NONLINEAR WAVES
Edited by
LOKENATH DEBNATH Professor and Chairman, Department of Mathematics University of Central Florida, Orlando, Florida and Professor of Mathematics and Adjunct Professor of Physics East Carolina University, Greenville, North Carolina
CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE LONDON NEW YORK NEWROCHELLE MELBOURNE SYDNEY
CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, Delhi Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www. Cambridge. org Information on this title: www.cambridge.org/9780521254687 © Cambridge University Press 1983 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1983 This digitally printed version 2008 A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data LOKENATH DEBNATH NONLINEAR WAVES (Cambridge monographs on mechanics and applied mathematics) Includes index. 1. Nonlinear waves. 2. Fluid dynamics. 3. Plasma (Ionized gases) 4. Inverse scattering transform. 5. Evolution equations. I. Debnath, Lokenath. QA927.N665 1983 531'.1133 83-15102 ISBN 978-0-521-25468-7 hardback ISBN 978-0-521-09304-0 paperback
CONTENTS PREFACE
PART I NONLINEAR WAVES IN FLUIDS 1. Towards the Analytic Description of Overturning Waves MICHAEL S. LONGUET-HIGGINS
1
2.
The Korteweg-de Vries Equation and Related Problems in Water Wave Theory R. S. JOHNSON
25
3.
Solitary Waves in Slowly Varying Environments: Long Nonlinear Waves R. GRIMSHAW
44
4.
Nonlinear Waves in a Channel M.C. SHEN
69
5.
Soliton Behaviour in Models of Baroclinic Instability IRENE M. MOROZ and JOHN BRINDLEY
84
6.
Waves and Wave Groups in Deep Water PETER J. BRYANT
100
7.
Two-and Three-Wave Resonance ALEX D. D. CRAIK
116
PART II NONLINEAR WAVES IN PLASMAS 8.
Nonlinear Electromagnetic Waves in Flowing Plasma N.E. ANDREEV, V. P. SILIN and P. V. SILIN
133
9.
Superluminous Waves in Plasmas P.C. CLEMMOW
162
10.
Electrostatic Ion Cyclotron Waves and Ion Heating in a Magnetic Field H.OKUDA
177
11.
Solitons in Plasma Physics P.K. SHUKLA
197
12.
A Theory for the Propagation of Slowly Varying Nonlinear Waves in a Non-Uniform Plasma R. J. GRIBBEN
221
PART HI SOLITONS, INVERSE SCATTERING TRANSFORM, AND NONLINEAR WAVES IN PHYSICS 13.
On the Inverse Scattering Transform in Two Spatial and One Temporal Dimensions A. S. FOKAS
245
14.
Linear Evolution Equations Associated with Isospectral Evolutions of Differential Operators ANTONIO DEGASPERIS
268
15.
Inverse Scattering for the Matrix Schrbdinger Equation with Non-Hermitian Potential PETER SCHUUR
285
16.
A General nth Order Spectral Transform P. J. CAUDREY
298
17.
Recurrence Phenomena and the Number of Effective Degrees of Freedom in Nonlinear Wave Motions A. THYAGARAJA
308
18.
Action-Angle Variables in the Statistical Mechanics of the Sine-Gordon Field R. K. BULLOUGH, D.J. PILLING and J. TIMONEN
326
INDEX
356
PREFACE The last two decades have produced major advances in the mathematical theory of nonlinear wave phenomena and their applications.
In an effort to
acquaint researchers in applied mathematics, physics, and engineering and to stimulate further research, an NSF-CBMS regional research conference on Nonlinear Waves and Integrable Systems was convened at East Carolina University in June, 1982.
Many distinguished applied mathematicians and sci-
entists from all over the world participated in the conference, and provided a digest of recent developments, open questions, and unsolved problems in this rapidly growing and important field. As a follow-up project, this book has developed from manuscripts submitted by renowned applied mathematicians and scientists who have made important contributions to the subject of nonlinear waves.
This publication
brings together current developments in the theory and applications of nonlinear waves and solitons that are likely to determine fruitful directions for future advanced study and research. The book has been divided into three parts. Waves in Fluids, consists of seven chapters.
Part I, entitled Nonlinear
Nonlinear Waves in Plasmas are
the contents of Part II, which has five chapters.
Part III contains six
chapters on current results and extensions of the inverse scattering transform and of evolution equations.
Included also is recent progress on statis-
tical mechanics of the sine-Gordon field. The opening chapter, by M.S. Longuet-Higgins, is devoted to recent progress in the analytical representation of overturning waves.
Among the forms
suggested for the fluid flow are, for the tip of the jet, a rotating Dirichlet hyperbola, and, for the tube, a "/T-ellipse" or a parametric cubic. these have been expressed in a semi-Lagrangian form.
All
The semi-Lagrangian
form for the rotating hyperbola is derived by a new and simpler method, and certain integral invariants are obtained which have the dimensions of mass, angular momentum and energy.
The relation of these to the previously known
constants of integration is discussed, and directions for further generalizations are indicated.
Also, a new class of polynomial solutions of the semi-
Lagrangian boundary conditions is derived.
These, or their generalizations,
may be of use when combining different solutions so as to form a complete description of the overturning wave.
In Chapter 2, R.S. Johnson describes
how the classical problem of inviscid water waves is used as the vehicle for introducing various forms of the Korteweg-deVries (KdV) and nonlinear Schrodinger (NLS) equations.
The appropriate equations in one and two
PREFACE dimensions are given with some discussion on the effect of shear and variable depth.
It is shown that KdV and NLS equations match in a suitable limit
of parameter space, and the various KdV solutions-notably similarity-are themselves matched to corresponding near fields.
Some other equations based
on more complicated physics are mentioned together with a brief comment on two-dimensional problems with shear or variable depth.
In Chapter 3, R.
Grimshaw discusses canonical equations for the evolution of long nonlinear solitary waves in slowly varying environments. type and include the effects of dissipation.
These equations are of KdV
The slowly varying solitary
wave is constructed as an asymptotic solution of these equations by a multiscale perturbation expansion, and is shown to consist of a solitary wave with slowly varying amplitude and trailing shelf.
The specific case of a solitary
wave decaying due to dissipation is described in detail.
Chapter 4, by
M.C. Shen, is concerned with some approximate equations for the study of nonlinear water waves in a channel of variable cross section.
He gives a sys-
tem of shallow water equations for finite amplitude waves, and a KdV equation with variable coefficients for small amplitude waves. more study are mentioned in this chapter.
Some problems deserving
Chapter 5, by I.M. Moroz and J.
Brindley, is concerned with the derivation of a system of evolution equations for slowly varying amplitude of a baroclinic wave packet.
The self-induced
transparency, sine-Gordon and nonlinear Schrbdinger equations, all of which possess soliton solutions, each arise for different inviscid limits.
The pre-
sence of viscosity, however, alters the form of the evolution equations and changes
the character of the solutions from highly predictable soliton solu-
tions to unpredictable chaotic solutions.
When viscosity is weak, equations
related to the Lorenz attractor equations obtain, while for strong viscosity the Ginzburg-Landau equation obtains.
P.J. Bryant, in Chapter 6, discusses
specific wave geometries which occur in deep water and are calculated by a numerical method based on Fourier transforms.
Examples are presented of
permanent waves and wave groups of permanent envelope in two and three dimensions without restriction on wave height.
Although the method is ap-
plied here only to gravity waves in deep water, it may be generalized to further forms of nonlinear wave motion.
Chapter 7, by Alex Craik, deals with
linear, or direct, resonance of two waves, and weakly nonlinear three-wave resonance.
Special attention is given to non-conservative three-wave sys-
tems, for which the mathematical theory is least developed.
In addition,
subharmonic resonance and further complications involving quadratic interaction of more than three waves are discussed.
PREFACE In Chapter 8, N.E. Andreev, V.P. Silin, and P.V. Silin discuss various aspects of the stationary theory of the interaction of an electromagnetic field with moving plasmas, with special attention to the field self-restriction phenomena in supersonic plasma.
The authors also suggest a direction
for further research and study on the theory.
In Chapter 9, P.C. Clemmow
discusses finite-amplitude plane waves travelling with uniform speed through a cold homogeneous plasma in a Lorentz frame of reference.
This problem can
be reduced to solving a single nonlinear ordinary vector differential equation.
