 1 and 1 − W = O() when K = O(2 ). The dispersion relation (2.195) can be approximated near the maximum by a quartic form similar to (2.165). Setting W = 1 − w,

K = 2 q,

(2.197) √ and rescaling k → k, λ → 2 λ, we balance the dispersion relation on the O(2 ) scale as (2.198) λ = −q + k 2 (w − βk 2 ). 2 The dispersion relation √ reaches maximum at kmax = w/2β and the instability threshold is w = 2 qβ. The linear growth of corrugations beyond the instability threshold is described by (2.163), suggesting formation of a transverse pattern with the wavelength 2π/kmax . A nonlinear equation describing evolution of long-wave transverse instabilities on a slow time scale can be constructed by expanding the original equations and boundary conditions in powers of the scale ratio (Sivashinsky, 1983). It is advantageous to work in the aligned coordinate frame. By construction of the aligned frame (Sect. 2.2.1), the x-axis is normal to the interface at each location, and the moving interface is always located at x = 0. Due to the lack of translational invariance in the directional solidiﬁcation problem, the local displacement in the laboratory frame ζ cannot be excluded; it is not anymore presumed to be inﬁnitesimally small, but has to be of O() to match orders of magnitude in further expansion, and is rescaled accordingly: ζ → ζ. As a consequence, the x-axis remains in the leading order directed along the temperature gradient as before, but a slight disalignment between the two axes has to be taken into account in higher orders of the expansion.

138

2 Fronts and Interfaces

Based on the results of linear analysis, we restrict to the parametric range (2.197) and scale √ the derivatives with respect to transverse coordinates and time as ∇⊥ = O( ) ∂t = O(2 ). Using the velocity of the ﬂat interface c0 to ﬁx the basic (short) length scale D/c0 and the short time scale D/c20 , we express the normal front propagation velocity in the aligned frame as (2.199) c = 1 + 3 ζt + 12 |∇⊥ ζ|2 . Successive approximations are constructed by expanding v = v (0) + v (1) + · · ·. Retaining the terms up to O(3 ), the diﬀusion equation (2.185) and the boundary conditions (2.186), (2.187) are rewritten in the rescaled dimensionless variables as (2.200) 2 vt = vxx + c − 2 ∇2⊥ ζ vx + ∇2⊥ v, −vx (0) = (c − 2 q)v(0), v(∞) = 2 q,

v(0) = − 1 − w − 2 q + 12 |∇⊥ ζ|2 ζ + 2 β∇2⊥ ζ.

(2.201) (2.202)

The equation and boundary conditions obtained in successive orders of the expansion have a general form (j) + vx(j) = Ψj (x, y, t), vxx

vx(j) (0) + v (j) (0) = Aj (y, t),

v (j) (0) = Vj0 (y, t),

(2.203) v (j) (∞) = Vj∞ . (2.204)

The problem is overdetermined, and a solution exists only when a certain solvability condition is satisﬁed. The general solution satisfying the boundary conditions at x = 0 is x 1 − eξ−x Ψj (ξ)dξ. (2.205) v (j) = Vj0 e−x + Aj 1 − e−x + 0

Assuming Ψj (∞) → 0, the solvability condition is therefore ∞ Vj∞ = Aj + Ψj (ξ)dξ.

(2.206)

0

The zero-order solution, in accordance with (2.188), (2.197), is v (0) = e−x . In the ﬁrst order, (2.200) – (2.202) reduce, in view of (2.199), to (1) + vx(1) = 0, vxx

−vx(1) (0) = v (1) (0),

v (1) (0) = −ζ,

v (1) (∞) = 0. (2.207)

As A1 = Ψ1 = 0, (2.206) is satisﬁed automatically. The solution v (1) = −ζe−x exists for an arbitrary ζ, so that the displacement remains indeﬁnite in this order. The second-order equation and boundary conditions are (2) + vx(2) − vx(0) ∇2⊥ ζ + ∇2⊥ v (1) = 0, vxx

(2.208)

2.6 Instabilities of Interphase Boundaries

−vx(2) (0) = v (2) (0)−qv (0) (0),

v (2) (∞) = q,

139

v (2) (0) = wζ +β∇2⊥ ζ. (2.209)

The inhomogeneous terms in (2.208) cancel, so that Ψ2 = 0, and, as A2 = q, (2.206) is satisﬁed again. This is, of course, a consequence of the particular choice of the values of K and W in (2.197). The solution is v (2) = q + e−x (wζ + β∇2⊥ ζ − q).

(2.210)

The evolution equation of ζ is ﬁnally obtained as the solvability condition of the third-order problem. The relevant inhomogeneities are + vx(1) ∇2⊥ ζ − vx(0) ζt + 12 |∇⊥ ζ|2 − ∇2⊥ v (2) = e−x ζ∇2⊥ ζ + 12 |∇⊥ ζ|2 − w∇2⊥ ζ − β∇4⊥ ζ , (2.211) (1) (0) 2 2 1 1 A3 = qv (0) − v (0) ζt + 2 |∇⊥ ζ| = − qζ + ζt + 2 |∇⊥ ζ| . (1)

Ψ 3 = vt

Using this in (2.206) yields the desired equation for ζ, which can be written in the form ζt + qζ + ∇⊥ · [(w − ζ)∇⊥ ζ] + β∇4⊥ ζ = 0. (2.212) The linear part of (2.212) yields the dispersion relation (2.198), while the nonlinear terms make the eﬀective diﬀusivity dependent on the “order parameter” ζ. The equation is reminiscent of the Kuramoto–Sivashinsky (KS) equation (1.23), but the substantial diﬀerence is the presence of the “order parameter” itself rather than only its derivatives. The dynamics is very much diﬀerent compared to the damped KS equation (with an added linear algebraic term), which was studied as a model of transition to spatio-temporal chaos (Chat´e and Manneville, 1987). Integrating (2.212) across the front yields simply ζ t = −q ζ ; thus, the average displacement ζ decays to zero with time, in contrast both to the translationally invariant form (1.23) and its damped version. The eﬀective diﬀusivity ζ − w increases with growing ζ, suggesting that the interface should be smoother on the segments convex in the direction of higher temperatures and sharper on concave segments left behind, as is common to solidiﬁcation proﬁles observed experimentally (Fig. 2.16b). When the diﬀusivity becomes negative, the interface is stabilized by the fourth-order term. In actual patterns, which are not faithfully modeled by long-scale equations, the instability on the back side may never saturate, leaving long grooves where the contaminant may be trapped. The standard KS equation (1.23) can be obtained, under rather restrictive conditions, for an unstable solidiﬁcation front in a translationally invariant system. A derivation in a more general context of reaction-diﬀusion systems with separated scales will be given in Sect. 3.3.5.

3 Systems with Separated Scales

3.1 Stationary Structures 3.1.1 FitzHugh–Nagumo System A reaction-diﬀusion model with separated scales suitable for generation of a variety of patterns is the two-component system (1.36) with the diﬀusivity ratio 1. The system includes two variables: a short-range “activator” u and a long-range “inhibitor” v. The time scaling becomes more transparent when the system is rewritten in the form (1.17) where the functions f (u, v) and g(u, v) are renormalized in such a way that the derivatives |∂f /∂u| and |∂g/∂v| be of O(1). We deﬁne the capacitance factors γj , which have the dimension of time, as the inverses of these derivative at some chosen values of u, v, and set f(u, v) = γ1 f (u, v), g(u, v) = γ2 g(u, v). Then (1.36) is rewritten as γ1 ut = γ1 D1 ∇2 u + f(u, v), γ2 vt = γ2 D2 ∇2 v + g(u, v).

(3.1)

Using the characteristic √ time and length scales of the long-range component (respectively, γ2 and γ2 D2 ) as basic units, and denoting the ratios of the characteristic time and length scales of the two variables γ = γ1 /γ2 , = γD1 /D2 , we write (omitting now the hats) γut = 2 ∇2 u + f (u, v), vt = ∇2 v + g(u, v).

(3.2) (3.3)

If the function f (u, v) has at a given value of v two stable zeros, say, u = u± (v), the short-scale variable may switch between these two values across a front of the characteristic thickness . The long-scale variable remains almost constant across the narrow front, and therefore the derivation of the normal front velocity in Sects. 2.1.3 and 2.2.2 remains in force. The front,

142

3 Systems with Separated Scales

generally, will propagate with a characteristic speed c∗ = /γ. This speed can be further reduced by adjusting v close to the Maxwell construction. The time scale ratio γ can be chosen in a particular way to make possible the formation of nontrivial static and dynamic structures involving immobilized or propagating fronts. The essential parameter is the ratio of the characteristic time that the front takes to advance at a distance of O(1) and the characteristic time of the inhibitor, τ ∗ = γ/. If τ ∗ 1, the inhibitor is fast and adjusts to the front motion quasistationarily. This case is conducive to the formation of stationary structures. In the opposite case τ ∗ 1, when the inhibitor is very inertial, immobilized fronts are unstable and mobile structures are typical. The transition between both types of behavior occurs at τ ∗ = O(1), i.e., under conditions when the activator is still fast as well as short-range. Another feature inﬂuencing the propagation speed is the strength of coupling between the activator and the inhibitor. The most interesting case for the study of stationary or slowly evolving structures in more than one dimension is that of weak coupling, |∂f /∂v| = O(). The nonlinear function in (3.2) can be written then, analogous to (2.139), as f (u, v) = fa (u) + vfb (u), where fa (u) is antisymmetric and fb (u) is symmetric. Then the system remains close to the Maxwell construction in a wide range of v, and the characteristic front propagation speed, slowing down by a factor of , becomes comparable to a correction due to interfacial curvature with a radius measurable on the long inhibitor scale.1 The essential time scale ratio is modiﬁed in this case to τ = γ/2 . This is the basic scaling to be used in a larger part of this chapter. A diﬀerent scaling, suitable for rapidly propagating fronts, will be introduced in Sect. 3.5. We shall explore in detail a particular form of (1.36) obeying this scaling, with f (u, v) cubic in u and linearly biased by v (i.e., fb (u) = 1), and a linear g(u, v):

τ

2 ut = 2 ∇2 u + u − u3 − v,

(3.4)

−1

(3.5)

vt = ∇ v − v − ν + µu. 2

The choice of the cubic function in (3.4) ensures the existence of two alternative HSS of this equation at v = O(1); other functions with three zeros would be suitable as well.2 Besides , the three parameters that cannot be eliminated by rescaling are the coupling parameter µ, the bias parameter ν, and the time scale ratio τ . This system has been ﬁrst suggested, in a somewhat diﬀerent form, as a model of nerve pulse propagation and is known as the 1

2

This justiﬁcation of a small bias is irrelevant in 1D, but an additional advantage of keeping the system nearly symmetric is to make the algebra easier without aﬀecting the results in a qualitative way. Another popular choice is a discontinuous piecewise linear function. It is still easier algebraically, but the presence of discontinuity may bring in nonphysical eﬀects, such as unlimited front propagation speed.

3.1 Stationary Structures

143

FitzHugh–Nagumo (FN) equation (FitzHugh, 1961). The scaling of the bias and diﬀerential terms in (3.4) is chosen to suit the eikonal equation (2.50), which is rewritten in the adopted scaling as c = bv − κ,

(3.6) √ where b is a numerical parameter, e.g., b = 3/ 2 for the cubic function in (3.4). The relevant range of parameters can be suggested by qualitative analysis based on the disposition of null-isoclines in the plane (u, v) along the lines of Sect. 1.4.2. At µ > 0, |ν| < µ, the system is weakly excitable, since both global stationary states u = ±1 + O(), v = ±µ − ν would retreat when placed in contact with an alternative HSS of the short-scale equation at the same value of v. With this choice of signs, v is depleted in the down state + u− s < 0 and produced in the upper state us > 0, while, in contrast, the lower state advances at v > 0 and the upper at v < 0. Such conditions are conducive to the formation of stable solitary structures.3 . Consider, for example, a ﬁnite region with u < 0 immersed within a continuum of positive u. If it starts to shrink, the level of v would raise and, as a result, the front may be immobilized again. If the dimension is higher than one, the question is whether this stabilizing action wins over curvature-driven shrinking, which is accelerated inversely proportional to its radius. At µ < 0, neither global stationary is excitable. Stationary solitary structures, which can be formally constructed also in this case, correspond to unstable critical nuclei of one of alternative states. Only one state is excitable at µ > 0, |ν| > µ. The parameter τ plays, as we shall further see, a special role, inﬂuencing the transition from stationary to mobile structures. Weak coupling in (3.2) is convenient for the analysis of stationary or slowly involving structures, but is unsuitable when propagation is fast. A convenient form of a strongly coupled system contains a quadratic symmetric part fb (u) = −(1 − u2 ), where v is constrained to |v| ≤ 1. This leads to f (u, v) = −(u + 1)(u − 1)(u − v). u± s

(3.7)

With this choice, = ±1, and only the unstable intermediate state = v is v-dependent.4 The disadvantage is, however, a singular zigzag form of the nullisocline replacing a smooth S-shaped curve in Fig. 1.5 (see also Sect. 3.5.2). u0s

3.1.2 Stationary Structures in 1D Various stationary structures can be obtained in the limit → 0 following a common strategy (Ohta et al., 1989). We solve the long-scale equation (3.5) replacing u by its quasistationary values u± s = ±1+O() in the regions separated 3

4

Localized solutions of nonlinear equations are now commonly called dissipative solitons. Unlike classical solitons in integrable conservative systems, such structures do not retain their identity upon collisions. Shifting and rescaling v leads to the form known as Barkley’s model.

144

3 Systems with Separated Scales

by a front, which is assumed to be of negligible thickness on the characteristic length scale of the long-range variable. The position of the front is further deﬁned by requiring the values of v to annul the propagation speed given by (3.6). A 1D solution corresponding to a single stationary front on the inﬁnite line veriﬁes the equation v (x) − v − ν − µ = 0 v (x) − v − ν + µ = 0

at at

x < 0, x > 0.

(3.8)

The matching conditions at the short-scale discontinuity x = 0 are given by requiring v to be continuous and smooth. The solution is µ(ex − 1) − ν at x ≤ 0, (3.9) v= µ(1 − e−x ) − ν at x ≥ 0. The stationarity condition is just v(0) = 0, which implies ν = 0. Thus, a single stationary front is possible only when the system is symmetric. A more interesting structure is a solitary band of width 2L bounded by two fronts of the opposite polarity. For the down state inside the band, the proﬁle of v should verify v (x) − v − ν − µ = 0 v (x) − v − ν + µ = 0

at at

|x| < L, |x| > L.

(3.10)

There are now two symmetric sets of matching conditions at x = ±L, and the solution is µ(2e−L cosh x − 1) − ν at |x| ≤ L (3.11) v= µ(1 − 2e−|x| sinh L) − ν at |x| ≥ L. −2L − ν = 0, which gives L = The fronts are stationary when v(±L) = µe ln µ/ν. The solutions exist only at µ > ν > 0. A regular striped pattern in the inﬁnite plane with the period 2L = 2(L+ + L− ) is formed by alternating fronts of opposite polarity: kinks at x = 2nL and antikinks at x = 2(nL + L+ ). The stationary proﬁle of the long-range variable is sinh L+ −1 −ν v = µ 2 cosh (x − 2nL + L− ) sinh L at 2nL − L− ≤ x ≤ 2nL, sinh L− v = µ 1 − 2 cosh (x − 2nL − L+ ) −ν sinh L (3.12) at 2nL ≤ x ≤ 2nL + L+ .

The equilibrium condition, which ﬁxes the relation between the lengths L± , is v(0) = v(2L+ ) = 0. A short computation yields ν = µ sinh(L+ − L− )/ sinh L.

(3.13)

3.1 Stationary Structures

145

Limiting the bias by the inequality µ > |ν| is necessary for the existence of solutions. Provided this holds, there is a continuum of solutions with arbitrary L and L+ − L− verifying (3.13). 3.1.3 Solitary Disk and Sphere In 2D, the front should acquire a circular shape due to the surface tension, and the stationary solution depends only on the long-range radial coordinate r. The stationary outer equation is vrr + r−1 vr − v − ν ± µ = 0.

(3.14)

The stationary solution for a disk with u < 0 immersed in a continuum with u > 0, satisfying the continuity and smoothness conditions at the front located at r = L, is µ[2LK1 (L)I0 (r) − 1] − ν at r ≤ L v= (3.15) µ[1 − 2LI1 (L)K0 (r)] − ν at r ≥ L, where In , Kn are modiﬁed Bessel functions. Using the identity K1 (L)I0 (L) + K0 (L)I1 (L) = L−1 , the common value of v(L) can be expressed as v(L) = µL [K1 (L) I0 (L) − K0 (L) I1 (L)] − ν.

(3.16)

Using v(L) and κ = L−1 in (3.6) yields cτ + L−1 + bν = bµL [K1 (L) I0 (L) − K0 (L) I1 (L)] .

(3.17)

A convenient form of the stationarity condition is µ−1 (b−1 + Lν) = L2 [K1 (L) I0 (L) − K0 (L) I1 (L)] .

(3.18)

The r.h.s. of (3.18), denoted as F (L), has a peak value of about 0.556 at L ≈ 1.6 and relaxes asymptotically to 1/2 at L → ∞ (Fig. 3.1). The equation can be solved graphically by ﬁnding an intersection of the universal function F (L) with straight lines dependent on the parameters of the problem. We expect the solution to be stable to perturbations preserving circular symmetry when the slope of the straight line exceeds at the intersection point the slope of F (L), and, consequently, the eﬀective repulsive action due to the long-range ﬁeld decreases with increasing radius faster than the shrinking action due to the surface tension. Whenever (3.18) has a single solution, this solution should be unstable according to the above criterion. In other parametric regions, there are no solutions at all. When the straight line is tangent to F (L) (Fig. 3.1), an additional pair of solutions bifurcates, one of which is stable to perturbations preserving the circular symmetry.5 The existence region in the parametric 5

This solution may be still subject to symmetry-breaking or oscillatory instabilities.

146

3 Systems with Separated Scales FL

0.54 0.52 1

2

3

4

5

6

L

0.48 0.46 0.44

Fig. 3.1. Plot of the r.h.s. of (3.18). Solutions are obtained as intersection points with straight lines presenting the l.h.s. of this equation

plane µ, ν is bounded by the axis ν = 0, µ > 2/b and the bifurcation line that can be obtained by diﬀerentiating (3.17) with respect to L: L−2 + 2bµL [K1 (L) I1 (L) − K0 (L) I0 (L)] = 0.

(3.19)

The existence boundary can be drawn (Fig. 3.2) as a parametric plot of the functions µ(L), ν(L) verifying (3.18), (3.19). Take note that the existence region penetrates into the half-plane ν < 0 as a narrow tongue terminating in a cusp. Combining the cusped plot in Fig. 3.2 with a mirror plot drawn for a disk with u > 0 immersed in a continuum with u < 0, we see that both may coexist within a narrow region at very small values of ν and µ slightly below 2/b. This feature is speciﬁc to 2D and is lost in both one and three dimensions. In 3D, solving the outer equation in spherical coordinates yields at r ≤ L µ[2(1 + L)e−L r−1 sinh r − 1] − ν (3.20) v= µ[1 − 2r−1 e−r (L cosh L − sinh L)] − ν at r ≥ L. The stationarity condition analogous to (3.18) is 1 − (1 + L)e−2L = µ−1 (2/b + Lν).

(3.21)

The left-hand side of this equation increases monotonically with increasing L. Again, a single stationary state is expected to be unstable, and a stable state bifurcates, paired with an unstable one, at a bifurcation point given by the tangency condition ν/µ = (1 + 2L)e−2L . The existence boundary in the parametric plane µ, ν is drawn in Fig. 3.2. Unlike the 2D case, stationary solutions are possible only at ν > 0. A regular assembly of spots in 2D or spheres or cylinders in 3D may form a pattern resembling a crystalline structure where symmetric solitary objects

3.1 Stationary Structures

147

(a) Ν 1.5

band

disk sphere

1.25 1 0.75 0.5 0.25 1

1.5

2

2.5

3

Μ

(b) Ν 0.2 disk 0.15 0.1 0.05

0.8

0.9

sphere

1

1.1

1.2

1.3

Μ

Fig. 3.2. (a) Existence domains of the solitary band, disk, and sphere in the parametric plane µ, ν. (b) Blow-up near the cusp point

serve the role of “atoms.” This analogy is not precise, since the environment of such an “atom” in the crystal is not isotropic, and its symmetric shape must be distorted to some degree.6 Nevertheless, both experiment and computations often show patterns (typically, hexagonal in 2D) closely resembling a crystalline arrangement of almost symmetric spots, as seen, for example, in Fig. 1.24. Moreover, one often observes a variety of “crystalline splinters” comprising ﬁnite clusters of spots. Like ﬁnite 1D clusters discussed in the previous subsection, these structures are a telltale sign of coexistence of a pattern with a stable stationary state (see Sect. 4.6.4). 6

This is true, incidentally, also for real atoms in real crystals.

148

3 Systems with Separated Scales

3.1.4 Migration-Enhanced Structures A system similar to the FN model can be constructed as a nonequilibrium extension of the phase ﬁeld model described in Sect. 2.5.1 (Hildebrand et al., 1999). The equations for the activator and the inhibitor are constructed in a way similar to (2.138) with addition of nongradient terms, which may describe, for example, a nonequilibrium chemical reaction. This way of constructing a system capable of generating nonequilibrium patterns has an advantage of being directly related to thermodynamics. As a consequence of equilibrium relations between the two species, the diﬀusional ﬂuxes are supplemented by migration, which enhances the gradient of the long-range inhibitor across the front of the short-range activator. The variational equations are constructed treating the activator as a nonconserved and the inhibitor as a conserved ﬁeld: ∂v δF ∂u −1 δF = −γ , = ∇ · Γ (v)∇ + g(u, v), (3.22) ∂t δu ∂t δv where γ −1 (presumed to be constant) and Γ (v) are kinetic coeﬃcients. The energy functional F is written in the form

(3.23) F= V (u, v) + W (v) + 12 2 |∇u|2 dx. The potential V (u, v) is such that f (u, v) = −∂V /∂u in (3.2). The v-dependent part of the potential, as well as the dependence of mobility on v, should be chosen to obtain the correct form of the diﬀusional term in (3.3): W (v) =

(v − v − ) ln(v − v − ) + (v + − v) ln(v + − v) , β(v + − v − ) Γ (v) = β(v − v − )(v + − v),

(3.24) (3.25)

where v ± bound the allowed range of v. With this choice of functions, the equation of u retains the form (3.2) and can be written, to emphasize its gradient structure, as ut = 2 ∇2 u −

∂V . ∂u

The equation of v takes the form ∂V 2 − + vt = ∇ v + β∇ · (v − v )(v − v)∇ + g(u, v). ∂v

(3.26)

(3.27)

The additional migration term, which causes transport of the inhibitor under the inﬂuence of the activator gradient, is dominant near the activator front and causes a jump of v across the interface. In the front region, the short-scale coordinate normal to the front should be used in (3.27). This renders the last

3.1 Stationary Structures

149

term negligible (∝ 2 ). For a stationary straight-line front, (3.27) reduces then to 2 ∂ V du ∂ 2 V dv d + v (x) = −β (v − v − )(v + − v) . (3.28) dx ∂u∂v dx ∂v 2 dx As an example, we take the particular form of f (u, v) in (3.7), which is obtained from the potential V (u, v) = 14 (1 − u2 )2 + uv(1 − 13 u2 ). Setting also v ± = ±1 brings (3.7) to the form d 2 2 du v (x) = −β (1 − u ) . dx dx

(3.29)

(3.30)

This equation can be integrated twice to obtain the relation between u and v:

(3.31) v = v0 − jx − βu 1 − 23 u2 + 15 u4 , where v0 , j are integration constants; the latter determines the ﬂux through the interface, which may be caused by the nonequilibrium part of (3.27) negligible in the vicinity of the front. The ﬂux should be measurable on the outer (long) scale and therefore must be small, so that the second term in (3.31) should be negligible within the interfacial region where u switches between its two equilibrium values. Taking into account that, on the short scale, u → ±1 at x → ∞, this gives the diﬀerence between the equilibrium values of v on the two sides of the interface 2∆ = 16 15 β. A stationary straight-line front is a solution of (3.26) when the values of the potential (3.29) are equal on both sides of the front. By symmetry, this requires v to be an odd function of u, so that v0 = 0. In the outer region, the migration term in (3.27) is negligible, and the equation of v reverts to the common form (3.3). Choosing again a linear function g(u, v) as in (3.5), we obtain the same equations for the stationary inhibitor ﬁeld as in Sect. 3.1.2; the only diﬀerence is that v is not required now to be continuous across the interface, but has to jump by 2∆. In particular, the solution for a solitary band with the half-width L analogous to (3.11) is obtained in the form 2(µ + ∆)e−L cosh x − µ − ν at |x| ≤ L (3.32) v= 2(µ + ∆)e−|x| sinh L) + µ − ν at |x| ≥ L. The ﬂux j = −v (±L) = ±2(µ + ∆)(1 − e−2L ) is continuous across the interface. The stationarity condition, following from the condition v(L + 0) + v(L − 0) = 0, is L = ln (µ + ∆)/ν. Other solutions in Sects. 3.1.2, 3.1.3 can be modiﬁed in a similar manner. Although the structure of solutions remains formally similar, comparing the expressions for the stationary band width with and without migration shows that the size of stationary structures can be signiﬁcantly aﬀected by the migration term.

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3 Systems with Separated Scales

3.1.5 Quasistationary Dynamics The stationary form of the long-scale equation (3.5) may also be applicable to dynamical problems when the scale ratio τ is very large. Under these conditions, the dynamics of the long-range variable is so fast that it follows quasistationarily the front displacements. In the leading order, the time derivative in (3.5) falls out, and it reduces to a stationary form ∇2 v − v − ν + µu = 0.

(3.33)

This equation can be resolved with the help of an appropriate Green’s function G(r) as v(x) = −ν + µ

G(x − ξ)u(ξ) d2 ξ.

(3.34)

In 2D, the Green’s function G of (3.33) is a modiﬁed Bessel function dependent on the distance between the two points r = |r| = |x − ξ|: G(r) = (2π)−1 K0 (r).

(3.35)

Using (3.34) in (3.4) brings it to the variational form (2.2) with the energy functional 1

2 −1 F= νu + 14 −2 (1 − u2 )2 dx 2 |∇u| + 1 −1 + 2 µ u(x)G(x − ξ)u(ξ) dxdξ. (3.36) If µ > 0, the nonlocal term is negative when u(x) and u(ξ) have opposite signs. This may compensate the distortion energy and make a nonuniform (patterned) state energetically favorable. Since the front displacement is fully determined by local values of v, it is possible to derive a closed nonlocal equation of the front motion by transforming (3.34) to a boundary integral (Petrich and Goldstein, 1994). Since u = ±1 + O(), (3.33) becomes ∇2 v − v − ν ± µ = 0. The general solution of this equation can be presented as v(x) = µ − ν − 2µ G(x − ξ) dξ,

(3.37)

(3.38)

S

where the integration is carried out over the area S occupied by the down state. This integral can be transformed into a contour integral along a boundary Γ with the help of the Gauss theorem. To avoid divergent expressions, the contour should exclude the point x = ξ. Clearly, excluding an inﬁnitesimal circle around this point does not aﬀect the integral (3.38), since its kernel (3.35) is only logarithmically divergent (K0 (r) ln r at r → 0). Replacing G(r) = ∇2 G(r) (r = 0), we transform the integral in (3.38) as

3.1 Stationary Structures

S

151

G(x − ξ) d2 ξ = − ∇ξ · H(x − ξ)d2 ξ S ! n(s) · H(|x − ξ(s)|)ds, =−

(3.39)

Γ

where H(r) = ∇G(r), and n is the normal to the contour Γ drawn to exclude the singularity. The vector Green’s function H corresponding to the kernel (3.35) is computed as r K1 (r). (3.40) H(r) = − 2πr If x is a boundary point, the contour Γ consists of the boundary Γ cut at this point and closed by an inﬁnitesimally small semicircle about x. The integral over the semicircle equals 1/2 and cancels with the ﬁrst term in (3.38). Deﬁning the external normal to Γ as the tangent vector t = x (s) rotated clockwise by π/2, the required value of the long-range variable on the spot boundary (parametrized by the arc length s or σ) is expressed, using the 2D cross product ×, as ! H(x(s) − x(σ)) × x (σ)dσ. (3.41) v(s) = −ν − 2µ Γ

The integral can be evaluated for the boundary points and used in (3.6) to obtain the local front velocity. The resulting expression depends nonlocally on the instantaneous front position. As an exercise, the stationary disk solution obtained in the preceding subsection can be reproduced using in (3.6) the radius vector r = L{cos φ − 1, sin φ} between the points on the circle of radius L separated by the angle φ and its length r = 2L sin(φ/2) and evaluating the integral π φ φ φ µL d µL π sin K1 L sin K0 L sin dφ = − dφ v+ν = π 0 2 2 π dL 0 2 d [I0 (L)K0 (L)] = µL[K1 (L) I0 (L) − K0 (L) I1 (L)]. (3.42) = −µL dL The result is identical to (3.16). The boundary integral method can be used to evolve the interface in time by discretizing the boundary and translating the points with the prescribed velocity along the normal n. Practical computations using this method are not easy, as they require adding and removing points in the course of evolution and watching for possible self-intersection of the evolving boundary, but the substantial advantage is that one only needs to keep in memory a 1D array of values of v at the front, rather than the entire 2D ﬁeld.

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3 Systems with Separated Scales

3.2 Symmetry Breaking Instabilities 3.2.1 Stability of a Straight-Line Front Stability of a stationary front to inﬁnitesimal displacements is checked in a standard way by linearizing the eikonal equation (3.6). We consider a straightline front parametrized by the coordinate y and denote the instantaneous displacement of the front from its unperturbed position toward the region occupied by the up state as ζ(y, t). As long as the amplitude of the perturbation is much smaller than its wavelength, the normal vector deﬁning the direction of the front propagation is almost parallel to the x-axis, and the propagation speed is expressed as ζt = c. In the same approximation, the curvature is given by κ = −ζyy . Expanding also the long-range variable in the vicinity of the unperturbed position, we reduce (3.6) to the form ζt = ζyy + b [ v (0, y, t) + vs (0)ζ] .

(3.43)

Here v is the perturbation of the long-range variable, and vs (0) = −j is determined by the stationary ﬂux at the front following from the stationary solution (3.9). The correction to v due to a weak perturbation of the front can be computed in a simple way by observing that any shift of the front position by an inﬁnitesimal increment ζ is equivalent to switching the sign of the source term in (3.5) in a narrow region near x = 0. This introduces a source localized at the unperturbed front position and proportional to its displacement in the equation for the perturbation v, which is written as τ −1 ∂t v = ∇2 v − v − 2µζ(y, t)δ(x).

(3.44)

Although the perturbation is habitually presumed to be arbitrarily small for the purpose of linear stability analysis, the method is actually also applicable to ﬁnite perturbations of the front position, provided they are small on the extended scale, i.e., ζ ≤ O(). As usual, we are looking for a solution in a spectral form ζ(y, t) = χ(k, λ)eλt+iky ,

v(x, y, t) = ψ(x; k, λ)eλt+iky .

(3.45)

The dispersion relation λ(k 2 ) determines the stability to inﬁnitesimal perturbations. Using (3.45) in (3.44) yields ψxx − q 2 ψ = 2µχ(k, λ)δ(x),

(3.46)

where q 2 = 1 + λ/τ + k 2 . The solution of this equation (presuming Re q > 0) is (3.47) ψ(x) = −µχ(k, λ)q −1 e−q|x| . Using (3.9), (3.45), (3.47) in (3.43) yields the dispersion relation

3.2 Symmetry Breaking Instabilities

λ = −k 2 + bµ(1 − q −1 ).

153

(3.48)

Since q is dependent on λ, this dispersion relation is still unresolved explicitly, but the result is explicit at τ → ∞. Then eigenvalues λ(k 2 ) are always real, as it must be in a system obeying a variational principle – see Sect. 3.1.5, and loss of stability occurs when λ passes zero. At λ = 0, however, the dependence on τ disappears. Therefore loss of stability at zero eigenvalue is independent of the time scale ratio and can be investigated assuming τ −1 = 0. We restrict to this quasistationary problem in this section, postponing the discussion of dynamic instabilities sensitive to the scale ratio to the next section. As expected, λ vanishes at k = 0, which reﬂects the translational symmetry of the front. The loss of stability to long-scale perturbations occurs when the derivative dλ/d(k 2 ) at k = 0 becomes positive. This happens at bµ > 2. Since the function λ(k 2 ) is convex, the loss of stability always occurs with increasing µ in the long-scale mode. We see that at µ > 2/b a straight-line front cannot exist, but is destroyed by transverse instability. 3.2.2 Stability of a Solitary Band In a similar way, we can write equations of motion for inﬁnitesimal displacements ζ (1) , ζ (2) of the kink at x = L and the antikink at x = −L that conﬁne a solitary band. We take note that the sign of the bias term should be reversed in the eikonal equation (3.6) when it is applied to the antikink. Using the expression for the stationary ﬂuxes following from (3.11), we obtain (j) (j) (3.49) + b ± v (±L, y, t) + µ(1 − e−2L )ζ (j) . ζt = ζyy It is advantageous to consider the symmetric and antisymmetric combinations of the displacements, ζ ± = 12 (ζ (1) ∓ ζ (2) ), that may lead, respectively, to varicose or zigzag instabilities. It is clear from (3.49) that ζ ± are coupled, respectively, to the symmetric and antisymmetric parts v± of the perturbation ﬁeld v:

± (3.50) + b v± (L, y, t) + µ(1 − e−2L )ζ ± . ζt± = ζyy The equations for both perturbation ﬁelds, v± , are identical: τ −1 ∂t v± = ∇2 v± − v± − 2µζ ± (y, t)δ(x − L). Setting as before v± = ψ ± eλt+iky , ζ ± = χ± eλt+iky yields −2µχ+ q −1 e−qL cosh qx at |x| ≤ L, ψ + (x) = −2µχ+ q −1 e−q|x| cosh qL at |x| ≥ L, −2µχ− q −1 e−qL sinh qx at |x| ≤ L, ψ − (x) = −2 sign(x) µχ− q −1 e−q|x| sinh qL at |x| ≥ L.

(3.51)

(3.52) (3.53)

The varicose and zigzag branches of the dispersion relation are given by

154

3 Systems with Separated Scales Ν 3 C

14

2.5

12 34

2

Z

1.5 1 0.5 1

2

3

4

5

Μ

Fig. 3.3. Existence boundary (C) and loci of zigzag (Z) and traveling instability for a solitary band. The loci of traveling instability are marked by respective values of τ

λ± = −k 2 + bµ 1 − e−2L − q −1 1 ± e−2qL .

(3.54)

Vanishing λ− at k = 0 reﬂects the translational symmetry of the band. Due to convexity of the function λ− (k 2 ), the loss of stability ﬁrst occurs at k = 0. The solution is unstable when the derivative dλ− /dk at k = 0 is positive. The relation deﬁning the critical point of long-scale zigzag instability is 2 = 1 − (1 + 2L)e−2L , bµ

(3.55)

or, using the stationarity condition e−2L = ν/µ, 2 µ ν = 1 − (1 + ln ) . bµ ν µ

(3.56)

At τ = 0 (when the eigenvalues are real and the dispersion relations are explicit), the varicose branch, λ+ (k 2 ), lies persistently lower than the zigzag one. Thus, stability is always lost in the zigzag mode. The most dangerous perturbation corresponds to the long-wave limit k → 0, i.e., q → 1. The locus of zigzag instability cuts oﬀ the existence domain 0 < ν < µ in the parametric plane (µ, ν) the area below the respective curve taking oﬀ from the point µ = 2/b, ν = 0 that corresponds to the limit L → ∞ (Fig. 3.3). Beyond the marginal instability locus, the maximum growth rate is observed at a ﬁnite wavelength. The zigzag instability is not saturated nonlinearly, but either destroys the band turning it into an array of spots of almost circular form or triggers spreading out into a labyrinthine pattern. 3.2.3 Stability of a Striped Pattern Perturbations of the long-range ﬁeld in alternating regions of a striped pattern with the period 2L due to displacements of respective kinks at x = 2nL by

3.2 Symmetry Breaking Instabilities

155

ζ (1,n) and antikinks at x = 2(nL + L+ ) by ζ (2,n) obey the equation τ −1 ∂t v = ∇2 v − v − 2µ (−1)n ζ (1,n) δ(x − 2nL) n

−ζ

(2,n)

δ(x − 2nL − 2L+ ) .

(3.57)

The equation of motion of the fronts is obtained, analogous to (3.43), by using in (3.6) v = vs (2nL)ζ (n) + v(2nL, y, t), where vs (2nL) ≡ J = 2µ sinh L+ sinh L− / sinh L is the stationary ﬂux following from (3.12); at the antikink locations, x = 2(nL + L+ ), vs (2nL) = −J, and the respective values enter the equation of motion with the reverse sign: (1,n) (1,n) ζt = ζyy + b Jζ (1,n) + v(2nL, y, t) , (2,n) (2,n) ζt = ζyy + b Jζ (2,n) − v(2nL + 2L+ , y, t) . (3.58) Presenting the perturbation v in each successive stripe with a prevailing positive or negative state of the short-scale variable as v(j,n) = ψ (j,n) eλt+iky and front displacements as ζ (j,n) = χ(j,n) eλt+iky , j = 1, 2, we write the equation of the spectral components as (j,n) ψxx − q 2 ψ (j,n) = 0.

(3.59)

The matching conditions, taking into account localized perturbations due to displacements of respective kinks and antikinks, are (1,n)

(2,n)

ψx − ψx = 2µχ(1,n) at x = 2nL, (1,n) (2,n+1) (2,n) ψx − ψx = 2µχ at x = 2(nL + L+ ) . (3.60) The solution of (3.59) is expressed as ψ (1,n) = ψ (2,n) , ψ (1,n) = ψ (2,n+1) ,

ψ (j,n) = A(j,n) eqx + B (j,n) e−qx .

(3.61)

We shall deﬁne the coeﬃcients in the above expression, as well as the displacements at particular locations, through respective generating functions: αj (z) =

∞

A(j,n) e(iz+2qL)n ,

βj (z) =

n=−∞

γj (z) =

∞

B (j,n) e(iz−2qL)n ,

n=−∞ ∞

χ(j,n) eizn ,

j = 1, 2.

(3.62)

n=−∞

Multiplying the matching conditions (3.60) by eizn and summing over n yields a set of equations for αj (z), βj (z):

156

3 Systems with Separated Scales

α1 (z) − α2 (z) + β1 (z) − β2 (z) = 0, α1 (z) − α2 (z) − β1 (z) + β2 (z) = 2µq −1 γ1 (z), e2qL+ α1 (z) − e−(iz+2qL− ) α2 (z) + e−2qL+ β1 (z) − e−(iz−2qL− ) β2 (z) = 0, e2qL+ α1 (z) − e−(iz+2qL− ) α2 (z) − e−2qL+ β1 (z) + e−(iz−2qL− ) β2 (z) = 2µq −1 γ2 (z).

(3.63)

Two more equations are obtained in the same way from (3.58): (λ + k 2 − bJ)γ1 (z) = α1 (z) + β1 (z), (λ + k 2 − bJ)γ2 (z) = −e2qL+ α1 (z) − e−2qL+ β1 (z).

(3.64)

The general dispersion relation, obtained as the condition that (3.63), (3.64) have a nontrivial solution, is quite cumbersome. As in the preceding subsection, it is convenient to work with zigzag and varicose modes, which correspond, respectively, to like and opposite signs of displacements of adjacent kinks and antikinks. These modes, obtained by introducing the symmetric and antisymmetric combinations γ ± = 12 (γ1 ± γ2 ), decouple, however, only at z = 0 or z = π. The symmetric (varicose) and antisymmetric (zigzag) branches of the dispersion relation at z = 0 are computed as sinh L+ sinh L− cosh qL+ cosh qL− + 2 − λ = −k + 2bµ , sinh L q sinh qL sinh L+ sinh L− sinh qL+ sinh qL− − . (3.65) λ− = −k 2 + 2bµ sinh L q sinh qL At τ = 0 (when the eigenvalues are real and the dispersion relations are explicit) the varicose branch λ+ (k 2 ) lies persistently lower than λ− (k 2 ) and therefore can be disregarded. Vanishing λ− at k = 0 reﬂects the translational symmetry of the entire pattern. The loss of stability in the long-scale zigzag mode occurs when the derivative ∂λ− /∂(k 2 ) at k = 0 vanishes.7 A simpler form of the dispersion relation is obtained at ν = 0 when the striped pattern is symmetric, with L+ = L− = L/2. The two branches of the dispersion relation are q −1 sinh qL L 2 λ = −k + bµ tanh − . (3.66) 2 cosh qL ± cos(z/2) One can see that λ is at maximum at z = 0 if one chooses the positive sign or at z = π if one chooses the negative sign. Either choice corresponds to zigzag deformation and brings λ to zero at k = 0, as it should do in view of the translational symmetry. Again, λ decreases monotonically with k, and therefore the loss of stability should ﬁrst occur at k = 0. Thus, when 7

A more convenient way to study long-scale instabilities of a pattern is provided by the phase dynamic method (Sect. 4.2). This approach has been applied to the FN equation by Hagberg et al. (2000).

3.2 Symmetry Breaking Instabilities

157

1bΜ 0.5 0.4 0.3 0.2 0.1

2

4

6

8

L

Fig. 3.4. Locus of marginal zigzag instability for a symmetric pattern (solid line) and a solitary stripe (dashed line). The instability region is to the right of the curve

µ increases, the onset of instability always occurs in a concerted long-scale zigzag mode. The marginal stability condition is computed, as in the preceding subsection, by diﬀerentiating the dispersion relation with respect to k and computing the derivative at k = 0. This yields L L L 2 = tanh − sech2 . bµ 2 2 2

(3.67)

At µ < 2/b, patterns with any wavelength are stable, while at larger µ instability sets on as the wavelength increases to the right of the curve drawn in Fig. 3.4. In the other limit, L+ L− , (3.65) reduces to the dispersion relation for a solitary band, (3.54); the corresponding stability limit (3.55) is shown by the dashed line in the same ﬁgure. Since bending eﬀectively reduces the wavelength, the zigzag instability can be seen as a way a pattern with excessive wavelength tries to relax back into the stable range. Since, however, the restructuring should be global, evolution to a regular striped pattern with a reduced wavelength may be very diﬃcult and indirect, and is apt to be disrupted due to the inﬂuence of faraway boundaries or preferred formation of other regular patterns. An unstable striped pattern may evolve then either into an array of spots of almost circular form arranged on a more or less regular grid, or into a disordered labyrinthine pattern. An alternative instability limiting the minimal wavelength, called the Eckhaus instability (see Sect. 4.1.5), is not revealed in this approximation, since it occurs on length scales comparable with the diﬀusional range of the short-scale activator. 3.2.4 Stability of a Solitary Disk and Cylinder In order to check the stability of a stationary solitary disk with the radius L in (Sect. 3.1.3), we use the cylindrical coordinates (r, φ) and set r = L[1+ζ(φ, t)],

158

3 Systems with Separated Scales

where ζ(φ, t) 1. The curvature of the front is expressed as κ = L−1 (1 − ζ − ζφφ ). Expanding, as before, the long-range variable in the vicinity of the unperturbed position brings (3.6) to the form

ζt = L−2 (ζφφ + ζ) + b L−1 v(0, φ, t) + vs (0)ζ , (3.68) where vs (0) = 2µLI1 (L)K1 (L) is related to the stationary ﬂux at the front following from (3.15). The equation for the perturbation ﬁeld v is τ −1 ∂t v = ∇2 v − v − 2µLζ(φ, t)δ(r − L).

(3.69)

We are looking for a solution in the form ζ(φ, t) = χn eλn t+inφ , v(r, φ, t) = ψ(r; n)eλn t+inφ , where n is an integer. The equation of ψ is 1 n2 ψrr + ψr − q 2 + 2 ψ = 2µLχn δ(r − L), r r

(3.70)

(3.71)

where q 2 = 1 + λ/τ . The solution of this equation is −2µL2 χn q −1 Kn (qL)In (qr) at r ≤ L, ψ(r) = −2µL2 χn q −1 In (qL)Kn (qr) at r ≥ L.

(3.72)

Using (3.70) and (3.72) in (3.68) yields the dispersion relation λn = −L−2 (n2 − 1) + 2bµL[I1 (L)K1 (L) − q −1 In (qL)Kn (qL)].

(3.73)

As expected, λn vanishes at n = 1, which corresponds to a shift of the disk without deformation. One can check numerically that, of all angular modes, 0.4 C 0.3 B 0.2 S

0.1

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

Fig. 3.5. Existence boundary (C) and loci of splitting (S) instability of a solitary disk and bending (B) instability of a cylinder

3.2 Symmetry Breaking Instabilities

159

Fig. 3.6. Splitting of a solitary spot (experimental sequence by Li et al., 1996; reproduced with permission. Copyright by the American Institute of Physics)

the dipole mode n = 2, deforming the disk into an ellipse, is the most dangerous one. Beyond the instability threshold, there is no nonlinear saturation, and the disk splits into two spots (Fig. 3.6). This may set oﬀ a continuing “multiplication” process (Reynolds et al., 1994), eventually generating a pattern ﬁlling the entire plane. The locus of splitting instability cuts oﬀ the existence domain in the parametric plane (µ, ν) the area below the curve S in Fig. 3.5. This curve starts at the point µ = 2/b, ν = 0, which corresponds to the limit L → ∞. Extending the solitary disk solution to 3D, we can consider the stability of a cylindrical domain r < L inﬁnitely extended along the z-axis. Using the expression for the Gauss mean curvature of a slightly distorted cylindrical surface, κ = L−1 (1 − ζ − ζφφ ) + ζzz , setting v = ψn (r)eλt+inφ+ikz , ζ = χn eλt+inφ+ikz , and solving the resulting equations as before, we arrive at the dispersion relation similar to (3.73), but with q 2 = 1 + λ/τ + k 2 and an added term due to the axial perturbations:

λ = L−2 (1 − n2 ) − k 2 + 2bµ K1 (L) I1 (L) − q −1 Kn (qL) In (qL) . (3.74) A cylindrical domain in 3D is subject, in addition to planar modes with k = 0, q = 1, to distortions along the z-axis. The most dangerous mode, which can lead to bending instability, is the translational mode n = 1, which is neutral at k = 0. The marginal stability condition for this mode is vanishing of the derivative dλ/d(k 2 ) at k = 0, n = 1. Diﬀerentiating (3.74) we express this condition as (bµ)−1 = L{L[I1 (L)K0 (L) − I0 (L)K1 (L)] + 3I1 (L)K1 (L)}.

(3.75)

Bending instability always prevails over planar distortion modes. The marginal stability curve in the parametric plane µ, ν is shown in Fig. 3.5.

160

3 Systems with Separated Scales

3.3 Dynamic Instabilities 3.3.1 Propagating Fronts Besides a stationary single front, one can construct a family of fronts propagating with a speed c. The long-range variable should verify then the equation in the comoving frame Cv (x) + v (x) − v − ν − µ = 0 Cv (x) + v (x) − v − ν + µ = 0

at at

x < 0, x > 0,

(3.76)

where C = c/τ . The general solution that satisﬁes the matching conditions at the short-scale discontinuity x = 0 and does not diverge at x → ±∞ is   µ 1 + s−1 e 12 (s−1)Cx − 1 − ν at x ≤ 0 (3.77) v=  µ s−1 − 1 e− 12 (1+s)Cx + 1 − ν at x ≥ 0, where s2 = 1 + (2/C)2 . The value of v on the front, v(0) = −ν + µ/s, must equal Cτ /b. This gives an equation for C, C 1 Cτ √ = ν+ . (3.78) µ b 4 + C2 The l.h.s. of this equation is an S-shaped curve (Fig. 3.7) that may have more than one intersection with the straight line representing the r.h.s. For example, in the symmetric case ν = 0 (Fig. 3.7a), besides the stationary solution C = 0, there are two symmetric nontrivial solutions C = ± (bµ/τ )2 − 4 corresponding to the fronts propagating in opposite directions. These solutions appear when τ drops below the critical value τc = bµ/2. By analogy with magnetic domain walls, stationary and propagating fronts have been labeled, (a)

(b)

0.75

-4

0.75

0.5

0.5

0.25

0.25

-2

2 -0.25

4

c

-4

-2

2

4

c

-0.25

-0.5

-0.5

-0.75

-0.75

Fig. 3.7. Plot of the l.h.s. of (3.78). Solutions are obtained as intersection points with straight lines presenting the r.h.s. of this equation. (a) One and three solutions in the symmetric case. (b) Asymmetric case

3.3 Dynamic Instabilities

161

ΤbΜ 0.5 0.4 0.3

IB

B

T

0.2 0.1 0.2

0.4

0.6

0.8

1

ΝΜ

Fig. 3.8. The Ising–Bloch bifurcation line (IB) for a single front in the parametric plane (ν/bµ, τ /bµ) and loci of traveling (T) and breathing (B) instabilities of a solitary band

respectively, as Ising and Bloch fronts, and the transition between them as Ising–Bloch (IB) bifurcation8 (Coullet et al., 1990; Hagberg and Meron, 1994). This is, clearly, a pitchfork bifurcation, which can exist only when the two propagation directions are symmetric. The trivial solution loses stability at τ = τc . The instability threshold can be computed most easily with the help of the implicit dispersion relation (3.48). This relation has at k = 0 a persistent zero root that corresponds to the translational Goldstone mode. At the threshold of traveling instability, another root crosses zero. Thus, the dispersion relation should be degenerate at this point, and the derivative of bµ(1 − q −1 ) − λ with respect to λ at λ = k = 0 should vanish. This method is also applicable to more complicated cases considered below where the propagating solution cannot be constructed explicitly. In the asymmetric case (ν = 0, Fig. 3.7b), no stationary solutions exist. At large τ , there is a single front propagating in a “natural” direction, e.g., with C > 0 (down-state advancing) at ν < 0. As τ decreases, a pair of solutions propagating in a “wrong” direction, one stable and another unstable, emerge at a saddle-node bifurcation. The bifurcation condition is determined by tangency: τ 4 . (3.79) = 2 3/2 bµ (4 + C ) The IB bifurcation line in the parametric plane (ν/bµ, τ /bµ) is drawn in Fig. 3.8. Propagating solutions exist for values of τ below this line, i.e., when 8

The term has been modeled after a similar transition described in Sect. 5.6.3, and has taken roots, although the speciﬁc structure of magnetic domain walls is not reproduced in the FN model and magnetic Bloch, as well as Ising, walls are stationary. The term “traveling instability” is more suggestive.

162

3 Systems with Separated Scales

the long-range variable is suﬃciently inertial and its distribution lags behind the displacement of the sharp front. In the asymmetric case, the critical value of τ required for bifurcation of a front going in a “wrong” direction decreases with growing bias. 3.3.2 Traveling Instability Other stationary structures, such as a band or a disk, also suﬀer a traveling instability, which can be located by diﬀerentiating the respective implicit dispersion relation for a branch of eigenvalues containing the Goldstone mode with respect to λ and computing the derivative at λ = 0. For a solitary band, the critical value of τ computed on the zigzag branch in (3.54) is (3.80) τc = bµ e−L sinh L − Le−L , approaching the value for a single front at L → ∞ (line T in Fig. 3.8). By construction, the locus of marginal traveling instability in the parametric plane (µ, ν) coincides precisely at τ = 1 with the locus of long-scale zigzag instability. At smaller values of τ , the traveling bifurcation precedes the static zigzag instability, progressively encroaching upon the static stability domain as τ decreases (Fig. 3.3). A traveling band solution can be constructed at values of τ below the instability threshold. A propagating band can be viewed as a bound state of a kink and an antikink. At ν = 0, one of the bound fronts propagates in the direction suggested by the imposed bias, and the other one, in the opposite direction. Naturally, the proﬁle of v is asymmetric, as seen in Fig. 3.9. For a ﬁxed L, a band moving with a faster speed has a steeper proﬁle and requires a larger bias and smaller values of τ . For a disk, the critical value of τ is computed by diﬀerentiating (3.73) with n = 1: v 0.8 0.6 0.4 0.2 -6

-4

-2

2

4

x

-0.2 -0.4

Fig. 3.9. A proﬁle of the long-range variable for a moving band

3.3 Dynamic Instabilities

163

Fig. 3.10. Evolution of a traveling spot (Schenk et al., 1997; reproduced with permission. Copyright by the American Physical Society)

τc = bµL{L[I1 (L)K0 (L) − I0 (L)K1 (L)] + 3I1 (L)K1 (L)}.

(3.81)

We shall see, however, in the next subsection that static spots become unstable already at larger values of τ due to a more dangerous oscillatory instability. As a moving disk loses its circular shape, a propagating solution cannot be constructed analytically. Numerical computations (Krischer and Mikhailov, 1994; Schenk et al., 1997) carried out at ﬁnite values of showed (Fig. 3.10) that a moving disk either shrinks and disappears or elongates in the direction normal to the direction of motion, bends at the ends, and eventually turns into a spiral pattern (see Sect. 3.4). Spreading or shrinkage can be suppressed by adding interaction with a global variable in a ﬁnite domain (Krischer and Mikhailov, 1994). This idea was reﬁned by Or-Guil et al. (1998), who replaced a rapidly reacting global variable by an additional fast long-range inhibitor w which, like the principal inhibitor v controlling the front propagation speed, is linearly coupled with the activator u, but serves a diﬀerent purpose of stabilizing the shape of the spot. Typical spot shapes and proﬁles of the variables along and across the direction of motion are shown in Fig. 3.11. The shift between the u and the v distribution is responsible for propagation; the distribution of w is broad and symmetric, and serves to contain lateral growth or shrinkage. 3.3.3 Oscillatory Instability Another possibility is an oscillatory (breathing ) instability that occurs when a pair of complex conjugate eigenvalues cross the imaginary axis, and signals

164

3 Systems with Separated Scales

Fig. 3.11. (a) Traveling spot in the three-variable model (Or-Guil et al., 1998; reproduced with permission. Copyright by the American Physical Society). The activator u is indicated by the isoline u = 0. The inhibitors v and w are shown as gray-scale plots, with dark gray representing high values. (b) Typical proﬁles of the variables u (solid line), v (dot-dashed line), and w (dotted line) along the direction of motion

bifurcation of a pulsating band or disk. This instability is more diﬃcult to detect analytically. The parameters at the bifurcation should verify a complex equation that can be obtained by setting in the dispersion relation λ = iωτ . For the varicose branch in (3.54), we obtain, after setting k = 0 and using the stationarity condition e−2L = ν/µ,

√ (3.82) iωτ = b µ − ν − µ(1 + iω)−1/2 1 + (ν/µ) 1+iω . The two equations obtained by separating the real and imaginary parts can be solved to obtain the rescaled frequency ω and a parameter deﬁning the bifurcation locus, say, τ . The latter enters in a very simple way; therefore, the best strategy is to ﬁnd ω by equating the real part to zero and then computing τ using the imaginary part. This yields the critical value τc at given values of µ and ν. One can see in Fig. 3.8 that breathing instability (line B) always

3.3 Dynamic Instabilities

165

takes place at smaller values of τ than traveling instability and therefore is irrelevant. The locus of the breathing instability of a solitary disk is obtained by setting in the dispersion relation (3.73) λ = iωτ and n = 0, bringing it to the form (3.83) iωτ = L−2 + bµF (L, ω), where

√ √ F (L, ω) = 2L I1 (L)K1 (L) − (1 + iω)−1/2 I0 L 1 + iω K0 L 1 + iω . (3.84) This relation can be resolved with respect to the parameters as bµ = −

Im F (L, ω) 1 , τ =− 2 . L2 Re F (L, ω) ωL Re F (L, ω)

(3.85)

This, together with the stationarity condition (3.54), deﬁnes the locus of marginal breathing instability as a surface in the parametric space µ, ν, τ parametrized by the disk radius L and frequency ω. This locus branches oﬀ the existence boundary on the line of double zero eigenvalue where the frequency vanishes. Close to this line, ω is small, and a Taylor series can be used. If the bifurcation relation (3.19) is substituted, the zero-order term vanishes and the ﬁrst-order term is purely imaginary. This gives the value of τ required to obtain the double zero as a function of the radius L: τd =

I0 (L)K0 (L) + L[I0 (L)K1 (L) − I1 (L)K0 (L)] . 2L2 [I0 (L)K0 (L) − I1 (L)K1 (L)]

(3.86)

Going along the existence boundary in the direction of increasing L, τd ﬁrst decreases, until it reaches the minimum τmin ≈ 1.757 at L ≈ 2.50, and then increases to the limiting value τd = 2 at L → ∞. The breathing bifurcation locus in the parametric plane µ, ν connects two double zero points on the existence boundary; at τmin < τ < 2 both points lie on the cusped curve deﬁned by (3.17), (3.19), while at τ > 2 one of these points lies on the axis ν = 0, µ > 2/b. The computations show that for the disk, unlike the band, breathing instability is always more dangerous than traveling instability, and no static spots can exist below τmin . Pulsating spots have been observed experimentally by Haim et al. (1996). 3.3.4 Phenomenological Velocity–Curvature Relation It is advantageous to derive closed evolution equations for fronts that would enable direct computation of front displacements without a need to solve a higher dimensional evolution equation for the long-scale ﬁeld. For the case when the latter is quasistationary, such a closed equation, albeit in a nonlocal

166

3 Systems with Separated Scales

form (3.41), has been derived in Sect. 3.1.5. We have noticed there that (3.41) is not very convenient for numerical computation. Trying to use a historydependent kernel for a general dynamic problem with a linear long-scale ﬁeld is still less practical. An alternative is to derive an approximate local equation valid under restricted conditions and faithfully reproducing, at least in a qualitative sense, essential features of the front motion, including both geometric factors and action of the long-scale ﬁeld. A simple relation of this kind has been suggested by Hagberg and Meron (1994). The starting point is (3.5) rewritten in the aligned coordinate frame (Sect. 2.2.1). This adds both the advective term τ −1 cv (x) as in (3.76) and the term κv (x), similar to the respective term in the equation of the shortscale variable that is responsible for the curvature dependence in the eikonal equation (3.6). The new term has exactly the same form as the advective term. Repeating the derivation in Sect. 3.3.1, we require the value of v on the front, computed now as v(0) = −ν + 2µ(C + κ)/s, where C = c/τ and s2 = 1 + 14 (C + κ)2 , to be equal to (c + κ)/b. In this way, we arrive at (3.78) with C replaced by C + κ on the l.h.s. and Cτ by Cτ + κ on the r.h.s.9 : C +κ 1 Cτ + κ = ν+ . (3.87) µ b 4 + (C + κ)2 This relation does not result from a rational expansion in small parameter and cannot be precise. One can see this, for example, by computing with its help the radius L = 1/κ of a stationary disk. The result is much simpler than in Sect. 3.1.3 and gives the same existence region, 0 < ν < µ, as for a solitary band, rather than a cusped region given by the precise computation. The reason is that the curvature, entering as a small correction in the short-scale equation, is not small on the extended scale of the long-range variable. The dependence of the long-range ﬁeld on the interface geometry is nonlocal and cannot be expressed precisely by any local relation. Nevertheless, (3.87) is useful as a simple phenomenological formula that captures in a qualitative way both traveling and transverse (zigzag) instabilities and reproduces therefore the essential features of front dynamics both in either slow or fast long-range ﬁeld.10 The loci of traveling (Ising–Bloch) and transverse instabilities (obtained, respectively, by diﬀerentiating (3.87) with respect to C and κ) coincide with those found for a single straight-line front11 in Sects. 3.3.1 and 3.2.1. Beyond the IB bifurcation, the dependence of 9

10

11

The scale ratio does not enter here because of the chosen scaling of the bias term in (3.4). Oscillatory instability is not directly reproduced in this way, but Haim et al. (1996) attribute it to transitions between the two branches of the multivalued velocity–curvature relations depicted by the S-shaped curve in Fig. 3.12. Since global shape dependence is missing in (3.87), distinctions in location of bifurcation points for diﬀerent shapes (band, disk, etc.) are, of course, not discerned.