Periodic solutions of this equation are investigated.
It is found
that some new results for propagation in a direction perpendicular to the ambient magnetostatic field go some way towards elucidating the conditions under which various types of wave can exist.
H. Okuda presents the results
of analytic theory as well as of numerical simulations on electrostatic ion cyclotron (EIC) waves in Chapter 10.
In Chapter 11, P.K. Shukla presents
an evaluative review on theories of solitons in plasma physics along with a discussion on some open questions and unsolved problems.
Chapter 12, by
R.J. Gribben, is concerned with uniformly-valid perturbations of uniform, monochromatic nonlinear, periodic wave solutions of the Vlasov and Poisson equations in one space dimension in the absence of a magnetic field.
Also,
a theory for the propagation of slowly varying nonlinear waves in a non-uniform plasma is presented.
Appropriate basic uniform wave solutions are re-
viewed, some general consequences of the theory given, and current work described, including solutions obtained for particular cases, and directions in which further study might proceed. In Chapter 13, A.S. Fokas describes some recent results and developments on the extension of the inverse scattering transform to solve nonlinear evolution equations in one time and two space dimensions.
Based on the
SchrOdinger partial differential operator as a simple mathematical model, A. Degasperis studies linear evolution equations associated with isospectral evolutions of differential operators in Chapter 14.
He also discusses how
to solve the corresponding initial value problem using the spectral properties of the Schrbdinger operator.
Then the scattering operator expression is
divided in the case of a linear evolution equation associated with a pure many-soliton solution. results are pointed out.
Some natural extensions and generalizations of these In Chapter 15, Peter Schuur develops an inverse
scattering formalism for the NxN matrix SchrOdinger equation with arbitrary, in general non-Hermitian potential matrix, decaying sufficiently rapidly for | x | -v oo . A general nth order spectral transform and a technique for inverting this transform are developed by P.J. Caudrey in Chapter 16.
The
PREFACE use of the whole procedure is illustrated by the solution of a system of nonlinear Klein-Gordon equations.
In Chapter 17, A. Thyagaraja gives an
elaborate account of recurrence phenomena and the number of effective degrees of freedom in nonlinear wave motion.
The relationships between re-
currence phenomena and different motions of stability due to Lagrange, Poisson, and Lyapunov are described.
The chapter concludes with a brief
discussion of some unsolved problems relevant to applications.
The final
chapter, by R.K. Bullough, D.J. Pilling, and J. Timonen is devoted to the statistical mechanics of the sine-Gordon (s-G) field.
Functional integrals
for the classical and quantum partition functions Z for the s-G field <J>(x,t) are calculated in different ways including methods which exploit the complete integrability of the classical s-G and its canonical transformation to a Hamiltonian involving action variables alone. poses no problems.
The free Klein-Gordon field
But discrepant results for the s-G kinks and anti-kinks
are explained by the observation that the functional integrals for Z are defined best by discretization to a lattice of spacing a on finite support L. The s-G problem then becomes that of a sequence of problems involving a finite number of degrees of freedom; and for L -> °° and a-> 0 kinks and antikinks are dressed by coupled K-G modes.
These dressings are calculated in
different ways both quantally and in classical limit, and connections established with kinks and anti-kinks are largely resolved, but quantum WKB methods, for example, pose problems of their own. I am grateful to the authors for their cooperation and contributions, and hope that this monograph brings together all of the most important, recent developments in the mathematical theory and physical applications of nonlinear waves and solitons in fluids and plasmas, besides describing all major current research on the inverse scattering transform.
I want the reader to share in
the excitement of present day research in this rapidly growing subject and to become stimulated to explore nonlinear phenomena. I express my grateful thanks to Dr. Carroll A. Webber for his help in improving the readability of several papers.
I am thankful to my wife for
her constant encouragement during the preparation of the book.
In conclu-
sion, I wish to express my sincere thanks to the Cambridge University Press for publishing the monograph. LOKENATH DEBNATH
CHAPTER 1 TOWARDS THE ANALYTIC DESCRIPTION OF OVERTURNING WAVES MICHAEL S. LONGUET-HIGGINS Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge, England and Institute of Oceanographic Sciences, Wormley, Godalming, Surrey
1.
INTRODUCTION. Till recently, one notable hiatus in the theory of surface waves was
the absence of any satisfactory analysis to describe an overturning wave. In this category we include both the well-known "plunging breaker" and also any standing or partially reflected wave which produces a symmetric or an asymmetric jet, with particle velocities sometimes much exceeding the linear phase-speed. A first attempt to describe the jet from a two-dimensional standing wave was made by Longuet-Higgins (1972), who introduced the "Dirichlet hyperbola", a flow in which any cross-section of the free surface takes the form of a hyperbola with varying angle between the asymptotes.
Numer-
ical experiments by Mclver and Peregrine (1981) have shown this solution to fit their calculations quite well.
The solution was further analysed
in a second paper (Longuet-Higgins, 1976) where a limiting form, the "Dirichlet
parabola", was shown to be a member of a wider class of self-
similar flows in two and three dimensions.
Using a formalism introduced
by John (1953) for irrotational flows in two dimensions, the author also showed the Dirichlet parabola to be one of a more general class of selfsimilar flows having a time-dependent free surface. All the above flows were gravity-free, that is to say they did not involve g explicitly; they are essentially descriptions of a rapidly evolving flow seen in a frame of reference which itself is in free-fall. A useful advance came with the development of a numerical time-stepping technique for unsteady gravity waves by Longuet-Higgins and Cokelet (1976, 1978).
As later refined and modified by Vinje and Brevig (1981), Mclver
and Peregrine (1981) and others, this has
given accurate and reproducible
results for overturning waves, with which analytic expressions can be compared. A further advance on the analytic front came with the introduction
M.S. LONGUET-HIGGINS
t=O
Figure 1 (after Longuet-Higgins 1980b).
Example of the free surface in a
rotating hyperbolic flow wheniSf • 0.30 (see equation (6.15)). 0 is in a free-fall trajectory.
The origin
ANALYTIC DESCRIPTION OF OVERTURNING WAVES
3
by Longuet-Higgins (1980a) of a general technique for describing free-surface flows, that is flows satisfying the two boundary conditions p = 0 at a free surface.
and
Dp/Dt = 0
(1.1)
Particular attention was paid to the parametric repre-
sentation of the flow in a form X - X(tf,t)
,
z = Z(uS,t)
(1.2)
where both the complex coordinate z = x + iy and the velocity potential x are expressed as functions of the intermediate complex variable w and the time t.
This was a generalisation of the formalism of F. John (1953), in
which a) was, however, assumed to be Lagrangian at the free surface, though not elsewhere in the fluid.
For this reason John's formalism was called
"semi-Lagrangian". The more general formalism was put to immediate use in a second paper (Longuet-Higgins 1980b) in which the "Dirichlet hyperbola" of previous papers was generalised to include "rotating hyperbolic flow".
Besides the
time-variation of the asymptotes, the principal axes were allowed to rotate, as shown in Figure 1.
This solution, in which the velocity potential x
was given in closed form, allowed for the first time a convincing possible description of the later stages of a plunging jet. of the jet, however, was not included.
The initial evolution
In a third paper (Longuet-Higgins
1981a) the author made use of the more general (non-Johnian) formalism to derive a plausible analytic description of the development of the flow, up to about the instant when the free surface first becomes vertical.
This
description introduced the approximate potential X = j ig tf3 + Uto2 + 2Aw
(1.3)
where U is a constant, A is a linear function of the time and z = o>2
.
(1.4)
The first term on the right of (1.3) by itself represents Stokesfs 120° corner-flow.
The third term represents a finite-amplitude perturbation of
the Stokes flow.
The expression (1.3) gives a rather convincing represen-
tation of the initial development of the breaking wave (see Figure 2 ) . However, the task of matching this flow to the later stages, including the time-dependent jet, remains still to be accomplished. In another direction New (1981) found empirically that in some of his
M.S. LONGUET-HIGGINS
Figure 2 (from Longuet-Higgins 1981).
Initial development of the over-
turning flow as given by equations (1.3) and (1.4) when g - 1, U • (-1,0.5), and A(t) is chosen so as to minimise /(Dp/Dt)2ds on p = 0. is in uniform motion; the solution includes gravity.