3.3 Dynamic Instabilities

167

c 1 0.5

-1.5

-1

-0.5

1

0.5

Κ

1.5

-0.5 -1

Fig. 3.12. Dependence of the velocity on the curvature at τ = 3/2, ν = 0 and several values of bµ (shown by the numbers at the curves). Take note of segments with dC/dκ > 0 appearing at bµ > 2 and multivalued S-shaped curves at bµ > 2τ

velocity on curvature, which can be obtained by resolving numerically (3.87), becomes S-shaped (Fig. 3.12). A propagating front is unstable to transverse perturbations when dC/dκ > 0. The derivative can be computed using the diﬀerentiation rule for implicit functions. The condition of marginal transverse instability obtained in this way is 4 1 . = 2 3/2 bµ (4 + C )

(3.88)

Τ 2

1.5

ZS

ZT

1

0.5

IB

0.5

1

1.5

2

2.5

3

Μ

Fig. 3.13. The bifurcation diagram in the parametric plane (µ, τ ), showing the loci of Ising–Bloch bifurcation (IB) and of transverse instability of static (ZS) and traveling (ZT) fronts. Stable propagating solutions exist for values of τ below the lines IB and ZT

168

3 Systems with Separated Scales

a

b

c

d

Fig. 3.14. A labyrinthine pattern developing from a single stripe when the inhibitor is fast (Hagberg and Meron, (1994); reproduced with permission. Copyright by the American Physical Society)

The bifurcation diagram in the parametric plane µ, τ for the symmetric case ν = 0 is shown in Fig. 3.13. The local relation (3.87) can be used for tracking the front motion numerically. Realistic disordered patterns are generated in this way (Hagberg and Meron, 1994): labyrinthine patterns develop at large τ to the right of the transverse instability locus ZT (Fig. 3.14), while spirals and traveling fragments prevail at smaller τ above the line ZS (Fig. 3.15). A simpliﬁed form of (3.87) can be obtained through a rational expansion near the IB bifurcation. The simplest formula retaining a multivalued velocity– curvature dependence can be obtained by setting in (3.87) bµ κ = 3 κ (1 − 2 τ), ν = µ3 ν , τ = (3.89) C = C, 2 and expanding in 1. This yields in the third order a cubic equation 1 1 C3 τC − − . (3.90) κ = − ν+ bµ 2 2 16 At τ < 1 when the IB (traveling) bifurcation precedes the zigzag instability the coeﬃcient at κ is positive. A longer but more convincing way to derive this formula is to expand (3.5), (3.6) for a weakly curved front in the vicinity of the IB bifurcation.

3.3 Dynamic Instabilities

a

b

c

d

169

Fig. 3.15. Spiral turbulence developing from a single stripe when the inhibitor is slow (Hagberg and Meron, (1994); reproduced with permission. Copyright by the American Physical Society)

A sequence of equations is obtained for functions vi in the expansion of the long-range variable v = v0 + v1 + · · · written in the comoving frame. Taking (3.89) as a scaling guideline, we assume C = O(), ν = O(3 ), κ = O(3 ). The respective scaling of transverse derivatives ∂y = O(3 ) eliminates them in the relevant orders of the expansion. The ﬁrst-order equation is v1 (x) − v1 + Cv0 (x) = 0,

(3.91)

where v0 veriﬁes (3.8) with ν = 0. The solution satisfying the matching conditions at the front located at x = 0 is  1  2 Cµ(1 + x)ex at x ≤ 0 (3.92) v1 = 1 −x Cµ(1 − x)e at x ≥ 0. 2 The ﬁrst-order eikonal equation is simply c = Cτ = bv1 (0) = Cbµ/2. This equation is veriﬁed trivially at any C at the IB bifurcation point τ0 = bµ/2. Fixing τ at this value, we proceed to the second-order equation v2 (x) − v2 + Cv1 (x) = 0, solved by

(3.93)

170

3 Systems with Separated Scales

 1 2  − 8 C µx(1 − x)ex at x ≤ 0 v2 =



− 18 C 2 µx(1 + x)e−x at x ≥ 0.

(3.94)

Since v2 (0) = 0, the eikonal equation is trivially veriﬁed also in the second order. Finally, in the third order, adding also O(3 ) bias and curvature and O(2 ) deviation of τ from the bifurcation value, we have v3 (x) − v3 + Cv2 (x) + κv0 (x) − ν = 0,

(3.95)

solved by   v3 =

1 x 2 µe

κ(1 + x) −

−x κ(1 − x) − 2 µe

1

1 3 24 C (3

− 3x + x3 ) − ν

1 3 24 C (3

at x ≤ 0

+ 3x − x3 ) − ν at x ≥ 0.

(3.96)

This leads to the third-order eikonal equation (τ − τ0 )C = −bν −

3 1 16 bµC

+κ

1

2 bµ

−1

(3.97)

identical to (3.90). This local velocity–curvature relation is especially useful since it can be combined with intrinsic equations of the front motion (Sect. 3.4.1). This will be used for construction of rotating spiral bands in Sect. 3.4.4. 3.3.5 Long-Scale Evolution Equations A straight-line propagating front (or a planar front in 3D) is neutrally stable to translations along the propagation direction. By continuity, the dynamics of transverse distortions is slow when their wavelength is large compared to the O(1) scale characteristic to the inhibitor ﬁeld. The dynamic equations on extended spatial and temporal scales can be obtained as in Sect. 2.6.5 through a regular expansion procedure, leading to a universal form independent of the underlying system. It is most convenient to √ work in the aligned frame. We assume that the curvature is scaled as O( ε) when measured on the inhibitor scale.12 The matching extended time variable is scaled as ∂t = O(ε). The inhibitor ﬁeld is expanded in the scale ratio ε: v = v0 + εv1 + · · ·, and the normal velocity (measured in the inhibitor units) is expanded as C = C0 + εC1 + · · ·. The zero-order equation in the moving frame is stationary and onedimensional, and coincides with (3.76). We denote its solution given by (3.77) as v0 (x). The ﬁrst-order equation is 12

Recall that the curvature was assumed to be small on a shorter activator scale when the eikonal equation (3.6) was derived. The notation for a small parameter is modiﬁed here in order to emphasize that it has nothing to do with the scale ratio used so far.

3.3 Dynamic Instabilities

v1 (x) − v1 + C0 v1 (x) + (C1 + κ)v0 (x) = 0.

171

(3.98)

Solving this equation subject to the matching conditions at x = 0, we obtain the value of v1 at the front: v1 (0) = 12 µ(1 + 14 C02 )3/2 (C1 + κ).

(3.99)

The correction to the normal velocity C1 is deﬁned by the ﬁrst-order eikonal equation obtained by expanding (3.6) rescaled to the inhibitor time units: τ C1 = bv1 (0) − κ. Using here (3.99) yields 2 2τ 1 2 3/2 1 2 3/2 1 + 4 C0 1 + 4 C0 − 1 C1 = − − 1 κ. bµ bµ

(3.100)

(3.101)

This equation is well posed when the bracketed coeﬃcients are both positive. Then it has the same form as (2.50) and can be reduced in a Cartesian frame to the Burgers equation as in Sect. 2.2.3. The coeﬃcient at C1 vanishes at the IB bifurcation; further expansion near this point has been carried out in the preceding subsection. Vanishing coeﬃcient at κ (“phase diﬀusivity”) marks the limit of long-scale zigzag instability. The expansion can be continued to the next order to describe slow dynamics of long-scale distortions near this point. Next, we ﬁx the parameter µ at the instability limit, allowing for a small deviation: 3/2 + εµ1 . (3.102) µ = 2b−1 1 + 14 C02 Then C1 = 0, and a higher order correction C2 has to be computed in the next order. The second-order equation should be written using the Laplacian in the aligned frame (2.42) or (2.45), expanded to the second order: v2 (x) − v2 + κv1 (x) + (C2 − xκ2 )v0 (x) + ∇2⊥ v1 = 0.

(3.103)

In 3D, κ2 should be replaced here by κ21 + κ22 . We need to use here the solution of (3.98). To avoid cumbersome expressions, we write this solution at the Maxwell construction (C0 = 0): −1 x b e (1 + x)κ + µ1 (ex − 1) at x ≤ 0 v1 = (3.104) −1 −x b e (1 − x)κ + µ1 (1 − e−x ) at x ≥ 0. Since we only need to compute the value of v2 at the front, some rules veriﬁed by direct computation can be set to save the eﬀort of solving this equation (subject to usual matching conditions). An inhomogeneity of the form Qex at x < 0, Qe−x at x > 0 contributes the constant Q/2 (which may depend on the transverse coordinates and slow time) to the value v2 (0) entering the eikonal equation. An inhomogeneity of the same form but with

172

3 Systems with Separated Scales

diﬀerent signs in the positive and negative domains gives no contribution. The contribution of an inhomogeneity of the form −Qxex at x < 0, Qxe−x at x > 0 is Q/4, while the same inhomogeneity with common signs in the two domains gives no contribution. Applying these rules, one can read the value of v2 (0) directly from (3.104) and use the result in the second-order eikonal equation τ C2 = bv2 (0) − κ.

(3.105)

The result turns out to be dependent only on ﬁrst-order terms in the expansion of the Laplacian in the aligned frame: (τ − 1)C2 = 12 µ1 κ + 14 ∇2⊥ κ.

(3.106)

The coeﬃcient at C2 is positive at τ > 1 when the zigzag instability precludes IB bifurcation. The ﬁrst term on the r.h.s. is destabilizing at µ1 > 0. Perturbations are stabilized by the last term that suppresses short-scale corrugations. This equation is transformed to the Kuramoto–Sivashinsky (KS) equation (1.23) by presenting the front position, as in Sect. 2.2.3, in the Cartesian frame (X, Y ) propagating with the speed C0 as X = ζ(Y , t) with ∇⊥ ζ 1 and using approximate expressions for the curvature κ = −∇2⊥ ζ and the displacement speed along the X-axis −1/2 1 C0 + 2 C2 − C0 ≈ − C0 |∇⊥ ζ|2 + 2 C2 . (3.107) ζt = 1 + |∇⊥ ζ|2 2 The nonlinear advective term turns out to be dominant, unless either the basic velocity C0 or the displacement is small. After rescaling C0 → C1 , ∂t → 2 ∂T , plugging (3.106) in (3.107) yields the KS equation balanced on an extended time scale: 1 ζt − C1 |∇⊥ ζ|2 + 12 µ1 (τ − 1)−1 ∇2⊥ ζ + 14 (τ − 1)−1 ∇4⊥ ζ = 0. 2

(3.108)

The original derivation of the KS equation for an unstable front in the FitzHugh–Nagumo system (Kuramoto and Tsuzuki, 1976) carried out in the “laboratory” rather than aligned frame is far more lengthy, as is an earlier derivation by Sivashinsky (1977) in the context of combustion fronts. We shall return to the KS equation in Sect. 5.1.3 where it appears as a long-scale evolution equation of phase instabilities.

3.4 Locally Induced Motion 3.4.1 Intrinsic Equations of Motion Kinematics of a planar curve can be described without reference to coordinates, in terms of its intrinsic characteristics: metrics and curvature. The starting point is the 2D version of the Frenet–Serret equations

3.4 Locally Induced Motion

X s = l,

ls = κn,

ns = −κl.

173

(3.109)

The two orthogonal unit vectors, tangent l and normal n, deﬁne at any point of the curve the Frenet dihedron. The curvature deﬁnes locally the rotation of the Frenet dihedron and therefore determines the shape of a smooth curve. If the function κ(s) is known, the curve can be reconstructed, in principle, by integrating (3.109), although this task is technically diﬃcult due to the implicit form of these equations. For practical computations, (3.109) should be transformed to a more convenient form involving a minimal number of variables. Suppose that the curve is set into motion in such a way that each point has a velocity v(s, t). For the time being, we allow v to be an arbitrary vector function but require it to be continuously dependent on s. In order to obtain an evolution equation of a curve in terms of its intrinsic parameters, it is advantageous to use projections of v on the Frenet dihedron; thus, we assume v = X t = χl − cn.

(3.110)

The sign of the last term follows the sign convention of Sect. 2.2.1. Since the arc length, generally, changes as the curve moves, the derivatives with respect to t and s do not commute. It might be therefore more convenient to choose a constant parametrization y, such that ∂t ∂y = ∂y ∂t . One can use for this purpose, for example, the arc length at a ﬁxed moment of time y = s(t0 ); this parametrization can be viewed just as a Lagrangian tag carried by each point as it moves. The derivative g = ∂s/∂y deﬁnes the metric of the curve in the chosen parametrization. Then the commutator between the derivatives with respect to t and s is computed as ∂t ∂s − ∂s ∂t = ∂t (g −1 ∂y ) − g −1 ∂y ∂t = −g −2 gt ∂y = −g −1 gt ∂s .

(3.111)

Using this commutation relation, we obtain, by diﬀerentiating the ﬁrst Frenet–Serret equation with respect to time and replacing the derivatives with respect to s with the help of other equations, the evolution equation of the tangent vector l: lt = ∂t ∂s X = ∂s ∂t X − g −1 gt ∂s X = ∂s v − g −1 gt l = (χs − g −1 gt + κc)l + (κχ − cs )n.

(3.112)

Since l is a unit vector, the projection l · lt should vanish. This yields the metric continuity equation gt = g(χs + κc).

(3.113)

Equation (3.112) thus reduces to lt = (κχ − cs )n.

(3.114)

174

3 Systems with Separated Scales

In the same manner, diﬀerentiating the second Frenet–Serret equation with respect to time and using (3.113) and (3.114) yields κnt = −(κt + g −1 gt )n + ∂s ((cs + κχ)n) = −(κt + κ2 c + css − κs χ)n − (κχ − cs )l.

(3.115)

Again, requiring the projection n · nt to vanish gives the evolution equation of the curvature: (3.116) κt = −κ2 c − css + κs χ. The tangential velocity χ is just a gauge variable that expresses an arbitrary reparametrization of the evolving curve. If the normal velocity component is expressed as a function of curvature κ, the system of two evolution equations (3.113), (3.116) for three variables κ, g, and χ can be closed by imposing an arbitrary gauge condition. Two gauges are most useful: the comoving gauge χ = 0 and the isometric gauge with time-independent metric g. Since the arc length is not conserved in the comoving gauge, it cannot be used as an independent variable, and the intrinsic equations of locally induced motion should be rewritten using a constant parametrization y related to s by the metric function, ds = g dy. Using the local equation of the front motion (2.50) with c0 = const, the intrinsic equations in the comoving gauge (3.113), (3.116) are written as gt = gκ(c0 − Dκ), κt = g −1 ∂y g −1 κy − κ2 (c0 − Dκ).

(3.117) (3.118)

Clearly, at Maxwell construction (c0 = 0) the curve shrinks as a result of motion along the normal, resulting in the decrease of energy roughly proportional to the length of the boundary. The line tension can be viewed, in fact, both (3.117) as the physical cause of the normal motion. If κ = R−1 = const, and (3.118) yield κt = Dκ3 or Rt = −D/R, integrating to R = 2D(t − t0 ), where t0 is the moment of collapse. Of course, the theory becomes inapplicable at the ultimate brief stage of collapse when the curvature radius becomes comparable to the width of the front. The diﬀusive character or normal motion becomes apparent in the limit of very small curvatures when the nonlinear terms in (3.118) can be neglected. As the rate of change of the metric is quadratic in κ, it can be considered constant in the leading order, and (3.118) reduces to the diﬀusion equation κt = κyy . This indicates the tendency of relaxation to circular shape. Conversely, at large curvatures, when the nonlinear terms prevail, the evolution is catastrophic, leading to rapid local collapse of warped segments. 3.4.2 Steadily Rotating Spiral Intrinsic equations of motion provide the most convenient means to study the front dynamics whenever a local velocity–curvature relation is available.

3.4 Locally Induced Motion

175

We shall now apply it to construct a rotating spiral wave. Development of a rotating spiral can be understood qualitatively starting from a semi-inﬁnite propagating band, or a “ﬁnger” (see Sect. 3.6.2). Near the end, the propagation speed tends to be lower due to the curvature eﬀects. As these segments are left behind, the band is bending. If the tip does not retreat in spite of its high curvature, the overall length of the band keeps increasing as it curves around the tip while propagating outward. The resulting structure is a rotating spiral. In a similar fashion, a pair of counter-rotating spirals would grow from a segment of ﬁnite length or out of any defect in a propagating wave pattern. The isometric gauge is most suitable for identifying a self-similar shape of an open-ended curve that remains invariant as it moves. The metric equation (3.113) turns in this case into the equation of the tangential velocity χs + κc = 0.

(3.119)

The curvature equation (3.116) can be rewritten then as a conservation equation κt + qs = 0, (3.120) where the ﬂux q is deﬁned as q = cs − κχ.

(3.121)

For an invariant curve, the ﬂux q must be constant. An invariant curve rotating with a constant frequency ω can be constructed using the polar frame (r, φ). In order to preserve the shape deﬁned by some function φ = Φ(r), the tangential velocity χ should be ﬁxed to ensure that each point be translated strictly in the angular direction while it propagates along the normal. The tangent and normal unit vectors are r + rΦ (r) φ rΦ (r) r − φ , n= , l= 1 + [rΦ (r)]2 1 + [rΦ (r)]2

(3.122)

are unit vectors in the radial and angular directions. The direction where r, φ of the normal is chosen to be opposite to the positive rotation direction. The radial and angular velocities are expressed now as χ − crΦ (r) = 0, cr = (χl − cn) · r = 1 + [rΦ (r)]2 = χrΦ (r) + c = ωr. cφ = (χl − cn) · φ 1 + [rΦ (r)]2

(3.123)

These conditions are resolved as Φ (r) =

χ , cr

χ = − ω 2 r2 − c2 .

(3.124)

176

3 Systems with Separated Scales

The choice of sign of χ ensures that the front is convex, i.e., propagates against the direction of the normal. The radial coordinate and the arc length are related by (3.125) ds/dr = 1 + [rΦ (r)]2 = 1 + (χ/c)2 = ±ωr/c. This can be used to diﬀerentiate the relation χ2 + c2 = ω 2 r2 with respect to the arc length. Substituting there the derivatives of χ and c given by (3.119), (3.121) yields q = ±ω. This relation holds for any steadily rotating spiral. The simplest case of propagation with a constant normal velocity c0 is known as the Wiener–Rosenblueth (WR) model. Clearly, (3.124) breaks down within an inner circle with the radius R = c0 /ω. Thus, the spiral can only rotate around a central hole of the radius R. This relation can also be interpreted in the opposite sense: the frequency ω = c0 /R is ﬁxed by the radius of the central hole. By (3.124), the tangential velocity vanishes on the inner circle, and the spiral kicks oﬀ normally to the circle. The actual shape of the spiral is readily obtained by integrating (3.124) starting from an arbitrary angular position φ0 on the hole perimeter: (r/R)2 − 1 + arctan (r/R)2 − 1 . (3.126) Φ(r) = φ0 − Asymptotically at large r this is an Archimedean spiral with the pitch 2πR equal to the circumference of the central hole (Fig. 3.16a). The unsettling feature of this solution is a singularity at the free end. The curvature κ ≈ Φ (r) diverges at r → R, as can be already seen by inspecting (3.124). One should also keep in mind that computing a single open-ended curve is not suﬃcient for constructing a realistic spiral pattern, since the front propagation velocity must change sign to create a rotating spiral band. The desired (a)

(b)

10 4 5 2

-10

-5

5

10

-6

-4

-2

2

4

6

-2 -5 -4

Fig. 3.16. (a) A Wiener–Rosenblueth spiral. The central hole is shaded. (b) A symmetric spiral band computed near the IB bifurcation

3.4 Locally Induced Motion

177

pattern can be imitated by adding an identical curve shifted by a diﬀerent angle φ0 . This construction is quite artiﬁcial, since the two curves become totally unrelated when bailed out of direct contact by the central hole. 3.4.3 Propagation into a Quiescent State A more realistic local model including curvature dependence for a curve moving around a circular obstacle is based on the eikonal equation (3.6). The value of the long-range variable v deﬁning the propagation speed of a straight-line excitation front is, generally, unknown and coupling it to the front position makes the problem severely nonlocal. Postponing the analysis of spiral waves coupled to the inhibitor ﬁeld till the next section, we consider here a simple case when v = v0 and hence c0 = bv0 is constant everywhere along the front. This may happen in an excitable system where an excitation pulse propagates into a “quiescent” state corresponding to its only HSS. An additional requirement imposed on a spiral excitation pulse is that the excited region should be narrow and the time interval between subsequent passages of the pulse through any point should far exceed the inhibitor relaxation time, so that the system would have suﬃcient time to relax back to the quiescent state after the passage of a preceding excitation. For any local velocity–curvature relation resolved with respect to κ, combining (3.119) and (3.121) yields c

q dc = −χ − . dχ κ(c)

(3.127)

Another form of this equation, which is more directly related to geometry, can be obtained if the normal velocity is replaced with the help of the second invariance condition (3.124) by the rescaled squared radial coordinate ρ = ω 2 r2 . It is convenient to use the latter as the independent variable:

dχ 1 (3.128) =− κ ρ − χ2 . dρ 2q The basic local velocity–curvature relation is given by the eikonal equation. We shall use here its original dimensional form (2.50). Substituting it in (3.127) yields a closed equation for c(χ), which is further reduced by using c0 as a velocity scale to Ω dc = −χ − , (3.129) c dχ 1−c where Ω = qD/c20 . We expect the curvature to decrease, and the normal velocity to grow along the curve parametrized by the arc length s. The tangential velocity must vanish at the free end s = 0, where the curvature should be maximal and the normal velocity minimal. Since at the free end c (s) > 0, while χ = 0, the ﬂux q is positive and equal to the rotation frequency ω (according to the result of the last subsection). As the ﬂux remains constant along the curve,

178

3 Systems with Separated Scales

χ must be negative at s → ∞ where c (s) vanishes. The dependence c(χ) can be obtained therefore by integrating (3.129) from χ = 0 in the negative direction, starting from some initial condition c(0) = η, 0 < η < 1. The rescaled normal velocity must approach unity asymptotically at χ → −∞. This happens only at a certain unique value of the rescaled frequency Ω = Ωc (η). This value is found iteratively with high precision as a border between trajectories c(χ) turning up to c = 1 or down to c = 0 with increasing |χ|. It is very diﬃcult to continue integration to large |χ|, and the curve should be matched there to an asymptotic solution c 1 − Ω/|χ| + O(χ−4 ) at |χ| → ∞.

(3.130)

The computed curve c(χ) can be translated to the radial velocity dependence with the help of the second invariance condition (3.124), and the dependence Ωc (η), transformed into the dependence of the rotation frequency on the hole radius R. Reverting to the dimensional variables, we observe that, for kinematic reasons, c = ωR on the hole circumference, i.e., η = ΩR/c0 . The relation between the frequency and the hole radius is expressed therefore as Pe = η/Ωc (η), where Pe = Rc0 /D is the Peclet number based on the hole radius. The numerical dependence is shown in Fig. 3.17. The curve Ω(Pe) approaches the WR limit, Ω = 1/Pe asymptotically at Pe → ∞. The other limiting value is Ω ≈ 0.331 at Pe, η → 0. The actual shape can be restored by integrating (3.124). It diﬀers from the WR spiral only close to the hole circumference, and the overall diﬀerence in form is hardly detectable visually. Clearly, what is missing in this kinematic description is the reverse transition to quiescent state. Mikhailov et al. (1994) interpreted the kinematic solution as a narrow excitation zone, without distinguishing between the dynamics of its front and back sides. Such a narrow zone can, indeed, be formed

0.2

0.1

0.05

0.02

Pe 0.5

1

5

10

Fig. 3.17. The dependence of the rescaled rotation frequency on the Peclet number obtained as the integrability condition of (3.129). The dashed line shows the WR limit

3.4 Locally Induced Motion

179

under certain conditions; the kinematic theory excludes, however, the crucial tip region of the excitation band. The vanishing tangential velocity at the free end of the spiral, which is arbitrary imposed as a boundary condition in kinematic theory, is actually ﬁxed by the tip curvature. More sophisticated treatment is needed to relate both tip curvature and the spiral core radius (replacing an artiﬁcial hole) to physical parameters of the excitable system (see Sect. 3.6.3). The kinematic theory of Mikhailov et al. (1994), though not mathematically precise, correctly predicted a general form of this relationship, which was later derived through rational expansion in the large core limit by Hakim and Karma (1999). 3.4.4 Spiral Band near Ising–Bloch Bifurcation In the plane lacking a contrived hole, the front propagation velocity must change sign to create a rotating spiral band. Both the front and back side can approach asymptotically the Archimedean spiral form with the curvature vanishing at large distances when the function κ(c) has two stable roots with opposite signs. The simplest local velocity–curvature relation suitable for generating such a structure is the cubic form (3.90) derived in the vicinity of the IB bifurcation. Using the symmetric rescaled version of (3.90), κ = c − c3 , we rewrite (3.128) as 1 dχ =− ρ − χ2 (1 − ρ + χ2 ). (3.131) dρ 2q It is convenient to integrate this equation backward starting from the large ρ asymptotics that can be expressed as a series q 1 − 2q 2 1 √ − ρ− √ − + ···. (3.132) 2 ρ 2ρ 8ρ3/2 The propagation speed c = ρ − χ2 should be less than unity (its asymptotic value at zero curvature). One can see from the above expansion that this condition holds when the ﬂux q is negative, and, using the general relation between the ﬂux and frequency from Sect. 3.4.2, we set q = −ω. The frequency should be chosen in such a way that the tangential velocity vanishes at the spiral tip. If this is achieved, the solution can be matched smoothly with the antisymmetric solution that has the normal velocity of the opposite sign and corresponds to the reverse transition front. For the symmetric case considered here, χ vanishes together with c and κ at ρ = 0 if one chooses ω ≈ 0.267. The normal and tangential velocities and curvature are plotted in Fig. 3.18. The actual form of the curve obtained by integrating the ﬁrst invariance condition (3.124) is shown in Fig. 3.16b. Asymptotically at large r, it is again an Archimedean spiral, since the normal velocity approaches a constant value. The two spiral branches joined at the origin divide the plane into two regions of identical shape occupied respectively by the up and down states. The visual |χ| =

180

3 Systems with Separated Scales Γ,c,Κ Γ

1 0.8

c 0.6 0.4 Κ

0.2

0.5

1

1.5

2

Ρ 2.5

Fig. 3.18. The normal and tangential velocities and curvature as functions of the √ rescaled radius ρ

√ “tip” of the spiral band is seen at ρ ≈ 0.62, where the curvature passes maximum, but its “mathematical” tip should be deﬁned as the point where the normal velocity vanishes, i.e., at the origin. In an asymmetric case (ν = 0), both branches of the spiral band should be integrated separately starting from an asymptotic expansion at large ρ where c approaches either of the two stable values with the opposite signs, and the frequency chosen to match both branches at the location of the unstable zero of the velocity–curvature relation. The “local induction” approximation gives the best chance to construct the rotating spiral almost analytically, as we did above. Near the IB bifurcation, this is indeed a rational approximation based on our choice of extra-long spatial scales. Since not only the curvature but also the spiral pitch will be in this case extremely long when rescaled back to “real” units, the nonlocal eﬀects due to the inﬂuence of faraway spiral segments should be negligible as well. The approximation may, however, break down when more subtle eﬀects such as stability are investigated. Returning to the kinematic equation (3.118), we can observe that the tip region is apt to be unstable, since close to the unstable zero of the velocity– curvature relation the propagation speed increases with curvature and (3.118) becomes ill-posed. This is related to the phenomenon of spiral tip meandering observed in numerous experiments and computations (more on this in Sect. 3.6.4). The instability due to the local negative diﬀusivity may either saturate nonlinearly upon acceleration or be advected by normal propagation to stable outward segments and decay there. Formally, the instability develops most rapidly on the shortest wavelength, but the approximation clearly breaks down on shorter scales. The short-scale instabilities are saturated nonlinearly when the implicit local velocity–curvature relation (3.87) is used in its full form, as in computations by Hagberg and Meron (1994), or replaced

3.5 Advective Limit

181

by a form cubic in both velocity and curvature. The latter form can be derived by bifurcation expansion near a higher codimension parametric point µ = 2/b, τ = 1, where the IB and zigzag instabilities are degenerate and both C and κ can be scaled as O() (Sect. 3.3.4). At shorter scales, however, nonlocal eﬀects mediated by the inhibitor ﬁeld are likely to become at least as strong as nonlinear curvature eﬀects, and should play an important role in the development or stabilization of emerging short-scale modulations.

3.5 Advective Limit 3.5.1 Inertial Scaling The scaling of Sect. 3.1.1 is not suitable for the analysis of oscillatory or excitable systems (see Sect. 1.4.2), since strong coupling of the two variables in the FN system is necessary to eliminate one or both ﬁxed points. Typical structures associated with oscillatory or excitable dynamics are propagative: excitation pulses and wave trains, which can be viewed as excitation spikes or oscillations of dynamical systems translated from the temporal to spatial domain. A wave train in an oscillatory system would combine alternating transitions from the lower to the upper state and back (Fig. 3.19a). For an excitable system (Fig. 3.19b), both wave trains and solitary spikes are possible. Unlike stationary or slowly evolving patterns where the characteristic length scale is set by the diﬀusional range of the long-scale variable, the wavelength of a propagating pattern is tied to the propagation speed and remains ﬁnite even when the inhibitor is nondiﬀusive. Returning to the dimensional equations (3.1), we redeﬁne the long scale on the basis of the characteristic relaxation time of the inhibitor γ2 and the characteristic propagation ∗ = D /γ . Using this “advective” length unit, speed of the activator front c 1 1 L∗ = γ2 c∗ = γ2 D1 /γ1 , we bring (3.1) to the dimensionless form (a)

(b)

0.4 0 0.2

front

0 0.2

0.4 1.5 1 0.5 0 u

back

0.2 v

v

0.2

0.4 back

0.4 0.5

1

1.5

1.5 1 0.5 0 u

front 0.5

1

1.5

Fig. 3.19. Null-isoclines of an oscillatory (a) and an excitable (b) system; dashed lines show transition fronts of the short-scale variable in a wave train (a) or a single excitation pulse (b)

182

3 Systems with Separated Scales

γut = γ 2 ∇2 u + f (u, v),

(3.133)

vt = δ ∇ v + g(u, v),

(3.134)

2

2

where δ = γ/. The “inner” scale of the transitional layer, where the system switches between the two alternative activator states, u = u± s , is now set exclusively by the capacitance ratio γ independently of diﬀusivities and, provided γ 1, remains small even when the inhibitor is less diﬀusive than the activator and (as deﬁned in Sect. 3.1.1) is not small. The parameters can be chosen in such a way that δ 1, so that the inhibitor diﬀusion is negligible. In terms of the original parameters of (3.1), this requires γ D1 /D2 . Under these conditions, the inhibitor diﬀusion can be neglected, reducing (3.3) to vt = g(u, v).

(3.135)

Although this equation contains no mechanism for healing discontinuities in v, the inhibitor ﬁeld should remain smooth in the course of evolution, barring freaky initial conditions or strongly localized perturbations. As before, (3.133) is replaced on the O(1) length scale by the eikonal equation (2.50) or (3.6), which is rewritten in the adopted scaling as c = c0 − γκ,

(3.136)

where c0 = c(v0 ) is the speed of the straight-line front governed by the local value of the controlling variable v0 . This formula is applicable only when the curvature radius far exceeds the inner scale, i.e. γ, in the dimensionless units adopted above. The front motion slows down when v0 is close to the “stall” value at the Maxwell construction v ∗ . Under these conditions, c0 may become of the same order of magnitude as the speed of curvature-driven motion. If both v0 and c0 are of the order O(γ n ), the two terms in (3.136) are balanced at κ = O(γ n−1 ). The characteristic dimensionless time τ of evolution of the controlling variable required to switch between the values of v at the “front” and the “back” sides of a propagating wave train or excitation pulse is reduced likewise to O(γ n ). Accordingly, the dimensionless wavelength or the length of the pulse, λ = c0 τ , becomes of O(γ 2n ). The curvature radius should be of the same order of magnitude as λ−1 ∝ γ −2n . Comparing this with the above estimate, we see that the orders of magnitude match when n = 1/3. This is known as Fife scaling (Fife, 1985). The curvature κ = O(γ −2/3 ) is still less than γ −1 , the inverse interface width, and therefore the sharp interface limit leading to (3.6) remains valid. After rescaling c→γ

1/3

c,

∂t → γ −1/3 ∂t , ∂x → γ −2/3 ∂x , v → v ∗ + γ 1/3 v, κ → γ −2/3 κ,

(3.137)

the eikonal equation (3.136) reverts to the form (3.6), which does not depend on the small parameter γ and only contains an O(1) model-dependent numerical parameter b = O(1), while (3.134) becomes

3.5 Advective Limit vt = δ 2 γ −1 ∇2 v + g± (v ∗ ) + γ 1/3 g± (v ∗ )v ,

183

(3.138)

where g ± (v) = g(u± (v), v). The diﬀusional term remains negligible after rescaling when δ 2 γ, i.e. γ 2 or, reverting to the original parameters, D2 D1 . As the last term is small, (3.138) reduces then to (3.135) in the leading order. 3.5.2 Dispersion Relation for Wave Trains An undistorted excitation pulse or wave train propagating with a constant speed c0 along the x-axis is deﬁned in the advective limit by (3.135) rewritten in the comoving frame: c0 vx + g ± (v) = 0,

g ± (v) = g(u± (v), v).

(3.139)

This equation is, generally, nonlinear even with a linear function g(u, v), in view of the nonlinear dependence of quasistationary activator values on v. This technical diﬃculty is usually removed by ﬁxing f (u, v) in such a way that u± (v) = const. A convenient choice is (3.7), leading to u± s = ±1, with only the unstable intermediate state u0s = v being v-dependent. The respective phase portrait is shown in Fig. √ 3.20. The propagation speed, following from (2.14), is c0 = bv0 with b = 2. With this assignment, and a linear g(u, v) in its usual form, (3.139) becomes linear and can be easily solved analytically: c0 vx − v − ν ± µ = 0.

(3.140)

Equation (3.140) admits a continuum of periodic solutions parameterized by the velocity c0 . The dependence between the velocity and wavelength L, i.e. a dispersion relation, can be obtained by ﬁxing the values of v at the transition fronts. Qualitatively similar solutions for nonlinear systems with phase portraits of the type shown in Fig. 3.19 can be routinely computed numerically. Consider an oscillatory system with the dynamical phase portrait shown in Fig. 3.20a. The only possible regular pattern in this system is a wave train combining alternating transitions from the lower to the upper state and back, as shown by the dashed line in this ﬁgure. To conform to the common usage, we shall call the upper state “excited.” On its advancing edge, called the “front” side and located, say, at x = nL, n = . . . , −1, 0, 1, . . ., the propagation speed, by our usual sign convention (Sect. 2.1.3), is negative. The reverse transition to a “quiescent” lower state takes place at the “back” side located at x = nL∓L± , where the propagation speed is positive; L± are the widths of the excited and quiescent regions. Clearly, the values of v0 at the front and back edges should be of the opposite sign and equal by absolute value. The solution of (3.140) satisfying the boundary conditions v(0) = −v0 = −c0 /b is (within a single period) µ − ν − (µ − ν + c0 /b)ex/c0 at −L+ ≤ x ≤ 0 v= (3.141) −(µ + ν) + (µ + ν − c0 /b)ex/c0 at 0 ≤ x ≤ L− .

184

3 Systems with Separated Scales

The widths of the excited and quiescent regions L± should be obtained from the boundary conditions v(−L+ ) = v(L− ) = v0 . This gives the relation between L± and the propagation speed c0 in the implicit form µ=

eL/c0 − 1 c0 , b (eL+ /c0 − 1)(eL− /c0 − 1)

eL+ /c0 − eL− /c0 c0 . b (eL+ /c0 − 1)(eL− /c0 − 1) (3.142) = L/2, one can obtain an explicit

ν=

In the symmetric case ν = 0 and L± dispersion relation L = 4c0 arc coth(bµ/c0 ).

(3.143)

The period diverges at c0 → bµ when the trajectory approaches a heteroclinic orbit passing through the corners of the zigzag null-isocline in Fig. 3.20. At c0 → 0, (3.143) yields formally a vanishing period; the formula is, however, inapplicable at small velocities, since diﬀusion has to be taken into account. The periodic solutions constructed in this way are stable to uncorrelated displacements of individual fronts, since any advance would bring it to an unrelaxed zone where the value of v is unfavorable, and thus cause deceleration. Conversely, any delay will shift the front further down the relaxation tail of the preceding transition and cause acceleration, restoring the original wavelength. Eckhaus instability limiting the minimal possible wavelength is observed, however, at ﬁnite values of γ. The instability threshold can be only obtained numerically (see, e.g., B¨ar and Or-Guil, 1999). The fastest-growing modes correspond to an alternating compression and expansion of subsequent pulses. Similar patterns can be constructed for an excitable system with the dynamical phase portrait shown in Fig. 3.20b. The only limitation is that the front edge should lie above the ﬁxed point. Generating such a pattern will require, however, special initial conditions. It could be sustained, in a more or less realistic setting, on a large ring, where the excitation waves of ﬁnite period would chase one another. A more natural structure for this system is a (b)

back v

v

(a) 2 1.5 1 0.5 0 0.5 1 1.5

front

1

0.5

0 u

0.5

1

2.5 2 1.5 1 0.5 0 0.5 1

back front

1

0.5

0 u

0.5

1

Fig. 3.20. Null-isoclines of an oscillatory (a) and an excitable (b) system with the nonlinear function (3.7). Dashed lines show transition fronts of the short-scale variable in a wave train (a) or a single excitation pulse (b)

3.5 Advective Limit

185

0.6 2

0.4

v

0.2

L bΜ

1.5 1 0.5

0

0 -0.2

0.2 0.4 0.6 0.8 Ν bΜ

1

-0.4 -8

-6

-4

-2

0

x

Fig. 3.21. Proﬁle of v in a typical propagating pulse in an excitable system. Inset: dependence of the scaled width of the excited zone on the ratio ν/µ

single excitation pulse propagating into the quiescent state v = vs . The speed of the excitation front is c0 = c(vs ). The reverse transition in the relaxation tail may be delayed, in the absence of perturbations, to the turning point of the positive branch in Fig. 3.20b, but the back edge forming there would move faster and catch up to the level of v where both speeds equalize. The √proﬁle satisfying the linear equation (3.140) should be computed with c0 = 2vs , vs = −(ν + µ). An excitable state on the lower branch with −1 < vs < 0 exists at −µ < ν < 1 − µ. The solution in the comoving frame with the front edge at the origin and the back edge at −L, satisfying the boundary condition v(−L) = −v(0) = ν + µ, is  −(µ + ν) at x ≥ 0    x at −L ≤ x ≤ 0 (3.144) v = µ − ν − 2µ exp b(µ+ν)   µ x  −(µ + ν) − 2µ(1 + ) exp at x ≤ −L ν b(µ+ν) L = −b(µ + ν) ln |ν/µ|.

(3.145)

A typical proﬁle is shown in Fig. 3.21. 3.5.3 Chaotic Wave Trains Replacing a generic system with the phase portrait in Fig. 3.19 by a system with a zigzag phase portrait as in Fig. 3.20 has its price, as it eliminates oscillatory approach to the stationary state in the wake of a propagating excitation pulse. Oscillatory tails aﬀect in a nontrivial way the interaction between successive pulses, which can lead to spatio-temporal complexity of wave train structure (Elphick et al., 1988). We return to the general FN system in the advective limit (3.133), (3.135). After rescaling the coordinate x → γx, the system is rewritten in the comoving

186

3 Systems with Separated Scales

frame as cux + uxx + f (u, v) = 0, cγ

−1

vx + g(u, v) = 0.

(3.146) (3.147)

Suppose the system is excitable, having a single stationary state (us , vs ). Linearizing (3.146) and (3.147) near this solution leads to a cubic equation for the eigenvalues λ, which characterize the exponential tail of the pulse. In a certain parametric region, this equation has a pair of complex conjugate roots, λ = λr ± iλi , in addition to a real root λs , where λr and λs have opposite signs. Under these conditions, the ﬁxed point (us , vs ) of (3.146), (3.147), viewed as a dynamical system with the coordinate x playing the role of time, is a saddle-focus. We have seen in Sect. 1.5.2 that trajectories passing near this point are chaotic. This indicates that a train of pulses with oscillatory tails may form a chaotic sequence. Elphick et al. (1988) described this sequence in terms of the interaction of well-separated pulses centered at x = Xj (see Fig. 3.22). Because of the exponential decay toward the ﬁxed point both ahead and behind each pulse, it is suﬃcient to take into account interaction between the nearest neighbors only. For a train propagating in the negative direction, λr < 0 and λs > 0, and each pulse is inﬂuenced by the oscillatory tail of the preceding pulse located at x = Xj−1 and the exponentially weak precursor of the trailing pulse located at x = Xj+1 . The corrections to the propagation speed due to these interactions can be computed by applying a solvability condition of perturbation equations. In this way, one arrives at the sequence of equations X˙ j = a+ e−λs (Xj+1 −Xj ) + a− eλr (Xj −Xj−1 ) cos [λi (Xj − Xj−1 ) − θ] , (3.148) where the coeﬃcients a± and θ can be related to the parameters of the original system by computing the above-mentioned solvability condition explicitly. For a steadily propagating train, X˙ j = ∆c = const, where ∆c is the speed increment relative to the comoving frame. Then (3.148) can be converted into

Fig. 3.22. Schematic illustration of a train of pulses with oscillatory tails propagating to the left (Elphick et al., 1988; reproduced with permission. Copyright by the American Physical Society)

3.5 Advective Limit

187

(a)

0.1 1 0.05

fZ

0.8

0.05

0.6

0.1

0.4

0.2

0

0.2

0.4

0.6 Z

1

0.8

1.2

(b) 0.4 0.35 0.3

fZ

0.25 0.2 0.15 0.1 0.05

0

0.1

0.2 Z

0.3

0.4

Fig. 3.23. (a) The pattern map (3.150) with r = 0.9 ω = 10, δ = θ = 0. Inset: blow-up near the origin. (b) The iterated pattern map

188

3 Systems with Separated Scales

Fig. 3.24. Spacings between subsequent pulses ∆Xj = Xj − Xj−1 (Elphick et al., 1988; reproduced with permission. Copyright by the American Physical Society)

a pattern map analogous to a Poincar´e map of a dynamical system. Denoting Zj = a+ e−λs (Xj −Xj−1 ) ,

a=−

a− , a+

r=

λr , λs

ω=

λi , λs

δ=

∆c , a+ (3.149)

we present the pattern map in the form Zj+1 = δ + aZjr cos (ω ln Zj − θ) ≡ f (Zj ).

(3.150)

The ﬁxed points of this map are solutions with equal spacing of the pulses. As the parameters change, these solutions appear or annihilate in pairs (one stable and one unstable) at saddle–node bifurcations. The ﬁxed points accumulate at Z → 0, which corresponds to the inﬁnite spacing (see Fig. 3.23a). The map (3.150) cannot be iterated to obtain chaotic sequences in the same way as Poincar´e maps were in Sect. 1.5, since it contains nonphysical segments, with f (Z) < 0 (not shown in Fig. 3.23). One can, however, obtain solutions with unequal spacing as ﬁxed points of iterated maps, e.g. period-two sequences are ﬁxed points of the iterated map in Fig. 3.23b. Numerical simulations of (3.148) starting from an unstable equally spaced solution showed the tendency of the wave train to break into groups of pulses separated by “defects” with very large spacing (Elphick et al., 1988). The sequence of spacings in a train consisting of several hundred pulses is shown in Fig. 3.24.

3.6 Rotating Spiral Waves 3.6.1 Advective Limit for a Rotating Spiral We have seen in Sect. 3.4 that purely kinematic or local theories of spiral waves are unable to describe both front and back sides of a propagating excitation

3.6 Rotating Spiral Waves

189

band connected by the spiral tip, unless a sign-changing velocity–curvature relation is used as in Sect. 3.4.4. Nonlocal theory involving direct computation of the inhibitor ﬁeld is far more diﬃcult, but becomes tractable when the inhibitor diﬀusion is neglected in the advective limit. The inhibitor ﬁeld v(r, ϕ − ωt) of a rotating spiral should be stationary in a polar coordinate frame (r, φ = ϕ − ωt) rotating with some frequency ω, so far indeterminate. Equation (3.139) in its usual linear form, rewritten in the rotating polar frame, is similar to (3.140): −ωvφ = −v − ν ± µ.

(3.151)

The values of v at the front and back sides of the “excited” region situated at φ = Φ± (r) should be ﬁxed to ensure propagation of both curves without change of form with a constant angular velocity. Integrating (3.151) yields the respective values 1 + e2π/ω − 2e∆/ω 1 + e2π/ω − 2e(2π−∆)/ω , v− = −ν + µ , 2π/ω e −1 e2π/ω − 1 (3.152) where ∆ = Φ+ − Φ− . Using the kinematic equations is, generally, not practical, since the geometrical separation of the two fronts, ∆, cannot be eliminated. One can compute instead the shape of both fronts directly after substituting into the eikonal equation the normal propagation speed obtained by eliminating χ from (3.124), ωr , (3.153) c = ± 1 + [rΦ (r)]2 v+ = −ν − µ

together with the expression for curvature in polar coordinates κ(φ) =

2Φ (r) + r2 [Φ (r)]3 + rΦ (r) (1 + [rΦ (r)]2 )

3/2

,

(3.154)

and the values of v± given by (3.152). Although the Fife scaling (Sect. 3.5.1) is most suitable for analysing the motion of curved fronts in the advective limit, we shall keep using the eikonal equation in its original form (2.50) as in Sect. 3.4.3; the diﬀusivity D should be set equal to γ in the inertial and to unity in the Fife scaling. The resulting coupled equations for Φ± are 1/2 dr [rΦ± (r)] bv± ωr = 1 + [rΦ± (r)]2 + Φ± (r) + . D D 1 + [rΦ± (r)]2

(3.155)

This system, coupled to (3.152), was integrated numerically by Pelc`e and Sun (1991). Fine-tuning of the rotation frequency and the tip radius where the two curves meet is needed to achieve proper convergence to the asymptotic Archimedean shape. The solution is singular at the spiral tip where the advective approximation breaks down.

190

3 Systems with Separated Scales

A special case is a symmetric spiral, which can be constructed with the help of intrinsic equations of motion even when no local equation of motion is available (Keener, 1994). In the symmetric case (ν = 0), the bands of the up and down state should be equal, so that the values v± at the branches where either state advances should be of the opposite sign and have the same absolute value v0 independent of r. The respective velocity of straight-line front is π . (3.156) c0 = b|v± | = bµ tanh 2ω Thus, we have a local relation between the curvature and propagation speed, which is the same as in Sect. 3.4.3, but with the maximum speed c0 dependent on frequency. As before, the spiral local velocity and eventually its shape can be found by integrating (3.127) or (3.128). The diﬀerence is that either equation is now applicable on both symmetric branches separated by π. According to our usual sign convention, both propagation speed and curvature change sign when the advancing states interchange, which leaves the eikonal equation invariant. Another substantial diﬀerence is that the solution should be extended to the origin, leaving no hole. By symmetry, the tangential as well as normal velocity should vanish at the origin. The solution for both branches is therefore identical to the limit R, η → 0 in Sect. 3.4.3, and is attained at the same unique value |Ω| = ωD/c20 ≈ 0.331. In view of (3.156), the frequency can be obtained as the root of ω tanh−2

(bµ)2 π = 0.331 . 2ω D

(3.157)

The l.h.s. of this equation is a monotonically increasing function of ω; hence, there is a unique solution with frequency monotonically increasing with growing µ (Fig. 3.25). Ω 3 2.5 2 1.5 1 0.5 1

2

3

4

5

6

bΜ D

Fig. 3.25. Dependence of the rotation frequency ω on the parameter µ for a symmetric spiral

3.6 Rotating Spiral Waves

191

The troubling feature of the solution obtained in this way, as well as of the above numerical solution, is a discontinuity of v at the origin where the two branches with v = v± = ±v0 of the opposite sign come together. This can be mended only by taking into account the inhibitor diﬀusion (Keener, 1994). The value of v on the spiral branches turns out then to be dependent on the radial coordinate; this dependence should be taken into account at least in the vicinity of the origin. 3.6.2 Propagating Finger If a straight-line excitation pulse propagating along the x-axis in the plane is truncated at y = 0, the curved tip of the remaining semi-inﬁnite “ﬁnger” will lag behind the faraway segments continuing to propagate with an unchanged speed. Generally, such a ﬁnger would either grow or retract, but a special solution corresponding to a steady translating “critical ﬁnger” can be obtained by adjusting the parameters of the inhibitor equation (3.135). This solution plays a special role in the dynamics of a spiral tip to be considered in the next subsection, and forms an essential part of the so far most comprehensive analytical theory of rotating spiral waves (Hakim and Karma, 1999). We shall use the Cartesian frame with the origin at the tip and coordinates x and y directed, respectively, along and across the propagation direction and deﬁne the front and back sides of the ﬁnger by some function x = X± (y). The front side propagates into the stationary state of the inhibitor ﬁeld, and the normal velocity is fully determined by local geometry. The shape of the interface can be therefore computed with the help of intrinsic equations of Sect. 3.4. To preserve a stationary shape, the tangential velocity χ should be ﬁxed to ensure that each point be translated strictly along the x-axis while it propagates along the normal. The tangent and normal unit vectors are X (y) y − x y + X (y) x , n= , l= 2 1 + [X (y)] 1 + [X (y)]2

(3.158)

and y are unit vectors in the respective directions. The velocity where x components along the axes are now expressed as χ − cX (y) = 0, cy = (χl − cn) · y = 1 + [X (y)]2 χX (y) + c = cx = (χl − cn) · x = c0 , 1 + [X (y)]2

(3.159)

where c0 is the propagation speed, which should coincide by the absolute value with the speed of a straight-line propagating band approached at y → ∞. This implies (3.160) X (y) = χ/c, c2 = c20 − χ2 .

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3 Systems with Separated Scales

Using the latter relation in (3.127) we see that the ﬂux q must vanish, which, in view of (3.121), yields the equation deﬁning the change of the normal velocity along the arc length s: c (s) = κ

c20 − c2 .

(3.161)

In accordance with the eikonal equation, which we take again in its basic form (2.50), the curvature is κ = (c0 − c)/D. Using this in (3.161) yields a closed equation for the normal velocity, which can be easily integrated to c c0 s/D (2 + c0 s/D) = . c0 2 + 2c0 s/D + (c0 s/D)2

(3.162)

The origin s = 0 is placed at the tip where the normal velocity vanishes and the curvature reaches the maximum value κt = c0 /D; thus, the tip radius is Rt = D/c0 . At s → ∞, where the front is ﬂat, c → c0 and the curvature vanishes. The actual shape of the front can be restored with the help of (3.160) (y)]2 dy: supplemented by the metric relation ds = 1 + [X+ dy c = , ds c0

dX+ = ds

1−

c2 . c20

(3.163)

It is convenient to combine these equations with (3.161): dy Dc = , dc c0 (c0 − c) c20 − c2

D dX+ = . dc c0 (c0 − c)

(3.164)

These equations are integrated to obtain the function X+ (y) in the parametric form c c0 + c D c D − arcsin − 1 , X+ (c) = − ln 1 − y(c) = . c0 c0 − c c0 c0 c0 (3.165) The normal velocity at the back side depends on the local inhibitor value, which, in turn, is dependent on the local ﬁnger thickness. This dependence can be obtained, as in the preceding subsection, by solving (3.140). However, since the ﬁnger is presumed to be narrow and the shift of v relative to its stationary value vs (lying close to the stall value v ∗ ) to be small, the solution can be linearized by approximating the change of v across the ﬁnger by a linear function of the separation, ∆v = g + (v ∗ )(X+ − X− )/c0 .