The origin 0
ANALYTIC DESCRIPTION OF OVERTURNING WAVES
5
numerical calculations of breaking waves the forward face, or "tube", of the breaker was remarkably well fitted by part of the circumference of an ellipse, with axes in the ratio /3:1.
Whereupon Longuet-Higgins (1981b,
1982) pointed out that the free surface was even better fitted (see Figure 3) by the cubic curve: z = itw3 + 3t2a)2 - it3w - j t ^
(1.5)
which is a limiting case of one of the self-similar flows found previously (Longuet-Higgins 1976).
Moreover the flow (1.5) contains another surface
p = 0 which comes close to the rear surface of the wave, though the second boundary condition Dp/Dt = 0 is not satisfied on it.
Nevertheless there
was perhaps some possibility that by suitably perturbing the flow (1.5) and by matching it to a rotating hyperbolic flow near the tip of the jet a complete solution might be found.
Since (1.5) is expressed in semi-
Lagrangian form a next step would be to express the rotating hyperbolic flow in semi-Lagrangian form also. This has in fact been done in a very recent paper (Longuet-Higgins 1983) where the rotating hyperbolic flow is shown to be expressible in the form z = F(t)cosho) + G(t)sinha)
(1.6)
the functions F and G being related to the solutions of a Kiccati equation. The corresponding particle trajectories have also been computed. Meanwhile in still unpublished work New (1983) has succeeded in finding a flow, in semi-Lagrangian representation, which is outside his elliptical free surface, and he has shown that the velocity field resembles that in numerically calculated waves, over about half the circumference of the ellipse.
Unlike the cubic (1.5), the elliptical model cannot of course de-
scribe the velocity discontinuity which must occur when the jet meets the forward face of the wave.
Greenhow (1983) has made further progress in
deriving a semi-Lagrangian expression, polynomial in a), which for large values of t approximates the hyperbolic jet on the one hand and New's ellipse on the other.
His expression also provides a "rear face" to the wave,
but is still gravity-free. The purpose of the present paper is twofold:
first, to derive the
semi-Lagrangian representation for the rotating hyperbolic flow in an alternative, and perhaps simpler, way than in Longuet-Higgins (1983).
The
present method has the advantage that it brings to light naturally some
M.S.
LONGUET-HTGGINS
101
(b)
Figure 3 (after Longuet-Higgins 1982).
Profile of the surfaces p = 0 in
the cubic flow (1.4)(a) when t = 1.0; (b) when t = 0.5. both p and Dp/Dt vanish.
On II only p vanishes.
indicates a possible perturbation.
On the curves I,
The broken curve III
ANALYTIC DESCRIPTION OF OVERTURNING WAVES
7
integral invariants K, y, and v which in turn provide constraints on the functions F and G.
It is shown how K, y, and v are related to the constants
of integration found in earlier papers.
Moreover the method suggests some
possible generalizations. A second purpose is to give some exact polynomial solutions to one of the problems investigated by Greenhow (1983).
The same methods may, in
turn, be employed in other, more general, problems occuring in the same context. 2.
SEMI-LAGRANGIAN COORDINATES. In the semi-Lagrangian representation of irrotational, free-surface
flows in two dimensions, the coordinate z - x + iy is expressed as an analytic function of a complex parameter w and the time t: z • z(o),t)
(2.1)
such that on the free surface w is real (w = w*) and Lagrangian (Dw/Dt • 0) . The condition that the pressure be constant along this surface can then be expressed as ztt - g = i « u
(2.2)
where g denotes gravity (the x-axis being vertically downwards) and r(w,t) is some function that must be real when a) is real.
If gravity is negli-
gible, or if the motion is viewed in a free-fall reference frame, then (2.2) reduces to z
tt -
±r
(2
\
-3)
In the interior of the fluid, the coordinate w is generally not Lagrangian, and the velocity is given by z (a>*,t), which of course equals z (o),t) on the boundary.
The vanishing of the derivative z
singularity in the flow field, unless at the same point [z vanishes also, hence z* (w,t) = 0.
implies a (to*,t)]*
In other words
OJt
z
- 0
U)
implies
z*_ = 0
(2.4)
U)t
everywhere in the interior. When equations (2.3) and (2.4) are satisfied we can, if necessary, find a velocity potential x(w>fc) throughout the fluid by calculating X(o),t) - / z*(w,t) zaj(o),t)do) for then
(2.5)
M.S. LONGUET-HIGGINS
Z
(2
t
'6)
as required. 3.
ROTATING HYPERBOLIC FLOW. As a very simple form of solution suppose that z - au> - bto
(3.1)
where a(t) and b(t) are some functions of the time, to be determined.
This
will satisfy equation (2.3) with r = o)R(t) provided a and R is real.
a) — b
(A)
— iR(aoo + bo) ) •
(3.2)
Also from equation (2.4) the two equations a + bw
=0
j
, * -2 a* + b*a)
=0
l
(3 3)
'
J
are to be satisified simultaneously, if the corresponding point z(ca,t) is to lie within the fluid. Starting from equations (3.2) and (3.3) we shall deduce a chain of results leading eventually to a differential equation for the unknown function R(t). On equating coefficients of GO and OJ
in equation (3.2) we have
iRa (3.4) -iRb where R is not necessarily real.
On eliminating R from these two equations
we get ab t t + a tt b - 0 .
(3.5)
Again, on eliminating a) from equations (3.3) we have ab* - a* b = 0 .
(3.6)
From (3.6) and its complex conjugate there follows (ab* - a*b) t - 0 ,
(3.7)
hence ab* - a*b =
constant
= IK
(3.8)
ANALYTIC DESCRIPTION OF OVERTURNING WAVES
9
say, where K is real. Next, if we differentiate (3.6) with respect to t and use equations (3.4), we get (atb* - a*bfc) + iR* (ab* + a*b) = 0
*
i(a b
(3.9)
it
- a b )
ab
+ a b
* Clearly R = R , so R is real, as required.
a
t
b
\ -
ab
*t - a tt b * • °
by equations (3.4)(and their conjugates). ab
- a b
=
Then we have (3
-n)
So
constant
= iy
(3.12)
and by (3.6) we have also a*b - a t b* = iy
(3.13)
(atb* + a*b t ) t = (attb* + .Jb tt ) + c.c.
(3.14)
= i R ( a b * - a*b ) + c.c.
(3.15)
so that y is real. Thirdly we have
by (3.4).
This vanishes by (3.6). (a b
Therefore
+ a b ) =
constant
= v
(3.16)
where v is real. The relations (3.8), (3.12) and (3.16) provide three constraints on the real and imaginary parts of a(t) and b(t), leaving these functions with one necessary degree of freedom. 4.
A DIFFERENTIAL EQUATION FOR R(t). From equation (3.6) we may write * a
t
it b
t
10
M.S. LONGUET-HIGGINS
where by (3.16) AA* (ab*+ a*b) - v
(ab*+ a*b) «
-~
(4.2)
.
AA
(4.3)
From (4.1) we have also * * * * * * (a b - a b ) • AA (a b - ab ) = -iKAA by (3.8).
So from (4.3), (4.4) and (3.10)
v/AA
V
We first derive a differential equation for A . From the conjugate of (4.1) it follows that (ab) t = (atb + ab t ) - A*(ab* + a*b) = j
(4.6)
by (4.3) so on differentiating with respect to t (ab) t t = - v A t / A 2
.
(4.7)
But from (3.5) (ab) t t = (afctb + abfct) + 2att>t - 2a t b t
(4.8)
so by (4.1) (ab) t t = 2A* 2 (ab)*
.
(4.9)
Comparing (4.7) and (4.9) we see that
2ab = -v
1— (AA ) 2
.
(4.10)
(ab* - a*b) 2 » -K 2
(4.11)
Now on squaring each side of (3.8) we have
and so 4 a b a V - (ab* + a*b)2 = K2 .
(4.12)
ANALYTIC DESCRIPTION OF OVERTURNING WAVES
11
Using (4.3) and (4.10) we find * A A
.
— ^ (AAV say.
2
1;—— (AA )
2
=P v
(4.13)
2
Hence
AfcA* - (AA*)2 = P ( A A V .
(4.14)
This equation involves both the amplitude and phase of A, whereas R(t) by (4.5) involves only the amplitude.
To eliminate the phase (or rather, its
rate of change with time), we note that by (3.10) and (4.1)
say.
A a b - A*a*b* = iy
(4.15)
(i/v)(A*a*b* - Aab) = y/v = X
(4.16)
From (4.10) this implies that * A A
* - AA
a (^*)2
=x
(4 17
->
that is i (A*Afc - AA*) = X(AA*) 2 .
(4.18)
A = aeia
(4.19)
Now write
where a and a are real.
Then (4.14) and (4.18) yield respectively (a2 + a 2 a 2 ) - ak = Pa 8
(4.20)
a2afc = \ak .
(4.21)
and
Eliminating a
from these two equations we obtain a 2 = 0^(1 - X 2 a 2 + Pa 4 )
an equation for a alone.
(4.22)
Since by (4.5) R = (
(4.23)
12
M.S. LONGUET-HIGGINS
equation (4.22) is in effect an ordinary differential equation for R(t). Equations (4.21) and (4.22) can be integrated immediately to give t a - / Xa 2 dt
(4.24)
and (4.25)
a 2 (l -
The last equation gives t in terms of a, and so inversely a as a function of t. 5.
EQUATION OF THE FREE SURFACE. At the free surface u) is real (see Section 2) and so from (3.1) we
have simultaneously z - aw - buT 1
* z
*
* 1
I
(5 1}
-
- a o j - b o
On eliminating a) from these two equations we get (az* - a*z)(bz* - b*z) = (ab* - a*b) 2 as the equation of the free surface.
(5.2)
Rearranging the left-hand side we
obtain *2 * * * **2 * * 2 ab z - (ab + a b)zz + a b z - (ab - a b) .
(5.3)
On using (3.8), (4.3) and (4.10) this becomes A z 2 + 2AA*zz* + A*z* 2 - -2S(AA*) 2
(5.4)
where S - -K2/v
(5.5)
This is the equation of a hyperbola (see Longuet-Higgins, 1980b). 6.
RELATION TO PREVIOUS SOLUTION. In an earlier paper ( Longuet-Higgins, 1983) a parametric represent-
ation for the rotating hyperbolic flow was given in the form z - F(t) coshft+ G(t) sinhft , Q being real at the free surface.
(6.1)
On writing this as
z - ~(F + G)en + |(F - G)e~^
(6.2)
ANALYTIC DESCRIPTION OF OVERTURNING WAVES
13
it becomes clear that (6.1) is equivalent to (3.1) if we take a) « e1 , 0) being real whenftis real.
(6.3)
In that case we must also take
a - |(F + G) -b = i(F - G) and inversely F = (a - b)
(6.5)
G = (a + b) We have then *
ic a
F
it
t • a -- b
—
A A
(6.6)
by (4.1), and similarly it
V TTnr
(6 7)
-
Hence it
— = k , z
(6.8)
showing that the function A i s the same as that defined previously (compare Longuet-Higgins 1983, Section 10).
In fact the velocity potential of the
motion (in Eulerian coordinates) i s X - Hkz2 + f(t)
(6.9)
so that z* = Xz - Az
.
(6.10)
From (4.21), (4.22) and (5.4) above it can also be seen that the constants X, P and S are the same as those defined earlier (see Longuet-Higgins 1980b, and 1983). The two functions F(t) and G(t) were shown in Longuet-Higgins (1983) to be related to the solutions of a Riccati equation.
However the three
14
M.S. LONGUET-HIGGINS
integral relations derived in Section 3 of the present paper lead to some prviously undiscovered identities between F and G.
Thus from (3.8) we have
(FG* - F*G) = 2(ab* - a*b) = 2iK .
(6.11)
Similarly from (3.12)
*+ + G tJ - c . c . - 4(a*b - a tb*) = 4iy t t
(F* -
(6.12)
and from (3.16) = _2v .
(6.13)
Other relations may be written down between F, G and A, or between F, G and a.
It can be verified that these relations are indeed satisfied by the
power series expansions for F(t) and G(t) which are given in Longuet-Higgins (1983), namely
2S
T + i -
• +
...
24T3
(6.14)
[X2j
[2T
6T2
6X3
where T • (t-tj)/X, t j a constant, and (6.15) We note also that from equations (4.3) and (4.5) _
KV
_
(ab* + a*b) 2
4KV
(FF* - GG*) 2
(6.16)
and so on substituting the series ( 6.14) we have, for large values of T,
R
h~\— "—
(6.17)
The three integrals K, y and v have the dimensions respectively of mass and of angular momentum and kinetic energy per unit mass (the density being taken as unity).
In the case of an elliptic boundary, the corre-
sponding integrated quantities would be finite, but for the rotating hyperbola it is easy to see from the Eulerian potential (6.9) that they must
ANALYTIC DESCRIPTION OF OVERTURNING WAVES be infinite.
15
Thus for the hyperbola K, y and v can be assigned no immediate
physical meanings. On the other hand for the rotating Dirichlet ellipse we may, by analogy with (3.1) and (6.3), write z
,
ue
in
- V e- ifi
(6.18)
where u and v are functions of the time only, and ft is to be real on the boundary.
Then from the boundary condition
we deduce « t t = -Ru
| (6.20)
vtt-
Rv
hence u
tt V
To annul the singularity at z
+
uv
tt" °
*
(6.21)
= 0 we must have also
uu* - w * = 0
.
(6.22)
Then proceeding by steps precisely analogous to Section 3 we may deduce that uu (uu
+ w u u
say.
- w
-
constant
= K
) - c.c. =
constant
= 2iy
constant
= v
+ v v
=
(6.23)
Writing
* * u v _L s _t = _ A v u so that (6.10) remains valid we can show by similar steps that
(6.24)
16
M.S. LONGUET-HIGGINS
uu
u
+ w
= v/AA
tUt " VtVt
=
~K
R - -(K/V)(AA ) 2
(6.25)
(uv) t = -v/A
2uv = v
(AA*) 2
leading to the differential equation k*
- (AA*) 2 =
for A (compare equation (4.14)).
-(K2/V2)(AA*
(6.26)
Hence for a and a (defined by (4.19))
we obtain Act2 (6.27) a2 and for the free surface of the ellipse A z 2 + 2AA*zz* + A z 2 = 2 S ( A A * ) 2
(6.28)
where as before A - y/v ,
P = K2/v2
S = K 2 /v .
,
(6.29)
Finally for the total mass M of the ellipse w e find by a straightforward calculation (6.30)
M = T T S / P ^ - TTK
and for the total angular momentum and kinetic energy a.m. - - i r r A S 2 / P 3 / 2 = - j My
(6.31)
and k.e.
= T T T S 2 / P 3 / 2 = yMv
4
4
.
(6.32)
Thus for the rotating ellipse, each of the quantities K,y and v does indeed have a precise physical interpretation in terms of the integrated flow.
17
ANALYTIC DESCRIPTION OF OVERTURNING WAVES 7.
A GENERALISATION OF (3.1)? To generalize the solution (3.1) let us try . -1 1 -3 z = ao)-bo) - real
Where a, b and c are all functions of time.
(7.1)
If we assume that in (2.3)
r = u)R(t) as before, then by equating coefficients of co, of"1 and ca~3
in
(2.3) we get the three relations a
tt
b
tt =
c
tt
=
=
iRa (7.2)
-iRb -3iRc
Also by (2.4) a + baT 2 + cuTk - 0
(7.3)
is to be satisfied in the interior only if *
*
9
0
+ b a)""z
a
at the same point, for all values of t.
The condition that
(7.4) (7.3) and
(7.4) have a common root is expressed by a
b
c
0
0 (7.5) 0
a
b
0
a
b
This can be expanded in the form (a,b*)(b,c*) = (a,c*) 2
(7.6)
where we introduce the abbreviated notation
(p,qt)
(7.7)
= P
t
q
t
Now if we differentiate (7.6) logarithmically with respect to t and use equations (7.2) we get
18
M.S.
2
(a,b*)
(b,c*)
LONGUET-HIGGINS
-iR
(a,c*)
* * * * ab +a b , 3bc +b c zr + (a,b*t) (b,c*)
6ac +2a c (a,c
(7.8)
t)
This expresses R directly in terms of the functions a,b,c and their first derivatives.
There is, however, no guarantee that R is real, as is required
by the relation r • u)R. To put this in another way, let us write
B,
at/a - A,
(7.9)
ct/c - C
where A, B, C are some functions to be determined.