(3.166)

This approximation works for the general form (3.135), and only the proportionality coeﬃcient is dependent on a particular form of the inhibitor equation. Accordingly, the normal velocity of a straight-line front corresponding to the

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193

local value of v at the back side is expressed as c = c0 − β(X+ − X− ). The parameter β = bg + (v ∗ )/c0 (where b is the derivative dc/dv computed at the stall value) serves in this context as a measure of excitability of the system, which increases as β decreases. In view of the position dependence, intrinsic equations of motion become insuﬃcient, and the equation for X− (y) should be obtained by using explicit expressions for the curvature and normal velocity in the Cartesian frame in the eikonal equation (3.6). Since this equation depends on X+ (y), which is deﬁned by the above analytic solution only implicitly, it is more convenient to solve the equations of both interfaces simultaneously. The system to be solved is 2 & 2 '3/2 dX+ dX+ D d2 X+ = 1+ − 1+ , (3.167) c0 dy 2 dy dy & 2 2 '3/2 dX− dX− D d2 X− β = 1+ + 1 − (X+ − X− ) 1 + . c0 dy 2 dy c0 dy The solution of this equation should satisfy the asymptotic condition X+ − X− = 2/β at y → ∞, since the propagation velocity far from the tip must be equal (by its absolute value) to c0 . On the other end, the curve X− (y) should match near y = 0 the front curve X+ (y) 2Dy/c0 . The problem is overdetermined, and the solution exists only at a particular value of the excitability parameter, β = βc . When (3.167) with some test value of β is solved numerically starting from the square root asymptotics at y → 0, the curve X− (y) turns away either upwards or downwards from the desired large y asymptotics, respectively, at β < βc or β > βc . The value βc /c0 = 0.535326 . . . can be found iteratively. The numerical solution is plotted in Fig. 3.26. The solution is weakly nonanalytic at y = 0, sincethe subleading terms in the asymptotics of X± (y) at y → 0 diﬀer: X+ (y) 2Dy/c0 + 13 , while X− (y) 2Dy/c0 + 13 (1 − 2β/c0 ). As in the preceding subsection, this is caused by a breakdown of the advective approximation at the spiral tip. The mismatch can be healed by taking into account the inhibitor diﬀusion near the tip (Hakim and Karma, 1999). A retracting ﬁnger solution can be constructed by allowing the propagation speed along the x axis ct to be larger than the normal speed of the straightline front c0 . The ﬁnger should be then inclined relative to the y axis, so that (y) (ct /c0 )2 − 1. The propagation speed ct asymptotically at y → ∞, X± should be used instead of c0 in (3.160) and (3.163). Accordingly, (3.161) is rewritten as c0 − c c2t − c2 , (3.168) c (s) = D and (3.164) takes the form Dc dy = , dc ct (c0 − c) c2t − c2

D dX+ = , dc ct (c0 − c)

(3.169)

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3 Systems with Separated Scales

7 X y

x

5 3

X– y

1 -1 0

5

10 y

15

20

Fig. 3.26. The critical ﬁnger obtained by integrating (3.167) with βc /c0 = 0.535326 (solid line) and a retracting ﬁnger satisfying (3.171) with ct /c0 = 1.05, β/c0 = 0.566907 (gray line). The coordinates are scaled by D/c0

which integrates to the parametric equation of the front interface:

  c0 c2t − cc0 + (c2t − c20 )(c2t − c2 ) D c0 c

y(c) = , ln − arcsin ct ct c2t − c20 ct (c0 − c) 1 + c2t − c20 D c X+ (c) = − ln 1 − . (3.170) ct c0 For simultaneous computation of the front and back interface, one has to integrate the equations generalizing (3.167): & 2 ' & 2 '3/2 dX+ dX+ ct D d2 X+ = , (3.171) 1+ − 1+ c0 dy 2 c0 dy dy & & 2 ' 2 '3/2 dX− dX− D d2 X− ct β = . 1+ + 1 − (X+ − X− ) 1 + c0 dy 2 c0 dy ct dy Retracting solutions exist at β > βc ; the dependence of the deviation β − βc on the ratio ct /c0 can be obtained as before by adjusting this parameter to arrive at the desired asymptotics when (3.171) is integrated numerically. Near β = βc , the dependence is linear: ct β − βc . =1+ c0 k

(3.172)

Hakim and Karma (1999) found the numerical value k ≈ 0.630 by expanding solutions of (3.171) near the critical ﬁnger and applying a solvability condition of the ﬁrst-order equation. This result is applicable as well at β < βc when

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195

the ﬁnger is advancing, and will be further used in Sect. 3.6.3 for the analysis of the rotating spiral band. At ct /c0 slightly deviating from unity, the tip of the retracting ﬁnger closely approximates the tip of the critical ﬁnger, as seen in Fig. 3.26. 3.6.3 Slender Spiral Band Analytical theory for a slender spiral with a large core radius was constructed by Hakim and Karma (1999). They analyzed the spiral shape by decomposing it into three overlapping regions where diﬀerent approximations can be applied. Close to the tip, the curvature of the tip trajectory can be neglected, and the shape of the excited region can be obtained by perturbing the critical ﬁnger solution described in Sect. 3.6.2. This yields a linear relation between the tip propagation velocity ct and deviation from the critical excitability ∆β = β − βc . Steadily rotating spirals exist when ∆β is negative. The shape of the critical ﬁnger in the inner region near the tip remains virtually unchanged as long as |∆β| is small. Far from the tip, the interfacial curvature is negligible, and the shape of the interface can be found by setting in the invariance conditions (3.124) c = c0 (v), the propagation speed of the straight-line front. The resulting equation is then the same as in the WR limit (Sect. 3.4.2), and the outer solution is given by (3.126), with R replaced by c0 /ω. This solution, as we know, breaks down at r → R. The inner radius R, which was interpreted as the radius of a circular obstacle (hole) in Sect. 3.4.2, has now a diﬀerent meaning of the core radius, and is deﬁned as the radial position of the tip. The inner and outer approximations do not overlap, but should be matched in an intermediate region where the interface is almost normal to the circle of rotation and its curvature is small. Recalling that the spiral end was normal to the hole circumference in the kinematic theory of Sect. 3.4.3, we observe that the radius of the “equivalent hole” should ﬁt, in fact, the radial position of this intermediate region rather than the tip position, and the rotation frequency is determined by the normal propagation speed in this region. The relation between the tip translation velocity ct , the core radius R, and the tip curvature κt = 1/Rt can be obtained by balancing the corrections to the short-scale and long-scale approximations in this region. The small parameter of the problem is the inverse of the Peclet number, Pe = Rc0 /D. In order to ﬁnd out a correct scaling in terms of ε = 1/Pe, we express tentatively the radial and angular variables as r = R(1 + εm y),

Φ = εn X,

(3.173)

with so far indeﬁnite exponents m, n > 0, and expand (3.155) in powers of ε. Retaining the leading terms only, we have ( )2 d2 X ε2(m−n) dX ct − c0 n + ε y = + εm−2n+1 . (3.174) c0 2 d y d y2

196

3 Systems with Separated Scales

All terms of this equation will be of the same order of magnitude if one chooses m = 1, n = 2/3, and sets ct /c0 − 1 = a ε2/3 ,

(3.175)

where a is an O(1) constant. Transforming now the variables y = 21/3 η − a, y ) = 2 ln ξ(η) reduces (3.174) to the Airy equation ξ (η) = ηξ(η). Of the X( two Airy functions solving this equation, one has to choose the one decreasing at η → ∞, Ai(η). The solution is therefore13 y ) = 2 ln Ai(2−1/3 (a + y)). X(

(3.176)

3/2 y ) − 1 (2 , matches the The asymptotics of this solution at y → ∞, X( 3 y) asymptotics of the outer solution (3.126) at r → R, as one can indeed see by expanding the latter near this point and rewriting the result in the rescaled variables (3.173). At y → 0, (3.176) should match the asymptotics of the ﬁnger solution (3.165) at y → ∞, 2D c0 y (3.177) ln √ . X+ (y) c0 2D

Since the argument of the logarithm in (3.176) has to be linear in y to enable the matching, 2−1/3 a must be a zero of the Airy function; this must be the largest zero, η0 = −2.3381 . . ., to ensure smooth connection to asymptotics at y → ∞. The solution for the back interface X− (y) is virtually identical in the intermediate and far regions to the solution for the front interface. Since in the tip region the curvature of the tip trajectory is negligible, the linear relation (3.172) remains valid and can be combined with (3.175) to obtain the dependence on excitability of both the ratio of the tip radius Rt = D/c0 to the core radius R and the rotation frequency ε=

ωD 1 Rt = 2 =√ R c0 2

βc − β k|η0 |

3/2 =

βc − β 1.856

3/2 .

(3.178)

The theory has to be further corrected by taking into account that the front interface of the rotating spiral band propagates not precisely into the quiescent state v = vs but rather into the tail of the preceding pulse where the relaxation of the controlling variable is still less than complete. The correction is necessary when the resulting velocity decrement is of the same order of magnitude as ct /c0 − 1 ∝ ε2/3 . The deviation of the controlling variable is computed by integrating (3.135) rewritten in the rotating polar frame. The result can be written precisely using the special linear form (3.151), but, since the shift ∆v across the narrow excitation band is always small in a 13

The other Airy function Bi(η), diverging at η → ∞, would describe a spiral rotating in the negative direction (clockwise).

3.6 Rotating Spiral Waves

197

weakly excitable system, the general form (3.135) can be linearized as well, and the only relevant constant characterizing the relaxation dynamics is the (vs ). Neglecting the width of a slender excited band, relaxation time τ = 1/g− the residual deviation is 2πR 2π βc exp − v(y) − vs = ∆v(y) exp − ≈ . (3.179) ωτ b c0 τ The latter expression is written taking into account that the shape of the perturbed band is close to that of the critical ﬁnger. The respective small velocity decrement, b(v−vs ), can be treated as a weak perturbation of the critical ﬁnger in the same way as ct /c0 − 1 was treated before. The relation (3.175) is modiﬁed in such a way that ct is compared not to the velocity of a single excitation pulse but with the velocity of a wave train with the wavelength 2πR (which coincides in the leading order with the asymptotic wavelength of the spiral): ct 2πR − 1 + 2 exp − (3.180) = 21/3 |η0 | ε2/3 . c0 c0 τ The relation modifying (3.178) obtained by perturbing the critical ﬁnger is 2πR 2/3 βc − β = k1 ε + k2 exp − , (3.181) c0 τ with k1 = 21/3 k|η0 | ≈ 1.856 and k2 ≈ 2.262. The solvability condition applied in this derivation eﬀectively averages the inﬂuence of the residual deviation ∆v(y) over the tip region. 3.6.4 Tip Meandering Interaction between subsequent excitation bands forming a rotating spiral wave may destabilize steady rotation. The instability arises at the spiral core and causes unsteady motion of the spiral tip. This phenomenon of tip meandering, observed both in experiment and numerical simulations, can be understood by extending the above large-core expansion to unsteady motion (Hakim and Karma, 1999). We continue to use the inverse Peclet number ε as the small parameter of the problem, and suppose that deviations of both the tip position and the rotation radius from their common stationary value, which we now denote as R0 , are of the same order of magnitude as the tip curvature radius, i.e. R − R0 = O(ε). Likewise, the instantaneous tip velocity ct should diﬀer from their steady-state value, now denoted as c0t = ωR0 , by O(ε). Thus, we set R = R0 + εR,

ct = c0t + ε ct .

(3.182)

In the leading order, the instantaneous rotation radius and tip velocity are related by (3.180) with the Peclet number based on R rather than R0 :

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3 Systems with Separated Scales

2/3 εR0 ct 2πR0 − 1 + 2 exp − . = 21/3 |η0 | c0 c0 τ R

(3.183)

Unsteady rotational motion or the spiral tip is most conveniently characterized by the tip velocity ct and the tip rotation rate ct /r, which is proportional to the vector product of the tip velocity ct and acceleration c˙t (the dot denotes the time derivative). The instantaneous tip position can be presented in polar coordinates as r = R[1 + ερ(t)],

φ = ωt + εψ(t).

(3.184)

It is advantageous to use complex representation for the tip position – written in the polar form as z = reiφ – and its time derivatives. Then we have (3.185) ˙ ct /R = Im z¨z¯˙ /|z| ˙2 , ct = |z|, where the bar denotes the complex conjugate. The right-hand sides of (3.185) are expanded to the ﬁrst order in ε as |z| ˙ = R[ω + ε(ψ˙ + ωρ)], Im z¨z¯˙ /|z| ˙ 2 = ω + ε(ψ˙ + ω −1 ρ¨). (3.186) Combining (3.182) and (3.186) cancels the zero-order terms, while the ﬁrstorder terms yield the dynamic equation for ψ, ρ: ct − ωρ, ψ˙ = R−1

ω −1 ρ¨ + ψ˙ = R−1 ( ct − ω R).

(3.187)

Both equations combine to a forced harmonic equation for the radial displacement ρ: (3.188) ρ¨ + ω 2 ρ = ω 2 R/R. The perturbation of the rotation radius can be expressed through the velocity perturbation with the help of a simpliﬁed relation (3.183) where the change of the stationary radius due to incomplete relaxation is neglected: R ct 3ε2/3 = 4/3 . R 2 |η0 | c0

(3.189)

Replacing also time by the angular coordinate brings (3.188) to the form d2 ρ ct 3ε2/3 + ρ = . dφ2 24/3 |η0 | c0

(3.190)

This equation, though looking innocuous, is in fact highly nonlocal because of the dependence of ct on perturbations of the inhibitor ﬁeld caused by perturbations of excitation pulses that had passed the same location in the past. Fortunately, in view of exponential decay of perturbations, it is suﬃcient to take into account only one immediately preceding pulse. Expressing ct through the latter’s position is the next task necessary for closing (3.190).

3.6 Rotating Spiral Waves

199

Since the origin of the coordinate y in the critical ﬁnger solution that serves as the basis for perturbations is located at the instantaneous tip position, the front of the propagating pulse feels at any given moment t, or at any given angular position of the tip φ, the residual value of the inhibitor that corresponds to its level v− at the back of the preceding pulse at the point y = y + ε[ρ(φ) − ρ(φ − 2π)]. The latter value, in its turn, depends on the level of v at the front of the preceding pulse at the same location and the local thickness of this pulse, Ξ = X+ − X− , as deﬁned by (3.166). This yields the relation between the inhibitor values at the front of the two successive pulses generalizing the stationary relation (3.179): β 2πR y ; φ − π) − vs exp − y ; φ − π) + Ξ( v(y; φ) − vs = v( . (3.191) b c0 τ The respective small velocity decrement, ct (y; φ) = b[v(y; φ) − vs ], can be treated as a weak perturbation of the critical ﬁnger in the same way as ct /c0 − 1 was treated before. A tedious but straightforward computation described in detail in the original publication by Hakim and Karma (1999) uses the solvability condition to express ct as a numerical function F (∆) of the difference between perturbations of the tip positions of the two successive pulses ∆ = ρ(φ) − ρ(φ − 2π). At large values of its argument, which bring either tip into interaction with the asymptotic (Archimedean) tail of its counterpart, this function should approach a constant value, while F (0) must vanish. The function is reasonably well approximated by a shifted hyperbolic tangent: F (∆) = tanh(∆ − α) + tanh α.

(3.192)

A qualitatively truthful picture of meandering transition and motion of the spiral in the supercritical region can be reconstructed by integrating the differential delay equation obtained by using (3.192) in (3.190): d2 ρ + ρ = mF [ρ(φ) − ρ(φ − 2π)]. dφ2

(3.193)

Beyond the instability threshold, the tip trajectory can be obtained by integrating (3.193) numerically. An example of an angular dependence of the deviation of the tip position is shown in Fig. 3.27. One can clearly see here a period-two orbit attained after a number of revolutions. As the parameter m increases, the trivial solution ρ = 0 loses stability. The instability threshold is detected in a standard way by linearizing (3.193) and seeking for solutions in the form ρ ∝ eλφ . This leads to the eigenvalue problem (3.194) 1 + λ2 = m 1 − e−2πλ . At any m, this equation is veriﬁed by λ = ±i. The respective eigenmodes correspond to simple translation of the spiral in the angular direction. Other solutions of (3.194) depend on the parameter m. At m 1, the r.h.s. of

200

3 Systems with Separated Scales Ρ 6 4 2 85

90

95

Φ 100 2 Π

-2 -4 -6

Fig. 3.27. Angular dependence of the deviation of the tip position ρ(φ) attained after a number of revolutions. Parameters: a = 0.2, m = 0.5

(3.194) may become of O(1) only when the real part of λ is negative, and therefore the trivial solution must be stable. As m increases, the roots of (3.194) move in the complex plane, and stability is lost when one of them crosses the imaginary axis. At the instability threshold m = mc , λ = iΩ is purely imaginary. Substituting this in (3.194) and separating the real and imaginary part yields the pair of equations deﬁning both the critical value mc and the incipient meandering frequency Ω: 1 − Ω 2 = mc (1 − cos 2πΩ)

mc sin 2πΩ = 0.

(3.195)

The second equation requires Ω = n/2 to be half-integer. The ﬁrst equation reduces then to 1 − (n/2)2 = mc [1 − (−1)n ]. For m > 0, the only solution of this equation is n = ±1, mc = 3/8. Thus, meandering sets on at this value of m with a frequency exactly twice less than the basic rotation frequency, indicating period doubling of the spiral rotation. 3.6.5 Phenomenology of Complex Spiral Motion Generally, meandering instability can be detected only numerically. For the FN model (3.133), (3.134) with the nonlinear function (3.7) and linear g(u, v) = −v − ν + µu, the computations were carried out by Barkley (1992). By symmetry, the spiral wave solution in the inﬁnite plane has three neutral eigenmodes: one with zero eigenvalue, corresponding to inﬁnitesimal rotations of the entire spiral, and two with imaginary eigenvalues, ±iω, corresponding to translations in the plane. Meandering instability arises as a result of Hopf bifurcation when another isolated pair of complex conjugate eigenvalues crosses the imaginary axis. The imaginary part of these eigenvalues, which we denote as ω , determines the frequency of the meandering motion near the bifurcation point. The eigenmode responsible for meandering instability has the form of

3.6 Rotating Spiral Waves

201

Fig. 3.28. Phase diagram for spiral-wave dynamics, showing the regions containing no spiral waves (N), stable rotating waves (RW), and modulated rotating waves (MRW). The latter two regions are separated by the locus of supercritical Hopf bifurcation. The dashed locus of modulated traveling waves (MTW) emerges from the resonant point on the Hopf curve and separates MRW states with inward (left) and outward (right) petals of spiral tip trajectories. Schematic tip paths are shown for six chosen parametric points (after Barkley, 1994; reproduced with permission. Copyright by the American Physical Society)

an attenuating wave propagating away from the zero curvature point at the spiral tip; the perturbation amplitude is maximal at the point of maximum curvature. Meandering instability in the system studied by Barkley (1992, 1994) is supercritical and saturates at ﬁnite amplitude, creating various meandering patterns determined by the ratio of the basic rotation frequency ω and the meandering frequency ω , which changes along the Hopf bifurcation locus in the parametric plane. A typical phase diagram in the parametric plane µ, ν for γ = 0.02 is shown in Fig. 3.28. This diagram can be reproduced qualitatively by an ODE model containing ﬁve equations for the frequency, coordinates, and the velocity of the spiral tip (Barkley, 1994). Following the meandering transition, steadily rotating waves (RW) with a circular tip trajectory give way to modulated rotating waves (MRW). The latter are quasiperiodic solutions, which are seen as periodic in the frame rotating with the basic rotation frequency ω. The tip trajectories form “ﬂowers” with either inward (for ω < ω) or outward (for ω > ω) petals. This causes mod-

202

3 Systems with Separated Scales

Fig. 3.29. Superspiral images and magniﬁed spiral tip trajectories for inwardly (left) and outwardly (right) meandering spirals (Sandstede and Scheel, 2001; reproduced with permission. Copyright by the American Physical Society)

ulation of the entire spiral wave structure, creating the “superspiral” seen in Fig. 3.29. At the boundary between these two types of motion, emerging from the resonance point ω = ω, the ﬂowers radii diverge, and the tip translates on the average in a certain direction; this intermediate structure can be characterized as a modulated traveling wave. More complex “hypermeandering” motion may arise at secondary bifurcations (Winfree, 1991). Besides meandering instability involving an isolated eigenvalue, spiral instability can originate in a continuous band limiting the range of stable wavelengths of plane waves. Since the wave emanated by the spiral core approaches a plane wave as its curvature decreases with distance, this instability is likely to arise in outlying spiral arms. The Eckhaus instability arising at short wavelengths never saturates, but results in nucleation of a pair of counterrotating spirals. In the ﬁnal state, the remaining spiral fragment is surrounded by a sea of vortex turbulence, as seen in Fig. 3.30d. Similar distinction between core and far-ﬁeld instabilities in spiral solutions of the CGL equation will be discussed in Sects. 5.1.2 and 5.3.4. A nontrivial interaction between both types of instabilities was suggested by B¨ ar and Eiswirth (1993) and further studied by B¨ ar and Or-Guil (1999) through simulations using a model with a somewhat more complicated function g(u, v). Tip meandering may destabilize outlying spiral arms through a large variation of the period caused by the Doppler eﬀect due to the tip motion. The wave trains are compressed in the direction of drift and expanded in the opposite direction, as seen in Fig. 3.29 or Fig. 3.30b. This may take the wavelength out of the stable range, initiating Eckhaus instability, which breaks the wave front through nucleation of vortex pairs as seen in Fig. 3.30c. The vortex turbulence of Fig. 3.30d may be the ﬁnal result of this process;

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203

Fig. 3.30. Spiral breakup in an oscillatory system. (a) Initial symmetric spiral; (b) development of modulational instability turbulence through spiral nucleation in the far ﬁeld; (c) spiral breakup and defect nucleation away from the tip; and (d) the ﬁnal state with a remaining spiral fragment surrounded by a sea of vortex turbulence (B¨ ar and Or-Guil, 1999; reproduced with permission. Copyright by the American Physical Society)

otherwise, it may develop directly as a result of a far-ﬁeld Eckhaus instability. B¨ar and Or-Guil (1999) argued that the Doppler mechanism is characteristic to excitable systems, while direct transition to spatiotemporal turbulence away from the core is prevalent in oscillatory systems. It should be noted that natural spiral wave patterns observed in numerous experiments as well as in model computations starting from random initial conditions are spatially disordered even under conditions when isolated spirals are stable. Typical multispiral patterns consist of domains of uneven sizes ﬁlled by rotating spirals and separated by shocks where the waves emanating from diﬀerent centers collide (more on this in Sect. 5.4.4). Interest to minor details of spiral motion has been sustained for decades by the supposed role of spiral excitations in heart ﬁbrillation. The notion of the heart tissue as a particular example of an excitable medium was powerful enough to persuade funding agencies to support studies based on universal models and lacking any physiological detail.

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3 Systems with Separated Scales

3.6.6 Scroll Waves The 3D extension of the spiral wave is a rotating scroll wave that has been observed in experiments with oscillatory chemical reactions (Winfree, 1987). The core ﬁlament of a scroll wave is a line vortex. A scroll wave with a straight-line core directed along the z-axis has identical spiral waves in each cross-section. Even then, the 3D extension can be nontrivial if the spiral phases are given a phase twist, i.e. are shifted along the z-axis. A curved core ﬁlament may also be closed into a ring or even form knots. A stable scroll structure evolves to decrease the ﬁlament curvature (Keener, 1988). This kind of dynamics is similar to curvature-driven motion of 2D fronts (Sect. 3.4.1), but may be reversed when the ﬁlament is unstable, as we shall see below. A steady scroll wave with a constant twist τ is a time-independent solution of applicable reaction-diﬀusion equations rewritten in rotating cylindrical coordinates r, φ = ϕ−ωt−τ z, z. In the advective limit, the equations stemming from (3.133) and (3.135) after rescaling the length are written as

(3.196) ut = ω∂φ + (∂z − τ ∂φ )2 + ∇2⊥ u + γ −1 f (u, v), vt = ωvφ + g(u, v),

(3.197)

where ∇2⊥ is the Laplacian in the plane transverse to z. The stationary ﬁelds u(r, φ) and v(r, φ) are also z-independent and are obtained by solving a 2D

Fig. 3.31. View of a restabilized core ﬁlament in the weakly nonlinear regime. (a) The instantaneous helical ﬁlament shown by the bold line. The dashed lines are the trajectories of the tip of the spiral in four regularly spaced horizontal planes. The dashed–dotted line is the axis around which the axis of the helical ﬁlament rotates. (b) Projections of the instantaneous ﬁlament on an horizontal plane at diﬀerent times are shown on the right by black bold circles. The bold radius in each circle shows the instantaneous ﬁlament point in the plane z = 0. The trajectory of this point is shown by the bold dashed–dotted line for several periods of rotation (evolution between circles 1 and 7) and by the gray line afterward (Henry and Hakim, 2002; reproduced with permission. Copyright by the American Physical Society)

3.6 Rotating Spiral Waves

205

problem. For γ 1, (3.196) can be replaced, as for a line front in the plane, by the eikonal equation (3.136), where the curvature κ should be now interpreted as twice the mean curvature of the activator front, in accordance with the expression (2.45) for the Laplacian in the aligned frame. Otherwise, the full system (3.196), (3.197) should be solved numerically. In either case, the rotation frequency ω should be determined as a function of the imposed twist and other parameters by solving a nonlinear eigenvalue problem. Detailed stability analysis of scroll waves was carried out by Henry and Hakim (2002). Since the linearized problem contains no explicit z-dependence, taking a Fourier transform eliminates the axial direction, so that the perturbation equations are purely two-dimensional but dependent on the axial wavenumber k. Thus, all 2D modes expand into k-dependent spectral bands. For zero twist, the eigenvalues are even functions of k. Owing to translational symmetry along the z axis, the most dangerous 3D perturbation modes are the long-scale modes in the spectral bands corresponding to the neutrally stable 2D modes speciﬁed in the preceding subsection. (a)

(b)

Fig. 3.32. Development of 3D turbulence due to core ﬁlament extension and breakup of scroll waves. (a) Snapshots of the core ﬁlament, starting from a closed loop. (b) Respective snapshots of wave patterns showing semitransparent visualization of the activator fronts (Alonso et al., 2004; reproduced with permission. Copyright by the American Physical Society)

206

3 Systems with Separated Scales

Instabilities may occur both in the meander band, which corresponds to z-dependent deformations of the 2D spiral meander modes, and in the translation band, which corresponds to z-dependent translations of the 2D spiral in the diﬀerent transverse planes. Henry and Hakim (2002) found that 3D meandering instability is most dangerous in the outward-meandering region to the right of the dashed line in Fig. 3.28, so that a scroll wave destabilizes through z-dependent meandering under conditions when a 2D spiral wave is stable. On the contrary, in the inward-meandering region on the left, meandering starts in the 2D mode. The most dangerous instability originates in this region in the translation band, causing spontaneous expansion and bending of the core ﬁlament. The meander band instability generally restabilizes as a distorted scroll wave with a twisted rotating core (Fig. 3.31). Instability in the translation band causes spontaneous bending of the scroll axis. This instability may occur also in the absence of twist. It is sometimes interpreted as “negative surface tension,” like that causing transverse instability of line fronts in the local approximation of Sect. 3.3.4. As a matter of fact, this eﬀect is nonlocal and is caused by interaction of diﬀerent wave fronts mediated by the long-range inhibitor ﬁeld. In contrast to the meandering instability, the bending instability does not saturate on the nonlinear level, but gives rise to a scroll wave with a continuously extending core (Fig. 3.32a). This leads to a turbulent state visualized as a tangle of breaking wave fronts (Fig. 3.32b). Bending instability of the core ﬁlament is more readily analyzed using the CGL model (see Sect. 5.5).

Fig. 3.33. View of a restabilized helical ﬁlament in the weakly nonlinear regime of spring instability. (a) The bold solid line represents the helical instantaneous ﬁlament, the dotted line the mean ﬁlament, and the thin solid lines the quasicircular trajectories of the instantaneous ﬁlament in horizontal planes of equally spaced z. (b) The thin solid line is the trajectory of the spiral tip in a horizontal plane and the bold dashed line is the trajectory of the mean ﬁlament in that plane. The mean ﬁlament rotates clockwise while the fast rotation of the spiral tip is counterclockwise. The position of the axis of the helix is marked by the dashed line (a) and the asterisk (b) (Henry and Hakim, 2002; reproduced with permission. Copyright by the American Physical Society)

3.6 Rotating Spiral Waves

207

For nonzero twist, the inversion symmetry along the z-axis is lost, and it only remains true that the eigenvalues λ(±k) are complex conjugate. Also, a neutrally stable eigenmode should combine translation along the z axis with rotation, and therefore, as follows directly from (3.196), it corresponds to k = ±τ rather than k = 0. These values remain extrema on the translation bands, and determine a ﬁnite marginal axial wavelength of a “sproing” instability which occurs in the translation band when twist exceeds a certain threshold. Unlike bending instability in the absence of twist, nonlinear development of this instability results in a restabilized helical wave, as seen in Fig. 3.33. When several unstable modes are present, simulations show core ﬁlaments taking a more complex shape, which is found to travel as a steady nonlinear wave in the vertical direction (Henry and Hakim, 2002).

4 Amplitude Equations for Patterns

4.1 Spatially Modulated Patterns 4.1.1 Ginzburg–Landau Equation A typical dispersion relation λ(k) in a spatially extended system just beyond a bifurcation point may have one of the forms shown in Fig. 4.1. In both cases, a narrow band of wavenumbers adjacent to the maximum of the dispersion relation (which may be reached either at k = 0 or at a ﬁnite k = k0 ) becomes unstable. Since the dispersion relation should be, generically, parabolic near the maximum, and the leading eigenvalue can be assumed to depend linearly on a chosen bifurcation parameter, say, µ, the width of the excited band scales as the square root of the deviation from the bifurcation point µ−µ0 . A spectral band of a ﬁnite width can be modeled by allowing the amplitude to change on an extended spatial scale, large compared to either k0−1 or any “natural” length scale characteristic to the underlying system. If the maximum of the dispersion relation is at k = 0 (curve 1 in Fig. 4.1a), the bifurcation does not break spatial symmetry, and the diﬀerential term supplementing the normal form (1.54) or (1.70) is symmetric. The equation of the lowest order in ∇ obtained near a bifurcation at zero eigenvalue is therefore just a diﬀusion equation with a polynomial nonlinearity (Sect. 2.1.1). In a similar way, the complex Ginzburg–Landau (CGL) equation (1.25) with a complex order parameter and complex coeﬃcients (to be studied in detail in Sect. 4.5) is obtained at a supercritical Hopf bifurcation when the dependence of the real part of the leading pair of complex conjugate eigenvalues on the wavenumber has the same form as curve 1 in Fig. 4.1a. If the dispersion relation is shaped as curve 2 in Fig. 4.1a, with the maximum attained at k 1, the representative equation is the Swift–Hohenberg (SH) equation (1.21), which has been originally introduced ad hoc as a simplest pattern-generating model. It can be derived formally in the course of expansion of a general reaction-diﬀusion system (RDS) near a bifurcation at zero eigenvalue when the components of the array of diﬀusivities D = D 0

210

4 Amplitude Equations for Patterns (a)

(b)

Λ

Λ k

1

k

2

Fig. 4.1. (a) Dispersion relations with an unstable band near k = 0 with the maximum either at k = 0 (curve 1) or at a small k (curve 2). (b) Dispersion relation with an unstable band at a ﬁnite k = k0

are adjusted in such a way that the amplitude diﬀusivity D = U † · D 0 U vanishes. For a cusp singularity, we can √ set D = D 0 + D 1 , and scale the long-scale spatial derivative as ∇ = O( ). Then the term D 0 U ∇2 a entering the second-order equation (1.60) in the formal procedure of Sect. 1.3.3 does not violate its solvability condition, but contributes to its solution u2 the term V ∇2 a with some vector V satisfying U † · V = 0. When this term is fed into the third-order equation (1.65), it contributes to its solvability condition the bi-Laplacian term K(∇2 )2 a with the coeﬃcient K = U † · D 20 V , as well as the diﬀusional term D1 ∇2 a with the coeﬃcient D1 = U † · D 1 U . The result, after proper rescaling, is the SH equation or, in the absence of inversion symmetry, its asymmetric extension with an added constant term. The same result can be obtained starting from a general operator with a similar dispersion relation. When the bifurcation occurs at a ﬁnite k = k0 , and the dispersion relation looks as in Fig. 4.1b, the amplitude, or “order parameter” characterizing the pattern, should be deﬁned as the envelope of a weakly distorted basic structure with the wavenumber k0 = O(1). Modifying the derivation of the amplitude equation for a regular pattern in Sect. 1.6.1, we express any mode satisfying the ﬁrst-order equation (1.110) as u1 = U a(t1 , t2 , x1 , . . .)eik·x0 + c.c.,

(4.1)

thereby adding to slow time dependence in (1.111) dependence on an extended coordinate x1 , which we distinguish from the short-scale coordinate, now denoted as x0 . The gradient operator is expanded tentatively as ∇ = ∇0 +∇1 +. . ., where ∇n acts on xn . This adds the term 2iD0 U (k·∇1 )a, stemming from the expansion of the Laplacian, to the second-order equation (1.112). The solvability condition of this equation is not aﬀected, since k0 = |k| is an extremum of the dispersion relation and, as a consequence, the scalar product U † · D 0 U vanishes. Similar to the above derivation of the SH equation, the second-order solution u2 now acquires the term 2iV (k · ∇1 )a with

4.1 Spatially Modulated Patterns

211

some vector V satisfying U † · V = 0, which contributes to the solvability condition of the third-order equation the term K(k · ∇1 )2 a with the coeﬃcient K = −4 U † · D 20 V . In 1D, the resulting amplitude equation reduces after rescaling and dropping the index of the extended coordinate to the real Ginzburg–Landau (RGL) equation (4.2) ut = uxx + u(1 − |u|2 ). This general form can be, of course, predicted out of symmetry consideration as the lowest order equation symmetric to an arbitrary constant phase shift of the complex “order parameter” u. The 2D extension of this equation (1.26) is applicable, however, only when the underlying system is anisotropic and admits a preferred direction of the wave vector k, say, along the x-axis.1 If the principal axes of D are directed along and across k, the x-component D x = D 0 satisﬁes the extremum condition U † · D 0 U = 0, while for the y-component D y this scalar product does not vanish and, as a result, the term U † · D y U ∂y2 a enters the solvability condition of the third-order equation alongside with K∂x2 a. The resulting amplitude equation is also anisotropic, but the scalings in both directions are of the same O(). If diﬀerent scales are chosen for the extended coordinates directed along and across k, the isotropic RGL equation is recovered: ut = ∇2 u + (1 − |u|2 )u.

(4.3)

4.1.2 Newell–Whitehead–Segel Equation Paradoxically, the amplitude equation must have an anisotropic form in an isotropic system, the source of anisotropy being the direction of the wave vector itself.2 Since the structure formed following the symmetry-breaking bifurcation is anisotropic, modulations of this amplitude along and across the chosen direction should be scaled diﬀerently. Indeed, if the direction of k is taken as the x-axis, adding a small longitudinal component, say, qx changes k = |k| by O(), while adding a transverse component of the same magnitude qy changes k by O(2 ) only. This points out that the derivatives √ in the x and y directions should be scaled strongly anisotropically: ∂y = O( ), as juxtaposed to ∂x = O(). With this scaling, the unidirectional derivative k · ∇1 in the derivation outlined in the preceding subsection is replaced by the operator k0 ∂x −(i/2)∂y2 . The solvability condition obtained in the third order is the Newell–Whitehead–Segel (NWS) equation (Newell and Whitehead, 1969; Segel, 1969). We write it in a rescaled form 1

2

A well-known example is convection in nematic liquid crystals, where the rolls can be given preferred orientation by treating the upper and lower surfaces of the convection cell (Bodenschatz et al., 1988). We restrict here to the case when a single mode is selected by nonlinear interactions, and the emerging structure is a striped pattern.

212

4 Amplitude Equations for Patterns

∂t u = ♥2 u + u − |u|2 u,

(4.4)

containing the NWS operator ♥ = ∂x −

i 2 ∂ . 2k0 y

(4.5)

The elaborate form of a highly nonisotropic mixed-order diﬀerential operator ♥ is, in fact, very natural. This is seen by considering its action on a √ plane envelope wave with the wave vector q = {qx , qy }: ( ) qy2 iq·x ♥e = i qx + (4.6) eiq·x . 2k0 The resulting factor coincides with the O() wavenumber increment due to the modulating wave: q = −1 (|k0 + q| − k0 ) = qx +

qy2 . 2k0

(4.7)

Thus, the NWS equation precisely accounts for the equivalence of all structures with identical wavenumbers, independently of the direction of the wave vector. Diﬀerent scalings in the directions along and across the stripes indicate that the latter are bent far more easily than they are compressed or extended. The NWS equation is not covariant and is very diﬃcult to analyze or compute. Indeed, even in a weakly distorted pattern, the orientation of the coordinate axes in the operator ♥ depends on the local phase gradient, so that this operator is in fact strongly nonlinear. The covariant formulation (Pismen, 1989) in a coordinate frame aligned with the pattern is also ill-suited for computations. 4.1.3 Variational Formulation Both RGL and NWS equations have a gradient structure ∂t u = −δF/δu, F = L dx,

(4.8)

with the Lagrangians LRGL = ∇u · ∇u + 12 (1 − |u|2 )2 ,

(4.9)

LNWS = ♥u ♥u + 12 (1 − |u|2 )2 .

(4.10)

We can observe that the complex amplitude is actually a shorthand for two real variables and can be presented in diﬀerent forms. We used so far the most economic symmetric form where the two variables are u and its complex conjugate u. Equivalent representations may be based, however, on the real

4.1 Spatially Modulated Patterns

213

and imaginary parts or the modulus ρ and the phase θ. The Lagrangian can be expressed in a way independent of the choice of a representation of the order parameter or, in geometric language, of coordinates ui in the order parameter space. For this purpose, we introduce the metric gij of the order parameter space by deﬁning an inﬁnitesimal interval ds as ds2 = gij dui duj

(4.11)

(summation over repeated upper and lower indices is presumed). The diﬀerential term possessing the required symmetry properties can be written as the scalar product gij ∇ui · ∇uj . Then the RGL Lagrangian is written as

(4.12) LRGL = 12 gij ∇ui · ∇uj + 12 (1 − |u|2 )2 . The NWS Lagrangian is written in the same way, with ∇ replaced by ♥ and its complex conjugate. The gradient dynamic equations are derived in a usual way by varying the energy integral F = L dx, and contain the inverse g ij of gij : (4.13) uit = −g ij δF/δuj . The RGL Lagrangian (4.9) is a particular case of the general covariant Lagrangian (4.12) based on the complex coordinates u, u and the metric tensor 01 gij = . Another useful expression employs polar coordinates ρ, θ and the 10 1 0 : metric tensor gij = 0 ρ2 LRGL =

1 2

|∇ρ|2 + ρ2 |∇θ|2 + 12 (1 − ρ2 )2 .

(4.14)

The RGL Lagrangian has the most basic form compatible with the symmetry of a system described by a complex order parameter, and is applicable in the vicinity of equilibrium phase transitions in a variety of physical systems. Pattern formation is largely a nonequilibrium phenomenon, and the RGL or NWS equation may be derived starting from an underlying system that does not possess gradient structure. Thus, this structure may be only approximate and disappear in higher orders of the expansion. Both RGL and NWS equations and the corresponding Lagrangians are invariant to shifting the phase of the complex amplitude by a constant increment. This invariance reﬂects symmetry of the regular pattern to translation along its wave vector. 4.1.4 Stationary Solutions It is usually advantageous to work with the polar form of (4.3) obtained by setting u = ρeiθ , removing the phase factor, and separating the real and imaginary parts:

214

4 Amplitude Equations for Patterns

ρt = ∇2 ρ + (1 − |∇θ|2 − ρ2 )ρ,

(4.15)

2 θt = ∇2 θ + ∇ρ · ∇θ. ρ

(4.16)

These equations can also be obtained directly by varying (4.14). Evident stationary solutions of (4.15), (4.16) are periodic structures θ = q · x, ρ = ρ0 = 1 − q 2 . (4.17) Recalling the origin of the RGL equation as an amplitude equation for the complex envelope of a striped pattern, we recognize q as a small addition to the wave vector k0 of the underlying periodic structure. The magnitude of q is restricted by the condition q = |q| < 1 deﬁning the (rescaled) width of the excited wavenumber band. Other stationary solutions can be detected by rewriting the phase equation (4.16) in the form (4.18) θt = ρ−2 ∇·j, j = ρ2 ∇θ. The phase is stationary when j = const. Using this in (4.15) reduces the stationary equation of the real amplitude to a single diﬀusion equation

∇2 ρ − V (ρ) = 0, V (ρ) = − 21 ρ2 − 12 ρ4 + (j/ρ)2 , (4.19) where j = |j|. This potential is monotonic at j ≥ 2/33/2 , but at lower values of j it has a minimum and a maximum (Fig. 4.2), which correspond to two stationary solutions. Replacing ρ = ρ0 , j = qρ20 , one can see that the minimum coincides exactly with the periodic solution (4.17); this solution, however, disappears at q 2 = 1/3, ρ20 = 2/3, i.e., j = 2/33/2 . We shall see in the 0

VΡ

-0.1 -0.2 -0.3 -0.4

0

0.2

0.4

0.6

0.8 Ρ

1

1.2

1.4

Fig. 4.2. Plots of the potential (4.19) with j 2 varying from 0 (upper curve) to 5/27 with the increment 1/27. The dashed line is the locus of the extrema

4.1 Spatially Modulated Patterns

215

following subsection that this coincides with the Eckhaus instability limit of the periodic solutions. One-dimensional inhomogeneous solutions of (4.19) correspond to quasiperiodic structures. They can be obtained in the same way as in Sect. 2.1.5. The solutions given by (2.25) can be presented as orbits in the phase plane ρ, p = ρ (x) surrounding the unstable homogeneous stationary state. They exist at the values of V lying between the minimum and the maximum of V (ρ) in Fig. 4.2. The lower bound of this interval corresponds to a homoclinic orbit approaching at x → ±∞ a stable periodic structure (4.17). These solutions constitute a family of dark solitons parametrized by the asymptotic wavenumber q: (4.20) ρ2 (x) = 2q 2 + 1 − 3q 2 tanh2 x (1 − 3q 2 ) /2 . The amplitude proﬁle spreads out and becomes more shallow with increasing q (Fig. 4.3). The simplest representative of this family is the solution with j = 0, V = −1/2, q = 0, which approaches at x → ±∞ the basic state ρ = 1, q = 0. It is identical to the stationary front solution (2.24), which can √ be written now as ρ(x) = tanh x/ 2. The phase turns by π at the singular point x = 0 where the real amplitude vanishes and the phase is indeﬁnite. Unlike vortices in 2D (Sect. 4.3), the singularity is not topological, and the solution, which is not protected by conservation of a topological charge, is not expected to be stable. According to (4.14), (4.17), the energy density of periodic solutions L = 1 2 1 2 1 − increases with q and exceeds the energy of the uniform state. q q 2 2 Nevertheless, they correspond to local energy minima and are stable to inﬁnitesimal perturbations as long as the wavenumber q is suﬃciently small. One can ask how it may be possible that the solutions belonging to a continuΡ 1 0.8

q0.5

0.6 0.4 0.2

q0

1

2

3

4

5

x

Fig. 4.3. Dark solitons (4.20); q varies from 0 to 0.5 with the increment 0.1

216

4 Amplitude Equations for Patterns

ous branch are all stable. The reason is that periodic solutions on the inﬁnite line cannot be deformed one into the other by inﬁnitesimal perturbations even when their wavenumbers are inﬁnitesimally close, since the diﬀerence between them accumulates at long distances.3 One can also observe that stable solutions should be separated by unstable solutions marking the boundaries of their attraction basins. Those are the quasiperiodic and homoclinic solutions found above. Though unstable, they play an important role in dynamics, as they correspond to saddle points of wavelength-changing transitions accompanied by disappearance of one or more periods (rolls, or stripes) constituting the pattern (Kramer and Zimmermann, 1985). Periodic stationary solutions of the NWS equation (4.4) have the same form (4.17) with q deﬁned by (4.7). Although the polar representation of (4.4) is quite clumsy, this follows directly from (4.4) in view of (4.6). The quasiperiodic solutions of RGL apply to NWS as well, provided the ﬂux j is directed along the basic wave vector k0 . 4.1.5 Stability of Stationary Solutions The study of stability of a stationary striped pattern with a spatial period L, u0 (x) = u0 (x + L), is a Floquet–Bloch problem, which is solved by presenting an inﬁnitesimally small perturbation u (x, y, t) = u(x, y, t) − u0 (x) in the form u (x, t) = U eλ(K)t+iK·x w(x),

(4.21)

where w(x) is L-periodic. The speciﬁc feature of this representation is that the basic periodicity is extracted from the perturbation. The component Kx of the perturbation wave vector K along the x-axis should lie within the limits 0 ≤ Kx < 2π/L (the ﬁrst Brillouin zone of the striped structure). As in the standard case of homogeneous states (Sect. 1.2.1), the stability condition is Re λ(K) ≤ 0 for all K. In a spatially homogeneous inﬁnitely extended system, there is always a neutrally stable (Goldstone) mode u (x) = u0 (x). This is a consequence of translational symmetry, implying neutral stability to a phase shift u0 (x) → u0 (x + φ) with φ = const. Most commonly, the most dangerous perturbation modes have vanishing wavenumber and can be viewed as long-scale distortions of neutrally stable translational modes. Loss of stability at K = 0 may cause formation of more complicated structures, e.g., period doubling when Kx is twice the wavenumber of the basic structure k. A distorted 2D structure would form when stability is lost to a perturbation mode with K noncollinear with the wave vector of the basic structure.4 Most commonly, instabilities of this kind do not saturate but lead to a spatially chaotic (turbulent) pattern. 3

4

On a ﬁnite interval, the paradox disappears, since the wavenumbers become discrete and no longer form a continuous band. It should not be mixed up with a “sideband” instability leading to excitation of a primary mode with a diﬀerent wave vector, which can be treated in the framework of amplitude equations without spatial dependence as in Sect. 1.6.1.

4.1 Spatially Modulated Patterns

217

Both RGL and NWS equations being gradient systems, their periodic stationary solutions exhibit only simplest kinds of instabilities. According to the general Floquet–Bloch method, stability of (4.17) is checked by presenting the perturbed solution in the form u = eiq·x 1 − q 2 + U (K) exp[λ(K)t + i K · x , (4.22) where the basic periodicity, expressed by the factor eiq·x , is extracted from the perturbation. Linearizing (4.3) leads to the eigenvalue problem U (K) U (K) (1 − q 2 ) + 2K · q + K 2 1 − q2 =− . 2 2 2 · 1 − q (1 − q ) − 2K · q + K U (−K) U (−K) (4.23) The determinant of the above matrix is K 2 {K 2 +2[1−q 2 (1+2 cos2 χ)]}, where K = |K| and χ is the angle between K and q. As the determinant increases with K, the instability ﬁrst sets on in the long-scale mode. The most dangerous perturbations are directed along the wave vector of the stationary pattern q. The determinant is negative at small K, indicating long-scale instability, at √ q > 1/ 3. This is the Eckhaus instability limit conﬁning the range of stable wavenumbers. Stability analysis of periodic solutions of the NWS equation is carried out in the same way. In spite of the complicated form of the ♥ operator, the algebra is made easy by using (4.6), (4.7), and (4.22). Then (4.23) is rewritten as λ

λ

U (K) U (K) 1 − 2(1 − q 2 ) + K+ −(1 − q 2 ) = · , 2 −(1 − q ) 1 − 2(1 − q 2 ) + K− U (−K) U (−K)

(4.24)

where K± = −1 |k0 + q ± K| − k0 . It is suﬃcient to assume q to be parallel to the basic wave vector k0 and to consider separately perturbations with K parallel and normal to this direction. For longitudinal perturbations, K± = q ±K, and the limit of Eckhaus instability is the same as in the RGL case. For transverse perturbations, (4.7) yields K± = q+K 2 /(2k0 ), and the determinant of the matrix in (4.24) is approximated at K 1 as 2(q/k0 )K 2 (1 − q 2 ). It is negative at q < 0, which sets the limit of the long-scale zigzag instability. This implies that all patterns with the wavelength longer than the basic wavelength 2π/k0 are unstable to long-scale transverse perturbations. Since both Eckhaus and zigzag instability occur at K → 0, their thresholds can be obtained in a simpler way by using the phase dynamics approach, as described in the following section. Recalling the basic scaling of the amplitude equation, where a parametric deviation from the bifurcation point is scaled as 2 and the spatial scale of distortion as , we observe that the band of stable wavenumbers scales as the square root of supercriticality. The eﬀective band width observed in experiments and simulations is more narrow, since selection of wavelengths close to the optimal one at “band center” is facilitated by motion of defects in a disordered pattern (see Sect. 4.3.1).

218

4 Amplitude Equations for Patterns

4.2 Phase Dynamics 4.2.1 Universal Form of Phase Equations The notion of a phase variable that appears in the polar form of the amplitude equations can be extended to large-amplitude striped patterns, which might be predominant away from the symmetry-breaking bifurcation point. A striped pattern is deﬁned as an ordered spatially periodic structure with translational symmetry broken along a certain direction. Although the basic structure cannot be presented in a simple harmonic form, it can be characterized locally by the wave vector k. The phase variable θ is introduced in such a way that k = ∇θ. In an isotropic system, the direction of k can be arbitrary, and diﬀerent orientations may prevail in diﬀerent far removed locations of a spatially extended system. As a rule, stable patterns can exist within a certain range of wavenumbers. Thus, besides changes of orientation, the wavelength of the striped pattern can vary between diﬀerent locations. A regular pattern with k = const and k = |k| lying within the stable range should be stationary, but a pattern arising naturally in an extended system is likely to be distorted. Distorted structures are, generally, nonstationary. It is also useful therefore to deﬁne the “frequency” ω = −θt . The wave vector and frequency are related by the integrability condition kt + ∇ω = 0.

(4.25)

A universal macroscopic description of weakly distorted and slowly evolving striped patterns, independent of either their short-scale structure or the nature of underlying equations, is provided by the phase dynamics formalism. The idea of phase dynamics has been ﬁrst put forward in the context of convective patterns by Pomeau and Manneville (1979), and formulated in a more general way by Cross and Newell (1984). The distortion can be treated as weak when the wave vector k is slowly varying on an extended scale far exceeding the wavelength of the underlying structure: k = k(X, T ), where X = x and T = t are extended spatial and temporal variables, respectively, and is a small parameter equal to the ratio of the prevailing wavelength to a characteristic macroscopic scale of pattern distortions. Equations of phase dynamics that describe evolution of a weakly distorted pattern on an extended time scale can be constructed by setting in original (where ∇ is the gradient with respect to the equations ∂t = 2 ∂T , ∇ = ∇ extended spatial variables), and expanding in powers of the scale ratio . Introducing also the extended phase variable Θ = θ and the rescaled frequency Ω = ω/, one can write = 0. k = ∇θ = ∇Θ, Ω = −−1 θt = −ΘT , kT + ∇Ω

(4.26)

If all dependent variables in the original system are scalars, the general form of the phase equation is determined by scaling and symmetry considerations alone. The frequency is a scalar and may depend only on scalars formed

4.2 Phase Dynamics

219

In the leading order, the only possible combinations with the help of k and ∇. and ∇·k, are n · ∇k where n = k/k is the unit vector normal to a line θ = const. Thus, the most general form of a rotationally invariant phase equation should be + D2 ∇·k, (4.27) −Ω = D1 n·∇k where D1 , D2 are phase diﬀusivities that both depend on a particular underlying problem and are, generally, functions of k. This equation can also be rewritten through the extended phase variable Θ: 2 Θ. 2 Θ + D2 ∇ ΘT = D1 (n·∇)

(4.28)

Another equivalent form is + kD⊥ ∇·n, −Ω = D n·∇k

(4.29)

where D = D1 + D2 and D⊥ = D2 are, respectively, the longitudinal and transverse phase diﬀusivities. The phase equation (4.28) is, in fact, strongly nonlinear due to the dependence of both the diﬀusivities and the direction of the unit vector n on the local phase gradient. It can be linearized, yielding an anisotropic diﬀusion equation, only when deviations from a prevailing wave vector k = k0 are arbitrary small. If the X- and Y -axes are drawn, respectively, along and across k0 , (4.29) is reduced to ΘT = D (k0 )ΘXX + D⊥ (k0 )ΘY Y ,

(4.30)

where the longitudinal and transverse phase diﬀusivities D , D⊥ both depend on k0 = |k0 |. The pattern with the wavenumber k0 is stable to long-scale perturbations when both phase diﬀusivities are positive. Vanishing D corresponds to the Eckhaus instability and vanishing D⊥ to the zigzag instability. 4.2.2 Variational Formulation Cross and Newell (1984) presented the phase equation as · [kB(k)]. C(k)ΘT = ∇

(4.31)

This form makes transparent the gradient structure of the phase equation, which can be presented in the form δF , F = L dx, (4.32) C(k)ΘT = − δΘ with the Lagrangian

k

L(k) =

kB(k)dk = k0

1 2

k2

B(k 2 )d(k 2 ). k02

(4.33)

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4 Amplitude Equations for Patterns

In this approximation, the energetic cost is totally determined by deviation from the optimal wavenumber k0 . Keeping in mind that variations of k are restricted by (4.26), the variation of (4.33) is computed as · [kB(k)]δΘ. δL = kB(k)δk = B(k)k·δk = −∇

(4.34)

Hence, the Euler–Lagrange equation of (4.33) reduces to (4.31) or (4.29), where the longitudinal and transverse phase diﬀusivities are related to the functions B(k), C(k) as D =

1 d2 L 1 d(kB) = , C(k) dk C(k) dk 2

D⊥ =

B(k) . C(k)

(4.35)

Longitudinal stability of the pattern, dependent on the sign of D , is determined by the sign of the second derivative of the energy density with respect to the wavenumber. This can be understood qualitatively in the following way. A deformation of the striped pattern changing locally its wavelength causes a local compression on the one side and local dilation on the other side. This causes an increase of the overall energy, indicating stability, when the dependence L(k) is concave, but a decrease, indicating instability, when the dependence is convex, as seen in Fig. 4.4. For a gradient system, the phase equation can be deduced directly from the variational formula (4.32), assuming that the entire spatial structure of the pattern can be expressed through the phase variable. A simple example is given by the RGL equation (4.3). For the time being, we can forget that the amplitude u is a complex envelope of an underlying short-scale pattern, 0.5 0.4 0.3 0.2 0.1

0.2

0.4

0.6

0.8

1

q

Fig. 4.4. Plot of the dependence of the energy density on the wavenumber L(k) (4.36) containing a concave and convex segments, respectively, to left and to the right of the dashed-dotted line corresponding to the Eckhaus instability limit. Dashed lines indicate the gain or loss of energy in the respective regions

4.2 Phase Dynamics

221

and identify q with k = ∇θ. Stationary solutions of (4.3) are given by (4.17). Neglecting in the leading order ∇q q 2 , the Lagrangian (4.14) is expressed through q only: (4.36) L(q) = 12 q 2 (1 − 12 q 2 ). The coeﬃcient C(q) = ρ2 = 1 − q 2 is found by comparing (4.32) to (4.13) and identifying it with the phase component of the metric tensor gij (i.e. inverse of g ij ). Comparing next (4.36) to (4.33), where we replace now k by q, yields B(q) = 2 dL/d(q 2 ) = 1 − q 2 ,

(4.37)

and, according to (4.35) , D =

1 − 3q 2 1 d(qB) = , C(q) dq 1 − q2

D⊥ = 1.

(4.38)

Thus, q 2 = 1/3 is the Eckhaus instability limit, in accordance with the results of Sect. 4.1.5. The phase equation following from the NWS Lagrangian (4.10) does not reduce to the form (4.31) due to the dependence on transverse derivatives of k generating fourth-order derivatives in the Euler–Lagrange equation. The correct leading-order expression for the NWS Lagrangian obtained by expressing the complex amplitude in (4.10) in the polar form and making use of (4.7), (4.17), is (4.39) L = 12 q 2 (1 − 12 q 2 ) + 18 (1 − q 2 )(∂qy /∂y)2 , where q is deﬁned by (4.7). The phase equation can be obtained in two different forms, depending on whether it is expressed through the phase θ of the amplitude u = ρeiθ or through the total phase of the underlying pattern, which we denote now as ϑ. The relation between the variations of q and θ is obtained in both cases from the two alternative forms of (4.7): q = −1 (|∇ϑ| − k0 ) = θx +

θy2 . 2k0

(4.40)

The last expression leads to a very complicated nonlinear fourth-order equation.5 Its awkward structure is, in essence, a consequence of the noncovariance of the NWS equation. Using the total phase yields a more transparent expression (see also Sect. 4.2.4). As follows from (4.7), (4.40), the variations of q and ϑ are related as (4.41) δq = |k0 + q|−1 (k0 + q) · δq = n · δ(∇ϑ), where n is the normal vector directed along the corrected wave vector k0 +q. If the last term in (4.39) is omitted, the variation δL retains the form (4.34): 5

A simpliﬁed form of this phase equation valid near the band center will appear in Sect. 4.3.3.