The necessary condition
(7.6) then becomes (A - B)(B - C) m ac (A - C ) 2
(7.10)
b2
At some initial instant, say t « 0, we can choose values of a,b,c,A,B,C, so as to satisfy (7.6). * * a A,
Then using the first-order differential equations bt - b B ,
we can determine a , b , c
c
* * = c C
(7.11)
and so R at time t « 0. Moreover we can deter-
mine (a*/a) t (b*/b) (c*/c)
t t
- -(AA
+ iR)a /a
-(BB*
- iR)b*/b
-(OC*
- 3iR)c*/c
\
j
also at time t = 0. By integrating step-by-step with respect to t we may then determine the seven functions a,b,c,A,B,C, and R, as functions of t. These will satisfy all the necessary conditions, except that R will not generally be real.
Only when c - 0 and C = 0 do we have A - B (from
(7.10)), leading to the equations of Sections 3 and 4. 8.
SOME POLYNOMIAL SOLUTIONS OF (2.3). If in the boundary condition (2.3) we let r - o>R(t) as in Section 3
and write (8.1) then (2.3) becomes
ANALYTIC DESCRIPTION OF OVERTURNING WAVES
19
then (2.3) becomes ztt = i r '
(8.2)
V
where rf - ^ R
.
(8.3)
Substituting for R from equation (6.17) we find in the limit as 73* + 0
5
(8.4)
If we choose the time origin so that t- = 0 and scale t so as to make A - 1, hence T = t, then for large values of t equation (8.4) becomes 1
(8.5)
(1 + t2 very nearly.
As pointed out by Greenhow (1983) this is essentially the
same function r(oa,t) as occurs in the representation of New's ellipse (New 1983).
Accordingly, any simple solutions of the boundary problem
tt are of particular interest.
(1 +
t2)2
(8.6)
0)
For, by combining such solutions linearly we
may be able to find a flow which incorporates the desirable features both of the rotating hyperbolic flow (for the tip of the jet) and New's ellipse (for the forward face of the wave). We shall now derive some simple solutions of equation (8.6). (a)
Circular functions of a)
Writing z
= eik(Vt)
(8.7)
in (8.6) we have
*tt
=
k
(i + t 2 ) 2
•
(8.8)
of which a solution is • - (1 + t 2 )
i Y arctan
(8.9)
provided Y2 = 1 - k
.
Hence if k ^ 1 we have solutions of the form
(8.10)
20
M.S. LONGUET-HIGGINS
]
(8.11)
where M and N are arbitrary constants and 0 = arctan t
.
(8.12)
In particular when k = 0, equation (8.11) becomes z = (1 + t2)4
(Me 10 + Ne" 1 9 )
(8.13)
and when k = -1 z = e- iw (l +
t
2 ) % (Me i / 2 Q + N e - i / 2 0 ) .
(8.14)
As k -»• 1 we find in the limit z = e ^ d + t 2 ) ^ (Mf + iN f 0)
(8.15)
where M1 and Nf are other constants; and when k - -2 z = e" 2lu (l +
t
2 ) % (Me l / 3 e + N e - 1 / 3 0 ) .
(8.16)
Similarly for values of k > 1 we have in general
[
^ 1 ^ + Ne"*-1^ |
(8.17)
and in particular when k = 2 z =
e
2ia)
(l +
t
2 ) % (Me0 + Ne" 9 ) .
(8.18)
New's ellipse corresponds to (8.11) with
(b)
Exponential functions of m
Writing k - -im
(8,20)
in (8.11) we have
[ Me i(1 + im)^9 + Ne- 1(1+ lm)%Q ] in which the free surface (w real) tends to infinity as mw -> °°.
(8.21) By adding
two such solutions, one with a positive value of m and one with a negative value, we get a solution with the boundary tending to infinity in two directions.
21
ANALYTIC DESCRIPTION OF OVERNTURNING WAVES (c)
Polynomials in o), t and 0
Writing (8.11) in the form z = eika)(l
N
.] <s.
22)
where M and N are arbitrary constants (not the same as before), let us differentiate with respect to the parameter ik and then set k = 0, remembering that (1 + t 2 ) ^ cos0= 1,
(1 + t 2 ) ^ sin0=* t.
(8.23)
Then we get a new solution z
- M(a) - %Lt0) + N(u)t + %i0).
(8.24)
Similarly if we differentiate (8.11) twice and set k = 0 we get z o = M[o)2 - it0u) - ^(t0 - 0 2 )]
N[ta)2 + lew + h(0 + t0 2 )]
(8.25)
and in general NQn(t,0,aj)
(8.26)
where _ P
n
n . n—1 , , . = a a ) + a nw + ... + a, a) + a n n-1 1 o n
(8.27)
11 1
Q = b o) + b -w " + ... + b-o) + b n n n-1 1 c
and
n!
R (t) (8.28)
n! n
with
"" m ) •
22
M.S.
R
o
LONGUET-HIGGINS
= 1
\
» te R0
= - -H02 - te)
L
2. • (8.29)
30 2 - 3t0)
R = (G>I+
4 A
" 6tG)3 "" 15G>2 + 15t0)
~ 45t03 " 105°2 + 1O5t0) etc., the (I + l)th coefficient in the series for R being m . 2~l
(m + I - 1 ) ! (m - I - 1) !
(8.30)
In the series for S (t), we simply replace 0 P by t0 P and t0 P by - 0 P in the series for R (t). m Finally we note that there exist solutions to (8.6) obtainable from (8.11) by differentiation with respect to k, in which k is not set equal to zero after differentiation.
These will in general have a spiral form.
To obtain physically valid flows, the above solutions must be combined in such a way as to satisfy equation (2.4), at least for large t. Such a procedure has been carried out by Greenhow (1983) in the case (8.31)
r = -
(namely the first two terms in the expansion of (8.5)) and it would be possible in principle to carry this process further. 9.
FURTHER SUGGESTIONS. We have shown how previous solutions, each representing part of the
flow in an overturning wave, can be represented very simply in semiLagrangian form, in particular the rotating hyperbola (for the tip of the wave) and the /3-ellipse (for the tube).
In the first instance we have
also found three integral invariants of the motion. By combining the methods of Sections 3 to 7 with that of Section 8 it may be possible to further generalize these solutions so as to obtain a rather complete representation of the flow, in semi-Lagrangian form. On the other hand it must be emphasised that all solutions to the homogeneous boundary condition (2.3) that do not contain g are essentially
ANALYTIC DESCRIPTION OF OVERTURNING WAVES
23
gravity-free, and can give only a local picture of the flow over a limited duration of time.
So far, the most promising solution incorporating
gravity has been that given by equations (1.3) and (1.4) above (see Figure 2) . This solution, however, is not in John's semi-Lagrangian form (the free surface does not correspond to u> real) but is in the more general parametric form suggested by Longuet-Higgins (1980a).
Quite probably it will prove
necessary to make use of the greater flexibility of the general formulation to represent the complete wave. One feature which the solution of equations (L.3) and (1.4) has in common with the semi-Lagrangian flow (1.5) is the existence of a branch-point, given by z
= 0 , outside the fluid domain.
It is this feature which can account
for the discontinuity in fluid velocity where the jet meets the forward face of the wave.
The reader will take note that the former solution (1.3),
(1.4) though approximate, is in some ways simpler that the cubic (1.5), since the velocity potential x corresponding to (1.5) is actually of the sixth degree in o>, as can be seen from (2.5).
Hence the further general-
isation of (1.3) and (1.4) may yet prove to be the most promising line for future investigation. REFERENCES JOHN, R. (1953) Two-Dimensional Potential Flows with a Free Boundary, Comm. Pure App. Math. £, 497-503. GREENHOW, M. (1983) Free Surface Flows Related to Breaking Waves, J. Fluid Mech. (in press). LONGUET-HIGGINS, M.S. (1972) A Class of Exact, Time-Dependent, Free-Surface Flows, J. Fluid Mech. 55, 529-543. LONGUET-HIGGINS, M.S. (1976) Self-Similar, Time-Dependent Flows with a Free Surface, J. Fluid Mech. 21» 603-620. LONGUET-HIGGINS, M.S. (1980a) A Technique for Time-Dependent, Free-Surface Flows, Proc. R. Soc. Lond. A.371, 441-451. LONGUET-HIGGINS, M.S. (1980b) On the Forming of Sharp Corners at a Free Surface, Proc. R. Soc. Lond. A.371, 453-478. LONGUET-HIGGINS, M.S. (1981a) On the Overturning of Gravity Waves, Proc. R. Soc. Lond. A.376, 377-400. LONGUET-HIGGINS, M.S. (1981b) A Parametric Flow for Breaking Waves, Proc. Symp. on Hydrodynamics in Ocean Eng., Trondheim, Norway, August 1981, pp. 121-135. LONGUET-HIGGINS, M.S. (1982) Parametric Solutions for Breaking Waves, J. Fluid Mech. 121, 403-424. LONGUET-HIGGINS, M.S. (1983) Rotating Hyperbolic Flow: Particle Trajectories and Parametric Representation, Quart. J. Mech. Appl. Math. 36, 247-270.
24
M.S. LONGUET-HIGGINS
LONGUET-HIGGINS, M.S. and COKELET, E.D. (1976) The Deformation of Steep Surface Waves on Water. I. A Numerical Method of Computation, Proc. R. Soc. Lond. A.350, 1-26. LONGUET-HIGGINS, M.S. and COKELET, E.D. (1978) The Deformation of Steep Surface Waves on Water. II. Growth of Normal-Mode Instabilities. Proc. R. Soc. Lond. A.364, 1-28. McIVER, P. and PEREGRINE, D.H. (1981) Comparisions of Numerical and Analytical Results for Waves that are Starting to Break, Proc. Symp. on Hydrodynamics in Ocean Eng., Trondheim, Norway, August 1981, pp.203215. NEW, A.L. (1981) Breaking Waves in Water of Finite Depth. Proc. Brit. Theor. Mech. Colloq., Univ. of Bradford, England, April 1981 (Abstract only). NEW, A.L. (1983) An Elliptic Class of Free Surface Flows, J. Fluid Mech. (in press). VINJE, T. and BREVIG, P. (1981) Breaking Waves on Finite Water Depths: a Numerical Study, Ship. Res. Inst., Trondheim, Norway, Rep. R-111.81; see also Adv. Water Resources 4, 77-82.
CHAPTER 2 THE KORTEWEG-DE VRIES EQUATION AND RELATED PROBLEMS IN WATER WAVE THEORY R.S. JOHNSON School of Mathematics The University, Newcastle upon Tyne NE1 7RU, United Kingdom
1.
INTRODUCTION. The last decade and more has produced an altogether unlooked-for im-
petus in the study of certain partial differential equations by use of the inverse scattering transform.
Two of the (now) standard equations which
are susceptible to this technique arise quite naturally in the study of water waves:
the Korteweg-de Vries (KdV) equation and the nonlinear Schro-
dinger (NLS) equation.
This suggests the possibility that there exist
other equations of a similar character which are also relevant in water wave theory.
The similarity may merely be that the equations are generali-
sations (more terms, variable coefficients, etc.) which convert the problem into a non-integrable one.
On the other hand a conceivable result is that
we generate other integrable equations which are extensions of the classical equations to different - possibly higher dimensional - co-ordinate systems.
The overall picture is that of a number of diverse equations which
describe various aspects of the same underlying problem.
This has the
virtue that we can more readily compare and contrast the equations, and in some cases specifically relate one to another. In this paper we shall collect together many of the varied forms of KdV (and to a lesser extent NLS) equations which arise in water wave theory. To emphasise the connecting themes the same variables and parameters will be used throughout.
We shall start from an appropriate set of basic equa-
tions and thence develop both KdV and NLS equations for one spatial dimension.
Since both equations describe alternative aspects of the same problem
(by employing different limits in parameter space), it should be possible to match these two equations:
this is readily demonstrated.
We then turn
to the two-dimensional problems which correspond to both the KdV and NLS equations.
Some properties of the relevant similarity solutions in one and
two dimensions are mentioned, together with matching to the near-field, i.e.
to initial data.
Finally, we briefly comment on a few other more
26
R.S. JOHNSON
involved equations of KdV-type which describe the role of other physical processes (such as viscous dissipation)• 2.
BASIC EQUATIONS. The fluid is assumed to be both incompressible and inviscid (although
not necessarily irrotational), bounded above by a free surface and below by a rigid surface.
The fluid is supposed to extend to infinity in all
horizontal directions so that, for example, the effects of beaches are ignored.
The free surface is characterised as a surface of constant pressure.
The appropriate equations are then non-dimensionalised by using an ambient depth (d), a typical wave length (A) and the acceleration of gravity.
The
non-dimensional equations and boundary conditions for an irrotational flow, with a flat horizontal surface below, are 4
+ 62V^(j) = 0; (j) - 0
on
z = 0;
6)22]] - 0, \ (V6)
V±±n)] n)] 62[nt +a(v±<|>) • (V
I
, ))
(2.1)
on z - 1 + an.
The vertical coordinate measured up from the bottom of the fluid is z, and the free surface is at z = 1 + ar\ where a is an amplitude parameter.
The
gradient operator perpendicular to z is represented by V , and the ratio of vertical to horizontal length scales is given by 6 = d/X. In addition to equations (2.1), we shall also require the equations (in one horizontal dimension only) which describe the perturbation of a main shear flow and which include variable depth.
If the flow has a velo-
city distribution u - U(z), moving in the x-direction, with a bottom topography given by z = b(x), then = 0 ; + (U + au)u + w(Uf + au ) + p t X Z X 2 6 [ w + (U + a u ) w + a w w ] + p = 0 ; x t X Z Z u + w = 0 ; with w/u = b f (x) on z = b(x),
u
and
P - ri, w = r| + ( U + au)r|
on
(2.2)
z = 1 + ari»
The prime denotes a derivative, p is the pressure and the velocity components are (U + au, aw). Equations (2.2) are therefore essentially the rotational counterparts of (2.1), being based on Euler's equation rather than Laplace's equation. Now, by suitable choice of the limits of the parameters and corresponding scaled variables, we may construct asymptotic expansions which
KdV EQUATION AND PROBLEMS IN WATER WAVE THEORY yield the KdV or NLS equations to leading order.
27
Other phenomena can be
incorporated by adjusting the size of the new parameter with respect to a and/or 6. 3.
ONE DIMENSIONAL PROBLEMS:
KdV.
The simplest theory for the KdV equation, which harks back to the 2 derivation of Korteweg and de Vries (1895), is obtained by choosing 6 =0(a) as a •> 0. §6). V
(More general scalings, valid for arbitrary 6, will be given in
For one-dimensional irrotational flow we use equations (2.1) with
= (3/3x,O); the wave is taken to be propagating in the positive x-direc-
tion only.
If we introduce £ = x - t,
x - at,
(3.1)
and expand both $ and r) in powers of a, then to leading order we obtain 2
i
?
= o.
(3.2)
This is the classical KdV equation, the solution of which exhibits the well-known soliton behavior. Corresponding equations can also be derived if the flow incorporates a basic shear (in the x-direction), or there is an appropriate slow variation in the ambient depth.
For both these analyses it is either necessary
or convenient to use equations (2.2).
In the case of the propagation over
a shear flow we take b = 0, i.e. the bottom is flat and horizontal.
The
characteristic and long time variables take the same form as in (3.1), and so we set £ = x - ct,
T
where c is a speed to be determined.
f
1
to leading order. speeds c:
(a -»• 0)
The expansion used abbve now yields
-2 [U(z) - c]
J
- at
Z
dz = 1,
0
This is the Burns integral (Burns, 1953) defining the
it is assumed here that U(z) ^ c for 0 £ z £ 1, and hence there
is no critical layer in the fluid.
At the next order we obtain the KdV
equation which now takes the form -2I 3 n T + 3I 4 nru + (62/a)J r W r - 0 where I n
-
(3.3)
f1 J
0
[W(z)]
-n
dz; J =
W 2 (y) y — dx dy dz; W - U(z) - c, J J J 0 z 0 for(z)fcr(x) f
(see Freeman and Johnson, 1970).
1
(3.4)
^
9
The special case of a solitary wave mov-
ing over a shear was given by Benjamin (1962).
The effect of a shear - at
28
R.S. JOHNSON
least in the absence of a critical layer - is merely to alter the numerical coefficients in the KdV equation.
It is clear that if we set U(z) = 0,
then c = ±1 and we recover equation (3.2) (with c = 1 ) . The case that arises when a critical layer is present can also be discussed.