222

4 Amplitude Equations for Patterns

· [nqB(q)] δϑ. δL = (dL/dq) δq = −∇

(4.42)

Hence, the Euler–Lagrange equation retains the form (4.31) with k replaced by q and B, C dependent on q only: · [nqB(q)] . C(q)ϑT = ∇

(4.43)

The expression for D in (4.38) deﬁning the Eckhaus instability limit remains without change. The expression for the transverse diﬀusivity D⊥ remains without change as well, but the factor q, which may be negative, replaces nonnegative k in (4.29). This signals zigzag instability at q < 0, which destabilizes patterns with the wavelength larger than optimal. Bringing back the last term in (4.39) generates higher order terms stabilizing the zigzag instability at shorter wavelengths and ﬁnite amplitudes, but does not aﬀect the instability threshold. One should keep in mind that when an amplitude equation, like (4.4) or (4.3), is taken as the “original” system, one should assume that the phase changes on the scale exceeding the spatial scale of this equation (already extended compared to the scale of the basic structure). Only under this condition, the real amplitudes are slaved to the phase, and the Lagrangian can be presented as above in the leading order as a function of q only. 4.2.3 Long-Scale Expansion A phase equation with the gradient structure (4.32) may also be valid when the original system is nongradient; then the gradient structure would be destroyed in higher order approximations. Consider a general reaction-diﬀusion system (1.18) with a constant diﬀusivity matrix D: (4.44) ut = D∇2 u + f (u). Suppose that stationary spatially periodic solutions of (4.44) can be written in the form of a 2π-periodic function u = u0 (θ; k) parametrized by the phase variable θ instead of a spatial coordinate and dependent on the wavenumber k. The function u0 (θ; k) veriﬁes the stationary equation D(k 2 )u0θθ + f (u0 ) = 0.

(4.45)

This function must be even, u0 (θ; k) = u0 (−θ; k), due to the symmetry to inversion of the wave vector k in an isotropic system. A weakly distorted and slowly evolving pattern is sought for as an expansion u(X, T ) = u0 (θ(X, T ); k(X, T )) + u1 (θ, X, T ) + · · · ,

(4.46)

where the small parameter is deﬁned, as in Sect. 4.2.1, as the ratio of the prevailing wavelength to a characteristic scale of long-scale distortions. The expansion uses the relations

4.2 Phase Dynamics

u0t = u0θ Ω + O(2 ), ∇u0 = ku0θ + u0k ∇k,

· k + 2u0kθ k · ∇k ∇2 u0 = ku0θ + u0θ ∇ + O(2 ).

223

(4.47)

The ﬁrst-order equation is a linear inhomogeneous equation of a general form Lu1 + Ψ (u0 ) = 0

(4.48)

containing the linear operator L = Dk 2 ∂θ2 + fu ,

(4.49)

where fu is the Jacobi matrix of the function f (u0 ) evaluated at u = u0 (θ), and the inhomogeneity

· k + 2u0 k · ∇k − u0θ Ω. (4.50) Ψ = D u0θ ∇ kθ The operator L has an eigenfunction with zero eigenvalue (Goldstone mode) u0θ which corresponds to the translational symmetry of the periodic solution. Generally, this operator is not self-adjoint, and one has to compute the eigenfunction ϕ† (θ) of the adjoint operator L† = D † k 2 ∂θ2 + fu † ,

(4.51)

containing transposed matrices D † and fu † .The adjoint eigenfunction should also be even in θ. The solvability condition of (4.48) is

2π

ϕ† (θ)Ψ dθ = 0.

(4.52)

0

Using here (4.50) yields the phase equation (4.27) with the phase diﬃsivities 2π 2π † 0 D1 = 2k ϕ (θ)Dukθ dθ ϕ† (θ)u0θ dθ , 0 0 2π 2π † 0 ϕ (θ)Duθ dθ ϕ† (θ)u0θ dθ . (4.53) D2 = 0

0

4.2.4 Covariant Phase-Amplitude Equation The reader may have noticed that the phase dynamics approach, when applied to a striped pattern in an isotropic system, yields symmetric covariant expressions, whereas the amplitude equation at a symmetry-breaking bifurcation point (4.4) is noncovariant and contains a higher order derivative. One can ask what is the root of this contradiction and whether it might be resolved. The expansion leading to the phase equation is essentially simpler than that leading to the amplitude equation, even though computation of the integrals (4.53) may be technically diﬃcult. Indeed, phase dynamics involves

224

4 Amplitude Equations for Patterns

only one dependent variable – phase – while in the amplitude expansion the real amplitude is an additional dynamical variable not slaved to the phase. This, however, has nothing to do with the loss of covariance, which is caused, essentially, by extraction of the basic structure, making it necessary to introduce the operator ♥ dependent on local orientation. In the following, we derive an alternative amplitude equation using a procedure similar to derivation of the phase dynamics equation in the preceding subsections but applicable near a bifurcation point where the dependence on the phase can be expressed in a simple harmonic form but the real amplitude is not slaved to the phase. The covariance will be retained at the price of keeping as a dynamic variable the total phase of the emerging pattern, rather than an envelope phase as in the NWS equation. Since the form of the amplitude equation should be universal, it can be obtained starting from any underlying system. We start therefore with a simple example taking as the original system the SH equation (1.21) with µ = 2 : ut = −(1 + ∇2 )2 u + u(2 − u2 ).

(4.54)

A striped pattern with the wavenumber k0 = 1 bifurcates at = 0. At small , we are looking for an expansion in the form (4.55) u = ρeiϑ + c.c. + 2 u2 + · · · . We assume that the phase obeys the relations ∇ϑ = k, |k| = k0 + 12 q.

(4.56)

Both the real amplitude ρ and the wavenumber mismatch q are allowed to change on an extended scale, so that ∇ = O() except when applied to the phase itself. The expansion is made easy by using the relation (1 + ∇2 )ρeiϑ = (−q + 2ik · ∇ + i∇ · k) ρeiϑ + O(2 ).

(4.57)

There are no terms of O() in the expansion of the r.h.s. of (4.54); hence, ∂t scales as 2 . The second-order equation can be written, after compounding the operator (4.57) and separating the real and imaginary part, in the form

ρt = 1 − q 2 − ρ2 + (∇ · k)2 + 2k · ∇(∇ · k) ρ + 4(∇ · k) k · ∇ρ + 2(k · ∇)2 ρ, (4.58) −2 2 (4.59) ϑt = 2ρ ∇ · kqρ . The phase equation (4.43) immediately follows from (4.59) when ρ = 1 − q 2 is slaved to the phase. Unfortunately, (4.58) has a very complicated form illsuited for numerical simulations, and the short-scale phase variation precludes using a sparse grid ﬁt to a slow change of the wave vector and real amplitude. The phase can be excluded by applying the gradient operator to (4.59):

4.3 Defects in Striped Patterns

2

kt = 2∇[ρ−2 ∇ · kqρ ].

225

(4.60)

This vector equation preserves the compatibility condition ∇ × k = 0 and can be solved on a long-scale grid together with (4.58). Having three equations replacing a single scalar SH equation that served as the basis for the derivation might be too high a price for universality and covariance. The SH equation remains the most handy tool for qualitative numerical modeling of striped patterns. An alternative equation, derived in an ad hoc way, was recently suggested by Qian and Mazenko (2004). This model equation, called a “nonlinear phase model” by the authors, can be written in the form of a vector Cahn–Hilliard equation for the gradient of the local wavenumber Q = ∇k: . (4.61) Qt = −∇ ∇ · ∇2 Q + 1 − |Q|2 This equation has a gradient structure

δF Q = −∇ ∇ · (∇ · Q)2 + 12 (1 − |Q|2 )2 dx. , F = 12 δQ

(4.62)

The algebraic term encourages relaxation to the “optimal” wavenumber |Q| = 1, so that this model, like SH, generates in the long run an ordered pattern with the optimal wavelength, unless restricted by boundary conditions, e.g., the condition requiring the wave vector to be parallel to the walls. Numerical simulations using the model (4.62) produce, however, at intermediate stages distorted striped patterns rather diﬀerent from those generated by the SH model and less similar to natural patterns observed in experiment.

4.3 Defects in Striped Patterns 4.3.1 Natural Patterns Natural patterns seen both in experiment and simulations have various defects: dislocations, disclinations, and domain walls. Proliferation of defects in striped patterns is, in essence, a consequence of rotational symmetry of the system. Diﬀerent orientations of stripes may be chosen at diﬀerent locations, either randomly or under inﬂuence of boundary conditions or local inhomogeneities. The discrepancies of local orientations can be reconciled through formation of disclinations and domain walls, while dislocations reconcile discrepancies of local wavelengths. Diﬀerent kinds of defects are seen in Fig. 4.5 (Bowman and Newell, 1998). Both dislocations and disclinations are codimension two topological defects localized at a point in the plane and characterized by circulation around a contour surrounding this point. A dislocation is a topological singularity of the phase ﬁeld. The total change of phase around any contour surrounding the singularity must be a multiple of 2π:

226

4 Amplitude Equations for Patterns

Fig. 4.5. Various forms of pattern defects. 1 – dislocation, 2 – concave disclination, 3 – convex disclination, 4 – amplitude domain wall, 5 – phase domain wall (Bowman and Newell, 1998; reproduced with permission. Copyright by the American Physical Society)

! dθ = 2πN.

(4.63)

The integer N is called a topological charge. The phase cannot be deﬁned globally as a continuous univalued function when dislocations are present. This does not cause any nonphysical discontinuities, inasmuch as physically relevant quantities – wave vector and frequency – are well deﬁned everywhere except the defect locations. A disclination involves rotation of the orientation of the underlying pattern. If one follows the direction of stripes continuously around a disclination, it rotates through an angle of ±π (more on this in Sect. 4.3.5). The rotation is positive for a convex and negative for a concave disclination. Topological charges of disclinations as well as of dislocations add up deﬁning total circulation of either direction or phase along a contour surrounding several defects. The topological charges are conserved when the pattern evolves in time. Codimension one defects (lines in 2D) are domain walls which separate regions with diﬀerent prevailing orientations of the wave vector. One can

4.3 Defects in Striped Patterns

227

distinguish between phase walls, which are lines where the phase is continuous, but the pattern wave vector changes abruptly, and amplitude walls where the phase is discontinuous and the amplitude vanishes. The point and line defects are related to each other in the overall texture of the pattern, as phase walls meet and terminate at disclinations, while amplitude walls disintegrate at a close look into dislocation chains. The phase theory is applicable in each domain, but full equations have to be reintroduced at domain walls, as well as at disclinations and dislocation cores. Manneville and Pomeau (1983) and later Passot and Newell (1994) and Newell et al. (1996) described disclinations and domain walls as singular solutions of equations of phase dynamics. This gives a qualitatively correct structure of the far ﬁeld of the singularities but is still inadequate for a dynamic description that would hinge upon the structure of the defect cores. Both point and line defects, though being elements of disorder, contribute to ordering of the pattern, as they facilitate selection of an optimal wavelength. As we have seen in Sect. 4.1.5, an ordered pattern is stable within a range of wavenumber deviations scaling as the square root of parametric deviation from the bifurcation point. When dislocations are present, their motion will eliminate patches with the wavelength deviating from the optimal one (see Sect. 4.4.1). Likewise, if a domain wall separates patches with diﬀerent wavelength, it will move in the direction reducing the overall energy of the pattern, thereby helping to eliminate nonoptimal domains. 4.3.2 Phase Field of a Dislocation Dislocations occur in patterns of almost parallel stripes and therefore are most amenable to analysis. Consider a pattern with a constant prevailing wavenumber k0 described by the phase diﬀusion equation (4.30). This nonisotropic equation can be transformed by rescaling the coordinates and reverting from the extended phase variable Θ to the actual phase θ into an isotropic diﬀusion equation 2 θ. (4.64) θT = ∇ This equation is veriﬁed by a stationary symmetric vortex solution centered at the origin, written in the polar coordinates as θ = N φ, where N should be integer to avoid multivaluedness of physical ﬁelds dependent on the phase modulo 2π. Since (4.30) is a linear equation, vortex solutions centered at any point can be additively combined one with the other: θ(x) = θi , where θi obeys the circulation condition (4.63) on a contour surrounding the ith defect. A naive solution corresponding to an assembly of static defects is (4.65) θ(x) = φi (x − xi ), where φi is the angle counted around the position of a particular defect located at x = xi (t). All phases can be counted from arbitrary reference angles, as

228

4 Amplitude Equations for Patterns (b)

X

X

(a)

Y

Y

(e)

X

X

(c)

Y

X

(d)

Y

X

(f )

Y Y

Fig. 4.6. Dislocations with a single (a) and double (b) charge and pairs of like (c) and unlike (d, e) dislocations. Shading shows levels of the real part of the complex order parameter (amplitude). (f ) Loci of vanishing real and imaginary part of the complex order parameter for the pair of dislocations in (d) with the basic stripe structure removed

4.3 Defects in Striped Patterns

229

the phase is deﬁned only up to a constant. This solution can be combined as well with any constant-gradient solution, say, θ = qX. A dislocation in the underlying striped pattern corresponds to a vortex in the envelope ﬁeld with the complex amplitude u. Adding the vortex solution with the unit charge to the underlying pattern θ = k0 x = −1 k0 X yields the phase ﬁeld of dislocation, such as seen in Fig. 4.6a. Dislocations with multiple charges (Fig. 4.6b) are also possible, but are apt to be unstable, and tend to break up into singly charged dislocations, such as seen in Fig. 4.6c, while keeping the total charge conserved. Pairs of dislocations of opposite charge, as in Figs. 4.6d, e, may annihilate restoring a nonsingular phase ﬁeld. The precise location of a dislocation in a natural pattern is revealed when the short-scale structure is removed by a spectral shift. The dislocations are located at the intersections of the level lines Re u = 0 and Im u = 0, as shown in Fig. 4.6f, which corresponds to the pair of dislocations in Fig. 4.6d. At these points, the real amplitude of the envelope ﬁeld vanishes, and its phase becomes indeﬁnite. The superposition (4.65) appears to be a stationary solution that, indeed, satisﬁes both (4.64) and the circulation conditions (4.63) for each defect – and this is for an arbitrary distribution of defects, subject only to the restriction that all mutual distances should be large. The trouble, of course, lies in the phase singularity at x = xi . Near the defect core, the long-scale approximation breaks down, and the full system (4.15), (4.16) should be solved. This system, being nonlinear, does not obey the superposition principle implied in (4.65). The full solution can be approximated as the product of solutions corresponding to isolated defects at x = xi : * * u(x) = ui (x − xi ) = ρ0 (|x − xi |) exp i φi (x − xi ) . (4.66) This is not an exact solution of the amplitude equation, since the circular symmetry of an isolated defect is broken by the phase and amplitude gradients induced by other defects. The phase gradient turns out to be the main source of an eﬀective force setting the defect into motion. We shall return to dynamics of dislocations in Sect. 4.4. 4.3.3 Dislocations in NWS Equation In a striped pattern with a nearly optimal wavelength described by the NWS equation (4.4), the simple symmetric solution θ = N φ does not hold, and obtaining its equivalent (the phase ﬁeld of a single static dislocation) requires some ingenuity. If the change of real amplitude is neglected, while the characteristic scale of phase distortions is assumed to be large even on the extended NWS scale, (4.4) reduces √ to the phase equation dependent on long-scaled variables X = x, Y = y, T = t. Unlike (4.64), this equation, obtained by applying the operator ∂T − ♥2 to u = ρ exp[iθ(X, Y, T )] and collecting the

230

4 Amplitude Equations for Patterns

imaginary part,6 is nonlinear : θT = θXX − 14 θY Y Y Y + θX θY Y + 2θXY θY + 32 θY2 θY Y .

(4.67)

Meiron and Newell (1985) showed that a stationary singular solution of (4.67) corresponding to a quiescent defect placed at the origin has a self-similar form, i.e., can be presented as a function of a single coordinate η = Y / 2|X| that satisﬁes the equation θ = 4η 2 θ + 12ηθ − 12ηθ θ − 8θ2 + 6θ2 θ .

(4.68)

The boundary conditions are stipulated by the requirements that the derivatives of θ(η) vanish at η → ±∞, and the total change of θ measured around a contour enclosing the defect be equal to 2πN , where N is the topological charge. The solution with the charge +1 is obtained by setting θ(−X, Y ) = −θ(X, Y ) and imposing the boundary conditions (at X > 0) θ(−∞) = 0, θ(∞) = π.

(4.69)

Nepomnyashchy and Pismen (1991) showed that the self-similar solution verifying (4.68) is a representative of a wider class of solutions of the phase equation (4.67) that also satisfy a linearizable equation of Burgers type. As a corollary, the self-similar solution can be presented in a remarkably simple analytical form. One can check by diﬀerentiation that all solutions of the equation (4.70) θY Y = ±(2θX + θY2 ) also verify (4.67). In each of the two domains separated by the axis X = 0, (4.70) is recognized as the integral form of the Burgers equation with the coordinate ±X playing the role of time. The expression in parentheses on the r.h.s. of (4.70) represents the wavenumber shift relative to the optimal value k = 1. Equation (4.70) can be linearized in a standard way through the Hopf–Cole transformation θ = ∓ ln f , yielding fY Y = ±2fX .

(4.71)

The upper and lower signs refer, respectively, to X > 0 and X < 0. Equation (4.71) can be integrated in the positive and negative half-planes with “initial” conditions ﬁxed at the axis X = 0. The self-similar solution to (4.71) f (η) satisﬁes (4.72) f + 2ηf = 0 with the boundary conditions (cf. (4.69)) 6

The real part of this expression contributes to the dependence of ρ on the derivatives or θ in the far ﬁeld where the real amplitude is slaved to the phase, ρ2 = 1 − Re ♥2 exp[iθ(X, Y, T )].

4.3 Defects in Striped Patterns

231

3

X

2

1

0 -1

-2

1

0

2

3

Y Fig. 4.7. Phase ﬁeld of a dislocation in the NWS equation shown in gray scale

f (−∞) = 1, f (∞) = e−π . After transforming back to the phase variable we obtain 1 + e−π 1 − e−π θ = −sign(x) ln − erf(η) . 2 2

(4.73)

(4.74)

The asymptotics of θ (η) at η → ±∞ is θ C± e−z , where 2

C+ =

eπ − 1 1 − e−π √ ≈ 12.5, C− = √ ≈ 0.54. π π

The diﬀerence between both asymptotic values reﬂects the strong asymmetry of the solution (see Fig. 4.7). Of course, solutions of (4.70) do not exhaust all possible stationary solutions of a higher order (4.67). Since (4.70) is solved as an “initial value” problem in X, one cannot satisfy boundary conditions that might be imposed in a ﬁnite domain (say, in a band parallel to the Y -axis). Self-similarity breaks down in ﬁnite domains as well. Equation (4.74) gives the analytical form of the self-similar singular solution corresponding to a solitary quiescent defect in a pattern with the optimal wavenumber. This solution is fundamental for pattern-forming systems characterized by a nonvanishing optimal wavenumber, playing the same role of a “basic nontrivial state” as the symmetric static solution Θ = φ of (4.64). Linearizability of (4.70) implies a superposition principle. One can construct a solution of (4.70) corresponding to an arbitrary array of defects centered at points Y = Yi along the axis X = 0 by superimposing solutions to (4.71). The solution, being a generalization of (4.74), is

232

4 Amplitude Equations for Patterns

& ( )' N −θi Y − Yi 1 −θ0 −θN −θi−1 e + e erf , θ = −sign(X) ln +e −e 2 2|X| i=1 (4.75) where θi − θi−1 = ±π for defects of unit charge. The existence of this stationary solution is contrary to our expectation that the defects, if initially arranged at some intervals along the X-axis, must start climbing under the inﬂuence of mutual interaction. This means that interaction of defects, at least in this linear conﬁguration, vanishes in the phase diﬀusion approximation – exactly the way it does in the RGL equation. As in the latter case, the phase equation alone cannot fully account for the defect dynamics, even though the nonlinear phase equation (4.67) does not look as innocuous as (4.64). 4.3.4 Dislocation Core The phase description of a dislocation fails at short distances where the phase gradients become strong. The structure of the dislocation core is modelspeciﬁc and has to be computed by solving the full underlying equations. As a simple example, we consider the RGL equation (1.26). A stationary vortex solution is obtained by rewriting (4.15) and (4.16) in polar coordinates r, φ and using the ansatz ρ = ρ0 (r), θ = N φ. Then (4.16) is satisﬁed automatically, while (4.15) becomes ρ0 + r−1 ρ0 + (1 − (N/r)2 − ρ20 )ρ0 = 0,

(4.76)

subject to the boundary conditions ρ0 (0) = 0, ρ0 (∞) = 1. Vanishing of the real amplitude at the origin is necessary to eliminate the divergence due to the phase singularity. Indeed, it can easily be seen that ρ0 (r) ∝ r|N | at r → 0. Equation (4.76) cannot be solved analytically, but the solution is easily found numerically (Ginzburg and Pitaevskii, 1958). The solution corresponding to a vortex of unit charge (N = ±1) is shown in Fig. 4.8. A useful analytical form of ρ0 (r) is a Pad´e approximant r2 (0.34 + 0.07r2 ) . (4.77) 1 + 0.41r2 + 0.07r4 This function has correct asymptotics both at the origin and at inﬁnity, and approximates the exact numerical solution (at |N | = 1) fairly well elsewhere. The energy can be computed by using the numerical function ρ0 (r) in (4.14). We shall ﬁrst compute the “potential” energy due to the last term. Since this integral is converging, the integration limit may be extended to inﬁnity. The integration can be done analytically, making use of (4.76) and integrating by parts: ∞ ∞ r(1 − ρ20 )2 dr = r2 (1 − ρ20 )ρ0 ρ0 (r) dr I1 = 12 0 0 ∞ d(rρ0 ) − N 2 ρ0 ρ0 (r) dr = 12 N 2 . =− (4.78) rρ0 (r) dr 0 ρ20 (r) =

4.3 Defects in Striped Patterns

233

Ρ 1 0.8 0.6 0.4 0.2

2

4

6

8

10

12

r

Fig. 4.8. The amplitude of a stationary vortex of unit charge as a function of the radial distance

The remaining distortion energy is computed most conveniently by extracting from the integrand the function N 2 r−1 H(r − 1), where H(x) is the Heaviside function: H(x) = 1 at x > 0 and H(x) = 0 otherwise. This function, integrating to N 2 ln L, compensates the logarithmic divergence. After the divergence has been extracted, integration may be extended to inﬁnity: ∞ r[ρ0 (r)]2 + N 2 r−1 [ρ20 − H(r − 1)] dr = −N 2 ln a0 . I0 = (4.79) 0

The result is expressed through a constant a0 dependent on the core structure. The numerical value for |N | = 1 is a0 ≈ 1.126. The total energy is √ L e . (4.80) F = π(I0 + I1 + N 2 ln L) = πN 2 ln a0 Since the energy is proportional to N 2 , only vortices with the unit topological charge are expected to be stable.7 The phase ﬁeld, due to its weak decay, gives the largest overall contribution to the energy integral and is solely responsible for the divergence at large distances. The total energy of the vortex within a circle with the radius L diverges logarithmically with the characteristic linear dimension L of the medium. This divergence is not dangerous by itself, since a sole defect in the inﬁnite plane is unphysical. It remains, however, a disquieting fact that may lead to paradoxes when one approaches a perturbation problem without due caution (see Sect. 4.4). 7

The energy argument still does not exclude metastable multiply charged vortices that might be stable to inﬁnitesimal perturbations.

234

4 Amplitude Equations for Patterns

4.3.5 Disclinations The description of disclinations poses more diﬃcult problems. On a naive level, the wave vector may be treated as a director , i.e., an “arrowless vector.” This is speciﬁc to stationary patterns, which are invariant to inversion k → −k. In this way, the pattern is viewed as an oriented medium similar to a nematic liquid crystal. A more precise mathematical form of the order parameter characterizing this medium is a traceless symmetric matrix (4.81) u = n ⊗ n − 13 I , where I is the unity matrix and n is a unit vector directed along k. Clearly, inversion of n does not aﬀect u, while using such a mathematically ambiguous term as an “arrowless vector” is avoided. In two dimensions, the topological singularities of the director ﬁeld are vortices with integer or half-integer circulation (Mermin, 1979). The latter are allowed in view of the inversion symmetry. Several types of disclination with circulation 1/2 and 1 are shown in Fig. 4.9. This approach is too simplistic, as it does not take into account that a striped pattern, unlike a director ﬁeld, is characterized locally by the phase, and not only by its gradient – the wave vector. If singularities of the phase, as well as of the wave vector ﬁeld, are taken into account, a striped pattern is recognized as a medium with a very complex topology that cannot be described in the framework of standard homotopy theory (Mermin, 1979). The diﬀerence between a director ﬁeld and a striped pattern is clearly seen when the applicable energy functionals are compared: for the former, the Lagrangian depends on the derivatives of the director only, while for the latter, the energy strongly depends on the absolute value of k. Disclinations in natural patterns, as well as in anisotropic ﬂuids, may be necessitated by boundary conditions. For example, a common boundary condition in convection patterns is orthogonality of the stripes to the bounding (a)

(b)

(c)

Fig. 4.9. Convex (a) and concave (b) disclinations with half-integer rotation, and convex disclination with integer rotation – a target pattern

4.3 Defects in Striped Patterns (a)

235

(b)

Fig. 4.10. Distorted striped patterns obtained (a) in simulations of the SH equation (Greenside and Coughran, 1984; reproduced with permission. Copyright by the American Physical Society) and (b) experiments (Croquette et al., 1983; reproduced with permission). Gray circles mark the disclinations

wall, as observed experimentally (see e.g. Fig. 4.10) and justiﬁed theoretically by Zaleski et al., 1984. This means that the wave vector parallel to the walls should rotate through 2π along the perimeter. In Fig. 4.10a, this rotation is compensated by two concave disclinations (marked by gray circles). One can also see in this ﬁgure domain walls going from the upper-left and lower-right corners toward the two disclinations. In Fig. 4.10b, the three target sources on the border rotate the wave vector by π, so that a single disclination is suﬃcient to compensate the remaining rotation. Dislocations, such as the one within the rectangular marker in the ﬁgure, do not aﬀect the rotation of the wave vector. 4.3.6 Domain Walls A change of orientation across a domain wall in patterns generated in simulations and experiments is usually eﬀected on a length comparable with the prevailing wavelength of the pattern. Nevertheless, using the amplitude equation may give a qualitatively correct picture of the transition. Malomed et al. (1990) described the structure of a domain wall by solving coupled NWS equations for the striped patterns with two alternative orientations. One can expect that a stationary solution exists only when the wavelengths are equal on both sides of the wall; otherwise, the wall would propagate in the direction decreasing the overall energy of the pattern. It turns out that an even stronger restriction is true, and both wavelengths should be optimal. The derivative with respect to the coordinate z normal to the wall should scale as O(), in the same way as the derivative along the wave vector (i.e., as the square root of the parametric deviation from the symmetry-breaking

236

4 Amplitude Equations for Patterns (a)

(b)

1 0.8 0.6

Ρ

0.4 0.2 0

6

4

2

0

z

2

4

6

z

Fig. 4.11. Change of real amplitudes across a domain wall (a) and the respective reconstructed pattern (b)

bifurcation point), unless the angles α± between this direction and the directions of the respective wave vectors are close to π/2. Under these conditions, the fourth-order derivatives in ♥2 can be neglected, and the amplitude equations reduce to two coupled nonlinear diﬀusion equations with the amplitude diﬀusivities cos2 α± . The stationary equations for the two amplitudes u± are (4.82) cos2 α± u± (z) + u± 1 − |u± |2 − ν|u∓ |2 = 0. Transforming to the polar form u± = ρ± eiθ± , removing the phase factor, and separating the real and imaginary parts yields the coupled equations for the real amplitudes and phases, 2 + ρ± 1 − ρ2± − νρ2∓ = 0, (4.83) cos2 α± ρ± (z) − ρ± K± cos2 α± d 2 ρ± K± = 0, 2 ρ± dz

(4.84)

where K± = θ± (z). It follows from the last equation that ρ2± K± = const, and since ρ± (∓∞) = 0, K± must vanish. This means that the wavelength on both sides is the optimal one, i.e., corresponds to the “band center”; thus, the domain wall strongly selects a unique wavenumber from the allowed stable range. We see that only real amplitudes ρ± , deﬁned by (4.83) with K = 0, change across the domain wall interpolating from zero on one side to unity on the other side; thus, it is an amplitude wall in terms of Sect. 4.3.1. The boundary conditions are ρ± (±∞) = 1, ρ± (∓∞) = 0. The coeﬃcient ν should be larger than unity to discourage coexistence of alternatively directed modes (see Sect. 1.6.1); in particular, for the SH model ν = 2. The result of a numerical computation, presenting a wall between domains with k turning from α− = −π/6 to α+ = π/3 relative to the normal to the wall, is shown in Fig. 4.11a; Fig. 4.11b presents the reconstructed pattern near

4.3 Defects in Striped Patterns

237

the wall. One can discern a slower decay of the amplitude on the left due to a larger amplitude diﬀusivity at a sharper angle. The two modes appear to be mixed near the wall where a patch of a rectangular pattern is observed. This is at variance with the natural patterns seen in Figs. 4.5, and 4.10 where the transition between alternative orientations is abrupt, and points out to limitations of the amplitude representation. The approximation breaks down when at least on one side of the wall |α± − π/2| = O(); then the fourth-order derivatives along the direction of stripes should be taken into account. In a particular case when the stripes are parallel to the wall on the one side and normal on the other side, the stationary equations for real amplitudes are (4.85) ρ+ (z) + ρ+ 1 − ρ2+ − νρ2− = 0, 2 2 2 (4.86) − ρ − (z) + ρ− 1 − ρ− − νρ+ = 0, 2k0 subject to the boundary conditions ρ± (±∞) = 1, ρ± (∓∞) = 0. An approximate solution of these equations is obtained in the following way (Manneville and Pomeau, 1983). The fourth-order derivative in (4.86) can be ne√ glected everywhere except a narrow O( ) boundary layer near some location z = z0 . Atz > √ z0 , one can set ρ− = 0, and the solution of (4.85) is simply ρ+ = tanh z/ 2 [cf. (2.24)]. At z < z0 , ρ2− = 1 − ρ2+ can be eliminated from (4.86), bringing (4.85) to the form ρ+ (z) − (ν − 1)ρ+ + ν 2 − 1 ρ3+ = 0, (4.87) √

which is solved by ρ+ = 2/(ν − 1) sech ν − 1(z − z1 ) . The elimination makes sense only when ρ2+ < 1/ν. Thus, the two solutions can be matched by requiring √ √

(4.88) tanh z0 / 2 = sech ν − 1(z − z1 ) = ν −1/2 , which ﬁxes both z0 and z1 . The solution exists only at ν > 1, since otherwise excitation of an alternative mode is preferred in each domain, and a quadratic pattern is formed. The solution obtained in this way is not smooth at z = z0 , but this can be repaired by restoring the fourth-order derivative in (4.86) within the boundary layer without aﬀecting the solutions in both outer regions in a considerable way. If the change of orientation across the domain wall is small, it can be incorporated in the phase. Then the a single amplitude is suﬃcient for the description of the entire pattern including the domain wall, being treated now as a phase wall (Sect. 4.3.1). No stationary solution exists in this case in an inﬁnite region, as the pattern oriented in a slightly diﬀerent way at z → ±∞ tends to straighten up with time; the last stages of this process can be described by solving a self-similar asymptotic form of the amplitude equation (Malomed et al., 1990).

238

4 Amplitude Equations for Patterns

An exception is the case when the stripes are almost normal to the weak domain wall. Then one can ﬁnd a stationary solution verifying the phase equation (4.68) and, more speciﬁcally, the Burgers equation (4.70). Directing the x-axis along the domain wall and replacing the transverse coordinate by z, this equation can be rewritten as dK/dz = ±(2C + K 2 ),

(4.89)

where K is the z-component of the wave vector and C is the correction to the x-component, assumed to be constant. This constant has to be set to C = −q 2 /2 to ensure that the pattern is at the band center far from the domain wall. A solution with K approaching a constant asymptotic value at z → ±∞ is obtained when the negative sign is chosen in (4.89): K = q tanh qx.

(4.90)

The existence of this solution is a consequence of the marginal zigzag instability at the band center (Sects. 4.1.5, and 4.2.2), which allows a weak bend in the pattern to remain stationary rather than straighten up as it does when the pattern is stable.

4.4 Motion of Dislocations 4.4.1 The Nature of the Driving Force A dislocation climbing across the direction of the wave vector of the underlying pattern eﬀects a change of the wavenumber over an extended region. Indeed, the phase gradient ∇θ due to the dislocation adds up to the prevailing wave vector k0 on one side of the vortex and is subtracted on the other side. The Peach–K¨ ohler force driving the dislocation is due to the wavenumber mismatch, i.e., deviation from the optimal wavenumber (the “band center”). In terms of the envelope ﬁeld, it corresponds to a nonvanishing constant wave vector q in (4.17). When a number of dislocations are present the action of their phase ﬁelds is superimposed on the inﬂuence of the prevailing wave vector. The ﬁrst attempts to understand dynamics of dislocations in striped patterns (Siggia and Zippelius, 1981; Pomeau et al., 1983; Tesauro and Cross, 1986) concentrated upon a more diﬃcult problem of patterns in isotropic systems near a critical point, which are described by the NWS equation (4.4). The analysis was based on a linearized version of the far ﬁeld equation derived from (4.4), and on the naive phase ﬁeld solution (4.65). We have seen in Sect. 4.3.3 that the actual situation is far more complex, and even the core structure of a static NWS vortex is not known to this day. In addition, patterns in isotropic systems are apt to contain disclinations and domain walls, and

4.4 Motion of Dislocations

239

are strongly inﬂuenced by lateral boundaries, which makes controlled measurement of defect motion all but impossible. Later on, attention focused on patterns in anisotropic systems, bolstered by precision experiments on electroconvection in nematics. The governing equations reduce in this case to the much simpler RGL model (4.3). The analytical theory to be described further in this section is restricted to this case. It is clear that diﬀerent approximations should be used in the far and near ﬁeld, as has been already done when evaluating the energy of an isolated dislocation in Sect. 4.3.4. In both regions, approximate solutions can be obtained by means of a rational expansion assuming that defects are at mutual distances far exceeding the core size (“healing length”) and are slowly moving. These solutions should be matched in an intermediate region, large compared to the core size but small compared to the interdefect separation or other relevant macroscopic scale. In this region, both near and far ﬁeld solutions must hold, and their simpliﬁed asymptotic forms should apply (Neu, 1990; Pismen and Rodriguez, 1990). Roughly speaking, the mechanism which sets the defects into motion is the incompatibility of the symmetries in the far and near ﬁelds. The far ﬁeld equation (4.64) is invariant to the gauge transformation θ → θ + k · x, i.e., adding an arbitrary constant phase gradient k; thus, it seems not to be aﬀected altogether by an imposed phase gradient that has to set the defect into motion. The far ﬁeld equation has solutions corresponding to steady motion of the defect with an arbitrary speed v. Finally, being linear, it obeys a superposition principle, so that defects do not interact at all in this approximation. It is clear, of course, that taking the core into consideration is essential. The nonlinear equation (4.3) lacks the symmetries of the far ﬁeld equation. On the other hand, the circular symmetry of the defect core is lost in the far ﬁeld and is distorted by the action of the driving force. 4.4.2 Phase Field of a Moving Defect A dislocation moving under the action of a constant force is expected to propagate with a constant speed. The direction of motion breaks the symmetry of the basic solution θ = φ, so that the phase ﬁeld of a static and moving dislocations might be qualitatively diﬀerent. We shall suppose that both the driving force due to externally imposed phase gradient k = K and the propagation speed v = V are small. Under these conditions, the changes of the real amplitude are negligible outside the dislocation core, and the phase ﬁeld can be computed as a stationary solution of (4.64) in a comoving coordinate frame. Without loss of generality, we assume that the motion is along the Y -axis, and the defect is at the origin. The far ﬁeld equation (4.64) is written in the comoving frame as V θY + θXX + θY Y = 0,

(4.91)

240

4 Amplitude Equations for Patterns

subject to the circulation condition (4.63) along any contour surrounding the origin. This equation is solved by introducing a dual function related to θ as follows (Dubois-Violette et al., 1983): θX = −N (ΦY + V Φ),

θY = N ΦX .

(4.92)

The integrability condition for Φ is (4.91), while the integrability condition for θ combined with the circulation condition (4.63) results in the equation of Φ containing a point source at the defect location: V ΦY + ΦXX + ΦY Y = 2πδ(X).

(4.93)

Unlike the phase, the dual function is univalued. The solution of (4.93) is expressed through a modiﬁed Bessel function: VR VY Φ = − exp − K0 , (4.94) 2 2 where R = r is the extended radial coordinate. The limit of the dual function at V → 0 and R ﬁxed is, up to a constant, Φ = ln R; this is a circularly symmetric function, which is the dual of the polar angle. The symmetry is broken in a singular way by any nonvanishing velocity. Indeed, (4.94) is scale-invariant and can be made independent of velocity X

-1

0 1

2

1

0 -2 -1 Y

0 1

Fig. 4.12. The rescaled dual function of a moving defect. Take note of the fast drop-oﬀ in the direction of motion

4.4 Motion of Dislocations

241

by rescaling the coordinate X → X/V . The rescaled function is drawn in Fig. 4.12. Its strong asymmetry is evident. The function falls oﬀ exponentially everywhere except the tail along the negative Y -axis; the steepest fall-oﬀ is in the direction of motion. Asymptotically at R → ∞, VR π exp − (1 + sin φ) . (4.95) Φ− VR 2 The components of the phase gradient are read directly from (4.92). The resulting expressions are scale-invariant and can be rewritten in short-scale variables without change: vr

vr vr Nv exp − sin φ K0 − sin φ K1 , 2 2 2 2

vr Nv vr exp − sin φ cos φ K1 . θy = 2 2 2

θx =

(4.96)

4.4.3 Dissipation Integral and Peach–K¨ ohler Force A velocity estimate can be obtained by comparing two equivalent expressions for the energy dissipation rate computed in a comoving frame, that both contain the unknown defect velocity (Pomeau et al., 1983; Bodenschatz et al., 1988). We consider a single defect steadily moving along the y-axis. After transforming y → y − vt, ∂t is replaced by −v∂y : δF ut + c.c. dx = −2 ut ut dx = −2v 2 uy uy dx ≡ −2v 2 I , Ft = δu (4.97) where I, by deﬁnition, is a dissipation integral . Another way to write the same expression is to transform to the comoving frame at an earlier stage: δF uy + c.c. dx Ft = −v δu 2

= −v uy ∇ u + (1 − |u|2 )u + c.c. dx ≡ −vF. (4.98) Comparing (4.98) with (4.97) deﬁnes the velocity, as yet unknown, as the ratio of the two integrals: v = F/2I. (4.99) Analogous to familiar equations of classical mechanics, F is interpreted as a driving force. This force is thermodynamic by its nature, as it originates in the energy integral and is called a Peach–K¨ ohler force. By the same analogy, the dissipation integral in the denominator is interpreted as a friction factor .

242

4 Amplitude Equations for Patterns

The integral (4.97) is mostly accumulated in the far ﬁeld, and the main contribution comes from the phase gradient. Thus, it is crucial whether a static (circularly symmetric) or a propagating solution deﬁned by (4.96) is used in the computation. A traditional approach was to take the symmetric solution θ = N φ; since, however, it leads to a diverging integral, it is necessary to assume a long-scale cut-oﬀ length equal to a typical interdefect distance L (Bray, 1995). Then the integral in (4.97) is ∝ N 2 ln L. A more quantitative result can be obtained by taking into account the fall-oﬀ of the amplitude in the defect core to eliminate the logarithmic divergence at r → 0. Using the numerical function ρ0 (r) computed in Sect. 4.3.4 yields

L r[ρ0 (r)]2 sin2 φ + N 2 r−1 ρ20 (r) cos2 φ dr = πN 2 ln . a 0 0 0 (4.100) This reduces after angular integration to the radial integral (4.79), so that we obtain the same numerical constant a0 = 1.126 for |N | = 1. If the motion is caused by the interaction of two defects at a mutual distance L, one can express the driving force directly as the derivative of the total energy with respect to L taken with the negative sign: I=

2π

dφ

L

Ft =

∂F ∂L ≡ −F v. ∂L ∂t

(4.101)

A rough estimate of the energy of an interacting pair can be obtained using the static solution and assuming the cutoﬀ length equal to a typical interdefect distance L (Bray, 1995): πN 2 ∂F = . (4.102) F = ∂L L Using (4.100), the velocity estimate is v=

1 . 2L ln(L/a0 )

(4.103)

The numerical factors should not, of course, be taken too seriously. In view of the result of the preceding subsection, the phase ﬁeld of a moving defect loses circular symmetry. Using the propagating, rather than stationary, far ﬁeld solution in the dissipation integral appears to be logical in view of the assumption of stationary propagation implied in (4.97). This assumption can, however, be expected to be valid quantitatively only when the driving force is also constant. This is not realistic in an ensemble of moving and interacting defects, but can be realized when the background solution has the form (4.17). In this case, there is a prevailing nonvanishing wave vector k that corresponds to a constant phase gradient acting upon a defect. This setup is common in the context of moving dislocations in a striped pattern. The defect moves in the direction normal to k in such a way that k = |k| ahead of it is larger than that behind, and therefore both the wavenumber

4.4 Motion of Dislocations

243

and energy decrease as a result. As the y-axis is taken as the direction of motion, of k can be taken as the x-axis. Let u = U eikx where √ the direction iφ U = 1 − k 2 e asymptotically at large distances. This corresponds to (4.17) with added phase circulation. Equation (4.98) now becomes ∞ ∞ 2

Ft = −v U y ∇ + (1 + 2ik∂x − k 2 − |U |2 ) U + c.c. dx dy. −∞

−∞

(4.104) All terms, except one containing the imaginary unit, can be presented as total derivatives and vanish upon integration. The remaining integral, deﬁning the Peach–K¨ohler force, is reduced by integrating by parts to a contour integral: ∞ ∞ F = 2ik U y Ux − c.c. dx dy −∞ −∞ ∞ ∞ y=∞ x=∞ = 2ik dx U Ux y=−∞ − dy U Uy x=−∞ . (4.105) −∞

−∞

The two terms on the right-hand side combine to a closed contour integral containing the derivative of the phase along the contour. Since the latter increases by 2πN around the contour enclosing the defect, we ﬁnally obtain ! ∂θ ds = 4πkN (1 − k 2 ). (4.106) F = 2k(1 − k 2 ) ∂s It remains to compute the dissipation integral. Plugging the propagating solution into (4.97) makes a lot of diﬀerence, as the long-distance divergence of the energy integral disappears at any v = 0. The far ﬁeld contribution to the dissipation integral is computed by assuming in (4.97) ρ = 1 outside a circle of radius 1 r0 v −1 , which is small on the extended scale. Denoting in (4.96) z = vr/2 yields I = N2

∞ 1 2 vr0

zK12 (z)dz

2π

e−2z sin φ cos2 φ dφ = πN 2

∞ 1 2 vr0

0

I1 (2z)K12 (z)dz.

(4.107) The integrand in the last expression decays asymptotically as z −3/2 at z → ∞, and therefore the integral is converging. Due to the logarithmic divergence at z → 0, I ∝ ln(1/v). A quantitative result is obtained by computing the inner part of the dissipation integral with due account for the fall-oﬀ of the amplitude in the defect core (Bodenschatz et al., 1988). This is done as in (4.100) with the cutoﬀ radius L replaced by r0 . The sum of the inner and outer integrals is independent of the cutoﬀ radius, provided it is large compared to the size of the defect core but small on the extended outer scale (more on this in the next section). The result is I = π ln

3.29 . v

(4.108)

244

4 Amplitude Equations for Patterns

Using (4.106) and (4.108) in (4.99) yields the mobility relation connecting the wavenumber mismatch with drift velocity: N v 3.29 ln = k(1 − k 2 ). 2 v

(4.109)

This result is, in fact, exact for the motion in a constant phase gradient. This is conﬁrmed by a more rigorous computation based on the method of matched asymptotic expansion in the next section. On the other hand, the “quasistatic” estimate (4.103) is mere qualitative and does not account for charges of defects and directions of motion in a proper way. The logarithmic divergence of the integral (4.100) was taken by some authors to imply a logarithmic decrease of mobility with the size of the system. This conclusion is ﬂawed because the far ﬁeld solution depends in a singular way on the defect velocity, so that the dissipation integral (4.107) evaluated in the comoving frame converges even when the velocity is arbitrarily small. The problem of far ﬁeld divergences is overcome by applying an asymptotic method based on the matching of solutions in the inner (in the vicinity of the defect) and outer (far ﬁeld) regions, as explained in detail in the next section. The dependence on the size of the system comes to be far more subtle and can be related to the inﬂuence of the shape of a planar region on the Green’s function of the heat equation (4.3) or to the action of images of defects. Nevertheless, (4.100) and (4.108) are not contradictory in a qualitative way. Both give two alternative estimates of the dissipation integral: one, provided by a nonvanishing velocity, and other by either the size of the region or a characteristic interdefect distance. In a realistic situation, when there are a number of moving defects in an extended region, the lowest of the two bounds might be most relevant. The velocities computed with the help of (4.103) and (4.109) are likely to be close to one another, since (neglecting logarithmic corrections) we expect both the typical defect velocity and the background wavenumber to be inversely proportional to the typical interdefect distance. 4.4.4 Matched Asymptotic Expansions The dislocation core, like its far ﬁeld, is distorted when the defect is set into motion. As long as the velocity is small, this eﬀect can be treated perturbatively. Matching the perturbed core solution to the far ﬁeld is the most precise way to determine the drift velocity. Consider a single defect steadily moving along the y-axis with a slow speed v = V , as yet unknown. Equation (4.3) is rewritten in a comoving coordinate frame centered at the defect as ∇2 u + (1 − |u|2 )u = −vuy .

(4.110)

The term containing the velocity is seen as a small correction, so that the solution can be sought for in the form of an expansion in either a dummy

4.4 Motion of Dislocations

245

small parameter , to be further connected to an externally imposed weak phase gradient, or in v itself: u = u0 + vu1 + · · · .

(4.111)

∇2 u0 + (1 − |u0 |2 )u0 = 0,

(4.112)

The zero-order equation is

satisﬁed by the symmetric static defect solution u0 = ρ0 (r)eiN φ . The ﬁrst-order equation can be rewritten in a compact form H(u1 , u1 ) + Ψ (x) = 0,

(4.113)

where Ψ (x) = ∂y u0 is the inhomogeneity and H is the linear operator H(u1 , u1 ) ≡ ∇2 u1 + (1 − 2|u0 |2 )u1 − u20 u1 .

(4.114)

It is not necessary to actually solve (4.113); it suﬃces to obtain the solvability condition of this equation. The operator H is self-adjoint and has two eigenfunctions ϕ(x) with zero eigenvalue that correspond to two translational degrees of freedom in the plane. These eigenfunctions are just the two vector components of ∇u0 ; indeed, taking the gradient of (4.112) yields H(∇u0 , ∇u0 ) = 0. The well-known Fredholm alternative states that if a homogeneous linear equation has a nontrivial solution, the respective inhomogeneous equation is solvable only if the inhomogeneity is orthogonal to the eigenfunction with zero eigenvalue. If the Fredholm alternative is applied to (4.113) in the inﬁnite plane, an apparent result is that it should have no solution: indeed, its righthand side is certainly not orthogonal to itself, i.e., to the y-component of the eigenfunction ∇u0 . This conclusion is erroneous since integration in the inﬁnite plane has no sense whatsoever: ﬁrst, because the integrals involving the static solution u0 diverge and, second, because this solution is inapplicable in the far ﬁeld, as the phase of a moving defect is strongly asymmetric at distances r = O(v −1 ) – see Sect. 4.4.2. The correct way to obtain the solvability condition is to apply the Fredholm alternative in a circle of radius r0 large compared to the core size but small on the far ﬁeld scale, i.e., 1 r0 v −1 . The resulting condition involves both area and contour integrals. It is formally derived by multiplying (4.113) by the eigenfunction ϕ(x) and integrating over a circle or, generally, any ﬁnite region. Since (4.113) is a shorthand for two complex conjugate equations of u1 and u1 , one also has to add the complex conjugate, and the resulting condition will be real. The diﬀerential term in H is transformed by twice integrating by parts: ! ϕ∇2 u1 dx = u1 ∇2 ϕ dx + (ϕ n · ∇u1 − u1 n · ∇ϕ) ds, (4.115)

246

4 Amplitude Equations for Patterns

where n is the outward normal to the integration contour, and s is the arc length. After the algebraic terms in H are added, the area integrand (excluding the inhomogeneous part) reduces to u1 H(ϕ, ϕ) + c.c., and vanishes identically since ϕ is an eigenfunction. The resulting solvability condition, restricting now to integration over a circle, is + r0 2π 2π r dr ϕΨ (r, φ) dφ + r0 (ϕ∂r u1 − u1 ∂r ϕ)r=r0 dφ = 0. Re 0

0

0

(4.116) With the particular inhomogeneity,

Ψ = ∂y u0 = ρ0 (r) sin φ + iN r−1 ρ0 (r) cos φ eiN φ ,

(4.117)

the left-hand side of (4.116) does not vanish upon angular integration only when ϕ is the y-component of ∇u0 given by the same expression (4.117). The contour integral depends on the asymptotics of the ﬁrst-order solution u1 . Assuming r0 = O(v −1/2 ), we have ρ0 (r0 ) = 1 − O(v). It follows that the contour integral can be expressed, to the leading order O(v), through the phase ﬁeld only, and is computed using the inner limit of the outer solution (4.96) with an arbitrary constant gradient k = vK added to θx :

N v vr N + γ − sin2 φ + vK, ln θx = − sin φ − r 2 4 Nv N cos φ − cos φ sin φ, (4.118) θy = r 2 where γ = 0.577 . . . is the Euler constant. The phase itself can be reconstituted from this asymptotic expression in the form containing a dipole component which contributes to the contour integral: N v vr γ−1−2K/N . (4.119) θ = N φ + rχ(r) cos φ, χ(r) = − ln e 2 4 Using the asymptotics of (4.117), ∂y u0 ≈ iθy eiN φ = iN r−1 cos φ eiN φ ,

(4.120)

the contour integral is computed now as N π[N − 2χ(r0 )]. The area integral in (4.116) is evaluated as in (4.100), yielding πN 2 ln(r0 /a0 ). When the area and contour integrals are added up, the auxiliary radius r0 falls out, and the solvability condition takes the form a0 v γ− 1 2K e 2 = . (4.121) ln 4 N This expression can be rewritten in the form of a mobility relation resolved with respect to the weak driving phase gradient k = vK: k=−

N v v0 ln , 2 v

v0 =

4 −γ+ 1 2 ≈ 3.29. e a0

(4.122)

4.4 Motion of Dislocations

247

0.175 0.15 0.125 k

0.1 0.075 0.05 0.025 0 0

0.02

0.04

0.06

0.08

0.1

v Fig. 4.13. Relation between the phase gradient k and the defect velocity v

This weakly nonlinear dependence is shown in Fig. 4.13. The numerical value relates to |N | = 1. The result coincides (at k 1) with (4.109), though evaluation of the integral (4.107) has been circumvented. Note that the analytical form of the mobility relation is universal, and only the constant a0 , hence, v0 , depends on the numerical integration in the core region. The motion is in the direction orthogonal to the imposed phase gradient. Since v must be small, the logarithm is always positive, and the direction of motion is turned clockwise or anticlockwise through π/2 with respect to the direction of the phase gradient when the defect is, respectively, positively or negatively charged. The vector form of (4.122) can be written with the help of the operator of rotation through the right angle J : v0 N 0 −1 ln J v; J = . (4.123) k= 1 0 2 |v| Since v0 = O(1), while |v| = O(), the argument of the logarithm is large. Thus, the ﬁnal result turns out to be weakly dependent on the ratio of the characteristic scales of problem. 4.4.5 Interaction of Dislocations In a disordered pattern, dislocations themselves serve as sources of phase gradients acting on other defects. Pick any defect, and place the origin of the polar coordinate system (r, φ) at its instantaneous position. Take tentatively (4.66) with the phase given by (4.65) and the real amplitude ρ0 (r) by the circularly symmetric stationary solution satisfying (4.76). Near the origin, the only signiﬁcant contribution is due to the “resident” defect; all other amplitudes can be presented by asymptotic expressions, and the corrections

248

4 Amplitude Equations for Patterns

are of O(L−2 ). Neglecting these terms and taking the phase of the resident defect to be equal to the polar angle, the trial solution becomes

θi (x − xi ) , (4.124) u(x) = ρ0 (r) exp i φ + where the summation is now over phases θi of all defects except the resident one. Using this in (4.15) and (4.16) leaves O(L−1 ) residual terms due to the phase gradients ∇θi . The combined gradient can be identiﬁed with an externally imposed gradient considered in Sect. 4.4.4, and the mobility relation (4.123) applied to compute the respective defect velocity. When the phase gradient is created by the phase ﬁeld of another defect, the result is, as expected, repulsion of likely charged defects and attraction of oppositely charged ones (see Fig. 4.14). The motion of an ensemble of interacting defects can be followed then by computing the instantaneous phase gradient at the defect location due to the action of all other defects, moving each defect with the speed determined by the mobility relation, recomputing the gradients, etc. The algorithm would fail only when two defects (say, mutually attracting defects of the opposite sign) closely approach each other, and the weak ﬁeld theory becomes inapplicable. This would be, however, only a very brief stage, resulting in annihilation of the colliding defects, while all others would not be considerably disturbed. In this way, the ﬁeld problem is reduced in eﬀect to a dynamical particle–ﬁeld problem with a ﬁnite number of degrees of freedom. Unfortunately, this approach is not precise due to the lack of a universally applicable (history-independent) mobility relation. Equation (4.123), strictly speaking, applies only to the motion in a constant phase gradient. Indeed, it relies on the near ﬁeld asymptotics of a particular far ﬁeld solution (4.96). The correct form of the asymptotics, (4.119), must be retained for any background phase ﬁeld slowly changing in time and space. The numerical constants in the argument of the logarithm in (4.119) might, however, be diﬀerent. The phase ﬁeld of a moving defect (4.96) is velocity-dependent, and, as the velocity changes, either by its absolute value or direction, it cannot adjust instantaneously to a new quasistationary distribution. Therefore, the phase gradient induced by each defect at the core of another one depends, strictly speaking, on the entire past history of accelerations. Since the characteristic time scale

k

k v

v

k v

v k

Fig. 4.14. Attracting and repelling defect pairs

4.4 Motion of Dislocations

249

of the phase ﬁeld evolution is the same as that of the defect displacement, one would need to solve the nonstationary phase equation (4.64) simultaneously with the equations of motion of defects. A rough estimate can be based on a “self-consistent” approximation that neglects these complications (Pismen and Rodriguez, 1990). It is assumed that each defect carries the quasistationary phase ﬁeld corresponding to its instantaneous velocity. Under this assumption, the dependence of the drift velocity on the instantaneous separation of a pair of dislocations is obtained in the following way. First, we compute the value of the phase gradient due to one of the defects at the core of the other one with the help of (4.96). The velocity is then given by the mobility relation (4.122) or (4.123). For oppositely charged defects moving toward each other along the y-axis, the phase gradient generated by one of the defects at the core of the other is θx (r, π2 ), while for like-charged defects, moving apart, it is θx (r, − π2 ). Quasistationary velocities are computed then as solutions of the equation v0 = e±vr/2 [K0 (vr/2) ± K1 (vr/2)] , ln (4.125) v where the positive sign should be taken for the repelling and the negative sign for the attracting pair of defects of unit charge. A stable quasistationary solution exists only beyond a certain separation; at shorter distances velocities are large, and the entire approximation scheme breaks down. At larger separations, two branches of solutions exist, of which the lower one, corresponding to velocities decreasing with separation, is physical. Quasistationary velocities in both cases are shown in Fig. 4.15. The velocities of mutually repelling defects are much larger, at a comparable distance, than of attracting ones, due to a slower fall-oﬀ of the phase gradient behind a moving defect. -2 -2.5 -3 ln v

-3.5 -4 -4.5 -5 -5.5 0

20

40 r

60

80

Fig. 4.15. Quasistationary velocities vs. separation for attracting (dashed line) and repelling (solid line) defects

250

4 Amplitude Equations for Patterns

The self-consistent solution should be viewed as an estimate rather than as a rational approximation, although it is expected to work well when the separation is very large even on the extended scale. An opposite, less congruous, assumption would be to use instead of (4.96) a “solid body” solution (4.65) neglecting the velocity-dependent deformation of the phase ﬁeld entirely. The truth may lie between these two extremes. As the vortex accelerates, the phase readjusts fast near the core but is retarded in far outlying regions. Thus, for example, when a circularly symmetric vortex at rest is suddenly accelerated, its phase ﬁeld will evolve to the asymmetric stationary form (4.96) only gradually, spreading the deformation from the center outward and fully readjusting only asymptotically at t → ∞. The solid body and self-consistent velocities correspond to the vortex motion in the respective limits of inﬁnite and zero acceleration. A better way to follow the evolution of the phase ﬁeld is to integrate the phase equation (4.64) subject to the circulation condition (4.63). Since the phase equation is linear, its solution can be presented as a superposition of phase ﬁelds θi generated by diﬀerent defects and subject to the respective circulation conditions. For the latter ﬁelds, it appears to be possible to write down solutions corresponding to arbitrary vortex trajectories with the help of an appropriate Green’s function. A diﬃculty arises, however, due to the multivaluedness of θ. A possible way out is to introduce, as in Sect. 4.4.2, a univalued function dual to θ which would obey the usual heat equation with a point source located at a mobile defect. This has been done consistently only for the case of rectilinear motion of a pair of interacting defects (Rodriguez et al., 1991). As before, the motion can be assumed, without loss of generality, to be along the y-axis. Then the dual function Φ(X, Y, T ) is deﬁned as ( ) Y

θX = N

ΦY −

ΦT dY

, −θY = N ΦX .

(4.126)

One can see that the equation ΦXX + ΦY Y − ΦT = 2πδ(X)δ(Y )

(4.127)

both serves as the integrability condition for θ in (4.126) and ensures that the circulation condition (4.63) be satisﬁed. The dual function can be expressed using the Green’s function of (4.127) as [Y + η(T ) − η(τ )]2 + X 2 1 t 1 exp − Φ(X, Y, T ) = dτ, (4.128) 2 0 T −τ 4(T − τ ) where η(T ) is an instantaneous position of the vortex moving along the axis X = 0. The phase gradient computed from (4.128), (4.126) is further used in the mobility relation (4.122) to obtain the velocities and subsequent positions of the vortices.

4.4 Motion of Dislocations

251

Fig. 4.16. The dependence of the velocity on the distance for a pair of interacting defects of opposite sign, starting at a certain distance from a self-consistent phase ﬁeld, and eventually annihilating (Rodriguez et al., 1991. Copyright by the American Physical Society). The solid line shows the computed v(r) curve. It is ﬂanked by the dotted lines showing the self-consistent approximation (below) and the “solid body” approximation with the initial phase unchanged (above)

Figure 4.16 (Rodriguez et al., 1991) shows the results of computations utilizing the above procedure for the case of a pair of unlikely charged defects moving toward each other in an unbounded domain from some initial distance and eventually annihilating at the point Y = 0. Due to the symmetry, it is suﬃcient to consider a single defect together with its image in a reﬂecting wall placed at Y = 0. Ideally, the interacting defects should be nucleated at an inﬁnite separation distance. For a practical numerical calculation the integration must start from a ﬁnite separation, and hence there is some uncertainty in the choice of initial conditions. The initial phase distribution in the computation shown in Fig. 4.16 corresponds to the self-consistent solution at the initial separation. The velocity dependence on the instantaneous separation, v(r), is shown by the solid line in the ﬁgure. For comparison, the dashed line in the same plot shows the quasistationary self-consistent dependence (4.125) and the dotted line a “solid body” solution computed under the assumption that one of the vortices moves in the unchanging phase ﬁeld of the other. The numerically computed velocity lies within these two bounds. The vortices accelerate steeply at close approach, and the computations have to stop shortly before the moment of annihilation, when the basic assumptions of slow motion and weak phase gradients are no longer valid. Computations are in reasonable agreement with experiments carried out on weakly distorted electroconvection patterns in nematics, where basic alignment of the pattern was ensured by treated solid boundaries (Braun and Steinberg, 1991), but

252

4 Amplitude Equations for Patterns

the precision is not suﬃcient to resolve the scaling factor of the logarithmic nonlinearity. 4.4.6 Motion in a Supercriticality Ramp Although the phase ﬁeld is the dominant cause of vortex motion in extended homogeneous systems with low density of defects, it is also possible to set vortices into motion by aﬀecting the background amplitude. In the framework of the RGL model, this can be done in the simplest way by making the linear growth rate position-dependent. Then the RGL Lagrangian (4.9) is rewritten as 2 , (4.129) L = 12 ∇u · ∇u + 12 µ(x) − |u|2 and the evolution equation (4.3) becomes ut = ∇2 u + (µ(x) − |u|2 )u.

(4.130)

The supercriticality parameter µ may be position-dependent due to either local inhomogeneities or a long-range ramp induced by boundary conditions. Since the energy of the vortex is lower where µ is small, stable conﬁgurations should correspond to vortices pinned at local minima of µ(x). It is natural to expect that, neglecting vortex interactions, the gradient dynamics would be reduced to a vortex drift down the gradient of µ(x). The eventual eﬀect of inhomogeneities might be to suppress the annihilation of vortices at some stage of the coarsening process, and thus freeze a disordered “vortex glass” state. The case when µ(x) varies on a characteristic scale far exceeding the size of the vortex core (the healing length) and the vortex drift is slow is treated perturbationally as above. The local landscape in the core region is expanded as µ(x) = µ0 (1 + M · x + · · ·), where µ0 = µ(x0 ) and its logarithmic gradient m ≡ M = ∇ ln µ(x0 ) are measured at the vortex vocation. We set v = V and expand in the dummy small parameter . In order to bring the core expansion equations to the same form as in the preceding section, it is convenient to eliminate µ0 by rescaling both the order parameter 1/2 −1/2 u and the temporal and spatial variables as u → µ0 u, r → µ0 r, t → µ−1 0 t. Then the rescaled zero-order vortex solution retains the form u0 = eiφ ρ0 (r), where the function ρ0 (r) veriﬁes (4.76). The ﬁrst-order equation has the general form (4.113) with the inhomogeneity Ψ (x) = V · ∇u0 + (M · x)u0 .

(4.131)

The solvability condition (4.116) is obtained exactly as before, but the area integral now includes a new term proportional to M . If V, M are the

4.4 Motion of Dislocations

253

respective projections on the y-axis, the area integral corresponding to the translational eigenfunction along the y-axis is r0 r0 2

r(ρ0 ) + N 2 r−1 ρ20 dr + πM πV r2 ρ0 ρ0 dr 0 0 r0 r0 π V ln + M ln . (4.132) a0 am Both integrals depend logarithmically on the upper limit r0 at large distances but with diﬀerent constants; for a vortex of unit charge am ≈ 0.67. This numerical discrepancy cannot be disregarded, and necessitates matching with the far ﬁeld, as ﬁrst-order distortions of the vortex core are not suppressed, in spite of the similarity of the asymptotics of the two integrals. The simplest large-scale setup is a single vortex moving along a supercriticality ramp. The amplitude in the far ﬁeld is no longer constant, but it remains √ a slaved variable, so that ρ = µ in leading order. Due to the variability, an · ∇θ = M · ∇θ is added to the far ﬁeld equation (4.64). extra term µ−1 ∇µ Clearly, V and M are collinear, and we can assume that both are directed along the Y -axis. Then the stationary far ﬁeld equation in the comoving frame becomes (4.133) (V + M )θY + θXX + θY Y = 0. This equation has the same form as (4.91), so we can write immediately the required inner asymptotics as a modiﬁed (4.119) (assuming there is no extrinsic phase gradient):

1 (4.134) θ = N φ − 12 N (V + M )r cos φ ln 14 (V + M )reγ− 2 . Using (4.132), (4.134), the solvability condition (4.116) becomes

1 −(v + ∇µ) ln 14 |v + ∇µ|eγ− 2 = v ln a0 + ∇µ ln am , (4.135) where we have rescaled to short-scale units and restored the vector notation. Note that, as long as am = a0 , the relation between the velocity and the local supercriticality gradient is nonlinear, although (at small |v|, |∇µ| when this formula is applicable) the deviation from the “naive” linear dependence v = −∇µ is small (Fig. 4.17). One can combine motion under the action of both amplitude and phase gradients. The mobility relation including the local phase gradient is just a merger of (4.123) and (4.135): vm v0 N J v + ln ∇×µ , (4.136) ln k= 2 |v + ∇µ| |v + ∇µ| where vm = (4/am ) exp(−γ + 12 ) ≈ 5.55.

254

4 Amplitude Equations for Patterns

0.1

m

0.08 0.06 0.04 0.02 0 0

0.02

0.04

0.06 v

0.08

0.1

0.12

Fig. 4.17. Relation between the gradient of the supercriticality parameter m = |∇µ| and the vortex velocity v

Even when no extrinsic inhomogeneities are present, vortices themselves may, in eﬀect, serve as a source of variable supercriticality in their neighborhood. If the real amplitude is presented by a product of vortex amplitudes as in (4.66), the action of other vortices eﬀectively reduces the linear growth rate at the vortex location. This should cause mutual attraction of all vortices, irrespectively of charge. The topological (charge-dependent) interaction, mediated by the phase ﬁeld, is, however, stronger, since the phase gradient decays at large distances as L−1 , compared with an L−2 decay of amplitude inhomogeneities. Topological repulsion of like vortices can be compensated by nontopological attraction, producing an unstable equilibrium at some critical separation. This conﬁguration, which corresponds to an incipient break-up of a double-charged vortex, would, however, be attained at distances where the perturbation theory is no longer applicable.

4.5 Propagation of Pattern and Pinning 4.5.1 Wavelength Selection in a Propagating Pattern When a stable pattern advances into an unstable homogeneous state, the problem of selection of propagation speed considered in Sect. 2.3 is augmented by the problem of selection of the wavelength of the emerging pattern. As a rule, there is a one-parametric family of patterned states parametrized by wavelength, which expands into a two-parametric family of propagating solutions. With all this, we expect that a unique propagation speed and a unique wavelength will be selected dynamically.

4.5 Propagation of Pattern and Pinning

255

The simplest nontrivial example of selection of a patterned state gives the Swift–Hohenberg model (1.21) that we shall write here in the 1D form ut = −(1 + ∂x2 )2 u + µu − u3 .

(4.137)

We assume, as in Sect. 2.3.2, that selection is determined by the leading edge of the propagating pattern and use the dispersion relation computed according to (2.72), (2.73):

(4.138) ω(k) = i µ − (1 − k 2 )2 . The group velocity is

c = ω (k) = 4ik(1 − k 2 ).

(4.139)

Using this expression, we ﬁnd that the condition ci = 0 in (2.75) has a nontrivial solution (4.140) kr = 1 + 3ki2 . The indices r and i denote here, as in Sect. 2.3, the real and imaginary parts, respectively. The function C(ki ) = ωi /ki − cr (k) deﬁned by (2.81) and computed with the help of this relationship is monotonically decreasing, and vanishes at

1/2 1 1 + 6µ − 1 . (4.141) ki = √ 2 3 This deﬁnes the slope of the marginally stable amplitude proﬁle. The propagation speed (4.139) is now evaluated as

1/2 4 c= c = 8ki (1 + 4ki2 ) = √ 2 + 1 + 6µ 1 + 6µ − 1 . 3 3

(4.142)

The wavenumber kr selected at the leading edge is not identical to the wavenumber of the full-grown pattern formed behind the front. The latter can be computed assuming that the number of wave crests does not change in a

b

c

k 1.35 1.3 1.25 1.2 1.15 1.1 1.05

8 6 4 2 0.5

1

1.5

2

2.5

3

Μ

0.5

1

1.5

2

2.5

3

Μ

Fig. 4.18. Dependence of the propagation velocity (a), of the wavenumber at the leading edge kr (b, upper line), and of the selected wavenumber k∗ (b, lower line) on the supercriticality parameter µ

256

4 Amplitude Equations for Patterns a 15

10

5

0 0

30

60

90

120

0

30

60

90

120

b 15

10

5

0

Fig. 4.19. Space-time plot of simulations results for pulled fronts in the SH equation with µ = 0.4 (a) and the modiﬁed equation (4.145) with µ = 0.4, ν = 3 (b). The lines denote the proﬁles of u at successive time intervals and are shifted upward relative to each other (van Saarloos, 2003; reproduced with permission. Copyright by the Elsevier Science)

the wake of the propagating front (Ben Jacob et al., 1985). The frequency of the emerging wave is determined by ωr that is Doppler-shifted in the moving frame to ω = ωr − ckr , and the wave crests are convected into the interior of the advancing pattern with the speed c. The actual wavenumber of the

4.5 Propagation of Pattern and Pinning

257

stationary pattern left behind the propagating front must therefore be 3/2 √ 3 3 + 1 + 6µ ω ωr . = k = − = kr − √ c c 8 2 + 1 + 6µ ∗

(4.143)

It is remarkable that the selected wavenumber coincides with the optimal value minimizing the energy functional of the SH model only in the limit µ → 0. The dependence of kr and k ∗ and c on µ is shown in Fig. 4.18. When the bifurcation point µ = 0 is approached, these values vary as ki ≈

1√ 3 1 √ µ, kr ≈ 1 + µ, k∗ ≈ 1 + µ, c ≈ 4 µ. 2 8 8

(4.144)

These predictions were conﬁrmed with high precision by direct computations (Ben Jacob et al., 1985). An example of the front advancement is seen in Fig. 4.19a. In spite of the success achieved in this and other simple cases, selection at the leading edge may have only limited inﬂuence on the properties of the full-grown pattern, since nonlinear interaction behind the front may modify the geometry and aﬀect the propagation speed. One can observe that, while the linear spreading dynamics of the SH equation is similar to that of the Cahn–Hilliard equation in Sect. 2.4.5, the dynamical nonlinear behavior of the patterns forming behind the front, which is determined by nonlinear interactions, is very diﬀerent in the two cases. The pattern formed in the SH equation does not coarsen as in Fig. 2.12, even though the selected wavelength is not optimal. It may happen, however, also in this case that the selected pattern is unstable, as seen, for example, in Fig. 4.19b. This particular computation was carried out for the SH equation modiﬁed by including a symmetry-breaking quadratic term: ut = −(1 + ∇2 )2 u + u(µ − u2 ) + νu2 .

(4.145)

The simulation results show that another front establishing a homogeneous nontrivial stationary state is formed behind the front of the advancing pattern. 4.5.2 Self-Induced Pinning In the case when a stable homogeneous solution coexists with a stable periodic pattern, Pomeau (1984, 1986) has described the mechanism of robust existence of stable stationary fronts between the two states. The motion of this front is aﬀected by the discrete structure of the pattern, which causes self-induced pinning hindering the retreat of a metastable state. There are two depinning transitions, corresponding to “crystallization” or “melting” of the pattern, shown schematically by thick lines in Fig. 4.20. To the right of the crystallization threshold C, the pattern advances by a periodic nucleation process which creates new elementary cells at the interface, while to the left

258

4 Amplitude Equations for Patterns 1-M

M

1-C

C

Fig. 4.20. A scheme of depinning transitions showing crystallization (C) and melting (M) thresholds for an inﬁnite cluster, as well as the corresponding limits for clusters of diﬀerent sizes, terminating in single-cell limits 1-C, 1-M

of the melting limit M, the pattern recedes as elementary cells at the interface are destroyed. A variety of metastable stationary structures including a diﬀerent number of cells can be observed within the interval between the melting transition M and the single-cell spreading limit 1-C lying to the right of the crystallization threshold . The minimal number of cells in a sustainable stationary structure increases as the melting threshold is approached; the corresponding melting limits, starting from the single-cell melting limit 1-M, are shown schematically by thin vertical lines in Fig. 4.20. The respective crystallization thresholds for clusters of ﬁnite size lie to the right of C approaching this point as the cluster size increases. Hilali et al. (1995) obtained metastable pinned structures numerically both in 1D and 2D for the modiﬁed SH equation (4.145). Another suitable system is the quintic SH equation (Sakaguchi and Brand, 1996) ut = −(1 + ∇2 )2 u + u(µ + νu2 − u4 ).

(4.146)

The pinned structures shown in Fig. 4.21 include a single-cell (“soliton”) solution (a); a ﬁnite patterned inclusion, sandwiched between semi-inﬁnite domains occupied by a uniform state (b); a semi-inﬁnite pattern, coexisting with a uniform state (c). A soliton can be viewed as an island where one of the uniform states is approached, immersed in the inﬁnite region occupied by an alternative state; the basic soliton “atom” is equivalent to a single period of the pattern immersed in a uniform state. A patterned inclusion can also be placed asymmetrically between the regions occupied by alternative uniform states. In 1D, the simplest system suitable for the study of pinned structures is the standard 1D SH equation (4.137). A small-amplitude striped pattern with the wavenumber k = 1 ﬁrst appears at the symmetry-breaking bifurcation point µ = 0. As the parameter µ further increases, the available wavenumber band widens, until it reaches the limiting value k√= 0. This signals the appearance of a pair of nontrivial uniform states, u = ± µ − 1. The two symmetric states are stable to inﬁnitesimal perturbations at µ > 3/2. Since the SH equation is a gradient system, one can use the expression for the energy functional

4.5 Propagation of Pattern and Pinning

259

Fig. 4.21. Metastable stationary 1D solutions of (4.145): a soliton (a); a ﬁnite patterned inclusion (b); a semi-inﬁnite pattern, coexisting with a uniform state (c). (Hilali et al., 1995; reproduced with permission. Copyright by the American Physical Society)

260

4 Amplitude Equations for Patterns

F=

1 2

[(1 + ∇2 )u]2 − µu2 + 12 u4 dx

(4.147)

to compute the energy of stationary solutions obtained in numerical computations, and characterize them as metastable states. The energy of the uniform state is higher than that of the regular pattern with the optimal wavenumber at µ < µM ≈ 5.7, and all above solutions containing uniform stretches should be metastable as well. In spite of the diﬀerence between the energies of the uniform and patterned state, the one-dimensional interface between them remains immobile at moderate and large values µ, and pinned fronts, as well as multiple localized states, remain stable to inﬁnitesimal perturbations. The general scheme of crystallization and melting thresholds is the same as in Fig. 4.20, with µ increasing from the right to the left. Numerical computation (Aranson et al., 2000) shows that the depinning transition for a single soliton (point 1-C) occurs at µ ≈ 1.74. The depinning threshold increases with the size of the patterned cluster, rapidly converging to the limiting crystallization threshold for the semi-inﬁnite pattern µ = µc ≈ 1.7574 . . . (point C). Self-induced pinning impeding propagation of a pattern into a metastable uniform state (“crystallization”) at moderate µ or the reverse “melting” process at µ 1 can be attributed to the oscillatory character of the asymptotic perturbations of the uniform state, as demonstrated by the phase space analysis in the next subsection. 4.5.3 Geometry of Pinned States The stationary solutions of (4.137) verify (1 − µ)u + u3 + 2uxx + uxxxx = 0.

(4.148)

This equation can be viewed as a dynamical system with the coordinate playing the role of time. The correspondence between spatial structures and trajectories of the respective dynamical system is summarized in Table 4.1. The spatial inversion symmetry of (4.137) is translated to reversibility of the dynamical system (4.148), which can be transformed to a nondissipative equation of motion of two coupled particles. The conserved quantity is Table 4.1. Correspondence between spatial structures and trajectories of a dynamical system. Spatial structure Regular pattern Symmetric inclusion Asymmetric inclusion Pinned front

Trajectory Periodic orbit Homoclinic orbit Heteroclinic orbit Heteroclinic connection between a ﬁxed point and a periodic orbit

4.5 Propagation of Pattern and Pinning

H = 14 (1 − µ + u2 )2 + u2x + ux uxxx − 12 u2xx .

261

(4.149)

Both homoclinic and heteroclinic orbits lie on the 3D hypersurface H = 0 in the 4D phase space of (4.148) spanned by u, ux , uxx , uxxx . At µ > 3/2, when the ﬁxed points are dynamically stable as solutions of (4.137), they are double foci when viewed as solutions of the dynamical system (4.148). The two pairs of complex eigenvalues obtained by linearizing (4.148) near the ﬁxed points have real parts of the opposite sign: 1 1 1 1 (4.150) k=± 2 (µ − 1) − 2 ± i 2 (µ − 1) + 2 . A homoclinic orbit exits the ﬁxed point along its unstable manifold spanned by the pair of eigenvalues with positive real parts and returns along its stable manifold spanned by the pair of eigenvalues with negative real parts. Both stable and unstable manifolds are two-dimensional, and, since they are conﬁned to the 3D manifold H = 0, they have to intersect generically along a line. Due to inversion symmetry, the homoclinic trajectories have to intersect the plane ux = uxxx = 0; the stable and unstable manifolds are mirror images of each other with respect to this plane. A heteroclinic orbit connects the unstable manifold of one ﬁxed point with the stable manifold of its counterpart. This trajectory is antisymmetric and has to intersect the plane u = uxx = 0. Heteroclinic orbits owe their existence to the symmetry of (4.137) or (4.148) to the inversion of u and disappear when this symmetry is broken as in (4.145). A periodic orbit must intersect the plane ux = uxxx = 0 exactly twice. At any chosen value of µ > 0, there is a continuum of periodic orbits with periods

Fig. 4.22. Phase portrait of (4.148) under conditions when pinned fronts exist (Coullet et al., 2000; reproduced with permission. Copyright by the American Physical Society). See the text for explanations

262

4 Amplitude Equations for Patterns

varying within a certain range. This family of orbits can be parametrized by the value of H. Since H is conserved, only a single orbit corresponding to H = 0 can connect to a ﬁxed point. Thus, a pinned front, like a pulled front in Sect. 4.5.1, selects a certain wavelength of the pattern, which does not need to coincide with the optimal wavelength minimizing (4.147). Genesis of pinned states can be understood (Coullet et al., 2000) by constructing a Poincar´e map in the phase space of (4.148). The appropriate Poincar´e section is the plane ux = uxxx = 0, denoted as Π in Fig. 4.22, which is crossed by all periodic trajectories. The family of periodic orbits parametrized by H is projected onto the line P in this plane. Each periodic orbit, under conditions when it is dynamically stable as a solution of (4.148), has a one-dimensional stable and a one-dimensional unstable manifolds. The stable and unstable manifolds of the family of periodic orbits span the surfaces W s (P ) and W u (P ) symmetrical with respect to Π (Fig. 4.22). The stable and unstable manifolds of the ﬁxed points are also symmetrical with respect to Π. Trajectories W s (A) and W u (A) lying in these manifolds intersect, respectively, W u (P ) and W s (P ) at the points marked by ﬁlled circles in Fig. 4.22 when a pinned front exists. The intersections of these curves with Π are marked in Fig. 4.22 by empty circles; the corresponding points on the pinned and localized structures are shown in Fig. 4.23. The evolution of the orbits corresponding to pinned and localized structures when a parameter of

Fig. 4.23. Correspondence between the marked points in Figs. 4.22, 4.24b and elementary cells of pinned fronts (upper picture) and localized structures with an even (lower right) or odd (lower left) number of elementary cells (Coullet et al., 2000; reproduced with permission. Copyright by the American Physical Society)

4.5 Propagation of Pattern and Pinning

263

Fig. 4.24. Parametric evolution of the orbits corresponding to heteroclinic and homoclinic point structures. In (a), a pinned front appears. In (b), the existence of the front induces the existence of localized structures; the number of pairs of intersections of Wu and Π equals to the number of elementary cells. In (c), the pinned front disappears. In (d), localized structure persists after the pinned front has disappeared. (Coullet et al., 2000; reproduced with permission. Copyright by the American Physical Society)

the problem changes as in Fig. 4.20 is shown schematically in Fig. 4.24, where the diagrams (a) and (c) correspond to the points M and C in Fig. 4.20. 4.5.4 Crystallization Kinetics The front propagation process can be described in terms of periodic nucleation events triggered by explosive growth of the localized zero-eigenvalue mode of the corresponding linear problem (Aranson et al., 2000). Figure 4.25 shows propagation of a semi-inﬁnite pattern into a uniform stable state at µc = 1.75. The front propagation takes the form of well separated in time periodic nucleation events creating new “atoms” of the “crystalline” state at the front. Between each successive nucleation events, the solution remains close to the stationary semi-inﬁnite pattern found at µc . This process resembles crystallization in equilibrium solids, with the important distinction that the new “atoms” are created directly from the metastable “vacuum” state. The time between consecutive nucleation events diverges as µ approaches the pinning threshold. Figure 4.26 presents the average front speed c as a function of √ µ−µc . This function can be approximated by c = c0 µc − µ, with c0 = 2.292.

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4 Amplitude Equations for Patterns

Fig. 4.25. The space-time plot of the stick-and-slip propagation of a large cluster (approximating a semi-inﬁnite pattern) into the metastable uniform phase at µ = 1.757. The horizontal axis is spatial coordinate, and the vertical axis is time (Aranson et al., 2000. Copyright by the American Physical Society)

At |δ| = µ − µc > 0, the immobile front solution u0 (x, µ) is linearly stable. Numerical stability analysis shows that there exists an exponentially decaying eigenfunction u ˜(x, µ) localized at the front with a negative eigenvalue λs , which approaches zero as |δ|1/2 . In addition, there is also an unstable front solution and a corresponding eigenfunction with a positive eigenvalue λu which also approaches zero as |δ|1/2 . At the pinning threshold, these two solutions collide and disappear via a saddle-node bifurcation. The stationary front solution U0 (x) ≡ u0 (x, µc ) and the corresponding eigenfunction U1 (x) ≡ u1 (x, µc ) with the eigenvalue vanishing at µ = µc are shown in Fig. 4.27. In the inset, the stable and unstable eigenvalues λs,u are shown as functions of µ. At µ < µc , the front solution becomes nonstationary. Nonetheless, the numerical study suggests that at |δ| 1 the solution remains close to the stationary front solution U0 (x) all the time except short intervals when a new cell nucleates. The front propagation can be analyzed perturbatively near the critical value µ = µc . At |δ| 1, the front solution can be presented in the form u(x, t) = U0 (x) + |δ|1/2 u1 (x, t), where U0 (x) is the stationary front solution at δ = 0 and u1 (x, t) is a small correction. This solution is uniformly valid at small positive δ; it is also valid during quasistationary phases (away from nucleation events) for small negative δ. Plugging this ansatz in (4.137), we obtain

(4.151) ∂t u1 = L[U0 ]u1 − 3|δ|1/2 U0 u21 − sign(δ)U0 + · · · , where L[U0 ] ≡ µc − 3U02 − (1 + ∇2 )2 is the linearized SH operator at µc . In the lowest order in δ, (4.151) yields the linear equation ∂t u1 = L[U0 ]u1 . This equation is always veriﬁed by the stationary translational mode U0 (x).

4.5 Propagation of Pattern and Pinning

265

Fig. 4.26. The average front speed c as a function of the deviation from the critical point δ = µ − µc . Points represent numerics, the dashed line is the best ﬁt c = 2.29|δ|1/2 and the dot-dashed line is the theoretical prediction c = 1.80|δ|1/2 (Aranson et al., 2000. Copyright by the American Physical Society)

In addition, the linearized operator L[U0 ] has a localized neutral eigenmode U1 shown in Fig. 4.27. Since all other eigenmodes have negative eigenvalues, the evolution of the system close to the bifurcation point can be reduced to single-mode dynamics near a saddle-node bifurcation point. Therefore, in the lowest order we can write u1 (x, t) = a(t)U1 (x), where a(t) is the amplitude of the zero mode. ˙ 1 can be treated as perturClose to the bifurcation threshold, ∂t u1 = aU bation. Therefore, in the second order we derive

(4.152) L[U0 ]u2 = aU ˙ 1 + |δ|1/2 3a2 U0 U12 − sign(δ)U0 . This equation has a bounded solution if its r.h.s. is orthogonal to the zero mode U1 of the operator L. This results in the solvability condition for the amplitude a:

(4.153) α a˙ = |δ|1/2 sign(δ)β − γa2 , ∞ 2 ∞ ∞ 3 where α = −∞ U1 dx, β = −∞ U1 U0 dx, γ = 3 −∞ U1 U0 dx. At δ > 0, a(t) reaches a stationary amplitude a0 = (β/γ)1/2 . This value corresponds to the diﬀerence between the stable front solutions at µ and µc . At small negative δ, (4.153) describes explosive growth of a, which passes from −∞ to ∞ in a ﬁnite time τe = πα/(|δ|βγ)1/2 . This explosion time gives an upper bound for the period between the nucleation events, after which the whole process repeats. The front speed is computed as c = Λ/τe , where

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4 Amplitude Equations for Patterns

Fig. 4.27. The stationary structure U0 (x) (solid line) and the corresponding localized zero mode U1 (x) (dashed line) for µ = µc . Inset: eigenvalues of the stable and unstable localized modes of the stationary front solutions for µ > µc (Aranson et al., 2000. Copyright by the American Physical Society)

Λ is the asymptotic spatial period of the pattern selected by the nucleation process, and τe is the time interval between nucleation events. The calculations give the value c = 1.8|δ|1/2 . This is in a good qualitative agreement with the results of the numerical simulations, see Fig. 4.26. However, the prefactor 1.8 is noticeably lower than the corresponding value c0 = 2.29 obtained by numerical simulation of (4.137). This discrepancy may be caused by distortions behind the moving front triggered by nucleation events. These distortions may eﬀectively “provoke” a consequent nucleation event by creating an initial perturbation of the zero mode U1 (x). This will lead to an increase of the front velocity. Although numerical computations give evidence for the importance of this eﬀect, its systematic study is very complicated and goes beyond perturbation theory. At very large µ√beyond the Maxwell construction µM ≈ 5.7, the homogeneous state u = ± µ − 1 has a lower energy than the periodic state. Nevertheless, the uniform state still does invade periodic state because of a strong self-induced pinning. In the presence of noise, propagation may be, however, triggered by thermally activated annihilation events at the edge of the periodic pattern, leading to “melting” of the periodic structure. The above results cannot be trivially extended to a regular 2D periodic structure (striped pattern) selected by the SH equation (1.21). By analogy with 1D dynamics, one could envisage depinning mechanism via local nucleation of new stripes of the patterned state, which subsequently spread out

4.5 Propagation of Pattern and Pinning

267

Fig. 4.28. Depinning of the striped pattern state in the SH equation at µ = 4 (Hagberg et al., unpublished)

sidewise. The actual mechanism of depinning seen in computations (Hagberg et al., unpublished) is, however, less straightforward. It is initiated by a zigzag instability of the pattern followed by the nucleation of convex–concave disclination pairs (Fig. 4.9a,b), with the convex disclinations moving toward the uniform state, as seen in Fig. 4.28. This generates stripes extending in the normal direction, turning eventually the original boundary into a domain wall separating striped patterns rotated by π/2. The pinning range shrinks drastically to the vicinity of µM due to the more eﬃcient 2D depinning mechanism. 4.5.5 Pinning of Defects The short-scale structure of the pattern may cause likewise pinning of defects – dislocations or domain walls – which appear to be mobile on the level of amplitude equations. Close to a primary bifurcation, this phenomenon can be understood as a “nonadiabatic” eﬀect dependent on oscillatory corrections to the amplitude equations (Pomeau, 1984; Bensimon et al., 1988; Malomed et al., 1990). We recall that the common procedure of derivation of amplitude equations (Sect. 1.6.2) based on computation of solvability conditions selects resonant terms only, while all terms oscillating with a period diﬀering from the period of the basic structure, e.g., the term proportional to the cube of the amplitude, vanish upon integration. When the amplitudes change on an extended scale, the integrals including nonresonant terms do not, however, vanish precisely, but leave a remainder, which depends exponentially on the small parameter measuring the ratio of the basic scale of the pattern to the characteristic scale of the amplitude change. The amplitude equations corrected to include nonadiabatic eﬀects retain these terms. This introduces a short-scale perturbation which acts as a shallow periodic potential on the motion of defects described in the leading order by common amplitude equations. For example, a striped pattern with the wavenumber k0 and the wave vector directed along the z-axis introduces through a cubic nonlinearity in underlying equations a potential proportional to cos 2−1 k0 x , where the large factor −1 appears when the coordinate x is measured in the long units of the amplitude equations. The corrected equation of motion of a

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4 Amplitude Equations for Patterns

Fig. 4.29. A domain wall conﬁguration with a curved set of stripes obtained by numerical solution of the SH equation (Boyer and Vi˜ nals, 2001; reproduced with permission. Copyright by the American Physical Society)

defect describing its position X(t) is obtained then in the form X(t) = c0 + c1 cos 2−1 k0 x ,

(4.154)

where c0 is the unperturbed speed determined on the amplitude equation level, and the parameter c1 is proportional to exp(−q/) with q = O(1). If c0 < c1 , (4.154) has alternating stable and unstable ﬁxed points, and the defect is pinned at one of stable positions. Otherwise, the propagation speed oscillates on a short time scale, and, while c0 exceeds c1 only slightly, the motion is of a “stick-and-slip” kind similar to that shown in Fig. 4.25. Bensimon et al. (1988) applied (4.154) to propagation and pinning of a front between a patterned and homogeneous states in the SH equation modiﬁed by adding a quintic term, which is, in essence, the same problem that was treated more precisely in Sect. 4.5.4. Malomed et al., (1990) applied a similar equation with a more complicated periodic term to a domain wall between a hexagonal and striped patterns. Boyer and Vi˜ nals (2001, 2002) considered in the same way a domain wall parallel to the direction of stripes on the one side and normal to the direction of stripes on the other side described by the stationary equations (4.85), (4.86). This conﬁguration was turned dynamic through long-scale perturbation of the parallel set of stripes, as shown in Fig. 4.29, and the resulting motion was analyzed with the help of nonadiabatic equations applicable to a weakly distorted structure. The same approach can be applied to motion of dislocations. The theory based on envelope equations does not distinguish between climbing of dislocations along the direction of stripes and their gliding in the direction of the wavenumber vector, so that interaction of dislocation pairs seen in Figs. 4.6d, e is indistinguishable in this approximation. Motion across the stripes is, however, aﬀected by the discrete short-scale structure of the pattern, which may cause either motion with a variable speed or pinning. The action of a periodic potential on interaction of dislocation pairs is illustrated schematically

4.5 Propagation of Pattern and Pinning

attract

repel

269

Fig. 4.30. Attracting and repelling dislocation pairs under the action of a periodic potential

in Fig. 4.30. The method used in Sect. 4.4.6 is not applicable in this situation, since the perturbation is short-scale, but one can expect that (4.154) is applicable, at least qualitatively, also to this situation. Generally, nonadiabatic corrections are not based on a ﬁrm ground of rational perturbation theory. Due to the exponentially weak dependence on the scale ratio, they can be substantial only when this ratio is not too small. Strictly speaking, all higher order algebraic terms, which are neglected in a usual amplitude expansion, decay at → 0 slower than exponentially, and retaining an exponentially small term while neglecting algebraically small ones cannot be formally justiﬁed. Nevertheless, adiabatic corrections give a faithful, at least in a qualitative sense, model of defect pinning conforming to direct numerical computations. Beneath the level of envelope equations, there is a direct connection between dislocations and domain walls, since the latter can be viewed as dislocation chains. This short-scale structure revealed in both experiments and simulations, as e.g. in Figs. 4.5 and 4.29, but, of course, remains hidden when the amplitude equations are solved as in Sect. 4.3.6. It is seen most clearly in Fig. 4.31 where the loci of Re u = 0 and Im u = 0 are drawn for the normal/parallel domain wall with ρ± solving (4.85), (4.86).

Fig. 4.31. Loci of Re u = 0 (solid lines) and Im u = 0 (dashed lines) intersecting at dislocations for the normal/parallel domain wall

270

4 Amplitude Equations for Patterns (a)

(b)

(c)

(d)

Fig. 4.32. A pattern in a circular cell (a) and its reconstruction (c, d) as a superposition of the phase ﬁeld of the array of defects shown in (b). The full circles mark the defects with positive topological charge and the open circles, mark the defects with negative charge. The reconstruction in (c) uses the symmetric phase ﬁeld, while that in (d) uses a phase ﬁeld with a pitch (La Porta and Surko, 2000; reproduced with permission. Copyright by the Elsevier Science)

La Porta and Surko (2000) used the representation of a domain wall as a dislocation chain for reconstruction of convection patterns observed experimentally. The pattern in a circular cell seen in Fig. 4.32a contains a number of dislocation chains forming domain walls, as well as isolated dislocations. All these defects, supplemented by compensating defects with the opposite topological charge placed on the cell boundaries, are extracted in Fig. 4.32b. It is remarkable that the entire pattern can be approximated by the superposition of the “amplitude ﬁelds” (4.66) of these defects. The reconstructed

4.6 Hexagonal Patterns

271

pattern presented in Fig. 4.32c has, however, a smaller mean wavenumber than the original pattern, and the orientations of stripes in individual domains are diﬀerent. The reconstruction becomes more faithful (Fig. 4.32d) if a pitch is added to the phase ﬁeld of the dislocations, i.e., the symmetric structure θi = φi in the polar coordinate system ri , φi centered on the ith defect is replaced by a spiral phase ﬁeld θi = φi + ψri . In a later work (Aegerter and Surko, 2002) additional compression of the phase ﬁeld with empirically adjusted parameters was introduced to further improve the ﬁt. This representation was motivated by a possibility of describing evolution of a distorted pattern in “particle-ﬁeld” approximation, which reduces the continuous “ﬁeld” problem to motion of a ﬁnite number of “particles” – defects in the manner of Sect. 4.4. This prospect is, however, illusory, since only widely separated defects rather than strongly coupled arrays are amenable to this technique. Pinning of both dislocations and domain walls may arrest evolution to an ordered state even when the system has a gradient structure. Computations of Boyer and Vi˜ nals (2002) using the SH model suggested that the characteristic scale of the structure (or the typical size of ordered domains), which increases with time as t1/3 at intermediate stages of a computation run starting from random initial conditions, saturates at long times, leaving a “glassy” state containing dislocations, disclinations, and domain walls, similar to that shown in Fig. 4.5.

4.6 Hexagonal Patterns 4.6.1 Triplet Amplitude Equations As we have already noticed in Sect. 1.6.1, the hexagonal pattern in 2D is supported by resonant interaction among a triplet of modes with wave vectors forming an equilateral triangle. The spatial extension of the amplitude equations (1.127) can be written in a rescaled form

u˙ j = (nj · ∇)2 uj + uj µ − |uj |2 − ν |uj−1 |2 + |uj+1 |2 + uj−1 uj+1 . (4.155) Here nj is the unit vector along the wave vector corresponding to the jth mode (indexed modulo 3); otherwise, the notation is the same as in (1.127). These equations are derivable from the Lagrangian L=

2

(nj · ∇)uj (nj · ∇)uj + V,

(4.156)

j=0

where the potential V is deﬁned by (1.128). The diﬀerential term is written in the form ﬁrst used by Pomeau (1986), including the derivative in the direction of the wave vector only. This is suﬃcient here since, unlike a striped pattern,

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4 Amplitude Equations for Patterns

a rigid structure of a hexagonal pattern is supported by resonant quadratic interactions, which makes operators of the NWS type (including higher order derivatives in transverse directions) superﬂuous. If the amplitudes and the spatial derivatives are scaled as u = O(), ∇ = O(), the orders of magnitudes in (1.127) formally match only when parameters of an underlying problem are chosen in such a way that the lowest order (quadratic) interactions are weak; omitting the cubic terms is, however, impossible, since the pattern cannot be otherwise stabilized at ﬁnite amplitudes (see Sect. 1.6.1). With this scaling, (1.127) can be supplemented by nonlinear diﬀerential terms of the same formal order of magnitude. The two additional terms in (4.155) admissible by symmetry are iβs (uj+1 nj−1 · ∇uj−1 + uj−1 nj+1 · ∇uj+1 ) + iβa (uj+1 tj−1 · ∇uj−1 − uj−1 tj+1 · ∇uj+1 ),

(4.157)

where tj is the vector normal to nj and βs , βa are coeﬃcients, which can be obtained by bifurcation expansion of a particular underlying system. The ﬁrst, symmetric term, introduced by Brand (1989), describes nonlinear action of a small change of the wavelengths of the constituent modes, while the second, antisymmetric term (Kuznetsov et al., 1995) describes a more subtle eﬀect of a change in their orientation. Both terms are incompatible with the inversion symmetry and are formally smaller than the algebraic quadratic term. Moreover, they cannot appear in a bifurcation expansion of a reactiondiﬀusion system with constant diﬀusivities, since they are dependent on the presence of nonlinear diﬀerential terms in underlying equations. The only known system where the second (but not the ﬁrst) term in (4.157) appears generically in the lowest order of the bifurcation expansion is convection in a ﬂuid layer heated from below (Kuznetsov et al., 1995). In the commonly used Boussinesq approximation, both the algebraic quadratic term and the symmetric term are forbidden due to the momentum conservation law in the underlying hydrodynamic equations (Gershuni and Zhukhovitsky, 1976), and the only surviving quadratic term is the antisymmetric diﬀerential term, which appears whenever the boundary conditions on the top and at the bottom of the ﬂuid layer are not symmetric. Computations of Kuznetsov et al., (1995) gave, however, a quite small value of βa for a realistic case of rigid bottom and free top boundary conditions. Generically, the orders of magnitude of all terms can match near a manifold of a higher codimension where the algebraic quadratic interaction term vanishes. Then both symmetric and antisymmetric terms can appear in the third order of the bifurcation expansion. For example, starting from the modiﬁed SH equation with two symmetry-breaking quadratic terms, ut = −(1 + ∇2 )2 u + u(µ − u2 ) + νu2 + β|∇u|2 ,

(4.158)

one can annul the quadratic term in the second order of the bifurcation expansion near the critical point µ = 0 by setting ν = λ/2. Then the nonlinear diﬀerential term added to (4.155) is

4.6 Hexagonal Patterns

−iβ (uj+1 nj+1 · ∇uj−1 + uj−1 nj−1 · ∇uj+1 ),

273

(4.159)

which can be split √ into symmetric and antisymmetric terms as in (4.157) with βs = β/2, βa = 3β/2. The polar form of (4.155) with added nonlinear diﬀerential terms is

ρ˙ j = (nj · ∇)2 ρj + ρj µ − (nj · ∇θj )2 − ρ2j − ν ρ2j−1 + ρ2j+1 + f (θj−1 , θj+1 )ρj+1 ρj−1 cos Θ, (4.160) 2nj · ∇θj ρj+1 ρj−1 θ˙j = (nj · ∇)2 θ + nj · ∇ρj − f (θj−1 , θj+1 ) sin Θ, (4.161) ρj ρj where Θ = θj is the total phase, and the function f (θj−1 , θj+1 ) = 1 + (βs nj−1 + βa tj−1 ) · ∇θj−1 + (βs nj+1 − βa tj+1 ) · ∇θj+1 (4.162) lumps contributions of the resonant quadratic terms. Out of the six real variables deﬁning the complex amplitudes of the resonantly interacting triplet, four (the three real amplitudes ρj and the total phase Θ) aﬀect the potential energy deﬁned by (1.128), while the remaining two correspond to the global translational symmetries. On a scale large compared to the characteristic scale of (4.155), it is suﬃcient to consider the two translational modes. Equations of phase dynamics describing their evolution on an extended scale can be obtained by adiabatic elimination of ρj and Θ. The general form of the phase equation is quite complicated and cannot be reduced to a simple form φ˙ = D ∇2 φ + D⊥ ∇(∇ · φ) postulated by Lauzeral et al., (1993). In the absence of nonlinear diﬀerential terms, the phase equation can be derived from the Lagrangian (4.156) using the same method as in Sect. 4.2.2 (Hoyle, 2000). A simpliﬁed version of the phase equation will appear in the following subsections. The nonlinear diﬀerential terms break the gradient structure of the amplitude equations and thereby make nontrivial dynamic behavior possible. This can be proved by trying to construct a suitable ﬁrst-order Lagrangian, which should have the form u1 u2 m3 · ∇u3 + (circular transpositions), where mj is a unit vector in some direction, so far unknown. Variation with respect to u1 gives u2 (m3 − m1 ) · ∇u3 + u3 (m2 − m1 ) · ∇u2 , and to obtain the required term, we need m3 − m1 = n3 , etc., altogether six incompatible conditions on three vectors mj . 4.6.2 Skewed Triplets Antisymmetric quadratic terms encourage formation of distorted “skewed” hexagonal patterns with the angles between the constituent modes diﬀering from π/3. This peculiar eﬀect can be understood in its purest form by considering the amplitude equation retaining the antisymmetric quadratic terms only:

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4 Amplitude Equations for Patterns

u˙ j = (nj · ∇)2 uj + uj µ − |uj |2 − ν |uj−1 |2 + |uj+1 |2 + iβa (uj+1 tj−1 · ∇uj−1 − uj−1 tj+1 · ∇uj+1 ).

(4.163)

We restrict to patterns with nearly optimal wavelength. Distorted patterns satisfying this condition can be constructed by setting uj = ρj eiθj with a phase distortion wave vector q j = ∇θj normal to the basic wave vector of the respective mode, nj · q j = 0. Under this assumption, the polar form of (4.163) becomes

ρ˙ j = (nj · ∇)2 ρj + ρj µ − ρ2j − ν ρ2j−1 + ρ2j+1 + βa ρj+1 ρj−1 (qj−1 − qj+1 ) cos Θ, (4.164) ρ ρ j+1 j−1 θ˙j = −βa (qj−1 − qj+1 ) sin Θ, (4.165) ρj where qj = |q j |, Θ = θj . Consider stability of a striped pattern with the optimal wavelength u1 = √ µ, u2 = u3 = 0 to excitation of the side modes u2 , u3 . Assuming the amplitudes of the side modes to be equal (ρ2 = ρ3 = ρ 1), the phase equations for the combinations θ± = θ2 ± θ3 derived from (4.165) become √ θ˙+ = βa µ(q3 − q2 ) sin θ+ ,

√ θ˙− = βa µ(q3 + q2 ) sin θ+ .

(4.166)

One can set here q2 = −q3 = q, θ− = 0. Then, provided q and βa have the same sign, θ+ = Θ relaxes to zero (modulo 2π), and the equation for the perturbation ρ following from (4.164) is ˙ = ρ [ µ(1 − ν) + qβa √µ ] , ρ

(4.167)

Ρ 1 0.8 0.6 0.4 0.2

1.2

1.4

1.6

1.8

2

qΒa

Fig. 4.33. Dependence of the amplitudes of the undistorted (ρ1 , solid line) and skewed (ρ2 = ρ3 , dashed line) modes on the wave vector rotation q for ν = 2

4.6 Hexagonal Patterns

275

Fig. 4.34. Reconstructed patterns with q = 1/12 (left) and q = −1/12 (right)

√ indicating sideband instability at qβa > µ(ν −1). The result does not change if either q or βa ﬂip the sign, since then θ+ in (4.166) relaxes to π (modulo 2π), while (4.167) remains unchanged. Above this instability limit, there is a stable regular pattern formed by three modes with ρ1 = ρ2 = ρ3 . The dependence of both amplitudes on q, which can be obtained analytically in an implicit form,8 is shown in Fig. 4.33. The reconstructed patterns for q = ±1/12, βa = 24 are shown in Fig. 4.34. More general cases including all quadratic terms and patterns with nonoptimal wavelength were considered by Echebarria and P´erez-Garc´ıa (1998), Nuz et al. (1998, 2000), and Pe˜ na and P´erez-Garc´ıa (2000). 4.6.3 Penta-Hepta Defects The resonant character of interactions among the modes forming a hexagonal pattern strongly aﬀects the interaction of topological defects – dislocations in these modes. The experiments by Ciliberto et al. (1990) and Bodenschatz et al. (1992), as well as computations by Rabinovich and Tsimring (1993) showed that dislocations in two modes constituting the hexagonal pattern, created originally at arbitrary locations, are always attracted to each other, eventually forming an immobile bound pair corresponding to a penta-hepta defect in the hexagonal pattern (see Fig. 4.35). This behavior can be explained by the following qualitative argument (Ciliberto et al., 1990; Newell et al., 1993). If one of interacting modes has a defect, so that the respective phase has a quantum of circulation around a certain point (e.g., θ2 = φ), the resonance condition Θ = θj = 0 following from the polar form of the amplitude 8

All computations in this section are carried out assuming the value of the interaction parameter ν = 2, which corresponds to a nonspeciﬁc (angle-independent) cubic interaction.

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4 Amplitude Equations for Patterns

Fig. 4.35. Hexagonal pattern containing a penta-hepta defect (left) and its three constituent modes obtained by Fourier ﬁltering of the initial image (Abou et al., 2000; reproduced with permission)

equations (1.120) will still be satisﬁed if another mode has a reverse circulation around the same point (e.g., θ3 = −φ). The defects in resonantly interacting modes thus become bound to one another and, consequently, conﬁned. A pure phase rotation (θj = ±φ) does not satisfy, however, the amplitude equations (4.155) describing spatial modulation of amplitudes on an extended scale, so that the phase ﬁeld of conﬁned dislocations must have a more complicated structure which may be obtained by solving these equations directly. Pismen and Nepomnyashchy (1993) computed the far ﬁeld of a dislocation by solving equations of phase dynamics derived from a simpliﬁed form of (4.161), omitting the nonlinear diﬀerential terms and restricting to the band center. Under these conditions, the gradients of the real amplitudes can be neglected, and rescaling the coordinates by a small factor δ reduces (4.161) to ∂θj = δ 2 (nj · ∇)2 θj − σj sin θ, ∂t

(4.168)

where σj = ρj+1 ρj−1 /ρj > 0. On the O(1) time scale, the total phase Θ relaxes to zero according to (1.120). The evolution equations for translationally invariant phase combinations, valid on an extended O(δ −2 ) time scale T , can be written as −1 ∂θj 2 − (nj · ∇) θj = idem. σj (4.169) ∂T Setting n1 = (1, 0), n2 = (−1/2,

√ √ 3/2), n3 = (−1/2, − 3/2)

(4.170)

and presuming the real amplitudes ρj in the far ﬁeld to be equal yields coupled equations for the phase combinations θ± = θ2 ± θ3 : √ 9 + 3 − 3 + θxx + θyy = θxy , 4 4 √2 1 − 3 + 3 − θxx + θyy θ . = (4.171) 4 4 2 xy

4.6 Hexagonal Patterns

277

These equations are subject to the circulation conditions θ− (2π) − θ− (0) = 4π, θ+ (2π) = θ+ (0). Quite miraculously, eliminating θ+ from this system of anisotropic equations yields an isotropic biharmonic equation (∇2 )2 θ− = 0. Transforming to the polar coordinates r, φ and assuming that the phases in the far ﬁeld do not depend on the radial coordinate, we obtain − − + θφφφφ = 0. 4θφφ

(4.172)

The general solution (up to an arbitrary constant phase shift) of this equation, satisfying the circulation condition for θ− and containing two integration constants a, b, is (4.173) θ− = 2φ + a sin 2φ + b cos 2φ. Using this in (4.171) one can see that both equations can be veriﬁed simultaneously if a = −1/2. The constant b is determined from the condition of zero circulation for θ+ . Adding also constant phase shifts to ﬁx θj (0) = 0 we cast the solution in the ﬁnal form 1 θ1 = −θ+ = √ (1 − cos 2φ), 2 3 1 2π 1 + 1 − + cos 2φ − θ2 = (θ + θ ) = φ − √ , 2 3 2 3 2 1 3

0.8

2

Θ2Π

0.6 3 0.4 2 0.2

1 0 0

0.2

0.4 0.6 Φ2Π

0.8

1

Fig. 4.36. The phases θ1 , θ2 , and −θ3 of the triplet of modes in the far ﬁeld of the penta-hepta defect. The curves are marked by the indices of the respective modes

278

4 Amplitude Equations for Patterns (a)

(b)

(c)

Fig. 4.37. Trajectories of interacting penta-hepta defects (Tsimring, 1996; reproduced with permission. Copyright by the Elsevier Science). Open circles indicate positions of the defect cores at constant time intervals. Arrows point toward directions of motion. The charges of dislocations in the individual modes (1,2,3) are (0, 1, −1) and (0, −1, 1) (a); (0, 1, −1) and (0, 1, −1) (b); (0, 1, −1) and (−1, 0, 1) (c). For each case, four sets of initial conditions are taken (labeled 1–4)

θ3 =

1 2π 1 + 1 (θ − θ− ) = −φ − √ + cos 2φ + . 2 3 2 3 2

(4.174)

The solution is plotted in Fig. 4.36. The phase ﬁeld of each constituent mode contains, in addition to the appropriate circulation, a quadrupole component. The solution (4.174) plays, essentially, the same role as the basic solution θ = φ for a vortex in the isotropic RGL equation, or the vortex solution in Sect. 4.3.3 for a dislocation in the roll pattern described by the NWS equation. The symmetry of the solution is compatible with the immobility of a sole bound pair of dislocations (penta-hepta defect) in the inﬁnite plane. The motion of the defect under the action of other defects or lateral boundaries must depend, as in the case of a striped pattern (Sect. 4.4), on the structure of the defect core where the real amplitudes and the total phase Θ change as well. It is clear that the real amplitudes of the modes containing dislocations (ρ2 and ρ3 ) should vanish at the origin, while the third amplitude (ρ1 ) remains ﬁnite, though is likely to decrease due to suppression of the resonant term. Mobility of interacting penta-hepta defects has been computed by Tsimring (1996). Interaction of defects is determined by the signs of dislocations in the individual modes: for example, the defects in Figs. 4.37a, c attract, while those in Fig. 4.37b repel. Trajectories of interacting defects are, generally, curvilinear, which can be attributed to the dipole component in the phase ﬁeld computed above. Colinet et al. (2002) included in their study the inﬂuence of nonlinear differential terms (4.157). A curious phenomenon detected in the latter work is multiplication of defects at higher values of the supercriticality parameter µ. A new pair of dislocations is generated in the dislocation-free mode. Following

4.6 Hexagonal Patterns

279

Fig. 4.38. Multiplication of a penta-hepta defect (Colinet et al., 2002; reproduced with permission). Plots (a),(b), and (c) show the contour lines of real amplitudes ρ1 , ρ2 , and ρ3 , respectively, together with the lines where their real and imaginary parts of the complex amplitudes u1 , u2 , u3 vanish (thicker solid and dashed curves, respectively); defects are located at the intersection of these lines. The plots on the right show the reconstructed patterns. The lines drawn by hand demonstrate that penta-hepta defects correspond to additional rows of hexagonal cells inserted in two of the three basic modes. Time increases from the top to the bottom row

this, the old penta-hepta defect separates into two dislocations, which recombine with the new pair of dislocations forming a pair of penta-hepta defects of the second generation (see Fig. 4.38). Multiplication can be repeated several times in a suﬃciently large region. 4.6.4 Domain Boundaries Both striped and hexagonal patterns are stable solutions of the amplitude equation (1.121) or (4.155) in the parametric region ν > 1, (ν − 1)−2 < µ < (ν + 2)/(ν − 1)2 (see Sect. 1.6.2 and Fig. 1.23). In the subcritical region the hexagonal pattern coexists with a stable homogeneous state. The structure of boundaries between alternative stable stationary patterns (stripes and hexagons), as well as between hexagons and a homogeneous state,

280

4 Amplitude Equations for Patterns

was studied by Malomed et al. (1990). In view of the resonant phase condition Θ= θj = 0, one can restrict to real amplitudes when the wavelengths are optimal. The velocity c of a front separating the hexagonal pattern from either a striped pattern, where only a single mode of the resonant triplet persists, or a homogeneous state is obtained by solving the amplitude equations rewritten in the comoving frame:

cos2 αj ρj (z)+c ρj (z)+ρj µ − ρ2j − ν ρ2j−1 + ρ2j+1 +ρj−1 ρj+1 = 0, (4.175) where αj is the angle between the axis z normal to the front and the wave vector of the jth mode. In the amplitude equation formulation, the boundary is stationary when the respective potentials (1.127) are equal. Straightforward computation using the solutions obtained in Sect. 1.6.2 places this hexagon-stripe Maxwell construction µ13 and the hexagon-homogeneous Maxwell construction µ03 at √ 1 + 3ν + 2(ν + 1)3/2 2 . (4.176) , µ03 = − µ13 = 2(2ν + 1)(ν − 1)2 9(1 + 2ν) The front is anisotropic, and its energy depends on its orientation relative to the constituent modes. The anisotropy is, however, very weak, as seen from the orientation dependence of the interfacial energy of the front between hexagons and homogeneous state depicted in Fig. 4.39. The energy is maximal when the front is normal to the wave vector of one of the constituent modes (α = 0) and minimal when the front is parallel to the wave vector of one of the constituent modes (α = π/6). The angular dependence is well approximated by a simple harmonic function E = 0.0578 + 1.12 × 10−4 cos(6α). The same (a)

(b) 0.0579

Α0

z

E

0.05785 0.0578 0.05775 ΑΠ6

0.0577 z

0

0.2

0.4

0.6

0.8

1

6ΑΠ

Fig. 4.39. The extreme conﬁgurations of the wave vectors of the constituent modes (a) and the dependence of the interfacial energy of the stationary front between the hexagonal pattern and homogeneous state on the orientation angle α for ν = 2 (b)

4.6 Hexagonal Patterns

281

1

ΡΡs

0.8 0.6 0.4 0.2 0 -6

-4

-2

0 z

2

4

6

Fig. 4.40. The proﬁle of a stationary front between the hexagonal pattern and homogeneous state at ν = 2, α = 0. The black line denotes the amplitude of the mode parallel to the front, and the gray line denotes the amplitudes of the two oblique modes

weak anisotropy persists in either stationary or propagating fronts between hexagons and stripes, where both energy of a stationary front and the propagation speed depend, in addition, on the orientation of the stripes (Malomed et al., 1990). It is notable that, due to the angular dependence of the eﬀective amplitude diﬀusivities in (4.175), the more diﬀusive mode spreads out further into the region occupied by the homogeneous state (see Fig. 4.40). As a result, a band of a striped pattern parallel to the front may appear as an intermediate stage of the decay of the hexagonal pattern into homogeneous state. Domain walls may separate as well hexagonal patterns with diﬀerent orientations. The short-scale structure of such a domain wall is resolved as a chain of penta-hepta defects (see Fig. 4.41). As in the case of striped patterns (Sect. 4.5.2), nonadiabatic eﬀects cause pinning of boundaries between the hexagonal pattern and homogeneous state within a ﬁnite parametric interval near the Maxwell construction. Moreover, either isolated spots or ﬁnite clusters which can be viewed as splinters of a hexagonal pattern can be stable in a ﬁnite parametric interval. The “crystallization” and “melting” thresholds for such clusters can be depicted schematically as in Fig. 4.20. Some pictures obtained in 2D simulations (Coullet et al, 2000) are shown in Fig. 4.42. In Fig. 4.42a, corresponding to the interval between points C and 1-C in Fig. 4.20 where the patterned state is preferred, a single cell remains stationary, while larger clusters grow. As the balance shifts in favor of the uniform state, as in Fig. 4.42b, corresponding to the interval to the left of 1-M in Fig. 4.20, a single cell disappears, while larger clusters remain stationary. A great number of stable stationary states in the interval between the melting and crystallization transitions for ﬁnite clusters makes it possible to use such “incomplete” patterns for information storage (Coullet

282

4 Amplitude Equations for Patterns

Fig. 4.41. Domain boundary between hexagonal patterns with diﬀerent orientations (Abou et al., 2000; reproduced with permission). Gray and white circles mark, respectively, heptagonal and pentagonal cells

et al, 2000). Isolated cells or clusters can be “written” and “erased” at will by localized perturbations, and remain stationary otherwise within a certain parametric range, e.g., between points 1-C and 1-M in Fig. 4.20 for single cells or between points C and 1-M for clusters of arbitrary size. (a)

(b)

Fig. 4.42. Localized structures obtained in two-dimensional simulations. In (a), the single cell is stable, but the larger structure is growing. In (b), one observes three-cell and seven-cell stationary clusters while a single cell cannot be stabilized (Coullet et al., 2000; reproduced with permission. Copyright by the American Physical Society)

4.6 Hexagonal Patterns

283

4.6.5 Propagation of the Hexagonal Pattern The angular dependence of the amplitude diﬀusivities also strongly inﬂuences the propagation of the hexagonal pattern into unstable equilibrium (Pismen and Nepomnyashchy, 1994). We shall again restrict to patterns with the optimal wavelength and use the real amplitude equations (4.175). For a pattern propagating along the direction of the wave vector of one of the modes, say, k1 , taken as the z-axis, the amplitudes of the other two modes can be presumed equal due to the symmetry. Thus, (4.155) is rewritten as ρ˙ 1 = ∂z2 ρ1 + ρ1 µ − ρ21 − 2νρ22 + ρ22 , (4.177)

2 2 1 2 ρ˙ 2 = 4 ∂z ρ2 + ρ2 µ − νρ1 − (1 + ν)ρ2 + ρ1 . (4.178) In the parametric interval (ν − 1)−2 < µ < (ν + 2)/(ν − 1)2 , the hexagonal pattern is the only stable stationary state. It seems natural to apply under these conditions the linear theory of propagation speed selection at the leading edge of the propagating front (Sect. 2.3). The linear theory gives, however, no advantage whatsoever to the hexagonal pattern, which is supported by quadratic interactions. Moreover, since the amplitude diﬀusivity of any particular mode is maximal in the direction of its wave vector, the mode with the wave vector parallel to the z-axis would have the largest propagation speed in this direction. According to the linear growth theory for a single reaction-diﬀusion equation in Sect. 2.3.4, the propagation speed of the front between the striped pattern and homogeneous state, denoted as (01), equals √ c01 = 2 µ. The striped pattern emerging behind the front is, however, unstable. The question is whether the front of the stripes–hexagons (13) transition moving in the wake is able to catch up with the advancing front of the stripes. The linear theory is also applicable to the leading edge of the hexagonal pattern propagating into the unstable striped pattern in the parametric range where hexagons is the only stable stationary state. The dispersion relation is obtained again by linearizing (4.178) in the vicinity of the single-mode solution √ √ ρ1 = µ, ρ2 = 0. Applying the general formula from Sect. 2.3.4, c = 2 αD, where α is the linear growth coeﬃcient of the side mode ρ2 and D = 1/4, yields √ c13 = µ − µ(ν − 1). (4.179) This speed exceeds c01 only at µ < (ν +3)−2 , i.e., within a very narrow interval close to the critical point µ = 0. Under these conditions, the hexagonal pattern √ advances directly into the homogeneous state with the speed c03 = c01 = 2 µ. Faster propagation is, of course, impossible before a striped domain is created; thus, the (01) and (13) fronts are bound into a direct (03) transition. At larger values of µ, the (01) front runs ahead, and, even though the striped pattern is unstable, a belt of stripes emerges as an intermediate state in the transition from the unstable homogeneous state to the hexagonal pattern. Moreover, this belt keeps widening with time as the hexagons lag behind.