Velthuizen and van Wijngaarden (1969) argue that the Burns inte-
gral (3.3), must be interpreted as a Cauchy principal value (with a corresponding contribution to the imaginary part of c ) . Using their approach it is then fairly straightforward to obtain (3.4) again, provided all the integrals are taken to mean principal values.
(A discussion of large ampli-
tude waves over a shear flow in the presence of a critical layer is given in Varley et al. 1977.) The problem of wave propagation over variable depth can be examined in terms of aKdV theory if the bottom varies slowly, specifically on the scale a.
If we note that the characteristic variable is to imply a varia-
tion in propagation speed, then we write 5 * g(x;a) - t,
X - ax
with b - b(X).
(3.5)
The leading order then serves to define g, g = I [D^x)]" 1 * dx,
D - 1 - b(X),
and the next order yields the dominant equation for r),
v
D?
^
^ = 0,
where the prime denotes a derivative with respect to X.
(3.6)
Equation (3.6) is
a variable coefficient KdV equation (Kakutani 1971, Johnson 1973a), which is often re-written under the transformation n
2H
2 = D Z ( X )H(x,S),
f x
=
J
0
V D 2 dX, D 2 dX,
+ 3HH r + 4(6 /a)H r r r - - f ~
H.
(3.7)
(3.8)
This equation predicts the soliton fission that occurs as a solitary wave moves into a shelving region (Madsen and Mei 1969, Tappert and Zabusky 1971; Johnson 1972b); see Figure 1.
The equation has also been used as the basis
for a discussion of the effects of a perturbation to the KdV equation, i.e. D'/D small.
This can be accomplished either by direct methods (Johnson,
1973b) or more satisfactorily via the inverse scattering transform (Kaup and Newell 1978; Karpman and Maslov 1978; Candler and Johnson 1981).
In
particular the phenomenon of the shelf that appears behind the solitary wave is now well-understood (Knickerbocker and Newell 1980).
29
KdV EQUATION AND PROBLEMS IN WATER WAVE THEORY
i
O-OI
D 71 ;. 1 1 1 1
r\ : / \:
10-
/ v / :'\ / * \ •
/ .•vyo
• 1 f
'•; ft / / /
I \
/ I // \\
1 • /
0-
-40 A « Figure 1. The geometry of the shelf (which changes between X=0, 0.01) and the two-soliton formation (do = 0.614) as predicted by equation (3.6). Initial solitary wave( ) ; solution at X=0.075 (••••); solution at X=0.25 ( ) ; solution at X = 0 . 5 ( — ) . Before we leave the classical KdV problem in one dimension, it is of interest to relate it to the so-called Boussinesq equation.
This single
equation, correct at 0(a), accommodates waves moving in both directions. From equations (2.1), with V
0(a), a direct expansion
= (9/3x,O) and 6
in powers of a yields a -* 0; n
n,
- n-,
lxx where 0 ot
+ 2n
ltt
2
+ n
2
0;
oxx - n ott
+ n n
- 20
n
+
1 2 T(5
/ct)n
fxn <•
i —OO
]
xx
+
3
n
'xxxx
= 0(a 2 ), "
which is not the conventional form for the Boussinesq equation. if we introduce the transformation
n(x,t) - o n , then (3.9) becomes
= 0,
ot ox o oxx ox oxt 3 oxxxx Expressed as a single equation for r\9 this gives
•r
(3.9) However,
r)(x,t)dx
1 6' TTxxxx x = °(«)
(3-10)
which is the equation written in a Lagrangian rather than an Eulerian frame
30
R.S. JOHNSON
(Ursell 1953),
Equation (3,10) possesses an n-soliton solution (Hirota
1973) which describes both overtaking and head-on collisions.
The KdV e-
quation (3.2) follows directly from (3.9) by using the co-ordinates (3.1) and again expanding in a; the error in the KdV equation is 0(a), as usual. 4.
ONE DIMENSIONAL PROBLEMS:
NLS.
The nonlinear Schrodinger equation describes the amplitude, of a harmonic wave profile, as a function of slow space and time variables.
The
wavelength of the carrier wave is taken to be 0(1) as a -*• 0, and this corresponds to 6 being fixed in the limit process.
The basic wave is there-
fore sort in the form n ~ A(£,T)e i k ? + c . c ,
a -> 0,
where c.c. denotes the complex conjugate and 5 - x - cpt,
C = a(x - c g t),
T
= a 2 t.
(4.1)
The expansion for r\ (and <j>) is so constructed that it be periodic (to all orders) in £, whence higher order terms must contain higher harmonics generated by the nonlinear coupling.
The carrier wave moves at the phase speed
(c ) and the amplitude modulation moves at the corresponding group speed (c ) , although the specific forms of c ,c O
C
respectively.
are not assumed a priori.
From
CO
equations (2.1), with V 2 p
= (3/8x,O), the leading and next order give
tanh 6k • — I k " '
1 ,. 26k °g " 2 C p ( 1 + sinh 26k >'
„ ON '2)
(4
At the very next order, which incorporates cubic nonlinear-
ity, we obtain -2ikc
AT + q A
+ r A|A|2 = 0
(4.3)
where q and r are involved functions of 6k (see Hasimoto and Ono 1972). Equation (4.3) is the NLS equation for water waves, but it has appeared earlier in a more general formulation given by Benney and Newell in 1967. The coefficient r(6k) has a zero at 6k = a
~ 1.363, and it is well-known
that the Stokes wave is then unstable if 6k > a (Benjamin and Feir 1967). o This suggests that the nature of the Benjamin-Feir instability could be examined via a suitable generalisation of (4.3) valid near r = 0.
It is
clear that new scales are now required since we can anticipate the appearance of quintic nonlinearity. Thus we set C = a 2 (x - c t), x = a t p and obtain the equation
with
6k = a o
KdV EQUATION AND PROBLEMS IN WATER WAVE THEORY 2iAT + a j A ^ + a2|A|2A^ + a^ A ( | A | 2 ) ^ + a4A|A|4 + where i|; - |A|
(see Johnson 1977).
&5
31
A* T = 0
(4.4)
Equations (4.3) and (4.4) can be com-
bined to yield an equation which is uniformly valid for all 6k, but with coefficients a
(i = 2,...,5) known only on 6k = o .
The inclusion of a background shear, U(z), in the x-direction follows rather similar lines to the corresponding problem for the KdV equation, However, the new coefficients are immensely more complicated and involve, in particular, the solution to a second order ordinary differential equation.
Let V(z) be the solution of (V f /W 2 ) f - (6k) 2 (V/W2) = 0
(W - U - c ) P
(4.5)
with V(l) = 1; V 1 (0) = 0, then using exactly the same expansion and variables as for the standard NLS equation, we obtain rl 2 (WI1) dz - 1
dz - 1,
c = c -I—
=
1
(4.6)
to the leading and next order, respectively, where I(z) -
VW
Jo
dz
Equations (4.6) serve to define c that c
= d(kc )/dk.
and
W
1
= U(l) - c .
P
and c , and it is comforting to find
The NLS equation is recovered at the next order as
8
2ik W (1 + W 1
f1 1 Jo
IIfWfdz)A
T
+ qA
where q,r are given in Johnson (1976).
^
+ rA|A|
2
- 0,
(4.7)
The condition for the onset of
Benjamin-Feir instability is still r = 0, although this is now a functional of U(z). 5.
MATCHING OF KdV AND NLS. Both the KdV and NLS equations are derived under the assumption that
a + 0; however, 6 is held fixed for the NLS equation whereas 6 -> 0 for the KdV equation.
This suggests that it should be possible to match these two
equations in the limit:
NLS as 6 -»• 0 with KdV as 6 •> °°.
We shall demon-
strate that this is indeed the case and, further, we shall apply the method to the equations valid for arbitrary shear, (3.4) and (4.7).
In the case
of the KdV equation, we must employ an appropriate expansion which allows a modulated carrier wave as the solution.
XAx, and write
Z - (5 + UAT)/A,
Let us therefore introduce
T = T/A,
A = 62/a
32
R.S. JOHNSON
11
~Z^Z 00
in equation (3.4), and
OO
+ c.c.
At successive orders we obtain JkZ __ 3Jk2 y _ A - - — , \i - - -fi—
(I ) 2 21 I 3 A ol . + 3kJA Q i e $ - | - A - - A o l |A o l | 2 = 0.