284

4 Amplitude Equations for Patterns

This picture has to be corrected, as in Sect. 2.3.4, by taking into account the competition between generic (linear, or “pulled”) and nongeneric (nonlinear, or “pushed”) propagating fronts. The nongeneric branch is obtained as a continuation of the branch of propagating fronts separating two stable stationary states into the parametric region where one of these states becomes unstable. A pushed front prevails when it is steeper and propagates with a higher speed than a pulled one. This necessarily happens in the vicinity of a marginal stability point where the speed of a pulled front vanishes, while that of a pushed one remains ﬁnite. The nongeneric propagating front connecting the hexagonal pattern with the homogeneous state is obtained by solving (4.177), (4.178) with the boundary conditions ρ1 = ρ2 = ρh at z → −∞, ρ1 = ρ2 = 0 at z → ∞, where ρh is the stable stationary amplitude satisfying (1.124). This solution determines propagation of the hexagonal pattern near the lower limit of its existence, just above the Maxwell construction µ03 . For the pushed front connecting hexagonal and striped patterns, the appropriate boundary conditions √ are ρ1 = ρ2 = ρh at z → −∞, ρ1 = µ, ρ2 = 0 at z → ∞. This front is relevant at higher µ, close to the lower stability limit of the stripes. Both analytical and computational results are compiled in Fig. 4.43. At low supercriticalities, the pushed (03) front moves faster than the pulled (01) front at µ < µln ≈ 0.016. In this parametric region, the transitional striped belt does not widen with time. An apparent narrow striped belt should exist, however, also in this region. The reason is that the mode with the wave vector parallel to the propagation direction is stronger than two others near the leading edge due to a higher amplitude diﬀusivity. This is clearly seen in Fig. 4.44 showing the computed proﬁles of both amplitudes at µ = 0.01. One can also note that, due to the same diﬀerence in amplitude diﬀusivities, the

Fig. 4.43. The speed of propagating linear (L) and nonlinear (N) equilibrium– stripes (01), equilibrium–hexagons (03), and stripes– hexagons (13) fronts as a function of the supercriticality parameter µ (for ν = 2). (a) Blow-up near the critical point µ = 0; (b) the supercritical region at large. The segments actually determining the propagation speed are shown by solid lines (Pismen and Nepomnyashchy, 1994)

4.6 Hexagonal Patterns

285

Fig. 4.44. The proﬁles of the amplitudes of the parallel (solid line) and inclined (dashed line) mode for µ = 0.01, ν = 2 (Pismen and Nepomnyashchy, 1994)

inclined modes slightly overshoot the parallel mode further in the transitional region. At µ > µln , the propagation speed selected linearly at the leading edge of the growing striped pattern is larger than the speed of the pushed (01) front. The striped pattern, however, cannot develop fully, as long as the speed of the (13) transition is higher, c13 > c01 . Apparently, the two side modes begin to grow at µ > µln before the amplitude or the leading mode approaches √ its stationary value ρ1 = µ. Computationally, one observes in this region propagation speeds corresponding to a smooth continuation of the nongeneric branch when the runs are not too long; the approach to higher values corresponding to the linear theory is very slow. At µ ≈ 0.04, the speeds of the (13) and (01) fronts equalize. Beyond this point, the (03) front splits into two fronts: the fast (01) front and the (13) front lagging behind. The disparity between the two speeds grows further with increasing supercriticality, especially after the speed of the (13) front passes a maximum at µ = 1/4. At µ >≈ 0.7, the pushed (13) front overtakes the pulled one, and the front keeps slowing down. Thus, one should observe a transitional striped pattern even under conditions when the stationary striped pattern is unstable. Csah´ok and Misbah (1999) carried out the same computation using the extended amplitude equation (4.157) and obtained qualitatively similar results, though the parametric region where the intermediate belt of unstable stripes is formed has shifted to higher values of µ. The front velocities obtained by direct numerical solution of the amplitude equations well approximated those obtained analytically. Strong discrepancies were revealed, however, when the solution of the amplitude equations was compared to the numerical solution of the simplest basic model – the modiﬁed SH equation (4.145) close to the symmetry-breaking threshold. It turned out that in the latter case the inter-

286

4 Amplitude Equations for Patterns

Fig. 4.45. Temporal evolution (from left to right) of the propagating hexagonal front in the modiﬁed SH equation for µ = 0.05 (Csah´ ok and Misbah, 1999; reproduced with permission). Note the stripes forming ahead of hexagons

mediate belt does not grow with time (Fig. 4.45). The overall front velocity is ﬁxed by the linear marginal velocity for the (01) front, while the (13) front goes faster than predicted by the linear theory, moving in a concerted fashion with the leading (01) front. This reﬂects limitations of the amplitude equations approach similar to those discussed in Sect. 4.5. The above discrepancy is, apparently, a nonadiabatic eﬀect dependent on the short-scale structure of the pattern. As shown in Sect. 4.5, the wavelength of the pattern at the leading edge of a pulled front does not coincide with the optimal wavelength; moreover, it changes in the wake of the propagating front until it reaches an asymptotic wavelength, which also diﬀers from the optimal one. Thus, the hexagonal pattern advances into the striped pattern with a variable wavenumber diﬀering from the optimal one (unity) implied by the above theory, which takes the real amplitude as the only variable characterizing the pattern. Deviations from the optimal wavelength of the striped pattern increasing its local energy density are likely to accelerate its displacement by the advancing hexagonal pattern. Experimentally, the peculiar structure of the boundary between a hexagonal pattern and homogeneous state was ﬁrst detected by Bodenschatz et al. (1992), both in the case when the hexagonal pattern spreads into an unstable homogeneous state and in the case when the border between the hexagons and equilibrium is kept stationary due to a parametric ramp. Formation of an extended transitional zone should be a general phenomenon observable under certain conditions in various systems possessing three or more stationary states (Glasner and Almgren, 2000).

5 Amplitude Equations for Waves

5.1 Plane Waves 5.1.1 Complex Ginzburg–Landau Equation The complex Ginzburg–Landau (CGL) equation generalizes the normal form (1.70) at the Hopf bifurcation to spatially extended systems. The general form of this equation reads ut = c2 ∇2 u + c1 u − c3 |u|2 u,

(5.1)

where the amplitude diﬀusivity c2 , the linear growth coeﬃcient c1 , and the nonlinear interaction coeﬃcient c3 are complex constants. The appellation, which is in common use among physicists, is not literally justiﬁed, since neither Landau nor Ginzburg had anything to do with the complex form (5.1) which makes its behavior so interesting. Its role is akin to that of the RGL equation in description of a second-order phase transition; the transition in question, leading to a persistent non-stationary state, is, however, possible only far from equilibrium. The complex amplitude u of a periodic solution emerging as the result of a supercritical Hopf bifurcation plays the role of an order parameter. Equation (5.1) describes the evolution of this amplitude on a slow time scale far exceeding the oscillation period, as well as its modulation on an extended spatial scale. Its coeﬃcients can be related to parameters of an underlying non-equilibrium system through a technically cumbersome but fully algorithmic procedure of multiscale expansion in the vicinity of a Hopf bifurcation. The equation for an unmodulated amplitude was derived by expansion of a general reaction-diﬀusion system (RDS) in Sect. 1.3.4. It is complemented by the diﬀusional term, in the same way as it was done for a bifurcation at zero eigenvalue in Sect. 2.1.1, when the complex amplitude is allowed to vary on an extended scale, with the spatial derivatives scaled as ∇ = O(). Starting from a general RDS (1.18), one then obtains the diﬀusional term with com† plex amplitude diﬀusivity c2 = U · DU added to the third-order solvability condition (1.81).

288

5 Amplitude Equations for Waves

As in the RGL equation, the real parts of all three coeﬃcients always can be reduced to unity by rescaling the amplitude, spatial coordinates, and time. Stable nontrivial solutions exist, provided the real parts of all three coeﬃcients are positive; this corresponds to a supercritical Hopf bifurcation. Thus, we set c1 = 1 + iµ, c2 = 1 + iη, c3 = 1 + iν. The imaginary part of the linear growth coeﬃcient is irrelevant, since it can always be removed by a transformation u → u eiµt , so that the added frequency is absorbed into the frequency of the underlying oscillations. Thus, we adopt the standard form ut = (1 + iη)∇2 u + u − (1 + iν)|u|2 u.

(5.2)

The polar form of (5.2) is obtained, as usual, by setting u = ρ(r)eiθ , removing the phase factor, and separating the real and imaginary parts: ρt = ∇2 ρ + (1 − |∇θ|2 − ρ2 )ρ − η(ρ∇2 θ + 2∇ρ · ∇θ),

(5.3)

θt = ∇2 θ + 2ρ−1 ∇ρ · ∇θ + η(ρ−1 ∇2 ρ − |∇θ|2 ) − νρ2 .

(5.4)

Another form of the CGL equation, particularly convenient for ﬁnding stationary solutions and their perturbations, is obtained by transforming u → ue−iωt and rescaling the amplitude, spatial coordinates and time: u→u

1 + ωη , 1 + νη

x→x

1 + η2 , 1 + ωη

t→

t . 1 + ωη

(5.5)

The resulting equation is (1 − iη)ut = ∇2 u + (1 + iΩ)u − (1 − iq)|u|2 u, where q=−

η−ν Im(c2 c3 ) = , Re(c2 c3 ) 1 + ην

Ω=

ω−η . 1 + ων

(5.6)

(5.7)

Equation (5.2) can be deﬁned through a “complex energy” F as ut = −δF/δu, ut = −δF/δu,

(5.8)

so that its r.h.s. is obtained as the Euler–Lagrange equation corresponding to a complex Lagrangian L(u) = (1 + iη) ∇u · ∇u − |u|2 + 12 (1 + iν) |u|4 . (5.9) The complex energy F = Ldx is neither evolving in a deﬁnite direction nor conserved. The CGL equation inherits therefore to a large extent sophisticated dynamic behavior of underlying non-equilibrium systems, and can be viewed as a prototype form of a reaction-diﬀusion system. Generalizations of the CGL equation have also received much attention in literature. Those are the quintic CGL equation (QCGL) with an added term

5.1 Plane Waves

289

|u|4 u (van Saarloos and Hohenberg, 1992) and the complex Swift–Hohenberg equation (CSH), with an added term ∇4 u, i.e. (1.21) with complex coeﬃcients at all terms (Lega et al., 1994). Both forms can be derived in the vicinity of codimension two bifurcation points: QCGL, at a border between a supercritical and subcritical Hopf bifurcation, and CSH, at symmetry-breaking Hopf (wave) bifurcation with a vanishing wavenumber (a dynamical Lifshitz point). We shall concentrate here on the basic generic form (5.2), which by itself generates rich spatio-temporal dynamics, as is emphasized in the title of a comprehensive review by Aranson and Kramer (2002): “The world of the complex Ginzburg-Landau equation.” 5.1.2 Perturbations of Plane Waves A plane wave solution of (5.2) or (5.3), (5.4) with the wave vector k is (5.10) u = ρ0 exp[i(k · x − ωt)], ρ0 = 1 − k 2 , ω = ν + (η − ν)k 2 . The waves are dispersive, and the group velocity is v = dω/dk = 2k(η − ν).

(5.11)

Stability of plane waves can be analyzed in a standard way as in Sect. 4.1.5. We take a perturbed solution in the form ρ = ρ0 + ρ,

θ = k · x − ωt + θ,

(5.12)

where ρ0 (k), ω are deﬁned by (5.10) and ρ, θ are inﬁnitesimally small, and linearize (5.3), (5.4): ρ) − 2ρ20 ρ, ρt = ∇2 ρ − 2ρ0 k · ∇θ − η(ρ0 ∇2 θ + 2k · ∇ 2 − 2νρ0 ρ. θt = ∇2 θ + 2ρ−1 ρ + η(ρ−1 − 2k · ∇θ) 0 k · ∇ 0 ∇ ρ

(5.13) (5.14)

Eigenvalues of this system dependent on the perturbation wave vector q = ∇θ can be found in a standard way by setting here ρ, θ ∝ exp(λ + q · x) to obtain a quadratic equation for λ:

λ2 + 2λ ρ20 + q 2 + 2iη k · q + (1 + η 2 ) q 4 − 4(k · q)2

(5.15) + 2ρ20 q 2 (1 + ην) + 2i(η − ν)k · q = 0. The resulting analytical expressions are quite cumbersome. The phase dynamics approach (Sect. 4.2.1) is far more eﬃcient here, since the most dangerous perturbation modes have a vanishing wavenumber, and can be viewed as longscale distortions of neutrally stable translational modes. The thresholds of Eckhaus and Benjamin–Feir (self-focusing) instabilities are identiﬁed, respectively, as points in the parametric space where the longitudinal and transverse phase diﬀusivities vanish.

290

5 Amplitude Equations for Waves

Assuming that the spatial scale of perturbations is large, the spatial derivatives are rescaled by a small parameter : ∇ → ∇. The dynamics on a slow time scale T is viewed in the coordinate frame moving with the group velocity (5.11). In this frame, the time derivative is scaled as 2 ; thus we transform ∂t → 2 ∂T − v · ∇. Under these conditions, perturbations of the real amplitude ρ are slaved to phase perturbations and can be expressed by expanding the transformed equation (5.13) in as −3 2 2 2 (5.16) − 12 ηρ−1 ρ = −ρ−1 0 k · ∇θ + 0 ∇ θ + νρ0 (k · ∇) θ . Using this in the transformed equation (5.14) yields in the leading O(2 ) order1 the phase diﬀusion equation in the comoving frame 2(1 + ν 2 ) θt = (1 + νη)∇2 θ − (k · ∇)2 θ. 1 − k2

(5.17)

In the coordinate frame aligned with k, this equation takes the form (4.30) with the longitudinal and transverse diﬀusivities D =

1 + νη − k 2 (3 + νη + 2ν 2 ) , 1 − k2

D⊥ = 1 + νη.

(5.18)

When the transverse phase diﬀusivity D⊥ is negative, all plane wave solutions are unstable. This is the Benjamin–Feir (BF) instability. Outside this selffocusing range, the Eckhaus stability condition D > 0 deﬁnes the largest stable wavenumber 1 + νη 2 kmax = . (5.19) 3 + νη + 2ν 2 A stationary localized perturbation of a stable pattern is screened upstream from the disturbance (i.e., against the direction of group velocity). The attenuation rate p can be obtained by setting in the dispersion relation (5.15) λ = 0, q = −ip; the direction of q is chosen parallel to the wave vector k and group velocity v. This leads to the equation for p:

(5.20) (1 + η 2 )p4 + 2 2k 2 (1 + η 2 ) − ρ20 (1 + νη) p2 + 2ρ20 vp = 0, where v = ±|v|. Besides the trivial root p = 0 (reﬂecting the translational symmetry), there are three roots, which should have real parts of diﬀerent sign, since their sum vanishes. Assuming v > 0, there is one root with a negative and two with a positive real part. The latter deﬁne the decay of the disturbance in the upstream direction. This pair of roots may collapse with growing k; at larger k, there will be then a pair of complex conjugate eigenvalues, and the decay will be oscillatory. 1

O( ) terms are purely advective and cancel exactly in the frame propagating with the group velocity (5.11).

5.1 Plane Waves

291

Both Benjamin–Feir and Eckhaus instabilities arising at the respective thresholds are convective. As perturbations grow, they are washed away with the prevailing group velocity. The time evolution of a small localized perturbation u (x, t) in a linear range is deﬁned through its Fourier transform u (q, t) as 1 u (q, t) exp[i(q · x + λ(q)t]dq, (5.21) u (x, t) = 4π 2 where the dispersion relation λ(q) is obtained by solving (5.15). In the limit t → ∞, the integral is dominated by the edge of the spectral band where Re(λ) is at maximum, and can be computed using the method of steepest descent in the complex q-plane (Aranson et al., 1992). Generally, one would need to integrate in two complex planes for each component of q, but it is more practical to consider separately the perturbations along and across the wave vector k of the basic pattern, of which the former are most dangerous. The integration contour is shifted to the saddle point q = q0 satisfying dλ(q0 )/dq = 0 (q now being a scalar), and the condition of absolute instability is Re[λ(q0 )] > 0, ∂q λ(q0 ) = 0, Re[∂q2 λ(q0 )] < 0.

(5.22)

This condition coincides with the marginal stability condition for linear selection of the front propagation speed – cf. (2.75) with ω = iλ. The absolute stability condition is less restrictive than the Eckhaus condition (see Fig. 5.1), and sometimes even less restrictive than the Benjamin–Feir

1.0

k

absolutely unstable

0.5

convectively unstable stable

0.0 0.0

0.5

1.0

ν

1.5

Fig. 5.1. Limits of convective and absolute instabilities in the plane (ν, k) for η = −3/2. The dot marks the limit of convectively unstable waves (Aranson and Kramer, 2002; reproduced with permission. Copyright by the American Physical Society)

292

5 Amplitude Equations for Waves

condition, which cuts oﬀ in this ﬁgure the region ν > 2/3 (see also Fig. 5.4). Numerical simulations by Aranson et al. (1992) showed that transition to turbulence indeed occurred only when the absolute stability condition was violated, but the system was very sensitive to noise in the convectively unstable region. 5.1.3 Phase Equations Nonlinear phase equations describing phase dynamics on the background of either plane waves or homogeneous oscillations can be obtained in a similar way by long-scale expansion of (5.3), (5.4) in the frame propagating with the group velocity (5.11). The phase perturbation is not assumed anymore to be small, and is now denoted as Θ(X, T ), but, due to rescaling ∇ → ∇, perturbations of both k and ρ do not exceed O(). Plugging θ = k · x+Θ, ρ = ρ0 + ρ1 + . . . in (5.3) transformed to the comoving frame and expanding in yields ρ1 = −ρ−1 0 k · ∇Θ, 1 2 −1 2 1 3 2 ρ2 = −ρ−1 0 2 η∇ Θ + 2 |∇Θ| + νρ0 k · ∇ρ1 + 2 ρ1 ν 1 1 1 2 2 2 2 1 =− η∇ Θ + 2 |∇Θ| − 2 (k · ∇) Θ + 2 (k · ∇Θ) . (5.23) ρ0 2 ρ0 ρ0 Using this in (5.3) transformed to the comoving frame yields in the leading second order of the expansion in the scale ratio the anisotropic Burgers equation describing relaxation of phase inhomogeneities on a slow O(−2 ) time scale T : ΘT = D ΘXX + D⊥ ΘY Y + (ν − η)|∇Θ|2 −

νk 2 Θ2 . 1 − k2 X

(5.24)

Here X, Y are extended coordinates directed, respectively, along and across k, and the parallel and transverse diﬀusivities are deﬁned by (5.18). The isotropic form of this equation at k = 0 is θT = (1 + νη)∇2 θ + (ν − η)|∇θ|2 ,

(5.25)

or, after rescaling (5.5), (1 + qη)θT = ∇2 θ − q|∇θ|2 .

(5.26)

At k = 0, the expansion can be continued near the self-focusing instability limit D⊥ = 1 + νη = 0. The long spatial scale should be shortened now from O(−1 ) to O(−1/2 ). To make the expansion consistent, the nonlinear term also has to be reduced to O(), which requires the phase deviations to be small. The r.h.s. of (5.24) vanishes after setting k = 0, η = −ν −1 (1 − η) and rescaling Θ → Θ. The real amplitude has to be computed now by continuing the expansion of (5.3) to the third order:

5.1 Plane Waves

ρ = 1 − 12 2 η ∇2 Θ − 12 3 |∇Θ|2 + 12 η∇2 Θ .

293

(5.27)

Expanding likewise (5.4) yields in the leading third order the Kuramoto– Sivashinsky (KS) equation describing phase dynamics on the O(−2 ) time scale: (5.28) ΘT = η ∇2 Θ + (ν + ν −1 )|∇Θ|2 − 12 (1 + ν −2 )∇4 Θ. A similar expansion near the limit of Eckhaus instability (Janiaud et al., 1992) is somewhat more problematic. In this case, only the longitudinal scale (along the X-axis) should be shortened, while the transverse axis retains the same scaling ∂Y = O() as in (5.24). The ﬁrst derivative with respect to the X-axis does not appear when the equation is written in the comoving frame, but the third derivative cannot be excluded, so the resulting equation is not properly balanced. The KS equation was ﬁrst derived by Kuramoto and Tsuzuki (1976) by direct long-scale small-amplitude expansion of a reaction-diﬀusion system in the vicinity of a spatially homogeneous periodic solution. The coeﬃcients of the KS equation derived in this way are related to the parameters of the underlying system through averaging over the oscillation period. The same equation describes weakly nonlinear dynamics of transversely unstable reaction fronts (Sect. 3.3.5). The parametric range where the KS equation can be obtained by rational expansion of underlying equations is very limited; nevertheless, it is important theoretically as a basic equation generating spatio-temporal chaos. Its appeal is in its universality: it can be obtained out of scaling and symmetry considerations alone, in the way it was introduced in Sect. 1.1.3. The KS equation has stationary periodic solutions, which are stable in a narrow wavenumber range (Frisch et al., 1986). These solutions, even when they do exist, have, apparently, small attraction basins, and spatio-temporally chaotic solutions are actually the relevant attractors. Chaotic solutions appear as well in the damped KS equation (with an added stabilizing linear algebraic term breaking translational invariance), which makes it possible to follow the parametric evolution from ordered cellular patterns immediately above the primary symmetry-breaking transition through intermittency to chaos with decreased damping (Chat´e and Manneville, 1987). Statistics of chaotic solutions both in the standard KS equation and its damped version has been studied in a number of publications (Egolf and Greenside, 1995; Elder et al., 1997; Bohr et al., 1998). Patterns generated by the KS equation seem to be similar to those appearing in simulations of the forced Burgers (KPZ) equation (Sect. 2.2.3), although ﬁne distinctions can be found in statistical characteristics of chaotic solutions (Boghosian et al., 1999). The KS equation cannot imitate the complexity of behavior of the CGL equation, which is not reduced to mere phase chaos, but is typically dominated by formation and motion of phase singularities, as we shall further discuss (Sects. 5.2.3, 5.2.4, and 5.4.4).

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5 Amplitude Equations for Waves

5.1.4 Propagation of Wave Pattern If a plane wave propagates into the unstable trivial state u = 0, the wavelength and propagation speed selected according to the marginal stability criterion of Sect. 2.3.2 are determined by (2.75), where the dispersion relation ω(k) = i 1 + (1 + iη)k 2 is obtained by linearizing (5.2) in the vicinity of the trivial state. Accordingly, the complex group velocity is c≡ cr + i ci = ω (k) = 2 [ki + ηkr + i(ηki − kr )] .

(5.29)

The value of the wavenumber verifying the stability condition ci = 0 is kr = of the phase and group velocities, ωi /ki = ηki . The condition of equality cr (k) is satisﬁed at ki = 1/ 1 + η 2 , which deﬁnes the slope of amplitude proﬁle in accordance to the principle of marginal stability. The corresponding propagation speed is (5.30) c= cr = 2 1 + η 2 . The wavenumber k ∗ of the pattern formed behind the propagating front does not coincide with kr but is computed with the help of the wave crests conservation condition as in Sect. 4.5.1. The entire pattern should be stationary in the comoving frame, except for a common factor eiΩt . Far behind the front, the relation between the frequency and the wavenumber is just the Doppler-shifted dispersion relation (5.10): Ω = ω − ck = ν + (η − ν)k 2 − ck.

(5.31)

On the other hand, near the leading edge the same frequency is computed as Ω = ωr − ckr = −η.

(5.32)

Equating the two expressions and solving for the root satisfying the inequality |k| < 1 yields the wavenumber k ∗ selected by the pulled front: 1 + η 2 − √1 + ν 2 ∗ (5.33) k = . η−ν The selected wavenumber may fall into the unstable range. In this case, the wave pattern may become chaotic in the wake of the propagating front, as seen in Fig. 5.2. Alternatively, the pattern can be stabilized further downstream through wavelength change facilitated by nonlinear interactions. As in Sect. 2.3.4, there are also non-generic nonlinear front solutions, which can be obtained in a way similar to that described in Sect. 5.2.2 (van Saarloos and Hohenberg, 1992). These fronts are, however, not selected by evolution starting from localized initial conditions.

5.1 Plane Waves

295

Fig. 5.2. Space–time plot of simulations results for pulled fronts in the CGL equation with η = 1, ν = −3 (van Saarloos, 2003; reproduced with permission from Elsevier Science). The lines denote the proﬁles of u at successive time intervals, and are shifted upward relative to each other. The plot illustrates an unstable wave pattern forming behind a pulled front and being invaded by a second front establishing a chaotic state in its wake. Take note that, due to a slower propagation speed of the trailing front, the segment occupied by the unstable state grows with time

5.1.5 Coupled CGL Equations We have seen in Sect. 1.6.3 that the amplitude equations for bifurcating wave patterns (at ω0 = 0, k0 = 0) are the same as for a Hopf bifurcation without spatial symmetry breaking (at ω0 = 0, k0 = 0), and the diﬀerence is only in degeneracy, since already in 1D there is competition between waves propagating in the opposite directions, which may either suppress one another or combine to a standing wave: u = u+ e−i(ω0 t0 −k0 x0 ) + u− e−i(ω0 t0 +k0 x0 ) ,

(5.34)

where we label the short-scale spatial coordinate and time by zero index in order to distinguish them from extended scales x, t of slow amplitude modulation. When weak spatial dependence of the amplitudes u± is taken into account, the situation is more complicated due to the presence of a nonvanishing group velocity c0 = dω0 /dk0 , which is oppositely directed for the two counter-propagating waves. Coupled space-dependent amplitude equations for u± , viewed now as wave envelopes, can be written, after usual normalization, as ± ± ± ± 2 ± ∓ 2 ± u± t ± c0 ux = (1 + iη)uxx + u − (1 + iν+ )|u | u − g(1 + iν− )|u | u , (5.35)

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5 Amplitude Equations for Waves

where the variables are scaled in powers of a formal small parameter as u± = O(), ∂x = O(), ∂t = O(2 ), and the deviation from the bifurcation point is of O(2 ). With this scaling, (1.134) is balanced only when c0 = O(), which can be true only on a manifold of higher codimension in the parametric space of the underlying system. Generically, c0 = O(1), and the advective term c0 u± x is dominant. For a single wave, it can be removed by transforming to the comoving frame. When both waves are present, each wave, viewed in its own frame ζ± = x ∓ c0 t samples the average amplitude of its counterpart, which is propagating in this frame with a fast speed, which is of O(−1 ) when measured on the extended scale (Knobloch and de Luca, 1990). The appropriate amplitude equations have then the form ± ± ± 2 ± ∓ 2 ± u± t = (1 + iη)uζ± ζ± + u − (1 + iν+ )|u | u − g(1 + iν− ) |u | u , (5.36)

These equations retain only global coupling carried by the spatial averages |u∓ |2 . Otherwise, they can be viewed as decoupled equations for each wave in its own comoving frame with a renormalized linear term. Nontrivial homogeneous solutions of (5.35) or (5.36) are either singlemode (propagating) or two-mode (standing) waves. The single-mode solution is deﬁned by (5.10) as before; the only diﬀerence is that the frequency is Doppler-shifted by ±c0 k. This solution is unstable to excitation of a counterpropagating wave at g < 1 + k 2 . For a two-mode solution with identical modulating wavenumbers k for both modes, the modulus and frequency of either wave are given by ν+ + gν− 1 − k2 ν+ + gν− , ω= − c0 k − η − ρ2 = (5.37) k2 . 1+g 1+g 1+g This solution is unstable to suppression of one mode at g > 1. Within the interval 1 < g < 1 + k 2 , excitation of a counter-propagating wave with a smaller modulation wavenumber can serve therefore as an eﬀective mechanism for reducing the deviation from the “optimal” wavelength (k = 0). The limits of Benjamin–Feir and Eckhaus instabilities for standing waves are determined in the long-wave limit in the same way as in Sect. 5.1.2. For k = 0, the stability condition is (Coullet et al., 1987) 1+

η(ν+ − g 2 ν− ) > 0. 1 − g2

(5.38)

The distinction between convective and absolute instabilities disappears in this case, since perturbation of one of the modes is transferred through the coupling term to its counterpart and propagates in the opposite direction; this exchange prevents it from being advected out of the system. The transition point between standing and propagating wave patterns shifts slightly above g = 1 when the average amplitude sampled by the competing wave is suppressed in a chaotic regime (Sakaguchi, 1996), As a result, the two modes do not suppress each other completely, as seen in Fig. 5.3.

5.2 One-Dimensional Structures

297

1.0 800

ρ

t

0.8

600

0.6

400

0.4

200 0 0

0.2 0.0 100

200

300

400 x 500

0

100

200

300

400 x 500

Fig. 5.3. Left: space–time plot showing the evolution of the amplitudes ρ± in the coupled CGL equations starting from random initial conditions in the parametric domain where bimodal patterns are weakly suppressed (g = 1.1). Other parameters: η = ν+ = 1, ν− = −1, c0 = 0.5. The gray shading is such that patches of ρ+ mode are light and the ρ− mode are dark. Right: amplitude proﬁles of the ﬁnal state showing that a single wave propagating in either direction prevails on alternating segments (van Hecke et al., 1999; reproduced with permission from Elsevier Science)

A simpler form of coupled CGL equations with c0 = 0 arises in other applications where u± are identiﬁed as intensities of two components of a light beam with opposite circular polarizations or densities of two mixed superﬂuids or Bose–Einstein condensates (Pismen, 1994, 1999; Amengual et al., 1997).

5.2 One-Dimensional Structures 5.2.1 Classiﬁcation of 1D Solutions There is a variety of non-uniform 1D solutions of the CGL equation with a constant frequency Ω and spatially varying modulus and wavenumber, which are stationary in a frame propagating with a certain speed c and depend on the comoving coordinate ζ = x−ct only. The solutions approaching asymptotically at ζ → ±∞ either plane waves or the trivial state are sometimes called coherent structures; they can be also viewed as defects separating domains where diﬀerent uniform states prevail. One can distinguish between several classes of 1D coherent structures: • pulses, approaching the trivial state at both extremes; • nonlinear fronts, describing the trivial state invaded by a wave train; • domain boundaries, separating plane waves directed in the opposite sense and, possibly, having diﬀerent wavelength. We shall further concentrate on the latter class of structures, which play a major role in dynamics of natural 1D patterns.

298

5 Amplitude Equations for Waves

An elegant form of the stationary equation in a comoving frame is obtained from (5.6) by introducing the complex variable z ≡ κ + ik = d ln u/dζ: ρζ = κρ, (1 + iη)(zζ + z 2 ) + 1 + iΩ + cz − (1 + iν)ρ2 = 0.

(5.39a) (5.39b)

Plane waves are ﬁxed points of this system with κ = Re(z) = 0, k = Im(z) = const = k0 , and ρ = ρ0 = 1 − k02 . The relation between the frequency and the wavenumber is given by the Doppler-shifted dispersion relation (5.31). Another ﬁxed point is the trivial state ρ = 0, κ, k = 0. The system (5.39) can be viewed as a dynamical system with ζ playing the role of time, and its coherent structures, as homoclinic or heteroclinic trajectories connecting the ﬁxed points. Homoclinic trajectories are also called holes or dark solitons when they involve depression of the modulus ρ due to strong phase rotation. Heteroclinic trajectories connecting domains with distinct wavenumbers are also called wavenumber kinks. Existence and robustness of these solutions can be estimated by “counting” argument used already in its simple version in Sect. 2.1.2. After linearizing (5.39) near a ﬁxed point, one can count the number of roots with positive and negative real parts deﬁning the dimensions of the stable and unstable manifolds. A family of trajectories starting from a ﬁxed point with n− -dimensional unstable manifold has n− −1 free parameters, which can be adjusted to choose a trajectory orthogonal to the n+ -dimensional unstable manifold of the target ﬁxed point. In addition, there are two free parameters, the frequency Ω and the propagation speed c. Thus, there is a n-parametric family of trajectories connecting the two points with n = n− − n+ + 1. Generally, if the equation contains np free parameters n = n− − n+ + np − 1.

(5.40)

With n < 0, no trajectories exist; n = 0 indicates existence of an isolated trajectory or a discrete set, while a continuous n-parametric family exists when n is positive. The counting argument is heuristic rather than strict. On the one hand, being based on necessary conditions, it gives an upper limit of multiplicity of solutions. On the other hand, symmetries of the dynamical system may increase the robustness of solutions, since fewer parameters may be needed in order to satisfy the constraints. A symmetry can be either obvious, e.g., related to a conservation law, or hidden. We shall see that some hidden symmetry of the CGL equation, apparently, causes the solutions to be more robust than suggested by counting argument (van Saarloos and Hohenberg, 1992). The eigenvalues p determining the dimensions of the stable and unstable manifolds are determined at c = 0 by (5.20). A generalized cubic equation for p at c = 0 (excluding the trivial root) can be obtained by linearizing (5.39) near a plane wave solution:

5.2 One-Dimensional Structures

299

(1 + η 2 )p3 + 2cp2 − 2 2k02 (1 + η 2 ) − ρ20 (1 + νη) + c(c − 4k0 η) p + 2ρ20 v = 0. (5.41) Domain boundaries can be either sources or sinks of waves, depending on the direction of the group velocity. Dynamically, sources are most important, since they create their own environment, while sinks play a passive role; perturbations travel away from sources and into sinks. The sign of the last term in (5.41) is opposite to the sign of the product of the three roots and coincides with the sign of the group velocity v, which, in the case of a source, is positive on the right and negative on the left. For a stationary source (c = 0), the sum of the real parts of all three roots is zero. At ζ → −∞, the product of the roots is positive, so that there should be one positive real root and two roots (either real or complex conjugate) with a negative real part. Thus, the dimension of the unstable manifold n− = 1. In a similar way, we establish that n+ = 2 for the ﬁxed point at ζ → ∞ where the product of the roots is negative; this can be also inferred by symmetry to inversion of the direction of the ζ axis: ζ → −ζ, z → −z, ρ → ρ. (5.42) Since the only adjustable parameter on hand is ω, it appears that no trajectories can generically exist, as n = n− − n+ < 0. This conclusion is, however, alleviated using the symmetry (5.42). Any trajectory starting from a point ρ, 0, k), and ρ0 = 0, κ = 0, k0 = 0 should hit the plane k = 0 at some point ( the latter value can be brought to zero by adjusting ω(k0 ). The remainder of the trajectory is continued symmetrically. This suggests that an isolated trajectory (or, possibly, a discrete set) can exist. The dimensions n± may change with growing propagation velocity |c|. Without loss of generality, we can assume c > 0 (placing the advancing domain on the left). Since the sum of the roots is now negative, while the product retains the same sign on both sides, real parts of all roots may become negative beyond some critical propagation speed, so that n+ reduces to zero, and a twoparametric family of trajectories may appear. For sinks at c = 0, the signs of the roots are inverted, so that n− = 2, n− = 1, and a two-parametric family of trajectories is predicted. This conforms to the above-mentioned passive role of sinks, which do not reject any approaching trajectory. 5.2.2 Holes and Wavenumber Kinks With good luck, non-uniform solutions can be obtained if a proper ansatz is found. The simplest particular case is a static solution with oppositely directed but otherwise identical waves on both sides. Nozaki and Bekki (1984) constructed this solution, which they called a hole, using the ansatz ρ0 ρ ρ − (5.43) κ(ρ) = κ0 , k(ρ) = k0 . ρ ρ0 ρ0 With this ansatz, (5.39) readily integrates to

300

5 Amplitude Equations for Waves

ρ(ζ) = ρ0 tanh κ0 ζ,

k = k0 tanh κ0 ζ.

(5.44)

Inserting (5.43) in (5.39) yields a complex expression linear in ρ2 , and the four unknowns κ0 , ρ0 , k0 , and Ω = ω can be chosen to cancel the four real coeﬃcients. Of course, ρ0 , k0 , and ω are related by (5.10); these relations should be satisﬁed automatically and can be used to reduce the four equations to just two, deﬁning κ0 and ω. Thus, we obtain k02 =

ω−ν , η−ν

ρ20 =

η−ω , η−ν

κ20 =

1 + ηω , 2(1 + η 2 )

(5.45)

where ω satisﬁes the quadratic equation 9(1 + η 2 )(ω − ν)(1 + ηω) − 2(η − ν)(η − ω)2 = 0.

(5.46)

The two alternative signs of k0 and κ0 should be chosen on the left and right the semi-axes in such a way that the group velocity (5.31) be directed away from the origin and the hole be a source which selects the wavelength and frequency of the waves propagating to both sides. The hole solutions are stable in a narrow interval bounded, on the one side, by the core instability limit and, on the other side, by the Eckhaus instability limit of the emitted waves (see Fig. 5.4). In a bounded region or in a system encompassing a number of holes separated by shocks (serving as sinks of emitted waves), the absolute instability limit is likely to be more relevant than convective Eckhaus instability. A heteroclinic trajectory connecting ﬁxed points corresponding to two distinct wavenumbers k± describes a wavenumber kink propagating with some speed c, which has to be chosen to equalize the corresponding Doppler-shifted frequencies Ω(c, k± ). Because of (5.31), this speed is expressed as c=

ω(k+ ) − ω(k− ) = (η − ν)(k+ + k− ), 2(k+ − k− )

(5.47)

where ω(k) is the dispersion relation for plane waves, (5.10). This formula can be interpreted as a phase conservation relation. Comparing the last expression with (5.11), we see that the kink propagates with the average group velocity of the two plane waves which are approached asymptotically at ζ → ±∞. The heteroclinic hole solution, also found by Bekki and Nozaki (1986) is not simply expressed in terms of polynomials κ(ρ) and k(ρ), but is written explicitly as 1 + b e−2κ0 ζ i k(ζ)dζ−iΩt e , 1 + e−2κ0 ζ k(ζ) = 12 [k+ + k− + (k+ − k− ) tanh κ0 ζ] ,

u(ζ) = u0

(5.48)

where k± are the two asymptotic wavenumbers, u0 and κ0 are real, and b is a complex parameter. Inserting (5.48) in (5.2) yields a cubic polynomial

5.2 One-Dimensional Structures

301

2.0

η

CI

EH MOH

1.0

AH

0.0

BF DC

−1.0 AI −2.0 0.0

2.0

4.0

6.0

ν

8.0

Fig. 5.4. Phase diagram of the 1D CGL equation in the parametric plane (η, ν) (Aranson and Kramer, 2002; reproduced with permission. Copyright by the American Physical Society). BF – Benjamin–Feir instability locus; AI – absolute instability limit; DC – boundary of defect chaos. The other lines pertain to standing Nozaki– Bekki hole solutions: CI – core instability line; EH – convective Eckhaus instability of the emitted plane waves; AH – absolute instability of the emitted plane waves; MOH – boundary between monotonic and oscillatory interaction

in e−2κ0 ζ , four complex coeﬃcients of which should be set to zero. This can be done by adjusting k± , c, Ω, u0 , κ0 , and b, altogether eight real constants. This suggests the existence of a discrete set of solutions. It turns out, however, that the solution procedure leads eventually to a relation which is satisﬁed identically, so that a continuous family of solutions parametrized by the speed c exists. These solutions have a dip of modulus, though not reaching zero as in the case of a standing hole, and are also sources. The constant κ0 decreases with c and vanishes at the maximum velocity cmax where the solution merges with a plane wave having the group velocity v = cmax . Here again, the solutions turn out to be more robust than suggested by the counting argument, which points out to a hidden symmetry of the CGL equation. It was found later that the family of moving holes is structurally unstable. If the CGL equation is slightly perturbed, e.g., by introducing a small quintic term, the solution family is destroyed, as the holes either accelerate or decelerate (Popp et al., 1995). A comprehensive study of 1D solutions of various generalizations of the CGL equation (with added fourth-derivative, quintic, or nonlinear derivative terms) was carried out by van Saarloos and Hohenberg (1992). Naturally, generalized equations are capable to sustain a still richer variety of behavior and generate diﬀerent kinds of stable coherent structures. For example, the quintic CGL equation allows for a non-monotonic dispersion relation, which

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5 Amplitude Equations for Waves

makes possible a stationary boundary between two wave domains with diﬀerent wavelengths but identical frequencies. It makes possible as well coexistence of stable trivial and non-trivial solutions, which naturally translates into existence of stable pulses. Some solutions can be constructed analytically in a way similar to to that described here. Doelman (1995) has shown that even small quintic perturbations create large families of traveling localized structures that do not exist in the cubic case. 5.2.3 Modulated Waves and Phase Turbulence Other 1D solutions of the CGL equation, which have been constructed numerically using continuation methods (Brusch et al., 2000, 2001) are called modulated amplitude waves (MAW). Those are P -periodic solutions of (5.39), which correspond to quasiperiodic states of an underlying physical system. The limiting form at P → ∞ is a “homoclon” (van Hecke, 1998) – a homoclinic solution approaching a plane wave at both extremes. Unlike a static Nozaki– Bekki hole, the direction of the asymptotic plane waves is now identical; a homoclon can be viewed therefore as a local phase twist that glides through a background plane wave. The amplitude has a peak as well as a dip, similarly to a moving hole-shock pair (Fig. 5.5); likewise, the wavenumber has a minimum and maximum, while the total change of the phase relative to the background wave is zero. A homoclon with a moderate propagation speed connects the 1D unstable manifold of the background plane wave with its 2D stable manifold. Since there are two adjustable parameters, c and Ω, the counting argument suggests, (a)

(b)

Fig. 5.5. (a): A typical amplitude proﬁle for a hole-shock pair (Aranson and Kramer, 2002; reproduced with permission. Copyright by the American Physical Society); (b): Shallow (solid curve) and deep (dashed curve) MAW amplitude proﬁles (Brusch et al., 2001; reproduced with permission from Elsevier Science)

5.2 One-Dimensional Structures

303

Fig. 5.6. Bifurcation diagrams for ﬁxed ν = −2 and P = 30, showing Hopf (ﬁlled square), Ising–Bloch (open diamond) and saddle-node (triangle) bifurcations. The dot-dashed line represents the homogeneously oscillating solution of the CGL equation, while the lower and upper branches of MAW are represented by solid and dashed curves, respectively. (a): Overview of the maximum phase gradient of the MAW as a function of η; (b): Close-up (Brusch et al., 2001; reproduced with permission from Elsevier Science)

according to (5.40), that a one-parametric family of homoclons should exist, and, adding the period as an additional parameter, a two-parametric family of MAW. The multiplicity of the MAW can also be estimated by considering the instability of the plane wave solutions from which the MAW emerge (Brusch et al., 2001). The plane waves form a one-parametric family; in the Eckhausunstable regime, a plane wave with a certain wavenumber is unstable to a whole band of perturbations parametrized by their wavenumbers, leading, in agreement with the former argument, to a two-parametric family of MAW. A typical bifurcation diagram of MAW constructed by Brusch et al. (2001) is shown in Fig. 5.6. The lower branch of shallow MAW bifurcates from unstable plane waves and terminates at a saddle-node bifurcation point where it joins the upper branch of deep MAW. When, as in Fig. 5.6, the branch bifurcates oﬀ a homogeneous oscillating solution, a non-equilibrium Ising– Bloch bifurcation (see Sects. 3.3.1 and 5.6.2) leading from a static to a pair of counter-propagating MAW takes place early on the lower branch.2 The bifurcation diagram is speciﬁc for a chosen period P . Generally, the saddle-node bifurcation shifts to smaller values of η and ν as the period grows. Shallow MAW with mild amplitude modulation are typical elements of phase turbulence, which can be alternatively described by the KS equation (Sect. 5.1.3). MAW may remain stable as long as the system size small, while for larger systems periodic sequences of MAW are unstable with respect to splitting or interaction. Splitting instability leads to the growth of a new peak in the homogeneous part of the MAW. As a result, two shorter-period MAW will appear. On the other hand, interaction between adjacent MAW shifts 2

Brusch et al. (2000, 2001) call it a drift pitchfork bifurcation.

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5 Amplitude Equations for Waves

adjacent peaks into opposite directions, thereby creating occasional largerperiod MAW, as well as causing peak collisions and merger of MAW. As a result of these instabilities, a perturbed unstable homogeneous state in a spatially extended system typically does not converge to a train of coherent MAW, but evolves instead to a state of phase chaos. Both the splitting and interaction mechanisms are similar to instabilities observed in the KS equation (Chang et al., 1993). Perturbations of shallow (lower branch) MAW lead, as a rule, to a mildly chaotic state, which remains close to the lower MAW branch (within the shaded area in Fig. 5.7). The exception is a perturbation pushing a MAW beyond the saddle-node bifurcation (arrow 5 in Fig. 5.7), which leads to defect chaos. Other arrows in this ﬁgure indicate typical ways of nonlinear evolution. Evolution starting from the unstable homogeneous state to the right of the bifurcation point (arrow 2) may lead to either stationary or moving MAW when the system size is small (Fig. 5.8a, b) or to a state of phase chaos (Fig. 5.8c). Deep (upper branch) MAW are always unstable due to the positive eigenvalue associated with the saddle-node bifurcation. Depending on a perturbation, they either evolve into shallow MAW (arrow 3) or form defects as a result of further growth of local amplitude and phase gradients (arrow 4). Some examples of evolution are given in Fig. 5.8d–f.

Fig. 5.7. Illustration of the relation between MAW structures and phase chaos for η = 3, ν = −0.65 (Brusch et al., 2001; reproduced with permission from Elsevier Science). Solid (dashed ) curves denote stable (unstable) solutions for the system size equal to the MAW period P . The shaded area indicates the typical values for nearMAW structures that occur in phase chaos. Arrows show typical evolution directions

5.2 One-Dimensional Structures

305

Fig. 5.8. Grayscale space–time plots of nonlinear evolution under conditions when no saddle-node bifurcation occurs for any P . The upper row shows evolution of an unstable homogeneous state toward a shallow MAW for η = 3, ν = −0.6. The ﬁnal state is a stable stationary MAW on a short segment (a), a stable drifting shallow MAW in a system of a larger size (b), and a state of phase chaos in a spatially extended system (c). The lower row shows evolution of a deep MAW. Slowing-down and spreading of the phase gradient characterizes the decay to the lower branch for η = 3, ν = −0.55 (d). For the same coeﬃcients, another perturbation leads to an increase in velocity and divergence of the phase gradient. A defect (hole) is formed, initiating a widening domain of defect chaos (e). For η = 3, ν = −0.6 a perturbed upper branch MAW leads to a defect as well (f), but defects do not percolate through the system. Darker shade indicates larger amplitude (Brusch et al., 2001; reproduced with permission from Elsevier Science)

5.2.4 Transition to Defect Chaos Several studies (e.g., Shraiman et al., 1992) tried to resolve the question whether phase turbulence is indeed a persistent state, and to draw a border of a parametric region where phase slips can be avoided altogether in a long-time evolution of an extended system. A combination of interacting shallow modulated waves and homoclons holds, apparently, a key to a mechanism that prevents phase slips in a well-deﬁned parameter range. This points out to the saddle-node bifurcation of MAW as the key point that deﬁnes the transition from the phase to defect chaos. More precisely, the conjecture is that the saddle-node line for P → ∞ provides a strict lower boundary for the transition from phase to defect chaos (Brusch et al., 2000, 2001). The bor-

306

5 Amplitude Equations for Waves

Fig. 5.9. Phase diagram of the CGL equation, showing the Benjamin–Feir limit curve (dot-dashed line), the Psn → ∞ limit (solid line), and the curves L1 (long dashes) and L3 (short dashes) indicating two types of transition to defect chaos. The dots correspond to the estimates based on direct simulations. The open circles correspond to the location in parametric plane where the maximum inter-peak spacing coincides with the maximum MAW period Psn (Brusch et al., 2001; reproduced with permission from Elsevier Science)

der separating the regions of phase and defect chaos is indicated in Fig. 5.4 and shown in more detail in Fig. 5.9 (take note of the diﬀerent orientation as well as a reversal of the sign of both coeﬃcients in the two ﬁgures). Estimates based on direct simulations and on the coincidence of the maximum inter-peak spacing with the maximum MAW period Psn (Brusch et al., 2001), shown, respectively, by the ﬁlled and open circles in Fig. 5.9, are practically overlapping, which justiﬁes the above conjecture. The inter-peak distance distribution is established as a result of competition between the two kinds of instabilities of MAW mentioned in the preceding subsection. Splitting instability tends to reduce peak-to-peak distances (or the period P ) and prevents MAW from crossing the saddle-node boundary; in the phase-chaotic regime, this tends to inhibit defect generation. On the contrary, interaction of MAW leads to an increase of the spacing between some peaks, thus enhancing the generation of defects. The “balance” point where the two instabilities are equally strong near the saddle-node bifurcation gives a rather good estimate of the apparent transition point from phase to defect chaos (Brusch et al., 2001). The transition between phase and defect chaos is reversible at large absolute values of |η| > 1.9 with the sign opposite to that of ν (Shraiman et al., 1992). There, the phase-slip rate goes smoothly to zero when the line DC in

5.2 One-Dimensional Structures

307

Fig. 5.10. Grayscale space–time plots of the amplitude ρ, phase θ and wavenumber k showing chaotic states in the spatio-temporal intermittent regime, for η = 0.6, ν = −1.4. (van Hecke, 1998; reproduced with permission. Copyright by the American Physical Society)

Fig. 5.4 or the long-dashed line in Fig. 5.9 is approached from the side of defect chaos. At smaller values of |η|, there is a region (between the long-dashed and short-dashed lines in Fig. 5.9) where phase and defect chaos coexist. The reason is hysteretic behavior due to a certain “memory” eﬀect: a phase slip causes a large amplitude perturbation making it easier for other phase slips to occur. Further up in Fig. 5.4, the region of defect chaos joins with the shaded region of stable Nozaki-Bekki holes. Unlike phase turbulence, defect chaos is dominated by formation, motion and interaction of holes. The intermittent regime where phase and defect chaos coexist were studied in detail by Chat´e (1994) and Ipsen and van Hecke (2001). Since holes are not topological singularities, they can appear spontaneously, as well as merge and disappear. Holes have to be separated by sinks (shocks). Although the latter pay only passive role in the dynamics, interaction between neighboring holes is sometimes described in terms of interaction between a hole and a shock. Hole-shock pairs are actually special cases of the deep homoclons, as seen in Fig. 5.5. The hole-shock interaction is always attractive in the large separation limit when the decay rates deﬁned by (5.41) are real. For static holes, this happens below the line MOH in Fig. 5.4, but generally, this limit is, of course, velocity-dependent. In the oscillatory range, and away from the core instability line, uniformly moving hole-shock arrangements were found to coexist with solutions with oscillating hole velocities (Popp et al., 1995). Oscillating zigzagging holes, such as shown in Fig. 5.10, can be an attractor in the intermittent regime, as shown in long-time simulations by van Hecke (1998). 5.2.5 Sources and Sinks in Coupled CGL Equations When the dynamics of counter-propagating waves is described by coupled CGL equations (5.35), sources and sinks arise naturally when one mode suppresses the other at g > 1. Then the system tends to form domains of either

308

5 Amplitude Equations for Waves 1.2 ρ

300 0.8

t 200

0.4 100 0 0

100

200

300

400 x 500

0.0 0

100

200

300

400 x 500

Fig. 5.11. Left: space–time plot showing the evolution of the amplitudes ρ± in the coupled CGL equations starting from random initial conditions in the parametric range where bimodal patterns are strongly suppressed (g = 2). Other parameters: µ = 1, η = 0.6, ν+ = −0.4, ν− = 0, c0 = 0.4. The gray shading is such that patches of ρ+ mode are light and the ρ− mode are dark. Right: amplitude proﬁles of the ﬁnal state showing a typical sink/source pattern (van Hecke et al., 1999; reproduced with permission from Elsevier Science)

left-moving or right-moving waves separated by domain walls or shocks, which should be classiﬁed as sources or sinks according to whether the total group velocity v = v ± c0 of the asymptotic plane waves, where the nonlinear part v = 2k(η − ν+ ) is deﬁned as in (5.11), points outwards or inwards. Generically, the linear part of the group velocity should dominate, so that the two wave trains meeting at a domain wall should belong to diﬀerent families, as in Fig. 5.11. In this case, the motion of a domain wall cannot be obtained from integral conditions as in (5.47). Single-mode sources or sinks identical (in the comoving frame) to those constructed in Sect. 5.2.2 can appear when c0 is small. Various coherent structures can be obtained by solving the system of coupled equations similar to (5.39): ± ρ± ζ = κρ ,

(1 +

iη)(zζ± −(1 +

(5.49a) 2 + z± ) + µ + iΩ ± + (c ± c0 )z ± iν+ )ρ2± − g(1 + iν− )ρ2∓ = 0.

(5.49b)

The coeﬃcient at the linear growth term µ is not rescaled here to unity. The reason is that, due to unbalanced scaling discussed in Sect. 5.1.5, the small parameter and hence, dependence on the deviation from the bifurcation point, is not eliminated from the problem.3 It was shown both numerically (Coullet et al., 1993) and through counting argument (van Hecke et al., 1999) that sources form a discrete set, and therefore select the wavenumber of the emitted waves; this, however, may be not 3

Upon rescaling, this dependence is transferred to the dependence on the linear group velocity c0 , which is more diﬃcult to monitor experimentally.

100

1.5

width

5.2 One-Dimensional Structures

1/width

309

1.0

10 0.5

1 0.0

0.2

0.4

0.6

0.8 µ 1.0

0.0 0.0

0.2

0.4

0.6

0.8

µ 1.0

Fig. 5.12. Width of sources (left) and inverse width of sinks (right) as a function of the linear growth parameter µ. Squares indicate numerical results obtained by solving (5.49). The parameter values are g = 2, η = 0.5, ν+ = −0.5, ν− = 0, c0 = 1 for sources and g = 2, η = 0.6, ν+ = −0.4, ν− = 0, c0 = 0.4 for sinks. The straight line shows the asymptotic inverse width of sinks at µ → 0 following from (5.52) (van Hecke et al., 1999; reproduced with permission from Elsevier Science)

true under “abnormal” conditions when |v| > c0 . Sinks, being passive, form a two-parametric family as in the case of a single CGL equation. Due to the advection direction, sources are, generally, wider than sinks. At small µ, or large c0 , there is a large interval where both amplitudes are close to zero. This can be easily understood, since at c0 = 0 the trivial state is only convectively unstable. Coullet et al. (1993) found that there is a ﬁnite threshold value µ = µc , below which no coherent sources exist. This can be understood (van Hecke et al., 1999) by viewing dynamics in the vicinity of a suﬃciently wide source as “upstream” propagation of a wave pattern into unstable trivial state against the linear group velocity. The motion of the front is governed by the single CGL equation in a frame moving with velocity c0 and its propagation speed is deﬁned by (5.30). Taking into account the rescaling and the change of the coordinate frame, the front propagation speed is c = 2 µ(1 + η 2 ) − c0 . (5.50) This speed vanishes at µc = 14 c20 /(1 + η 2 ), and no stationary source can exist below this value. The width of coherent sources diverges when µc is approached, as seen in Fig. 5.12. Non-stationary sources which cannot be obtained as solutions of (5.49) remain, however, possible also below µc . Sinks also become wider in the limit µ → 0, and their width can be estimated by solving (5.49) with the diﬀusional term neglected (Coullet et al., 1993). Then (5.49b) turn into algebraic equations. Assuming that the sink is stationary and therefore symmetric, we set c = 0, ζ = x. After separating z± = κ± + ik± , the real part of (5.49b) is easily solved to obtain 2 ρ± + gρ2∓ − µ . κ± = ±c−1 (5.51) 0

310

5 Amplitude Equations for Waves

The two ﬁrst-order equations obtained by using this in (5.49a) are integrated to yield µ

, ρ2± = (5.52) 1 + exp ±2c−1 0 µ(g − 1)x which satisﬁes for g > 1 the asymptotic conditions ρ± → 0 at x → ±∞. The asymptotic inverse width is linear in µ.

5.3 Spiral Waves 5.3.1 Symmetric Spiral Formation of holes is a purely 1D phenomenon, since in 2D a locally initiated singularity of the wave front would develop into a spiral wave. The substantial diﬀerence lies in the topology of the phase ﬁeld: in 2D the phase circulation must be conserved on an arbitrary contour surrounding a singularity. A circularly symmetric vortex solution of (5.2) is a rotating spiral wave. The N -armed spiral wave solution (or a spiral vortex with the topological charge N ) is obtained (Hagan, 1982) by assuming an ansatz u = ρ(r)eiθ , θ = N φ + ψ(r) − ωt.

(5.53)

This form diﬀers from the vortex solution of RGL (Sect. 4.3.4) by the presence of an explicit time dependence, as well as of a radial dependence of the phase ψ(r). This radial dependence is responsible for a peculiar appearance of a phase-resolved rotating spiral wave seen in Fig. 5.13. Moreover, the radial wavenumber k = ψ (r) generally approaches at r → ∞ a constant nonvanishing value k∞ , so that the vortex appears to be radiating a wave with

(a)

(b)

Fig. 5.13. A one-armed (N = 1) spiral wave. (a) Levels of constant phase. (b) Levels of constant Re u (levels Re u = ±2/3 are shown)

5.3 Spiral Waves

311

r, k 1 0.8

r

0.6 k 0.4 0.2

2

4

6

8

10

r

Fig. 5.14. Radial dependence of the wavenumber k and real amplitude ρ (q = 2)

a certain asymptotic wavelength. The frequency ω should ﬁt the asymptotic wavenumber, and is given by (5.10) with k replaced by k∞ . It is most convenient to use the stationary CGL equation in the form (5.6). The rescaling (5.5) is legitimate provided 1 + ην > 0, since then, with ω deﬁned by (5.10) and k∞ < 1, we have also 2 2 2 1 + ωη = 1 + νη + η(η − ν)k∞ = (1 + νη)(1 − k∞ ) + (1 + η 2 )k∞ > 0.

The rescaled asymptotic values of the real amplitude and wavenumber are related by (5.10), and the expression for Ω in (5.7) can be reduced using 2 ). (5.10) to Ω = −q(1 − k∞ The stationary polar form of (5.6) is ∇2 ρ + (1 − |∇θ|2 − ρ2 )ρ = 0,

(5.54)

∇2 θ + 2ρ−1 ∇ρ · ∇θ = q(ρ2∞ − ρ2 ).

(5.55)

More explicitly, these equations are rewritten using the ansatz (5.53) in polar coordinates as ρ (r) + r−1 ρ (r) + (1 − k 2 − N 2 /r2 − ρ2 )ρ = 0, k (r) +

2 k 1 d + kρ (r) ≡ 2 (rkρ2 ) = q(ρ2∞ − ρ2 ). r ρ rρ dr

(5.56) (5.57)

where k = ψ (r) is the radial wavenumber. Equations (5.56) and (5.57), subject to the boundary conditions ρ(0) = k(0) = 0, k(∞) = k∞ , ρ(∞) = ρ∞ ,

(5.58)

were solved numerically by Hagan (1982). As the problem is overdetermined, the numerical solution deﬁnes in a unique way the asymptotic wavenumber

312

5 Amplitude Equations for Waves

Fig. 5.15. Dependence of the asymptotic wavenumber k∞ on the parameter q (Hagan, 1982; reproduced with permission. Copyright by the Society for Industrial and Applied Mathematics. All rights reserved). The exact curve is ﬂanked by the small q and large q asymptotics, expressed, respectively, by (5.67) and (5.68)

k∞ , dependent on the parameter q. A typical dependence of ρ and k on the radial distance is shown in Fig. 5.14. The function k∞ (q) is plotted in Fig. 5.15. Take note that (5.57) is symmetric to the sign reversal of q and k; since both are of the same sign, the radial component of the group velocity (5.11) is always positive, and therefore all perturbations are swept from the core to the periphery of the spiral domain.4 Standard stability analysis of plane waves in Sect. 5.1.2 should apply to far regions of spiral waves; one could expect therefore a transition to a turbulent state to occur under conditions when the selected asymptotic wavenumber k∞ falls into the range where the corresponding plane wave solution of (5.2) is unstable. The respective limits of Benjamin–Feir, Eckhaus and absolute instability in the parametric plane (η, ν) are presented in Fig. 5.16. 5.3.2 Asymptotic Relations At q = 0, the standard vortex structure obtained in Sect. 4.3.4 is recovered, so that k∞ (0) = 0. One can attempt therefore to obtain the dependence k∞ (q) at q 1 perturbatively (Hagan, 1982). Assuming k = O(q), (5.56) gives just an O(q 2 ) correction to the standard vortex solution ρ0 (r), verifying (4.76), while (5.57) is readily integrated to yield, in the leading order r q k(r) = 2 [1 − ρ20 (s)]ρ20 (s)s ds. (5.59) rρ0 (r) 0 4

This has nothing to do with an apparent direction of motion of spiral branches, which is determined by the phase velocity ω/k. The latter would also depend in an observable underlying system on an added basic oscillation frequency. As it has no physical signiﬁcance, there is no sense in making a distinction between “spirals” rotating outwards and “antispirals” rotating inwards.

5.3 Spiral Waves

313

Fig. 5.16. Stability limits of a spiral wave solution in the parametric plane (η, ν). The curve EI shows the limit of convective instability and AI, of absolute instability for the waves emitted by the spiral; OR is the boundary of the oscillatory spatial decay for the emitted waves, q = 0.845 (bound states exist to the right of this line). BF indicates the Benjamin–Feir limit νη = −1, L is the limit of 2D phase turbulence, and T corresponds to the transition to defect turbulence for random initial conditions (Aranson and Kramer, 2002, based on Chat´e and Manneville, 1996; reproduced with permission. Copyright by the American Physical Society)

The wavenumber is well behaved at r → 0, as both ρ0 (r) and k(r) are linear in r in this limit. At r → ∞, the integral (5.59) is, however, logarithmically divergent, so that k∞ cannot be computed in such a straightforward manner. The asymptotic expression can be reduced by using (4.76) and integrating by parts to the familiar integral (4.100): q r d N 2 s−1 ρ20 (s) − ρ0 (s) ds(sρ0 (s)) ds lim k(r) = lim r→∞ r→∞ r 0 ds r q q r 2 −1 2 = lim N s ρ0 (s) + s(ρ0 (s)]2 ds = ln , (5.60) r→∞ r 0 r a0 where a0 ≈ 1.126 for |N | = 1. Matching with an outer solution, which is valid at distances r = O(q −1 ), is required. Rescaling to the extended radial coordinate R = qr, we see that the real amplitude is slaved in the far ﬁeld to the wavenumber, so that (5.54) or (5.56) reduces in the leading order to an algebraic relation ρ2 = 1−|∇θ|2 = 1 − k 2 − N 2 /r2 . Using this in (5.55) yields the far ﬁeld equation 2 ∇2 θ + q(k∞ − |∇θ|2 ) = 0.

(5.61)

314

5 Amplitude Equations for Waves

The far ﬁeld equation is now nonlinear; it is recognized as the stationary version of the Burgers equation. The nonlinearity is weak: since the radial component of the wave vector in the far ﬁeld of a spiral wave is small only for |q| 1, the terms containing q are formally of a higher order of magnitude under conditions when the radial wavenumber is small. The nonlinear terms, responsible for the screening eﬀect, play, however, a crucial role in the theory, since their action accumulates at growing distances from the spiral core, and cannot be treated as a mere small perturbation. Equation (5.61) can be linearized through the Hopf–Cole transformation θ = −q −1 ln G. The resulting equation is ∇2 G − (k∞ q)2 G = 0.

(5.62)

In view of the circulation condition (4.63), G is not a univalued function, but the singularity can be removed by setting G(r, φ) = e−N qφ χ(r).

(5.63)

Clearly, χ(r) is univalued, while the phase circulation equals 2πN as required. In polar coordinates, the equation of χ(r) reads 1 N 2 q2 2 2 χ (r) + χ (r) − k∞ q + χ = 0. (5.64) r r2 The symmetric solution corresponding to the far ﬁeld of an isolated vortex is expressed by a modiﬁed Bessel function with an imaginary index: χ(r) = Ki|N q| (k∞ qr).

(5.65)

In the following, we set |N | = 1. The outer limit of the inner solution (5.60) should be matched to the inner limit of the outer solution (Pismen and Nepomnyashchy, 1992): d 1 qγ sin q ln 12 k∞ qr − cos q ln 12 k∞ qr 1 , (5.66) lim ln Ki|q| (qk∞ r) = q r→0 dr r sin q ln 12 k∞ qr + qγ cos q ln 12 k∞ qr where γ = 0.5772 . . . is the Euler constant. The asymptotic expression (5.66) rapidly oscillates at r → 0, and can match (5.60) only when the argument 1 of the trigonometric functions is close to −12 π sinq. When (5.66) is expanded −1 1 to O(q), it reduces to r 2 π + γ + q ln 2 k∞ qr . Matching with the inner solution (5.60) yields the asymptotic wavenumber π 2 exp − −γ . (5.67) k∞ = a0 q 2|q| This expression has been obtained in a somewhat more complicated way by Hagan (1982). It is plotted along with the “exact” numerical curve in Fig. 5.15.

5.3 Spiral Waves

315

Hagan also computed the asymptotics at q 1: 2 q(1 − k∞ ) ≈ 2.34 at |N | = 1. q→∞ k∞

lim

(5.68)

This range is, however, not relevant, since (5.68) can be balanced only when k∞ close to unity, i.e., under conditions when the radiated waves are Eckhausunstable. The asymptotic wavenumber at |q| 1 turns out to be exponentially small. Such a quantity can be set to zero when a regular expansion in powers of a small parameter is carried out. Thus, a fair approximation to k(r) can be obtained by solving (5.64) with k∞ set to zero. The solution is r χ(r) = cos q ln , (5.69) a where an indeﬁnite constant a ﬁxes a phase shift. Choosing a = a0 yields r r 1 1 , k(r) = tan q ln . (5.70) ψ(r) = ln cos q ln q a0 r a0 The last formula matches the asymptotics (5.60) when r 1 but ln r = O(1). The approximation remains valid even at exponentially large distances ln r = O(q −1 ) but breaks down when the argument of the trigonometric function approaches π/2. Matching (5.70) to (5.65) uses the asymptotics of the Bessel function at ξ = k∞ qr 1 expanded in powers of q: iq (ξ/2) π ξ ξ Im = −q −1 sin q ln − γ cos q ln sinh(πq) 2 2 Γ (1 + iq) π ξ = −q −1 cos q ln + qγ + . (5.71) 2 2

Kiq (ξ) −

The cosine function in the last expression coincides with that in (5.70) after substituting (5.67). The radial wavenumber k(r) computed using Eq. (5.65) is very well approximated by (5.70) up to exponentially large distances r ≈ a0 exp(π/2q). The radial wavenumber gradually decreases with r, but, although k∞ → 0 beyond all orders at q → 0, this constant asymptotic value is approached at distances so huge that they are all but irrelevant. The approximations to k(r) on moderate, i.e., O(q −1 ), and large, i.e., O(e1/q ), distances for a moderately small value q = 0.1 are compared in Fig. 5.17. The diminutive asymptotic wavenumber k∞ ≈ 1.5 · 10−6 is not reached even at exponentially large distances shown in Fig. 5.17b, and the deviation from the asymptotic value reduces to a few percentage points only at r > 108 .

316

5 Amplitude Equations for Waves (a)

(b) k

k 0.035 0.03 0.025 0.02 0.015 0.01 0.005

0.01 0.001 0.0001 0.00001 1. · 10-6

2

4

6

8 10 12 14

r

100

10000 1. · 106

r

Fig. 5.17. Comparison of the approximations to k(r) for q = 0.1 on moderate (a) and large (b) distances (Pismen, 2003; reproduced with permission from Elsevier Science). Solid line: near ﬁeld ﬁrst-order solution (5.59) (a) and far ﬁeld solution (5.65) (b); dashed line: near ﬁeld asymptotics (5.60); dotted line: intermediate asymptotics (5.70)

5.3.3 Nondissipative Limit The non-dissipative limit of the CGL equation, −iut = ∇2 u + (1 − |u|2 )u,

(5.72)

is known as the Gross–Pitaevskii (GP) equation in the theory of superﬂuids (Donnelly, 1991) and Bose–Einstein condensates (BEC), or as the defocusing nonlinear Schr¨ odinger (NLS) equation in nonlinear optics (Newell and Moloney, 1992). It is derived from a Hamiltonian

δH , H= ∇u · ∇u + 12 (1 − |u|2 )2 dx, (5.73) ut = −i δu which is conserved in the course of evolution. The linear term in (5.72) is, of course, inessential, since a transformation u → ue−iωt adds the term ωu to the r.h.s. The choice of a coeﬃcient at the linear term is actually dictated by the desired properties of a “background” ﬁeld, i.e., a spatially homogeneous solution of the ﬁeld equations. When (5.72) is taken as the standard form, the background ﬁeld has zero frequency and a non-vanishing amplitude, normalized to unity. The NLS equation (5.72) is invariant to the Galilean transformation to a frame moving with an arbitrary velocity v < 2 (not exceeding the speed of sound), combined with a suitable gauge transformation of the ﬁeld variable: x → x − vt, ∂t → ∂t − v · ∇, u → u 1 − 14 v 2 exp [(i/2) v · x] . (5.74) This hints at an intimate connection between (5.72) and hydrodynamic equations of ideal (inviscid) ﬂuids. The NLS equation can be presented in a ﬂuidmechanical form with the help of a modiﬁed version of polar representation,

5.3 Spiral Waves

317

√ often called the Madelung transform: u = $eiϑ/2 , where $ = |u|2 is interpreted as the ﬂuid density, and ϑ = 2θ as the velocity potential deﬁning the superﬂuid velocity v = ∇ϑ. Then the non-dissipative counterparts of (5.3) and (5.4) are rewritten as the continuity equation $t + ∇ · ($v) = 0,

(5.75)

ϑt + 12 |v|2 + p = 0,

(5.76)

and the Bernoulli equation

where p is pressure introduced via the “equation of state” p = 2 ($ − 1) − $−1/2 ∇2 $1/2 .

(5.77)

Taking the gradient of (5.76) yields the Euler equation – the basic equation of ideal ﬂuid mechanics: v t + (v · ∇)v + ∇p = 0.

(5.78)

Static vortex solutions of (5.72) are exactly the same as those of the RGL equation (4.3) – see Sect. 4.3.4. The dynamics is, however, completely different; in this respect, both equations are polar opposites. In view of the Galilean invariance, the vortex remains symmetric when placed in a constant phase gradient and viewed in the respective Galilean frame, or, in other words, when advected with a constant velocity v = 2∇θ. The change in ρ = |u| is a second-order eﬀect when the velocity is small, and, for v = O(), the static solution remains invariant to O(2 ). This can be contrasted to the O() distortions of a dissipative vortex discussed in Sect. 4.4, which led there to a mobility relation containing a nonlinear logarithmic factor. This implies a remarkably simple law of vortex motion in a slowly varying ﬁeld: the vortex is embedded in the ambient superﬂuid and moves with the local velocity; the latter can be considered constant as long as it does not change considerably on distances comparable with the size of the vortex core. For v = O(), the continuity equation (5.75) reduces in the leading order to incompressibility condition ∇ · v = 0, so that the phase ﬁeld obeys the Laplace equation ∇2 θ = 0. This equation can be solved exactly for any timedependent conﬁguration of vortices. The solution corresponding to a single vortex located at a point xj (t) in the inﬁnite plane is proportional to the polar angle counted relative to the vortex location: θj = Nj φ(x − xj ); the respective velocity vector is v = 2Nj rj−2 J r j , where r j = x − xj , rj = |r j |, and the operator J denotes rotation through π/2. The total phase ﬁeld is obtained by superposition of the ﬁeld induced by all extant vortices (and in a ﬁnite region, also of their images). This gives the equations of motion of vortices xj − xk dxj =2 Nk J . (5.79) dt |xj − xk |2 k=j

318

5 Amplitude Equations for Waves

The ﬁeld problem has been reduced thereby to a dynamical problem with a ﬁnite number of degrees of freedom. This is the ultimate success of the particle–ﬁeld approximation, which has not been achieved in any other problem of vortex motion. Solutions of the dynamical system (5.79) have been extensively studied in the context of classical ideal ﬂuid mechanics (Batchelor, 1967); the motion can be chaotic when the number of vortices is larger than three. The CGL equation approaches the NLS equation in the weakly dissipative limit η 1, ν 1. The transition is most transparent when the form (5.6) is used. Requiring also ω 1, we see that both Ω and q, as deﬁned by (5.7) are small in this limit, and the NLS equation is recovered in the limit η → ∞. After rescaling t → ηt, this leads to the standard form (ε − i)ut = ∇2 u + (1 + iΩ)u − (1 − iq)|u|2 u = 0,

(5.80)

where ε = 1/η. Clearly, the basic vortex structure of the NLS and RGL equations with the vanishing radial wavenumber exists at q = Ω = 0 for any η; this allows us to span the entire transition from conservative to dissipative vortex motion in a simple way. 5.3.4 Acceleration Instability Aranson et al. (1994) suggested that a spiral core may suﬀer an acceleration instability, which is analogous to the meandering instability of the spiral tip in reaction-diﬀusion models with separated scales (Sect. 3.6.4). The acceleration instability can be studied perturbatively when the real part of the amplitude diﬀusivity c2 is small. When c2 is purely imaginary, as in the NLS equation (5.72), the system is Galilean invariant. The symmetry is broken due to both real correction to amplitude diﬀusivity and acceleration. A possible strategy is therefore to apply the transformation (5.74) and then balance the extra terms that appear when Galilean invariance is weakly broken. Let X(x) be a variable position of the phase singularity (i.e., of the zero of the complex order parameter). A linear relation between the velocity v = X t and the acceleration, presumed to be small and expressed as εa = v t , must have the form a = F · v, where F is a “friction” matrix. In an isotropic system, the diagonal elements of the matrix F should be equal, and the oﬀdiagonal ones should be antisymmetric. Acceleration instability is observed when F has an eigenvalue with a positive real part. At the instability threshold, the diagonal elements vanish. Since the oﬀ-diagonal elements, generally, diﬀer from zero at this point, the matrix has a pair of imaginary eigenvalues. This means that acceleration is normal to velocity, as it should be in rotational motion. Meandering motion is possible in the supercritical region, provided the instability is saturated nonlinearly; otherwise, transition to turbulence is imminent. The Galilean transformation (5.74) is applicable as well in the case when the velocity is variable, and the coordinates are transformed as x → x −

5.3 Spiral Waves

319

X(t). Starting with the weakly dissipative CGL equation (5.80) and applying the transformation (5.74), we obtain two non-vanishing O(ε) terms lacking circular symmetry: one due to acceleration, and the other due to the real part of the amplitude diﬀusivity. A solution of (5.80) corresponding to a slowly moving and weakly accelerated spiral vortex can be computed by expanding in ε in the same way as in Sect. 4.4.4. Assuming v = O(ε), a = O(ε2 ), the dipole inhomogeneity is of O(ε2 ). The second-order equation obtained by linearizing the CGL equation has the general form (4.113) with the dipole inhomogeneity (5.81) Ψ (x) = − 12 a · x u0 (x) + v · ∇u0 (x). The ﬁrst term is contributed by the time derivative of the transformed amplitude in (5.74).5 The caveat is that the operator H is now not self-adjoint. The solvability condition requires (5.81) to be orthogonal to the eigenfunctions of the adjoint operator H† with zero eigenvalue. Unlike the translational eigenfunctions of H, the respective eigenfunctions of the adjoint operator cannot be obtained simply by diﬀerentiating the circularly symmetric solution, and should be computed numerically. One may ﬁnd computing the eigenfunctions with the sole purpose of using them in the solvability condition impractical. The alternative is to solve the ﬁrst-order perturbation equation directly. This was the approach adopted by Aranson et al. (1994). Generically, the perturbation equation with the inhomogeneity (5.81) diverges exponentially at r → ∞, but this behavior is suppressed, leaving a slower (power-like) growth, when the relation between the velocity and acceleration is chosen in a unique way. The numerical results (Fig. 5.18) show that the spiral core is stable only when the real part of the amplitude diﬀusivity exceeds a certain ﬁnite positive value εc , going to zero in the limiting case ν = Im(c3 ) → ∞ that corresponds to the defocusing NLS equation. The eigenvalues of the matrix F acquire an imaginary part when ν drops below a certain level, indicating a meandering instability similar to that observed in reaction-diﬀusion models (Sect. 3.6.4), where it typically stabilizes at a ﬁnite rotation amplitude. Numerical simulations of the CGL equation have not, however, produced such behavior. The state at ε < εc was typically turbulent, but intermittent dynamics was observed near the instability threshold (Aranson et al., 1994). Analytical computation is possible in the limit q → 0, i.e., ν ≥ ε−1 . Then the translational eigenfunction ∇u(x) is used in the solvability condition. Comparing (5.81) with (4.131) we see that the problem is formally identical to that of motion in the supercriticality ramp, with the replacement a → −2M . The area integrals of both terms in (5.81) diverge logarithmically, albeit with diﬀerent constants. The divergence is eliminated by choosing a = 2v. The vortex turns to be always unstable under these conditions, and, as the oﬀ5

An additional circularly symmetric O(ε2 ) term stemming from this transformation modiﬁes the frequency but does not aﬀect the motion.

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Fig. 5.18. Stability limits of a spiral wave in the large η limit. CI is the core instability limit (unstable to the left); the area below EI is the Eckhaus unstable region, ST denotes the transition line to strong turbulence, and the area below OR is the oscillatory range (after Aranson et al., 1994; reproduced with permission. Copyright by the American Physical Society)

diagonal elements of F vanish, it is accelerated along a straight line. Non-zero oﬀ-diagonal elements appear only at larger values of q.

5.4 Interaction of Spiral Vortices 5.4.1 Nonradiative Limit The structure of a symmetric spiral vortex can be perturbed by external phase and amplitude gradients, which would set the spiral core into motion. This problem appears at a ﬁrst glance to be similar to that of perturbation of RGL vortices considered in Sect. 4.4, and, likewise, external gradients may be induced by interaction with neighboring vortices. Proceeding in the same way, one could try to impose on Hagan’s solution a perturbation breaking the circular symmetry, and compute the drift velocity that would compensate this perturbation. This approach fails for two reasons. The ﬁrst obstacle is merely technical, and is connected to the fact that the linearized CGL equation is not self-adjoint, and the translational eigenfunction of the adjoint problem is not readily available, as already discussed in Sect. 5.3.4. This obstacle can be overcome at the cost of numerical computation of the adjoint eigenfunction.

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321

Another impediment is more substantial, as it places the entire activity under a big question mark. Unlike the situation in Sect. 4.4, an external phase gradient, which may be generated, for example, by the phase ﬁeld of another spiral vortex, generally, does not penetrate into the spiral wave domain, much less into the spiral core, due to eﬀective screening caused by interference of nonlinear waves. This screening phenomenon is the principal reason for the stability of spiral domain patterns with spiral domains separated by shocks, which are observed in experiments and simulations (see Sect. 5.4.3). Moreover, all perturbations are washed out to the shocks, where they are absorbed, and never travel inwards. Clearly, a simple superposition of phase ﬁelds corresponding to isolated spiral wave solutions far from their cores cannot be a good choice of a basic solution in a pattern dominated by shocks, and the appropriate zero-order approximation should reﬂect the predominance of either vortex on the respective side of the shock. A simple superposition of phase ﬁelds is possible only in the limit of small asymptotic wavenumbers, i.e., at |ν − η| 1, when shocks weaken and disappear. The non-radiative limit ν − η = 0 encompasses the dissipative dynamics of Sect. 4.4 as well as the conservative dynamics brieﬂy outlined in Sect. 5.3.3 (see Pismen, 1999 for a comprehensive review). In this limit, the CGL equation, written in the moving frame associated with a steadily propagating vortex, can be reduced by a gauge transformation to the real GL equation (Pismen and Rodriguez, 1990). This transformation does not aﬀect the topological repulsion or attraction, but adds a velocity component normal to the straight line connecting a vortex pair, so that the overall direction of motion becomes oblique. We start with the CGL equation in the form (5.6) rewritten in the comoving frame (with q and Ω set to zero): (1 − iη)v · ∇u + ∇2 u + u − |u|2 u = 0.

(5.82)

Assuming the velocity to be small, we apply a combined Galilean and gauge transformation similar to (5.74): x → x − vt, ∂t → ∂t − v · ∇, u → u exp [(iη/2) v · x] + O(v 2 ).

(5.83)

Unlike the NLS or GP equation (5.72) in Sect. 5.3.3 which possesses Galilean symmetry, the CGL equation retains after the gauge transformation the dissipative part of the advective term with dipole symmetry. Choosing the direction of motion as the y axis, we recover the stationary equation in the comoving frame (4.110). The perturbation expansion and matching to the far ﬁeld proceeds exactly as in Sect. 4.4.4, leading eventually to the relation (4.123) between the velocity and the phase gradient k. Recalling the phase transformation (5.83), one has to extract from k the added phase gradient. Thus, (4.123) is modiﬁed to N v0 η ln J v. k− v = 2 2 |v|

(5.84)

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phase gradient velocity

Fig. 5.19. Oblique motion of a positively charged vortex under the action of the phase gradient due to the presence of either a positive vortex on the left or a negative vortex on the right. According to (5.84), the external phase gradient is the sum of two vectors directed along and across the velocity vector. The motion is directed toward the negative vortex or away from the positive vortex

At η 1, when the NLS limit is approached, the r.h.s. can be neglected, so that the vortex is simply advected by the imposed phase gradient. If the latter is caused by another vortex removed at a distance large compared to the size of the vortex core, this causes translation of a pair of oppositely charged vortices and rotation of a pair of likely charged vortices. The dissipative interaction on the r.h.s. causes oblique motion, with likely and oppositely charged vortices, respectively, repelling and attracting each other (see Fig. 5.19). 5.4.2 Weakly Radiative Vortices Interaction of weakly radiative vortices at q 1 can be considered as a perturbation of the non-radiative limit. Since, however, according to the results of Sect. 5.17, the limit q → 0 is singular, the strategy, as well as chances for success, should depend on the range of separation between the interacting vortices. At “near” O(q −1 ) distances seen in Fig. 5.17a, a weakly radiating spiral is similar to a non-radiating vortex. Since ψ ≈ O(q), this range of distances still falls within a single wavelength from the core, even though the separation is large compared to the core size. One can expect that the radial component of the phase ﬁeld of one vortex acting upon the core of the other one should cause weak repulsion. An interesting question here is whether a bound vortex pair can be formed when this non-topological repulsion balances topological attraction of a pair of unlikely charged vortices. The relative strength of the two eﬀects should be proportional to the ratio of the radial to the angular components of the wave vector. Since this ratio of O(q), balancing them may be possible at q 1 only when the system is close to the non-dissipative (NLS) limit, i.e., at η 1. It is advantageous therefore to use the “weakly dissipative” scaling of the CGL equation in the form (5.80). The phase gradient acting on the vortex core, which deﬁnes the velocity according to the mobility relation (5.84), should be obtained by solving the far

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323

ﬁeld equation. The appropriate form is (5.26), modiﬁed in the current scaling to (5.85) ( + q)θt = ∇2 θ − q|∇θ|2 . This form can be derived directly from the polar representation of (5.80) in a way similar to that described in Sect. 5.1.3. The asymptotic wavenumber k∞ has been neglected here as an exponentially small quantity. On the other hand, although at |q| = O() 1, ∇ = O() the nonlinear term is formally of a higher order in the small parameter of the problem, it should be retained as the only source of non-topological interaction. Consider a vortex pair with the positive and negative charges located, respectively, at x = ±L/2, y = 0. Assuming L−1 as well as q to be of O() 1, rescale q → q, and L → −1 L. Up to O(2 ) corrections, the vortex propagates, as in NLS equation, along the y-axis with the speed v = −2/L. Introducing this book-keeping small parameter explicitly, we rewrite the far ﬁeld equation (5.85) in the comoving frame as (5.86) ∇2 θ − q|∇θ|2 + 22 L−1 (1 + q)θy = 0. n (n) This equation is solved by expanding in as θ = θ (Pismen, 2003). We shall use the bipolar coordinate system with foci at the vortex positions. The coordinates σ, τ (respectively, circular and unbounded) are related to the Cartesian coordinates x, y as x=

L sinh τ sin σ L , y= . 2 cosh τ + cos σ 2 cosh τ + cos σ

(5.87)

The zero order solution satisfying the Laplace equation ∇2 θ(0) = 0 and the topological conditions of ±2π circulation around the foci at x = ±L/2 is θ(0) = π − σ. The ﬁrst-order radiative correction stems from the nonlinear term in (5.86). The inhomogeneity in the ﬁrst order equation written in the bipolar coordinates turns out to be constant: (1) θτ(1) τ + θσσ − q = 0.

(5.88)

The boundary conditions should take into account radiation to inﬁnity. A circle of inﬁnitely large radius is projected in bipolar coordinates into a single point τ = 0, σ = ±π. If (5.88) is solved in the interval −π ≤ σ ≤ π, the appropriate boundary condition should contain two symmetric “point” sinks placed at τ = 0 and σ = ±π. The strength of the sinks will be further adjusted to verify the asymptotic matching condition (5.60) applied at τ → ±∞. The obvious inhomogeneous solution independent of σ, θ(1) = 12 qτ 2 , does not match in the limit τ → ∞ the far ﬁeld asymptotics of the core solution. It should be complemented therefore by a suitable solution of the Laplace equation with sinks at τ = 0, σ = ±π which accounts for radiation by the spiral pair. It is easier to express the required solution as a symmetric combination of two sources at the foci, c [ln(r/L) + ln(r /L)], where r, r are radial

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distances from the foci. The adjustable constant c is chosen to satisfy the matching condition, which should be rewritten using the asymptotic relation between the bipolar and focus-centered polar coordinates. Near the positive focus, we have, in accordance with (5.87), τ = − ln

r r r + cos φ + . . . , σ = π − φ + sin φ + . . . , L L L

(5.89)

where the terms of second and higher order in r/L are omitted. The circularly symmetric component of θ(1) matches in the limit r → 0 (i.e., τ → ∞) the far ﬁeld asymptotics of the core solution deﬁned by (5.60) if one sets c = −q ln(a0 /L). Then θ(1) is expressed in the vicinity of the focus as θ(1) =

r a0 r ra0 q 2 r ln − q ln ln − q ln 2 cos φ + . . . . 2 L L L L L

(5.90)

The inhomogeneity in the second-order equation is contributed by the advective term: ∂τ2 θ2 + ∂σ2 θ2 + (1 + q)

cosh τ cos σ + 1 = 0. (cosh τ + cos σ)2

(5.91)

The inhomogeneity vanishes when averaged over σ; therefore there is no radiation to inﬁnity that would necessitate a singular boundary condition as in the ﬁrst-order problem. In view of the symmetry σ → −σ, the solution is sought for in the form of a cosine Fourier series: θ(2) (τ, σ) =

∞

ψk (τ ) cos kσ.

(5.92)

k=1

We only have to compute the dipole component ψ1 , which veriﬁes the equation ψ21 (τ ) − ψ21 + 2(1 + q)e−|τ | = 0.

(5.93)

The unique solution is deﬁned by the conditions of continuity and smoothness at τ = 0 and absence of exponential growth at τ → ±∞: ψ21 = (1 + q)(1 + |τ |)e−|τ | .

(5.94)

Using (5.89), this yields near the vortex location the dipolar term (1)

θ2 =

r r (1 + q) ln cos φ. L eL

(5.95)

The dipole contributions of both advective and radiative terms computed above are of the same O(2 ). Collecting the dipole terms from Eqs. (5.90), (5.95), and adding the correction to the topological phase following from (5.89) yields the inner limit of the dipole component of the far ﬁeld near the positive focus:

5.4 Interaction of Spiral Vortices

325

r r ra0 r (5.96) (1 + q) ln − q ln 2 cos φ. θ1 = − sin φ + L L eL L The ﬁrst term in the r.h.s. can be removed by the gauge transformation (5.74) corresponding to the non-dissipative velocity component vy = 2/L. The second term can be rearranged to the form (4.119): 1 a0 r +1+q . (5.97) + (1 − q) ln θ1 = rχ(r) cos φ, χ(r) = ln L a0 L This is the inner limit of the outer solution, which can be used to evaluate the contour integral in the solvability condition (4.116) as in Sect. 4.4.4. The contribution of the logarithmic term in χ to the contour integral cancels with the area integral in (4.116). The remainder deﬁnes the dissipative velocity component vx : 1 a0 vx = 2(1 − q) ln + 1 + 2q , (5.98) L L or, returning to the original parameters, 1 a0 + 1 + 2ηq . (5.99) 2(1 − ηq) ln vx = L L Since L (in the original units) must be large, the ﬁrst term is dominating, unless at q close to 1/η. At q < 1/η, vx is negative, so that the interaction is attractive, as it is in the non-radiative case q = 0. At q > 1/η, repulsion prevails at large distances. The stationary separation, 1/2 + qη Ls = a0 exp , (5.100) qη − 1 falls in the required range L 1 only when qη − 1 is negative and small. Under these conditions, the interaction becomes repulsive at large distances, but the equilibrium is obviously unstable. Thus, oppositely charged spirals cannot form bound pairs in this limit, although topological attraction does switch to repulsion with growing separation in a certain parametric range. A remaining possibility is that a stable equilibrium might be realized at exponentially large distances where screening becomes important, and the weak interaction limit is applicable. Aranson et al. (1991) and Pismen and Nepomnyashchy (1991, 1992) computed a correction to the vortex mobility due to a ﬁnite but small asymptotic wavenumber. The perturbation theory was based on the usual core expansion in the comoving frame matched to a solution of the far ﬁeld equation. The radial and the angular components of the wavenumber equalize when ln r = O(|q|−1 ). In this range, the far ﬁeld equation (5.86) cannot be solved perturbatively as above. The phase equation can be linearized through the Hopf–Cole transformation as in Sect. 5.3.2. We have seen, however, there that the transformed function becomes multivalued when the phase ﬁeld is topologically non-trivial. Pismen and Nepomnyashchy (1991, 1992) constructed an approximate solution by computing a correction

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to the circulation condition at small q. Remarkably, this topological correction resulted in a screening eﬀect, so that the vortex interaction turned out, indeed, to be exponentially decaying with separation. Another way (Pismen, 2003) is to extract ﬁrst the topological part by setting, say, θ = Θ + ψ (e.g., Θ = φ for the circularly symmetric case in Sect. 5.3.2), and to apply the Hopf–Cole transformation ψ = −q −1 ln χ to the non-topological part only. Starting from (5.26) rewritten in the comoving frame, this transformation yields a linear equation with variable coeﬃcients dependent on Θ:

(5.101) (1 + qη)χt = ∇2 χ − 2q∇χ · ∇Θ + qχ q|∇Θ|2 + (1 + qη)Θt . Using as before the bipolar coordinate system, the extracted topological phase is Θ = π − σ. Allowing for propagation along the y axis with the speed v, we write the far ﬁeld equation in the comoving frame in the form χτ τ + χσσ + 2qχσ + q 2 χ + 12 Lv(1 + qη)F[χ] = 0,

(5.102)

where the coordinate dependence is contained only in the advective operator F[χ] =

(cosh τ cos σ + 1)(χσ + qχ) − χτ sinh τ sin σ . (cosh τ + cos σ)2

(5.103)

The order-of-magnitude estimates in (5.102) diﬀer from those for the near ﬁeld, since 1/η should not be presumed small, while velocity is expected to be exponentially small (though still far exceeding k∞ ). The velocity of a hypothetical bound pair has to be determined from the solvability condition, since it is not approximated anymore by vortex dynamics in the NLS equation. Using a power expansion in q to detect an exponentially small eﬀect is precarious. This puts a question mark on the earlier analytical results. If the perturbation parameter is of O(e−|q| ), all O(q n ) quantities should be formally treated as being of O(1), which introduces insurmountable diﬃculties, and even invalidates the approximation of the phase equation by an isotropic Burgers equation. Detecting an exponentially weak eﬀect in this limit (at exponentially large separations which cannot be realized in practice) has little value, besides a purely intellectual challenge. 5.4.3 Strong Radiation and Shocks The essential feature determining the interaction of strongly radiating spiral vortices at q = O(1) is the formation of shocks – domain boundaries where waves with diﬀerently directed wave vectors emanating from diﬀerent centers collide. The shocks are, generally, mobile. Their speed is determined by (5.47), where k± should be interpreted now as the normal components of the wave vectors on the two side of the shock. For a shock separating two counter-propagating waves, the r.h.s. of (5.47) is the diﬀerence of the normal

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327

Fig. 5.20. Evolution of an unlike-charged spiral pair into an antisymmetric state for η = 0, ν = 1 (Aranson and Kramer, 2002; reproduced with permission. Copyright by the American Physical Society)

components of the group velocities. According to this formula, the domain with a higher group velocity (i.e., a larger wavenumber k) advances at the expense of its rival. A shock separating two extended symmetric spiral domains is expected to be stable, as long as the radial wave vector component decreases with the distance from the core at large distances.6 Under these conditions, a symmetrybreaking shift is counteracted by an increase of the group velocity on the receding, and a decrease on the advancing side, which returns the shock to its original symmetric position. If, however, the shock shifts very close to the core, into a region where the radial wavenumber still increases, stability is lost, resulting in a drastic symmetry-breaking eﬀect. This is seen in Fig. 5.20, where, as a result of symmetry breaking, only one extended spiral domain remains, whereas the other spiral, while protected by conservation of topological charge from disappearing altogether, is reduced to a “naked core”. The shocks eﬀectively screen diﬀerent spiral domains from radiation emitted by other spiral cores. The perturbations due to a neighboring spiral decay behind the shock with the attenuation rate deﬁned by (5.20). An important threshold is the critical value |q| ≈ 0.845 beyond which the decay becomes oscillatory. This limit, computed using in (5.20) the asymptotic wavenumber k∞ selected by the spiral wave, is shown by the line OR in Fig. 5.16. At larger values of |q| (to the right of this line), formation of bound spiral pairs is possible, as conﬁrmed by computations of Aranson et al. (1993). Both likely and unlikely charged spirals can form bound pairs, as seen in Fig. 5.21. The oscil6

This is clearly seen in Fig. 5.17 drawn for q 1; at larger q (Fig. 5.14), the decrease persists, though is not as pronounced.

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latory decay at large |q|, creating eﬀective traps for interacting spiral cores, is essential for this eﬀect, and there is no wonder that bound pairs were not found in the analysis for |q| 1 (Sect. 5.4.2). Bound pairs are always mobile. Since the velocities, being induced by interactions decaying exponentially with the distance, are slow, perturbation analysis is applicable here, but the diﬃculty lies in the non-Hermitian nature of the linearized problem at q = 0. Lacking a readily available translational eigenfunction, one has to solve the adjoint problem numerically to compute the adjoint eigenfunction with zero eigenvalue necessary for evaluation of the solvability condition. For q = 0, the adjoint eigenmode decays exponentially at large distances, thereby ensuring convergence of the area integral in the inﬁnite plane. Therefore it is no longer necessary to compute the solvability condition in a ﬁnite region as in Sect. 4.4.4. Although this seems to make the computations easier by eliminating the need in asymptotic matching of the near and far ﬁeld solutions, it eﬀectively disconnects the motion of the spiral core from its far ﬁeld background. As a result, it becomes, for example, unclear whether the phase gradient added through transformation (5.83) does indeed induce drift velocity as in Sect. 5.4.1 or is advected away from the core to be absorbed at a shock.

Fig. 5.21. The bound states of oppositely (left) and likely (right) charged spirals for η = 0, ν = 1.5. The images show the modulus ρ(x, y) (top) and Re(u) (bottom) (Aranson and Kramer, 2002; reproduced with permission. Copyright by the American Physical Society)

5.4 Interaction of Spiral Vortices

329

Aranson et al. (1993) used another approach based on direct solution of the perturbation problem. The idea is that solutions exist only with the “correct” velocities. The solution is sought for in the form u = u(0) (x) + u(1) (x), where is a formal small parameter scaling the velocity v, and u(0) (x) is the symmetric spiral solution within a limited domain, e.g., in a half-plane for a symmetric spiral pair. For a pair of spirals separated by a shock, this solution does not satisfy the boundary conditions at the shock taking into account the neighboring spiral, e.g., the symmetry condition ux (0, y) = 0 when the symmetry axis is x = 0, as in the left panel of Fig. 5.21. Assuming the discrepancy to be of O(), the corrections are determined by the linearized ﬁrst(1) (0) order problem with the applicable boundary condition, e.g., ux = −−1 ux in the symmetric case. The advective term v · ∇u(0) comes in as an inhomogeneity. The ﬁrst-order problem is solved directly, and the velocity is adjusted in a unique way to avoid exponential divergence at large distances, which, in essence, is a numerical equivalent of computing a solvability condition. The interaction of spirals with widely separated cores is only weakly dependent on their polarity, since the oscillatory asymptotic decay is the same for spirals of opposite charge. The only diﬀerence is in tangential motion deﬁned by symmetry. For a pair of spirals separated by the shock at x = 0, the symmetry is u(x, y) = u(−x, −y) for likely charged spirals and u(x, y) = u(−x, y) for unlikely charged spirals. This leaves the symmetry condition ux (0, y) = 0 used in the above computation identical in both cases, but the directions of motion of the two spirals in the bound pair should be identical when the charges are opposite (resulting in translation along the y axis) and opposite when the signs are identical (resulting in rotation). In aggregates of more than two bound spirals, each spiral core may move along a more complicated and possibly irregular trajectory. 5.4.4 Multispiral Patterns The shock formation and its inﬂuence on spiral interactions are crucial for generic long-time evolution of large systems containing a number of spirals. A typical example of a spiral domain pattern in a stable parametric range obtained in a simulation run starting from random initial conditions is presented in Fig. 5.22. At the initial stage, the system tends to relax locally to the stable state with unity real amplitude, but, as the phases are random, the relaxation is frustrated, and a large number of defects – vortices of unit charge – are formed. At the following coarsening stage, oppositely charged vortices annihilate, so that the density of defects decreases. The coarsening process, however, stops halfway, leaving a certain number of single-charged spiral vortices. The resulting stable spiral domain pattern is called vortex glass (Huber et al., 1992; Chat´e and Manneville, 1996). A rotating spiral wave of either sense emanating from each vortex ﬁlls a surrounding domain; the domain boundaries are shocks where waves with diﬀerently directed wave vectors emanating from diﬀerent centers collide. Vortices that failed to conquer a suﬃciently large domain and

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Fig. 5.22. Spiral domains. Left: levels of constant phase. Right: grayscale amplitude map showing enhanced amplitudes at the shocks (Chat´e and Manneville, 1996; reproduced with permission from Elsevier Science)

were reduced to a “naked core”, as explained in the preceding subsection, may survive at shock junctions as phase singularities tightly bound by converging shocks. In the ﬁnal pattern, neighboring spiral domains may have any sense of rotation, i.e., carry topological charge of any sign. Since the circulation of phase around any domain should be equal to ±2π, a phase slip on a remnant vortex might be necessary to close the circulation balance. The waves always

Fig. 5.23. Development of vortex glass in the convectively unstable range. Left: large spirals nucleating in turbulent sea, η = −2, ν = 0.75. Right: developed vortex glass, η = −2, ν = 0.7. Dark shades shows lower amplitude (Chat´e and Manneville, 1996; reproduced with permission from Elsevier Science)

5.4 Interaction of Spiral Vortices (a)

331

(b)

Fig. 5.24. (a): A family of domain boundaries deﬁned by (5.105). (b): Domain boundaries in a vortex glass pattern approximated with the help of this formula (Bohr et al., 1997; reproduced with permission)

propagate outwards from the vortex cores; in fact, the entire domain structure is generated when local order spreads out from centers to the periphery as seen in Fig. 5.23. Perturbations, also traveling outwards with the prevailing group velocity, are absorbed at shocks, and therefore the pattern may survive beyond the convective instability threshold. The turbulent state takes over only when the waves become absolutely unstable, i.e., when some perturbations grow locally in the laboratory frame. The shapes of domain boundaries can be determined from the condition of phase continuity across the shocks (Bohr et al., 1997). Far away from the core of an unperturbed spiral, the phase is given by θi = k∞ ri ± φi + Ci ,

(5.104)

where ri , φi are the polar coordinates measured from the center of the ith spiral and Ci is a constant. At distances ri 2π/k∞ , the angular term in (5.104) can be neglected; then the phase balance condition θi = θj for two neighboring spirals yields ri − rj = (Cj − Ci )/k∞ = const,

(5.105)

which deﬁnes a hyperbola in the x, y plane (Fig. 5.24a). This simple formula was found to reproduce the structure of the spiral domains with high accuracy (Fig. 5.24b). The overall structure of the pattern is diﬀerent in the monotonic and oscillatory ranges separated by the line OR in Fig. 5.16. In the monotonic range above this curve, stable domains have to be of about the same size, as in Fig. 5.25a, since shocks are immobile when the normal components of the wave vector, which weakly decrease with distance, are equal. On the contrary,

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(b)

Fig. 5.25. Spiral domain pattern in (a) monotonic range (η = 0, ν = 0.6) and (b) oscillatory range (η = 0, ν = 1.2). Oppositely charge vortices are distinguished in the left panel by the size of the dots indicating the core locations. Shock are shown schematically by straight lines (Brito et al., 2003; reproduced with permission. Copyright by the American Physical Society)

in oscillatory range domains may have various sizes, as in Fig. 5.25b, since the balance can be achieved at diﬀerent separations. Multispiral domain patterns were usually considered stationary, but very long computation runs by Brito et al. (2003) have shown that “frozen” patterns actually evolve on a very long time scale. The ultraslow evolution was characterized in this study by “activity” ρt , where the average was taken over the entire computation domain. Since the amplitude ρ is almost constant everywhere except the core region, this quantity is related to the velocity of spiral cores. In the monotonic range, the spiral cores perform very slow diﬀusive motion, and ρt ﬂuctuates around a certain value, while the pattern retains the general form of Fig. 5.25a; the apparent diﬀusivity increases with vortex density. In contrast, in the oscillatory range, the spiral population spontaneously segregates after a very long transient into two distinct phases seen in Fig. 5.25b: large and almost immobile spirals and clusters of trapped small vortices. When the “liquid fraction” is small, the resulting pattern exhibits slow intermittent dynamics: bursts of activity separated by long quiescent intervals. The system keeps evolving on an extremely slow scale, which is consistent with exponentially weak repulsion between well separated spiral cores. Another possibility, realized in a diﬀerent parametric region, is a dynamic chaotic state that shows no persistent features. This state is attained under conditions when either spiral waves or vortex cores, or both, are unstable.

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As in 1D, one can distinguish between mild phase turbulence when no phase singularities occur and defect chaos characterized by persistent creation and annihilation of vortex pairs. In 2D, phase turbulence is always metastable with respect to transitions to either defect turbulence or vortex glass, but may persist in the parametric region between the Benjamin–Feir line and the line L in Fig. 5.16 (Chat´e and Manneville, 1996; Manneville and Chat´e, 1996). The range is somewhat smaller than in 1D. Beyond the line L, defects are created spontaneously, leading to defect chaos. The transition from vortex glass to defect turbulence in simulations starting from random initial conditions occurs at the numerically determined line T in Fig. 5.16 (Chat´e and Manneville, 1996). The transition occurs somewhat prior to the absolute instability limit determined by the linear stability analysis of plane waves emitted by spirals. The absolute instability limit can be approached, however, by starting from carefully prepared initial conditions in the form of large spirals. Prior to the transition, one can observe transient defect turbulence which is unstable to spontaneous nucleation of spirals from the “turbulent sea”, leading eventually to a vortex glass state as in Fig. 5.23. The nucleation time diverges at the line T. 5.4.5 Period Doubling in Spirals A special kind of spiral wave patterns arises when the underlying dynamical system undergoes a period doubling transition. The period doubling causes

Fig. 5.26. (a): A pair of period two spiral waves with the fundamental period τ and the average wavelength λ. The white solid lines are the synchronization defects (SD). (b): A period two time series measured at the point marked by the white ﬁlled square (Park and Lee, 2002; reproduced with permission. Copyright by the American Physical Society)

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(b)

Fig. 5.27. Core dynamics of period two spirals in the extended R¨ ossler model (Davidsen et al., 2004; reproduced with permission. Copyright by the American Physical Society). (a): dynamics in laminar regime λ = 3.5. Left: snapshot of a rotating period two spiral. The SD line originating from the spiral core is shown in black. Right: Magniﬁcation of the rectangular region in the left panel showing the SD line at two moments of time; α is the angle between the spiral core trajectory (black line) and the attached SD line. (b): Spiral core trajectories in the turbulent region at λ = 4.6 (left) and λ = 5.8 (right)

the appearance of synchronization defect (SD) lines, which serve to reconcile the doubling of the oscillation period with the period of rotation of the spiral wave (see Fig. 5.26a). These lines are deﬁned as the loci of those points in the medium where the two loops of the period two orbit exchange their positions in local phase space. The period two oscillations on the opposite sides of a SD are shifted relative to each other by 2π (i.e., a half of the full period), so that the dynamics projected on the rotation direction is eﬀectively of period one, while it is of period two locally at any point in the medium (Fig. 5.26b). In a similar way, period four local dynamics generates two types of SD lines, with the phase jumping either by 2π or by 4π. SD lines of this kind were detected

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335

Fig. 5.28. A spiral wave with a sinusoidal (left) and irregular (center ) SD line. The snapshot on the right shows development of a chaotic domain with tangled SD lines (Park et al., 2004; reproduced with permission. Copyright by the American Physical Society).

in experiments in a continuously fed gel reactor (Park and Lee, 1999). The local dynamics of the CGL equation is, of course, too simple to reproduce this phenomenon, but it has been successfully modeled using the R¨ossler model (1.104) complemented by diﬀusional terms with equal diﬀusivities (Goryachev et al., 1999). SD lines break the rotational symmetry and thus, on general grounds, one expects the spiral core to move. In the “laminar” regime at smaller values of the parameter λ in (1.104), the observed motion is very slow, and proceeds, after a transient, with a constant velocity directed at a certain angle to the SD line (Davidsen et al., 2004, see Fig. 5.27a). With growing λ, the velocity increases, while the angle changes from almost π at the period doubling transition to close to π/2 at the transition to turbulent regime. Close to the onset of turbulence, the core trajectories take the form of “ballistic” ﬂights with sporadically changing directions. Deeper in the turbulent region, the ﬂights shorten, and the trajectory resembles Brownian motion (Fig. 5.27b). Transition to turbulence can be also initiated by bending SD lines, as seen in Fig. 5.28 (Park et al., 2004). Spiral domain patterns in extended systems undergo a drastic change following a period doubling transition. As discussed in Sect. 5.4.4, multispiral patterns in simply periodic systems typically relax to a frozen “vortex glass” state consisting of spiral domains with comparable areas separated by shock lines, such as shown in Fig. 5.29a. This changes under the period two local dynamics (Fig. 5.29b): spiral pairs connected by SD lines now repeatedly rearrange, since the motion at an obtuse angle to the connecting line pushes them away one from the other. In multispiral disordered patterns, the spiral cores move almost independently, stretching the SD line to which they are attached, until they meet another core, leading to reconnections and creation or annihilation of SD lines. This process may be frustrated when spiral cores become attached to more

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Fig. 5.29. Snapshots of typical multispiral patterns in the period one (a, λ = 2.5) and period two (b, λ = 3.5) regime. SD lines are shown in black. The direction of the cores motion is highlighted by the white arrows in the blowup (c) showing a single pair of spirals. (d): Trajectories of spiral cores leading to the conﬁguration shown in (b); shading changes indicate elapsed time (Davidsen et al., 2004; reproduced with permission. Copyright by the American Physical Society)

than one SD line. The rearrangements occur mainly when two cores are close and may initiate a cascade of reconnection events. Repulsion of vortices of opposite sign connected by an SD line overcomes their topological attraction and prevents their annihilation at close approach. Thus, vortex glass gives way to a “vortex liquid” state where not only the total topological charge, but also the number of spiral cores is preserved due to the complex dynamics of the connecting SD lines (Davidsen et al., 2004).

5.5 Line Vortices and Scroll Waves 5.5.1 Curvature-Driven Motion CGL spirals, like spirals in the FN system (Sect. 3.6.6), can be extended in 3D into a scroll wave. We have seen in Sect. 3.6.6 that dynamics of a scroll

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337

wave is largely determined by the motion its core ﬁlament. This motion can be analyzed most readily using the CGL model where the core is identiﬁed in a most natural way as a line vortex – the locus of the zero of the complex ﬁeld. Consider a curve C deﬁned at any moment of time t in a parametric form X(s; t), where s is the arc length. As in 2D, the tangent to C at any point is the derivative X s = l. Since l, by construction, it is a unit vector, the vector ls must be orthogonal to l. By deﬁnition, the direction of ls is the normal n, and its absolute value7 is the curvature κ. The vector ns measuring the rate of change of the normal must again be orthogonal to n; its projection on the tangent must be equal to −κ to insure the identity ∂s (l · n) = 0. The projection of ns on the binormal b, which is the unit vector orthogonal to both l and n, is the torsion τ . Finally, the projections of the derivative bs on l and n are ﬁxed by the identities ∂s (l · b) = ∂s (n · b) = 0. This yields the Frenet–Serret equations X s = l, ls = κn,

ns = −κl + τ b, bs = −τ n.

(5.106)

The three orthogonal unit vectors l, n, b deﬁne at any point of the curve the Frenet trihedron. Curvature and torsion are the two intrinsic parameters that deﬁne locally the rotation of the Frenet trihedron, and therefore determine the shape of a smooth curve. If the functions κ(s) and τ (s) are known, the curve can be reconstructed, in principle, by integrating (5.106), although this task is technically diﬃcult due to the implicit form of these equations. For practical computations, the Frenet–Serret equations can be transformed to a more convenient set of intrinsic equations of motion in a way similar to Sect. 3.4.1, provided the normal and binormal velocities are known (see e.g. Pismen, 1999). The latter should depend in 3D, in addition to other factors, on the local geometry of the line vortex. When the curvature radius is much larger than the core size, it can be treated perturbatively by setting up an aligned coordinate frame in a way similar to Sect. 2.2.1. The plane normal to the curve at any point is spanned by the local normal and the binormal vectors. The aligned frame is built up, as in 2D, by shifting points of the line vortex along the local normal (Fig. 5.30). It is convenient to work with the CGL equation in the form (5.6), where we can set Ω = 0 (absorbing it into the background frequency) Assuming both the curvature κ and the local normal velocity v to be small, we set κ = K, v = V and rewrite (5.6), to the ﬁrst order in a book-keeping small parameter , as a stationary equation in the aligned frame ∇2⊥ u + u − (1 − iq)|u|2 u + [(1 − iη)V − Kn] · ∇⊥ u = 0,

(5.107)

where ∇⊥ is the 2D gradient operator in the local normal plane. 7

Unlike Sect. 2.2.1 where the sign of the curvature was related to the direction of change of the order parameter, the curvature, as deﬁned here, is always nonnegative.

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n

b

l

Fig. 5.30. Aligned coordinate frame in 3D. A sequence of coordinate lines (shown in gray), obtained by shifting points of the line vortex (shown in black ) along the local normal, develops a singularity at an O( −1 ) distance when κ = O( −1 ), while displacement along the binormal is without distortion. The Frenet trihedron and two circles in the local normal plane are also shown

The perturbation term is independent of q, which creates an illusion that both radiative and nonradiative vortices can be treated in the same manner. If η = 0 (which at q = 0 corresponds to the gradient RGL equation), the ﬁrst-order correction cancels exactly by setting v = κn. Thus, the local law of dissipative motion is utterly simple: each point on the line vortex is translated along the normal with the speed proportional to the local curvature, while the vortex core remains undeformed in the ﬁrst order. Clearly, the curve shrinks as a result of motion along the normal, resulting (in the gradient case) in a decrease of energy integral F in (4.8), which is roughly proportional to the length of the line vortex. For example, a ring vortex of radius L should shrink with the speed dL/dt = L−1 , yielding upon integration the total lifetime t0 = L20 /2 for a ring with the initial radius L0 . Generally, the ﬁnal result of dissipative evolution is collapse of a closed curve or straightening of a curve with ﬁxed ends, but intermediate conﬁgurations of a curve evolving from smooth initial conditions may involve singularities formed as a result of the local collapse of strongly curved sections. The local law of motion based on the expansion in small curvature becomes, of course, invalid shortly before the collapse when the curvature radius becomes comparable to the core size. With η = 0, the imaginary part of the advective term can be removed, as in 2D, by applying the transformation (5.83). Indeed, it is easy to see that

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339

the velocity and the added phase gradient can be chosen in such a way that the inhomogeneity vanishes exactly: k = 12 κηn,

v = (1 + η 2 )κn.

(5.108)

Gabbay et al. (1997) interpreted this as an indication that binormal motion is suppressed by the applied phase gradient. The phase gradient cannot be, however, chosen at will, but has to be obtained by solving applicable far ﬁeld equations, e.g., the Burgers equation (5.26) or the diﬀusion equation in the nonradiative case q = 0. In the latter case, nonlocal induction by removed segments of the line vortex should be signiﬁcant, as discussed below. At q = 0, nonlinear interference on shocks may serve to suppress distant action. Simulations by Frisch and Rica (1992) and Gabbay et al. (1998) indicated no binormal velocity component, which may be taken as a sign that these poorly understood interactions conspire in a miraculous way to create the compensating phase gradient. 5.5.2 Nondissipative Motion Lacking a freely adjustable phase gradient, there is no exact compensation in the non-gradient case, and a solvability condition has to be applied. As in Sect. 4.4.4, integration is carried out over a large circle of radius 1 r0 −1 , and the solvability condition involves the contour integral dependent on the gradient of the phase ﬁeld k⊥ = ∇⊥ θ induced by the entire line vortex. As in 2D, the relevant constituent of the phase ﬁeld, which is also presumed to be of O(), should be non-singular, i.e., exclude the local phase circulation. The notion of “local” is not as obvious in 3D, where we deal with a continuous line vortex, as it was in in the case of isolated point vortices in 2D. Separating local and nonlocal contributions necessitates, in fact, an additional asymptotic matching procedure, which can be carried out consistently in the non-dissipative case η → ∞ when the NLS equation is recovered upon rescaling (Pismen, 1999). Starting from the NLS equation (5.72) rewritten in the aligned frame, we remove the term containing the drift velocity v with the help of the Galilean transformation (5.74). The resulting equation, ∇2 u + (1 − |u|2 )u = K n · ∇⊥ u,

(5.109)

coincides with (4.110) if the y-axis is directed against n, and κ = K is replaced by v. Thus, the expansion and asymptotic matching procedure from Sect. 4.4.4 leading to the solvability condition (4.116) is again applicable, with v replaced by κ as a small parameter. Next, the phase ﬁeld on the bounding circle entering the solvability condition has to be evaluated by solving the far ﬁeld equation. The phase in the far ﬁeld satisﬁes the Laplace equation ∇2 θ = 0 (Sect. 5.3.3), which can be solved exactly for any given conﬁguration of the line vortex X(s; t). The solution

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, should satisfy the circulation condition dθ = 2π on any contour surrounding the line vortex.8 The phase gradient, which, in view of the Galilean symmetry, is equivalent to the velocity v = 2∇θ, is expressed by the Biot–Savart integral ! R(s; t) × l(s) v(x, t) = − ds, (5.110) R3 C where R(s; t) = x − X(s; t), R = |R|. Small curvature presumes large length of the line vortex; thus, both R and the arc length s should be measurable on an outer scale extended by −1 . If the inner (core) scale is used, the estimate v = O() follows from R, s = O(−1 ). The Biot–Savart formula is applicable far from the vortex core but diverges on the line vortex itself. One cannot compute therefore the vortex velocity in a straightforward way by applying the Galilean transformation (5.74) as in 2D. The matching procedure is, however, not impaired, since the integral is well deﬁned on a surrounding circle lying well outside the vortex core. We shall use now the outer (extended) scale to measure distances. Picking an arbitrary point P on the curve C, we take it as the origin of both the arc length s and the 3D coordinates x. A point X on C in the vicinity of P has the representation valid as long as the arc length is small when measured on the outer scale: (5.111) X = sl + 12 Ks2 n + O(s3 ). A point lying on a circle of radius r (small when measured on the outer scale) centered at P in the plane perpendicular to C which passes through P (Fig. 5.30) is presented as x = r(− sin φ n + cos φ b),

(5.112)

where φ is the angular coordinate in the (n, b) plane (counted from the direction of b). In order to evaluate the line integral in (5.110), we divide it into two parts, V 1 + V 2 , where V 1 is computed on the segment |s| < δ, where r δ 1, and V 2 , on the rest of the line vortex. The near integral V 1 expresses the local contribution of the adjacent segment of the curve, diverging at r → 0. On the near part of the contour, we rescale s = rσ, and compute the normal components of (x − X) × l using the representation (5.111):

(x − X) × X s = r −σl − (sin φ + 12 Krσ 2 )n + cos φ b × (l + rKσn)

⇒ r cos φ n + (sin φ − 12 Krσ 2 )b + O(r2 ) . (5.113) In the last expression, the projection on l has been discarded. Using also the expansion

3/2 (x − X)3 = s2 + r2 + rKs2 sin φ + O(s3 ) 3rKσ 2 sin φ 2 + O(r = r3 (1 + σ 2 )3/2 1 + ) , (5.114) 2(1 + σ 2 ) 8

Throughout this section, we presume the charge N = 1; negative charges are superﬂuous, since the sign of circulation changes with the reversal of l.

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341

we compute

δ/r 1 3 rKσ 2 sin φ 1 − − (sin φ b + cos φ n) dσ 2 3/2 δ→∞ r 2 (1 + σ 2 )5/2 −δ/r (1 + σ ) - δ/r σ 2 dσ 2 1 = − + K sin φ n cos φ + Kb 2 3/2 2 r −δ/r (1 + σ ) r 2 2δ + − sin φ − K cos2 φ + K ln . (5.115) b+O r r δ

V 1 = lim

When the induction V 2 by the far part of the line |s| > δ is evaluated, one can neglect the core size and set r = 0. The logarithmic singularity in (5.110) at s → 0 is removed by subtracting from the integrand the term 1 −1 f (|s|)b, where f (s) is any suitable function with correct asymptotic 2 K|s| behavior, f (0) = 1. The remaining convergent integral expressing the contribution of the far part of C is denoted as −V s . This yields |s|−1 f (|s|) ds. (5.116) V 2 = V s + Kb |s|>δ

The limit of the last integral integral at δ → 0 can be expressed as λ |s|−1 f (|s|) ds = Kb ln , Kb lim δ→0 |s|>δ δ

(5.117)

where λ is a constant dependent on the choice of the compensating function and the geometry of the vortex. A simple example is a ring vortex with radius L. The normal vector is directed against the radial coordinate r. If the direction of the tangent l (deﬁning the sign of the circulation) is chosen as anticlockwise, the binormal is directed upwards along the axial coordinate z. The arc length is proportional to the angular coordinate, ds = Ldφ, and the curvature is L−1 . The nonlocal integral (5.116) yields in this case a purely binormal induction; it is evaluated most easily using the Cartesian components X = L{1−cos φ, − sin φ, 0}, l = {− sin φ, cos φ, 0}. Using this in (5.110) yields π dφ 1 1 4L b lim b, (5.118) = ln V2= 2L δ→0 δ/L sin φ2 L δ which ﬁts the form (5.116), (5.117) with λ = 4L, V s = 0. After the near and far integrals (5.115) and (5.116) are combined, the auxiliary parameter δ falls out: 2 V = V s + − cos φ + K sin φ cos φ n r 2λ 2 − cos2 φ b. + − sin φ + K ln (5.119) r r

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Using this expression, the phase ﬁeld in the vicinity of the line vortex can be reconstituted in a form similar to (4.119). One should also extract from the resulting expression the phase gradient 12 V · x removed by the Galilean transformation (5.74). The result is 1 2λ + (V s − V ) · x . (5.120) θ =φ+ Kr cos φ ln 2 r The coordinates should be rescaled to the inner units before the solvability condition is evaluated. The entire expression is scale-invariant, except the argument of the logarithm, where one should set r = r0 to obtain the expression valid on the matching circle: 1 2λ + (V s − V ) · x . (5.121) θ = φ + θ1 , θ1 = Kr0 cos φ ln 2 r0 Using u1 = iθ1 together with the asymptotic expression (4.120), the solvability condition (4.116) is evaluated as in Sect. 4.4.4. Replacing r cos φ by b · x, the resulting mobility relation is expressed in the vector form v = v s + κ b ln

2λ √ . a0 e

(5.122)

The small parameter is retained in the argument of the logarithm. In the particular case of a ring vortex, the propagation velocity is evaluated as v=

λ0 L 1 b ln ; L

λ0 =

8 √ ≈ 4.31. a0 e

(5.123)

This is an exceptional case when no normal motion is induced nonlocally, and the ring steadily propagates along its axis. Take note that the dependence of velocity on curvature κ = L−1 turns out to be weakly nonlinear. A somewhat unsettling presence of a small parameter is simply explained: the ring radius should be measured in short scale (core) units to get a correct answer. Generally, the non-locally induced velocity contained in the vector v s would have both normal and binormal components. The binormal component is formally larger, since it is multiplied by the logarithm of a large number. Neglecting v s altogether leaves only locally induced binormal motion. This localized induction approximation (LIA) has been extensively studied due to its simplicity and mathematical elegance (for review see Ricca, 1996). The small parameter enters, however, only logarithmically, and therefore neglecting the nonlocal terms makes a very poor approximation. Even more serious objection to LIA is its structural instability. Binormal motion conserves the length, while normal motion due to accumulated action of even weak nonlocal induction causes the vortex ﬁlament to shrink or stretch at diﬀerent locations, and typically leads to highly convoluted or even singular shapes.

5.5 Line Vortices and Scroll Waves

343

5.5.3 Dissipative Nonradiative Line Vortex In a dissipative nonradiative case ν = η = 0 when the asymptotic wavenumber vanishes, the motion of a line vortex is expected to interpolate, as in 2D (Sect. 5.4.1), between dissipative motion described by the RGL equation and conservative motion described by the NLS equation. The 3D and 2D equations in comoving frame, (5.107) and (5.82), have identical structure, diﬀering only by the added curvature-dependent term, which can be absorbed by replacing the real part of the advective term V → V − Kn. One might be tempted therefore to use the results of Sect. 5.4.1 straightforwardly, and to adopt as the local mobility relation (5.84) with the matching replacement in r.h.s. v → v − κn. Taking note that the planar rotation operator J is replaced in 3D by the vector product l×, we rewrite (5.84) as k−

1 v0 η v = l × (v − κn) ln . 2 2 |v − κn|

(5.124)

An impediment is that (5.84) has been obtained by matching with the solution of the far ﬁeld equation (4.91) obtained under the assumption of stationary motion. The nonlinear logarithmic correction in (5.84) is the result of this matching. Generally, the argument of the logarithm obtained by matching with a non-stationary solution of the dissipative far ﬁeld equation would be history-dependent, in the same way as the argument of the logarithm in (5.122) depends on the global shape of the line vortex. The formal analogy between the terms containing the velocity and curvature ends at this point, since, on the one hand, motion with a constant curvature is not a reasonable assumption, and, on the other hand, making the local mobility relation dependent on the history of motion is not practical. The remaining alternative is to treat the constant in the argument of the logarithm as an adjustable parameter, keeping in mind that it must be an O(−1 ) quantity. An additional coup de force is to allow for diﬀerent logarithmic factors in the velocity and curvature terms, and identifying the latter with the respective factor in (5.122). Then (5.124) is modiﬁed to a linear mobility relation describing dissipative motion under combined action of the local phase gradient k and the vortex line curvature: λ0 λ1 − κb ln . (5.125) 2k = ηv + l × v ln One can separate here the non-dissipative part v 0 = 2k + κΛ0 b,

Λ0 = ln

λ0 ,

(5.126)

which has the same structure as (5.122) and describes the combined action of local curvature and “advection” by the phase gradient induced by nonlocal action of the line vortex. The parameter λ0 remains indeﬁnite, but so is also the parameter λ in (5.122), as long as the global shape of the vortex is not

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deﬁned. The mobility relation (5.125) can be further resolved with respect to the dissipative part of the vortex drift velocity, v − v 0 : v=

ηv 0 − Λ l × v 0 , η 2 + Λ2

Λ = ln

λ1 > 0.

(5.127)

An equivalent mobility relation was postulated by Schwarz (1985) for the purpose of numerical computation of vortex motion in superﬂuids taking into account dissipation due to friction with a normal ﬂuid component. The formula gives correct limiting behavior (up to logarithmic corrections) in both weakly and strongly dissipative limits (η 1 or η 1). Applying (5.127), for example, to a ring vortex with v 0 directed along the binormal b, as deﬁned by (5.123), we see that dissipation adds a velocity component directed along the normal n = −l × b. As the ring velocity acquires a normal component, the ring shrinks as it propagates, and eventually collapses. In the gradient case, the simple result from Sect. 5.5.1 remains valid, since the total phase gradient k induced by an undistorted ring is zero. 5.5.4 Instability of Line Vortices Line vortices in the NLS equation are prone to instabilities and tend to develop into convoluted mazes, which repeatedly rearrange through reconnection events. Stability analysis taking proper account of nonlocal induction can be

Fig. 5.31. Snapshots of a line vortex showing the initial helix and hairpin structures near the peak of the maximum stretching rate (Klein and Majda, 1991; reproduced with permission from Elsevier Science)

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carried out for simplest shapes, such as a straight line, a ring, and a helix (for review, see Pismen, 1999). The former two turn out to be unstable to perturbations at short wavelengths only, i.e. under conditions when the applicability of the multiscale analysis lying at the basis of analytical theory becomes questionable. On the other hand, a helix, which is always unstable even in the LIA limit, can be viewed as a ﬁnite-amplitude wave on a straightline vortex. This suggests that perturbations with small but ﬁnite amplitude may cause destabilization also on longer wavelengths. The nonlinear development of these instabilities seen in Fig. 5.31, leading to vortex stretching and hairpin formation, can be followed, short of direct simulation, with the help of a nonlocal ﬁlament equation derived by Klein and Majda (1991). This instability persists in the weakly dissipative case, where it can be viewed as a 3D extension of the core acceleration instability of a point vortex (Sect. 5.3.4). Simulations by Aranson and Bishop (1997) and Aranson et al. (1998) in the unstable regime revealed a spectacular eﬀect of this instability on a scroll vortex. A series of snapshots in Fig. 5.32 shows how a slightly perturbed scroll vortex breaks down into a turbulent maze. Scroll vortices, as topological singularities of the phase ﬁeld, play an important role in organizing the overall structure of the turbulent ﬁeld, and may be called, in the ﬂuid mechanic’s idiom, “sinews of turbulence”. Nevertheless, the local structure of individual vortices crowded in a turbulent maze becomes so distorted that vortex dynamics ceases to be a useful computational concept under these conditions, and simulation of the full CGL equation remains the only way to reproduce structures of this kind. Dissipation stabilizes a straight-line vortex, which is the only stationary shape unaﬀected by curvature-driven normal motion. This can be checked with the help of the mobility relation (5.127) after presenting a perturbed vortex ﬁlament as (5.128) X = {eλt+ikz ξ, z},

Fig. 5.32. Instability of a straight-line vortex ﬁlament (η = 50, ν = −0.03). Isosurfaces of the modulus ρ = 0.1 are shown. (Aranson and Bishop, 1997; reproduced with permission. Copyright by the American Physical Society)

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where the last component indicates the unperturbed position along the z axis, and ξ is an inﬁnitesimally small 2D vector in the normal plane. Computing the vectors forming the Frenet trihedron and curvature according to Sect. 5.5.1 yields, to the leading order in ξ, l = {ikeλt+ikz ξ, 1}, κ = −k 2 eλr t |ξ|,

n = {ei(λi t+kz) ξ/|ξ|, 0}, b = {ei(λi t+kz) J ξ/|ξ|, 0},

(5.129)

where J is the operator of rotation by π/2 in the plane normal to the z axis; separating the real and imaginary parts of the eigenvalue λ = λr + iλi is necessary for computation of absolute values. The action of the phase gradient k in (5.127) is negligible for inﬁnitesimally small perturbations, and the stability analysis reduces after plugging (5.129) and (5.126) in (5.127) to computation of the eigenvalues of the matrix + k 2 Λ0 Λ −η M =− 2 . (5.130) η + Λ2 η Λ Evidently, the vortex is stable, as both Λ and Λ0 are positive. Stability can be, however, impaired in the presence of a phase twist when the phase ﬁeld of the unperturbed ﬁlament changes linearly along the z-axis. In the nonradiative case, the unperturbed phase ﬁeld is θ = φ − ωt + Qz. Including the phase gradient k = {0, 0, Q} in (5.126) modiﬁes (5.130) to (a)

(b)

Fig. 5.33. (a): A restabilized helical vortex (η = 1, ν = 0, γ = 0.098). (b): A doubly periodic “superhelix” (η = 1.7, ν = −0.5, γ = 0.27). Isosurfaces of the modulus ρ = 0.6 shaded by phase ﬁeld are shown (Rousseau et al., 1998; reproduced with permission. Copyright by the American Physical Society)

5.6 Resonant Oscillatory Forcing

M =−

k 2 Λ0 + Λ2

η2

Λ −H H Λ

+ ,

H =η−

2iQΛ . kΛ0

347

(5.131)

The eigenvalues of this matrix are k 2 Λ0 λ± = − 2 η + Λ2

2QΛ Λ ± iη ± . kΛ0

(5.132)

The straight-line vortex turns out to be always unstable to perturbations with a suﬃciently long wavelength (which, of course, can be limited by the total length of the ﬁlament anchored at bounding walls in a ﬁnite system). The growth rate of the perturbations is maximal at k = Q/Λ0 . Numerical simulations (Rousseau et al., 1998) show that the ﬁlament can be restabilized as a ﬁnite-amplitude helical structure seen in Fig. 5.33a. A secondary bifurcation leading to a doubly periodic “superhelix” (Fig. 5.33b) has been also detected in these simulations.

5.6 Resonant Oscillatory Forcing 5.6.1 Broken Phase Symmetry The symmetry of the CGL equation to phase rotations is broken when the system is parametrically forced at a frequency ωc resonant with the basic frequency at the Hopf bifurcation ω0 . For an integer ratio ωc /ω0 = n, the amplitude equation amending (5.2) can be written by adding the forcing term possessing the required symmetry: ut = (1 + iη)∇2 u + (µ + iω)u − (1 + iν)|u|2 u + γun−1 ,

(5.133)

where γ is the forcing amplitude and 2 ω is weak eﬀective detuning, due to both parametric deviations from the Hopf bifurcation point and weak mismatch between ωc /n and ω0 . We retain here the linear growth coeﬃcient µ, which can be normalized to ±1, respectively, in the supercritical and subcritical region, since, in the presence of forcing, nontrivial solutions exist also for µ < 0. Recalling the basic u = O() scaling of (5.2), the orders of magnitude in (5.133) match in this case at a forcing amplitude γ = O(3−n ). The polar form of (5.133) useful for computations is ρt = ∇2 ρ + (µ − |∇θ|2 − ρ2 )ρ + γρn−1 cos nθ − η(ρ∇2 θ + 2∇ρ · ∇θ), (5.134) θt = ∇2 θ +2ρ−1 ∇ρ·∇θ +η(ρ−1 ∇2 ρ−|∇θ|2 )+ω −γρn−2 sin nθ −νρ2 . (5.135) The forcing term in (5.133) reduces the invariance to arbitrary global phase shifts to a discrete symmetry u → eiπm u, m = 1, . . . , n − 1. For n > 1, this necessarily causes phase multiplicity of homogeneous solutions to (5.133) and changes the character of defects: instead of vortices, one can observed fronts separating alternative phase states.

348

5 Amplitude Equations for Waves

Fig. 5.34. Diﬀerent frequency-locked regimes of light-sensitive BZ reaction under periodic optical forcing (Petrov et al., 1997, reproduced with permission). The axis above shows the ratio of the forcing to basic frequency. the natural frequency of the system. Patterns are shown in pairs, one above the other, at times separated by the forcing period 2π/ωc , except for the 1:1 resonance where the interval is π/ωc

Various patterns at diﬀerent forcing frequencies, which can be modeled by (5.133), were observed both in experiments and simulations (Petrov et al., 1997; Lin et al., 2004). Some typical patterns are shown in Fig. 5.34. One can see that behavior changes dramatically when the frequency ratio is changed. Forcing at the same frequency (n = 1) is least interesting. At suﬃciently large O(2 ) forcing intensities, it just ﬁxes a homogeneous state oscillating in phase with the forcing. Finite minimal intensity is, however, required for synchronization under conditions when the unforced system is unstable, i.e., at νη < −1 (Chat´e et al., 1999). For n = 2, two homogeneous phase states exist, separated by a front. Under certain conditions, the front destabilizes, giving rise to a chaotic labyrinthine pattern. For n = 3, there are three phase states and three kinds of fronts, which can rotate around their common point, such as one seen in Fig. 5.34. Resonances at fractional ratios require frequencies lower than the basic underlying frequency ω0 of the CGL equation. In this case, no homogeneous states exist; thus, typical dynamics observed for 3:2 resonance is chaotic “bubbling” with the temporal spectrum at each point having peaks at ω0 /2. 5.6.2 Forcing at Double Frequency The strongest nontrivial resonance occurs at the frequency ratio 2:1 (Coullet and Emilsson, 1992; Yochelis et al., 2004). For n = 2, the residual symmetry is inversion, u → −u, and the only possible singularity is a front separating

5.6 Resonant Oscillatory Forcing

349

3.5 3 2.5 Γ

2 1.5 1 0.5 0 -3

-2

-1

0 Ω

1

2

3

Fig. 5.35. The resonance tongue in the parametric plane (ω, γ) for µ = ν = 1 bounded by the solid lines. The dashed line is the Hopf bifurcation locus. Dots mark the crossover between the two existence limits (5.137) and the double zero point. The gray line is the limit of the resonance domain for µ = −1, ν = 1

the two alternative states. The formal expressions deﬁning nontrivial homogeneous stationary solutions of (5.134), (5.135) are ω − νρ2± µ + ων ± (1 + ν 2 )γ 2 − (ω − µν)2 2 . (5.136) , sin 2θ± = ρ± = 2 1+ν γ To ensure existence, the expression for ρ2± should be real and positive. This gives the bounds9 |ω − µν| γ > γc = √ , 1 + ν2

γ > γ0 =

µ2 + ω 2 .

(5.137)

Both limits meet tangentially at ω = ωc = −µ/ν; the ﬁrst limit is relevant at ω > ωc and the second, at ω < ωc . Together, they form an Arnold resonance tongue (Fig. 5.35) where the system responds exactly at half the forcing frequency, or the basic frequency at the Hopf bifurcation ω0 . The tip of the resonance tongue descends to γ = 0 in the supercritical region µ > 0, but rounds up at a ﬁnite value of γ when µ is negative (the gray line in Fig. 5.35). With increasing γ, a pair of solutions with moduli ρ± appear as a result of a saddle-node bifurcation at γ = γc , ω > ωc . The upper solution ρ = ρ+ is stable, and the lower one is unstable. At ω < ωc , a single stable branch emerges at γc = γ0 . Each solution exists in four phase states shifted by π/2. On the stable branch, these phase states are alternately stable and unstable. Some solution branches are shown in Fig. 5.36. A secondary Hopf bifurcation leading to quasiperiodic motion may occur on the stable solution branch. Its locus in the parametric plane ω, γ is shown by the dashed line in Fig. 5.35. The 9

To be deﬁnite, we assume ω > 0.

350

5 Amplitude Equations for Waves 2.5 2

2

1

0

Ρ

1.5 1 1 2

0.5

0.5

1

1.5

2 Γ

2.5

3

3.5

4

Fig. 5.36. Dependence of the modulus ρ on the forcing amplitude γ for µ = ν = 1. The stable and unstable branches are shown, respectively, by the black and gray lines. The numbers at the curves show the values of ω. The line ω = 1 is also the locus of the saddle-node bifurcation

two periods are of diﬀerent orders of magnitude, since the periodic change of the amplitude and phase takes place on the slow time scale of (5.133) extended by a factor of O(−2 ) relative to the time scale of the underlying oscillations. 5.6.3 Ising and Bloch Fronts If two alternative stable phase states are attained at diﬀerent locations, a front is formed at the boundary of phase domains. The phase diﬀers in the two domains by π, and can change across the front in two distinct ways. The ﬁrst possibility is that it changes abruptly at a point where the modulus ρ vanishes. The alternative is that the phase rotates continuously, while the modulus remains everywhere ﬁnite (see Fig. 5.37). These two kinds of the fronts are called, respectively, Ising and Bloch fronts by analogy with fronts between domains with diﬀerent magnetization in solids (Coullet et al., 1990). The structure of both Ising and Bloch fronts can be obtained analytically in a simpliﬁed case when all coeﬃcients in (5.133) are real, i.e., ω = ν = η = 0. In this case, the system has a gradient structure, so that (5.133) is derivable according to the dissipative dynamics principle (4.8) from an energy functional F = L dx with the Lagrangian L = ∇u · ∇u + 12 (µ − |u|2 )2 − 12 γu2 .

(5.138)

The fronts separating the two symmetric phase states with identical energy must be stationary. The Ising front is obtained in a straightforward way by assuming u to be real and solving the stationary 1D equation uxx + u(µ + γ − u2 ) = 0,

(5.139)

5.6 Resonant Oscillatory Forcing

351

1

ΘΠ,Ρ Μ Γ

0.8 Ρ 0.6 0.4 0.2 Θ

Θ

0 -4

-2

0 x

2

4

Fig. 5.37. Normalized amplitude and phase proﬁles of the two symmetric Bloch fronts in the gradient case ω = ν = η = 0 for γ = 1/6 (solid lines). The dashed line is the amplitude proﬁle of the Ising front at the bifurcation point γ = µ/3

which reduces by rescaling to the standard cubic reaction-diﬀusion equation. The solution is [cf. (2.24)] ) ( √ µ+γ . (5.140) u(x) = µ + γ tanh x 2 The solution for the Bloch front is more elaborate and can be best obtained by separating the real and imaginary parts of u (Coullet et al., 1990): √ u(x) = µ + γ tanh x 2γ ± i µ − 3γ sech x 2γ . (5.141) The two Bloch fronts with the opposite sense of phase rotation break the inversion symmetry of (5.133). Their existence region is bounded by the inequality γ < µ/3. The Ising front loses stability when γ decreases past the Ising–Bloch (parity breaking) bifurcation point γ = µ/3. In the non-gradient case the front solution can be, generally, obtained only numerically, but perturbation analysis can be carried out when the imaginary parts of all coeﬃcients are small (Coullet et al., 1990). Because of the symmetry, the Ising front remains stationary, but the Bloch fronts become mobile. Scaling the velocity c as well as the imaginary parts of all coeﬃcients with a dummy small parameter and expanding in this parameter, we can present the ﬁrst-order equation in a general form (4.113) with a linear operator obtained by linearizing (5.139) and an inhomogeneity dependent on the front proﬁle for the gradient case u0 deﬁned by (5.141): Ψ (x) = cu0 (x) + i(ω − ν|u0 |2 + ηd2x )u0 .

(5.142)

Because of the translational symmetry of the front, u0 (x) is a Goldstone mode, and the inhomogeneity should satisfy the solvability condition

352

5 Amplitude Equations for Waves (a)

(b)

Fig. 5.38. Evolution starting from random initial conditions in the parametric regions where either Ising (a) or Bloch (b) fronts are stable (Yochelis et al., 2004, reproduced with permission from Elsevier Science)

∞

Re −∞

u0 (x)Ψ (x)dx = 0.

This gives the value of the propagation speed 3π (µ − 3γ)(µ + γ) √ c=± [µν − ω + (η − ν)γ]. 2 2γ(3µ − 3γ)

(5.143)

(5.144)

Since the velocity can have either sign, either phase state can advance into an alternative phase domain. This generates phenomena similar to those discussed in connection to the non-equilibrium Ising–Bloch (IB) bifurcation in reaction-diﬀusion systems with separated scales (Sect. 3.3.1 and 3.3.4). Propagating pulses can be constructed in 1D (Elphick et al., 1997); in 2D they

5.6 Resonant Oscillatory Forcing

353

(a)

(b)

Fig. 5.39. Development of a labyrinthine pattern within (a) and outside (b) the resonance tongue, respectively, through transverse instability of the Ising front and stripe-by-stripe growth (Yochelis et al., 2004, reproduced with permission from Elsevier Science)

turn into spiral waves composed of pairs of counter-propagating fronts, which are similar to those discussed in Sect. 3.6 rather than to spiral waves with a phase singularity at the core characteristic to the unforced CGL equation. The Ising–Bloch bifurcation causes in a non-gradient case a qualitative change of dynamics. When the Ising front is stable, evolution starting from random initial conditions leads, after formation of alternative phase domains, through roughening to a single straight-line front, as seen in Fig. 5.38a. Beyond the IB bifurcation, when counter-propagating Bloch fronts are stable, the result of evolution is a spiral wave pattern, as seen in Fig. 5.38b.

354

5 Amplitude Equations for Waves

The Ising front becomes transversely unstable at large detuning close to the boundary of the resonance tongue (Skryabin et al., 2001; Yochelis et al., 2004). This leads to the development of a labyrinthine stationary patterns,10 such as shown in Fig. 5.39a. It turns out that these resonant labyrinthine patterns persist also outside the resonance tongue where Ising front solutions no longer exist (Yochelis et al., 2004). The asymptotic pattern is very similar to that obtained inside the resonance tongue, but the formation mechanism is diﬀerent. Initial nuclei expand through successive stripe-by-stripe growth into the surrounding domain of unlocked homogeneous oscillations, as seen in Fig. 5.39b. This phenomenon is explained by Turing instability the basic homogeneous state, which may lead to preferential growth of perturbations at a ﬁnite wavenumber. The dispersion relation obtained from (5.133) is (5.145) λ(k) = µ − k 2 ± γ 2 − (ω − ηk2 )2 . The positive branch exhibits the fastest growth at ηkc2 = ω − γ/ 1 + η 2 , and the Turing bifurcation λ(kc ) = 0 occurs at k02 =

µ + ηω , 1 + η2

ω − µη γ= . 1 + η2

(5.146)

The latter value may be lower than the resonance limit (5.137). A rich variety of dynamic behavior can found in the vicinity of the degenerate Hopf–Turing bifurcation point µ = 0, γ = ω/ 1 + η 2 where the homogeneous Hopf bifurcation occurs simultaneously with the Turing bifurcation at the wavenumber k02 = ηω/(1 + η 2 ). 5.6.4 Higher Resonances For the 3:1 resonance (n = 3) the three homogeneous solutions of (5.134), (5.135) are ν 2µ + γ ± γ 4 + 4µγ 2 − 4µ2 ν 2 2 , (5.147) sin 3θ = − ρ± , ρ± = γ 2(1 + ν 2 ) where we set for simplicity ω = 0. The solutions exist at

γ 2 > γc2 = 2µ 1 + ν2 − 1 .

(5.148)

For ν = 0, the solutions appear at a ﬁnite forcing intensity. Of the three pairs of solutions that appear as a result of a saddle-node bifurcation at γ = γc , the triplet with ρ = ρ− is always unstable. The other triplet √ is stable in monotonous mode, but becomes oscillatory unstable for ν > 3 in a certain range of γ (Gallego et al., 2001). 10

Taking into account the underlying oscillations, these patterns are actually standing waves.

5.6 Resonant Oscillatory Forcing

355

Although all three phase states are equivalent, the front separating them is, generally, nonstationary. The front solution cannot be obtained analytically in a general case, but at small forcing intensities can be constructed in the phase ﬁeld approximation (Gallego et al., 2001). To make the phase approximation consistent, we scale, as in Sect. 5.1.3, ∇ = O() and, to match the orders of magnitude, γ = O(2 ). In view of (5.148), we have to require then also √ ν = O(2 ) to make nontrivial phase-locked states possible above γc ≈ ν µ. Then the modulus is slaved to the phase and is expressed by solving (5.134) as ρ2 = µ + O(2 ). Using this in (5.134) and retaining O(2 ) terms only yields the phase equation on a slow O(−2 ) time scale: θt = ∇2 θ − η|∇θ|2 − νµ2 − γµ sin 3θ.

(5.149)

A propagating front solution to this equation satisﬁes cθx + θxx − ηθx2 − νµ2 − γµ sin 3θ,

(5.150)

where c is the front velocity and θ(±∞) are two distinct phase states: 1 νµ + 2πj . (5.151) θj = − arcsin 3 γ The phase changes across the front by 23 π. It is clear that the sign of c switches when the two states interchange; thus, one can assume here θx > 0. By symmetry, all such fronts are identical. Multiplying (5.150) by θx and integrating over the inﬁnite line yields the expression for the propagation speed: ∞ ∞ 2 θx3 dx + πνµ2 θx2 dx . (5.152) c= η 3 −∞ −∞ If η and ν have the same sign, this ﬁxes the respective sign of c. When the signs of η and ν are opposite, c may vanish “accidentally” for a certain combination 2 1 0 −1 −2 −2

−1

0

1

2

Fig. 5.40. Modulus (left) and phase (center ) of the complex amplitude u, and the corresponding phase portrait in the complex plane taken at a ﬁxed location (right). Parameters: µ =, 1 ν = −0.2, η = 2, γ = 0.25. (Gallego et al., 2001, reproduced with permission. Copyright by the American Physical Society)

356

5 Amplitude Equations for Waves

of parameters. Thus, we see that, as the phase changes monotonically, each front in a circular sequence (deﬁned modulo 3) advances into its rival. In the presence of a phase singularity where the three fronts meet, this triplet forms a kind of a rotating spiral, such as seen in the right panel of Fig. 5.34 or in Fig. 5.40. It diﬀers from the spiral wave in the unforced CGL equation in uneven pace of the phase change: the phase remains almost constant within the alternative phase domains and changes abruptly at the fronts. Accordingly, the phase measured at any particular point remains for the most time close to one of the three values θj in (5.151) switching between these values as a front passes, as seen in the right panel of Fig. 5.40. For the 4:1 resonance (n = 4), there are two distinct kinds of fronts separating the four homogeneous phase states, with the phase changing across the front by either π or π/2. The π-fronts are similar to the fronts in the system forced at double frequency, and can be of both Ising and Bloch type; the Ising front must be stationary due to symmetry. The π/2-fronts are, generally, mobile, and can form rotating spiral bands similar to those at 3:1 resonance. Elphick et al. (1999) studied the special case ω = ν = η = 0 when the system has gradient structure and its solutions minimize an energy functional with the Lagrangian similar to (5.138), as well as the inﬂuence of weak deviations from gradient structure. They obtained conditions of instability of a stationary π-front to splitting into a pair of traveling π/2-fronts. This provides a scenario of transition from stationary to moving structures additional to the Ising–Bloch bifurcation discussed in Sect. 5.6.2. This scenario can operate even in the gradient case, since a moving front between two equivalent states does not change the energy of the system. The splitting instability can be interpreted in terms of interaction between π/2-fronts: a π-front is stable when the interaction is attractive and unstable when it is repulsive. Forcing can come in many other varieties, including spatially as well as temporally variable inputs, and be in spatial, as well as temporal resonance with structures arising spontaneously in a nonlinear system . Studies of inﬂuence of external forcing usually come under the heading of synchronization, which is, in some sense, opposite to symmetry breaking, as it emphasizes imposing order by external control rather than natural development of spatiotemporal structures or chaos in a non-equilibrium system. External control may give predictable (bland even if useful) results if the imposed structure is faithfully reproduced, but the examples in this section show that external inputs, by enhancing complexity of the underlying system, can create a greater variety of outcomes and, as a result, reduce rather than promote order. This could be, perhaps, a good lesson for those trying to control “from above” such complex and opaque systems as civil society and economy.

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Index

adsorption, 11, 12 aligned coordinate frame, 94, 113, 124, 137, 166, 170, 212, 337 amplitude diﬀusivity, 83, 210, 237, 281, 283, 287, 318 amplitude equation, 19, 27, 37, 70, 75, 77, 210, 223 amplitude wall, 227, 236 Archimedean spiral, 176, 179 Arnold tongue, 39, 349 Arrhenius law, 12 asymptotic stability, 20 attraction basin, 20, 49 attractor, 20, 21, 39, 52 attractor, chaotic, 20, 54, 56–58 baker’s transformation, 52 Barkley’s model, 143 Belousov–Zhabotinsky (BZ) reaction, 6, 72 Bernoulli equation, 317 Bernoulli shift, 52 bifurcation manifold, 20, 27, 29, 32 bifurcation, degenerate, 21, 39, 42, 56, 354 bifurcation, drift pitchfork, 303 bifurcation, fold, 23, 30–32 bifurcation, Hopf, 23, 29, 35, 39, 40, 42, 47, 60, 61, 81, 200, 209, 287, 347 bifurcation, Ising–Bloch (IB), 161, 168, 172, 179, 303, 351, 353, 356 bifurcation, monotonic (at zero eigenvalue), 29, 40, 83, 209

bifurcation, pitchfork, 32, 61, 80, 90, 161 bifurcation, saddle-loop (sl), or homoclinic, 40, 47 bifurcation, saddle-node, 303 bifurcation, saddle-node (sn), 23, 30, 39, 47, 161, 188, 264, 265, 349 bifurcation, saddle-node of periodic orbits (snp), 26, 37, 40, 47 bifurcation, secondary, 25, 80, 349 bifurcation, Shilnikov, 54 bifurcation, sniper, or Andronov, 40, 51 bifurcation, subcritical, 32, 37, 42, 69, 76 bifurcation, symmetry-breaking, 25, 211, 223 bifurcation, symmetry-breaking Hopf (wave), 289 bifurcation, Takens–Bogdanov (TB), 38, 45 bifurcation, transcritical, 30, 34 bifurcation, Turing, 24, 66, 67, 354 Biot–Savart integral, 340 Bloch front, 350, 351, 353 Bose–Einstein condensate, 297, 316 Brillouin zone, 216 Brusselator model, 6, 10 Burgers equation, 16, 99, 230, 238, 292, 314, 339 Cahn–Hilliard equation, 112, 117, 225, 257 capacitance, 13, 14, 36, 125, 141, 182 catastrophe theory, 32

366

Index

center manifold, 40 chaos, 2, 39, 51 chaos, spatio-temporal, 2, 293 chemical potential, 111, 112 chemotaxis, 14 codimension, 20 coherent structure, 297, 301, 308 combustion, 6 comoving frame, 85, 97, 186, 239, 241, 290, 292, 296, 323 comoving gauge, 174 complex energy, 288 continuation method, 302 continuity equation, 317 core ﬁlament, 204, 206, 337 counting argument, 85, 298, 301, 308 critical ﬁnger, 191, 195, 197, 199 critical nucleus, 91 cubic reaction-diﬀusion equation, 16, 90 defect chaos, 304, 306, 307, 333 dendrites, 128, 133 depinning, 257, 260, 266 diﬀusivity matrix, 14, 23, 222 director, 234 disclination, 226, 234, 235, 267 dislocation, 225, 227, 229, 239, 275, 278 dislocation chain, 227, 269, 270 dislocation core, 227, 232, 244 dislocation, climbing, 238, 268 dislocation, gliding, 268 dispersion relation, 17, 18, 24, 105, 117, 133, 152, 153, 156, 158, 161, 165, 183, 209, 283, 290, 291, 294 dissipation integral, 46, 47, 49, 86, 89, 94, 97, 115, 241, 243 dissipation rate, 46, 92, 241 distortion energy, 84, 89, 110, 150, 233 distribution coeﬃcient, 134 domain boundary, 297, 299 domain wall, 226, 235, 236, 267, 268, 270, 281, 308 drift velocity, 244, 249 dual function, 240, 250 eikonal equation, 97, 98, 100, 116, 127, 128, 143, 152, 166, 169–172, 177, 182, 189, 190, 192, 193, 205 electroconvection, 239, 251

energy functional, 17, 83, 98, 110, 148, 150, 258, 350, 356 Euler equation, 317 excitability, 193, 195 excitation pulse, 177, 181–183, 185, 191, 197, 198 Faraday instability, 6 Fife scaling, 182, 189 ﬁngering, 121, 128, 134 FitzHugh–Nagumo (FN) model, 10, 43, 143 ﬁxed point, 9, 15, 30, 32, 40, 42 Floquet exponents, 26 Floquet multipliers, 26 Floquet–Bloch problem, 216 Fourier integral, 24, 105 fractal dimension, 20, 52, 53 Fredholm alternative, 29, 245 free energy, 3, 9, 84, 110, 123 Frenet dihedron, 173 Frenet trihedron, 337, 346 Frenet–Serret equations, 172, 337 frequency locking, 26, 348 friction coeﬃcient, 85 friction factor, 241 front interaction, 91, 119 Galilean invariance, 130, 316–318 Gibbs–Thomson relation, 98, 115, 117, 121, 126, 130, 133, 135 Ginzburg–Landau equation, complex (CGL), 17, 209, 287–289, 297, 302, 311, 318, 319, 321, 337 Ginzburg–Landau equation, complex (CGL) coupled, 295, 307 Ginzburg–Landau equation, complex quintic (QCGL), 288, 301 Ginzburg–Landau equation, real (RGL), 17, 211, 220, 232, 252, 317, 321 Goldstone mode, 69, 73, 94, 97, 115, 125, 161, 216, 223, 351 gradient structure, 30, 70, 84, 148, 212, 219, 222, 225, 271, 273, 350, 356 granular media, 7 Gray–Scott model, 10 Gross–Pitaevskii (GP) equation, 316 group velocity, 106, 117, 289, 294, 308, 312, 327

Index Hamiltonian, 12, 13, 316 healing length, 239 heart ﬁbrillation, 203 Hele–Shaw limit, 121 homoclon, 302, 307 homotopy theory, 234 Hopf–Cole transformation, 99, 230, 314, 326 hypermeandering, 202 information storage, 281 instability, absolute, 25, 291, 312, 333 instability, acceleration, 318, 345 instability, bending, 159, 206 instability, Benjamin–Feir (BF), or self-focusing, 289–292, 296, 312 instability, breathing, 163, 165 instability, convective, 25, 291, 309 instability, Eckhaus, 157, 184, 202, 215, 217, 219, 221, 222, 289, 291, 293, 296, 300, 312, 315 instability, meandering, 200–202, 206, 318, 319 instability, sideband, 216, 275 instability, splitting, 159, 303, 356 instability, sproing, 207 instability, symmetry-breaking, 23, 128, 152 instability, transverse, 130, 137, 153, 167 instability, traveling, 161, 166 instability, varicose, 153 instability, zigzag, 153, 154, 156, 157, 166, 171, 172, 217, 219, 222, 238, 267 interaction coeﬃcients, 66, 68, 77 intermittency, 66, 293 Ising front, 350, 351, 353, 354, 356 isometric gauge, 174, 175 Jacobi matrix, 21, 67 Jordan form, 38 Kardar–Parisi–Zhang (KPZ) equation, 100, 293 Karhunen–Loewe method, 18 kink/antikink pair, 92 Kuramoto–Sivashinsky (KS) equation, 6, 16, 139, 172, 293

367

leading band, 22 leading edge, 25 leading eigenvalue, 28 leading mode, 22, 26 Lennard–Jones potential, 111 Lifshitz point, 289 line vortex, 204, 337–340, 343 liquid crystal, 15, 211, 234 localized induction approximation (LIA), 342 logistic map, 58, 60, 64 Lorenz model, 6, 18, 61 Lyapounov functional, 3 Madelung transform, 317 marginal stability, 22, 23, 26, 106, 291, 294 mass action, 10, 11 Maxwell construction, 89, 97, 99, 110, 125, 142, 171, 174, 182, 266, 280, 281, 284 mean ﬁeld, 5 migration, 14, 148 mobility coeﬃcient, 98, 111 mobility relation, 244, 246, 248, 250, 253, 322, 343, 344 modulated amplitude waves (MAW), 302–305 monodromy matrix, 26 Mullins–Sekerka instability, 127, 129 multiscale expansion, 16, 27, 66 Navier–Stokes equation, 99 near-identity transformation, 27 Newell–Whitehead–Segel (NWS) equation, 211, 221, 224, 229, 235 Newton law, 12 nonadiabatic eﬀect, 267, 286 nonequilibrium structures, 6 nonlinear optics, 7, 15, 77, 316 nonlinear Schr¨ odinger (NLS) equation, 17, 316, 318, 319, 339 normal form, 10, 19, 27, 35, 45 Nozaki–Bekki hole, 299, 300, 302, 307 null-isocline, 43, 143, 184 orbit, chaotic, 20 orbit, heteroclinic, 74, 76, 85, 184, 261, 298

368

Index

orbit, homoclinic, 39, 40, 45, 47, 54, 62, 90, 215, 261, 298 orbit, periodic, 20, 25, 40, 42, 90 orbit, quasiperiodic, 20, 26, 215 order parameter, 5, 10 oscillations, chemical, 6 Ostwald ripening, 101, 121 Pad´e approximant, 232 parameter array, 13, 28 parametric space, 20, 28 parametrical forcing, 347 pattern map, 188 pattern selection, 66 pattern, dodecagonal, 73 pattern, hexagonal, 69, 74, 147, 271, 275, 279, 281, 283 pattern, labyrinthine, 154, 157, 168, 348, 354 pattern, multispiral, 203 pattern, regular, 66 pattern, skewed hexagonal, 273 pattern, striped, 68, 70, 144, 154, 214, 216, 218, 224, 225, 229, 234, 258, 266, 267, 274, 279, 281 patterns, chemical, 6, 77 patterns, convective, 6, 77, 218, 234, 270 patterns, disordered, 20 patterns, quasiperiodic, 20 patterns, stationary periodic, 20, 90, 94 Peach–K¨ ohler force, 92, 238, 241, 243 penta-hepta defect, 275, 278, 281 period doubling, 6, 26, 57, 58, 61, 200, 333, 335 phase chaos, 293, 304, 306 phase diﬀusion, 227, 232, 290 phase diﬀusivity, 171, 219, 220, 223, 289 phase dynamics, 218, 223, 273, 276, 289, 292, 293 phase ﬁeld model, 123, 129, 148 phase space, 9, 261 phase twist, 204, 346 phase wall, 227, 237 pinned structures, 258 pinning, 257, 260, 263, 266–268 plane wave, 289, 294, 312 planforms, 66, 77 Poincar´e map, 58, 63, 188, 262

Poincar´e section, 58, 63, 262 population dynamics, 11, 30, 103 potential, 10, 12, 14, 30, 45, 70, 71, 84, 85, 123, 148 Proctor–Sivashinsky equation, 16 propagating band, 162, 175 propagating pulse, 90 propagation speed, 84, 85, 87, 97, 104, 106, 108, 109, 118, 119, 129, 142, 184, 190, 191, 239, 254, 283, 294, 302, 309, 352 pulled front, 105, 108, 284, 286, 294 pushed front, 108, 284 quasicrystals, 72 R¨ ossler model, 56, 61, 64, 335 Rayleigh–B´enard convection, 18, 68 reactions, autocatalytic, 11 reactions, catalytic, 11 reactions, enzyme, 11 reactions, exothermic, 12 relaxation oscillations, 44 renormalization group, 61 resonance, 39, 66, 77, 348 rigidity, 84 ring vortex, 341 rotating polar frame, 189 rotating spiral, 175, 180, 189, 195, 356 saddle-focus, 54, 56, 57, 62, 186 scale separation, 18, 19, 26 Schl¨ ogl model, 10 scroll vortex, 345 scroll wave, 204, 205, 336 self-organization, 6 self-similarity, 61 Shilnikov snake, 56 shock, 100, 203, 300, 307, 308, 321, 326, 327, 329, 331, 339 singularity, butterﬂy, 32, 35 singularity, cusp, 30, 34, 35, 83, 210 singularity, swallowtail, 32 slaving principle, 19 slow mode, 18, 24, 25, 27, 35 Smale horseshoe map, 53, 54, 56–58 solidiﬁcation, 6, 15, 123, 129 solitary band, 144, 153 solitary disk, 145, 157, 165

Index soliton, 3, 258 soliton, dark, 215, 298 soliton, dissipative, 143 solvability condition, 29, 83, 97, 115, 125, 138, 139, 186, 194, 210, 211, 245, 246, 252, 253, 319, 339, 351 spectral band, 22, 24, 26 spinodal decomposition, 112 spiral domain pattern, 321, 329, 335 spiral pair, 323, 329, 335 spiral wave, 6, 175, 197, 310, 312, 321, 334, 353 stable manifold, 40, 261, 298, 302 state array, 9, 13, 28 states, absolutely stable, 9 states, homogeneous stationary (HSS), 15, 21, 83, 142 states, metastable, 9, 84, 260 states, stationary, 9, 20 states, unstable, 9 steepest descent, 105, 291 Stefan problem, 116, 126 Sturm–Liouville problem, 91 supercriticality ramp, 253, 319 superﬂuid, 297, 316, 344 superlattice, 77 superspiral, 202 surface tension, 89, 97, 110, 128, 130, 134 Swift–Hohenberg (SH) equation, 6, 16, 209, 224, 255, 257, 258 Swift–Hohenberg (SH) equation, modiﬁed, 257, 258, 272, 285 Swift–Hohenberg equation, complex (CSH), 289 symmetry, hidden, 298, 301 symmetry, inversion, 10, 16, 32, 34, 68, 73, 222, 234, 260, 261, 272, 348 symmetry, rotational, 225 symmetry, to phase shifts, 35, 211, 216 symmetry, translational, 16, 97, 213, 216, 218, 223, 273, 290, 351 synchronization, 348, 356

369

synchronization defect (SD), 334, 335 synergetics, 6 systems, closed, 3, 9 systems, conservative (Hamiltonian), 2, 13, 38, 45 systems, dissipative, 3, 46, 53, 58 systems, distributed, 1, 19, 21 systems, dynamical, 1, 9, 28, 186, 260, 298 systems, excitable, 44, 143, 177, 184 systems, gradient, 10, 17, 83, 217, 220 systems, integrable, 3 systems, open, 4, 10 systems, reaction-diﬀusion (RDS), 13, 14, 17, 21, 23, 209, 222, 287, 293 tip meandering, 180, 197, 202 topological charge, 226, 230, 310 topological defect, 225 transition, ﬁrst-order, 32 transition, second-order, 32 turbulence, 2, 6, 18 undercooling, 129 unstable manifold, 40, 261, 298, 299, 302 van der Waals ﬂuid, 17 vertical structure, 4, 15, 68 Volterra–Lotka system, 13 vortex, 227, 229, 232, 234, 310, 317, 320, 322 vortex core, 322 vortex glass, 252, 329, 333, 335 vortex liquid, 336 vortex pair, 322, 323 vortex turbulence, 202 wave pattern, 74, 76 wave train, 20, 90, 181–183, 185, 197, 202 wavelength selection, 254 wavenumber kink, 298, 300 Wiener–Rosenblueth (WR) model, 176

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