(5.1)
The NLS equation, (4.7), is to be approximated for 6 -• 0 which involves estimating the coefficients in this limit.
So, for example, we use
V(z) ~ 1 - (6k) 2 f f W2(x)w"2(y)dydx; c - c - -(6k) 2 y- , whence
eventually we obtain 21 6 2 I 3 A T + 36 4 kJA^ - | — £ j - A|A|2 = 0.
It is clear that (5.1) and (5.2) match, i.e. k
(5.2) 2
= A, since T=6 f, C=62£.
For more details of the matching procedure, see Freeman and Davey (1975) and Johnson (1976). 6.
TWO DIMENSIONAL PROBLEMS: KdV. The various equations which describe propagation on the whole surface,
rather than on the line, will be developed for irrotational flow over a flat horizontal surface. Thus we deal here only with equations (2.1), although the special cases of nearly plane waves over a variable depth and ring waves over a shear will be mentioned briefly in §10. The KdV equations which apply for nearly plane waves, concentric waves and nearly concentric waves will be given in terms of general scalings valid for arbitrary 6, as a •> 0. The nearly plane (or 2-D) KdV requires V
= O/8x,8/9y) with
? - (o*/6)(x - t); Y = (a/6)y; T - (a3/2/6)t; $ = (ofV6)cf> and then, as a -»• 0, we obtain to leading order
(2n + 3nn +
3-n
+ n Y Y - 0.
(6.1)
This reduces to equation (3.2), upon one integration, if r| is independent 2 of Y and we re-define £> T with 6 = 0(a). The equation was first suggested by Kadomtsev and Petviashvili (1970) and its inverse scattering representation was given by Dryuma (1974).
The oblique solitary wave interaction
solution was given by Satsuma (1976), and Miles (1977a) has shown in detail how the equation becomes relevant as oblique waves become nearly parallel. The equation also exhibits resonant soliton solutions (see Miles 1977b, Anker and Freeman 1978).
KdV EQUATION AND PROBLEMS IN WATER WAVE THEORY
33
The problem of concentric waves on the surface is a little more involved since the amplitude decays as the radius increases.
I f V = (8/8r,O)
then we define £ « (a/6)2(r - t); R = (a6/64)r; $ = <j>/a; H = (62/a3)T?
(6.2)
4 2 and the equations (2.1) now contain the single parameter A = a /6 • As A -> 0, the leading order problem for H yields 2H R + | H + 3HH ? + j H ^ =
0.
(6.3)
This is the so-called cylindrical KdV equation which was first derived in another context by Maxon and Viecelli (1974).
The inverse scattering trans-
form for (6.3) involves a linearly increasing potential which generates eigenfunctions based on the Airy function (see Calogero and Degasperis 1978).
A discussion of the solutions of both the nearly plane and concen-
tric KdV equations is to be found in Freeman (1980). As the classical KdV equation has a two-dimensional counterpart (in the nearly plane equation), so does the concentric equation. V
We introduce
E O/^r, r^a/aS) and use the variables (6.2) together with 6 = Q/&^
whence, as A -> 0, the dominant equation defining H(£,R,G) becomes iH + 3HH ? + iH^) 5 + i I H e e =0. R
(6.4)
It is clear that as (6.1) corresponds to (3.2), so (6.4) corresponds to (6.3).
There is, at present, no inverse scattering transform for this
equation when written in all three independent variables (£,R,O), although some special transformations will be mentioned below which reduce (6.4) to a standard problem but in one few variables.
One simple interpretation of
this new equation is that it applies to a small angular sector (of 0(A 2 )) over which the wave may vary i n 0 :
since the wave is nearly straight in a
small angular region it is nearly plane, and in this sense it is analogous to (6.1). The three equations developed above take differing forms depending on whether cartesian or polar coordinates are employed.
It is therefore worth
investigating the possibility that transformations exist between the equations.
Thus, for example, we have that 2 r - t ~ x(l + j ^ ) ~ t x 2 h
= (6/a ) (5 + £ 2 _ )
as y/x + 0
34
R.S. JOHNSON
1 2 which suggests that if n = n(T,£ + j Y / T ) in (6.1) then we would obtain (6.3) (after one integration).
This is easily confirmed (if we interpret
T as R ) . Similarly, if we set H = H(T,£ - ~ 10 ) in (6.4), then we recover the classical KdV equation, (3.2) (again, after one integration).
The
transformation from the nearly plane (2-D) equation to the cylindrical equation can also be used in a generalised form to construct the inverse scattering transform for this latter equation (see Johnson 1979).
Since we
have shown that the nearly concentric equation can be transformed into the KdV equation, (3.2), it follows that a limited class of solutions is now available to equation (6.4). 7.
TWO DIMENSIONAL PROBLEMS:
NLS.
The nearly plane KdV equation was obtained by allowing the variation in the y-direction to be slow (or weak); the same approach can be adopted for the NLS equation.
From equations (2.1), using V
= (3/8x, 8/9y), with
variables (4.1) and Y = ay, we now let A = A ( £ , Y , T ) (see §4). The expressions for c and c are unaltered (see (4.2)), and for A we obtain the P g coupled system -2ikcp A T + q A ? c - c ^ ^
= r A|A|2
o
(1- V * C C
+
*YY
where q,r,s,$ are involved functions of 6k (see Davey and Stewartson 1974; also Benney and Roskes 1969).
Equations (7.1) recover the NLS equation
(4.3) upon the assumption that there is no dependence on Y.
Solutions of
(7.1) in the long wave length limit (6 -*• 0) are discussed by Anker and Freeman (1978), and the deep water limit discussed by Lake et al. (1977) corresponds to 6 •> °° (in (7.1) and (4.3)). 8.
SIMILARITY SOLUTIONS:
KdV.
In more recent years the studies in inverse scattering theory have brought to prominence the role of Painleve equations generated by seeking similarity solutions of the evolution equations.
In particular both the
classical and concentric KdV equations possess similarity solutions of the 2 For the plane KdV equation (3.2), with 6 = a (or from
appropriate form.
(6.1) in the absence of any dependence on Y) we set
n = i(2/x) 2/3 F(X), x = c/(2x) 1/3 whence V" - XV - V 3 = 0, where
/ T F '- F
« V and V + 0 (exponentially).
(8.1) Equation (8.1) is a
35
KdV EQUATION AND PROBLEMS IN WATER WAVE THEORY
Painleve of the second kind (P-2) and its solutions are discussed at some length by Berezin and Karpman (1964) and Rosales (1978). (8.1) which decays exponentially as X •+• +°° grows like |x|
The solution of as X -* -°° (see
Figure 2 ) .
-10
Figure 2. Similarity solutions, F(X). For the concentric KdV equation: A - - 8 ( — ) ; A - 0 with F(O)=1 (•••). For the plane KdV equation with F(0)=l( ). The concentric KdV equation (6.3) gives rise to a far more acceptable similarity solution.
First we introduce
H - (2/(3R 2 )) 1/3 F(X),
X = £/(2R) 1 / 3
and so FF" - j F | 2 + 2(F 3 - XF 2 ) = where A is an arbitrary constant.
(8.2)
If, as above, we seek the solution for
which F -* 0 (exponentially), then A = 0 and we can write v" - Xv + v 3 = 0,
F - v2.
(8.3)
This is another P-2 equation, simply related to (8.1) by an elementary (although non-real) transformation (i.e. V = iv). Equation (8.3), which has been discussed by Ablowitz and Segur (1977), Miles (1978b), Rosales (1978), has a solution which decays like |X| ^ as X •*• -°° and might therefore describe a realizable phenomenon.
A particularly simple exact solution
to equation (8.2) can be obtained by working from the nearly plane KdV equation and then using the transformation between (6.1) and (6.3) (for details, see Johnson and Thompson 1978, Johnson 1979); this yeilds
36
R.S. JOHNSON F(X)
2 r°° 2 i y [£a( A 2 dX) ], dX^ Jx 1
which corresponds to A = -8 (see (8.2)).
(8.4)
Solution (8.4) decays algebraical-
ly as X •> +°°; both (8.4) and the solution of (8.3) are depicted in Figure 2. This special closed-form solution is directly related to one of the set of similarity solutions found by Airault (1979) for the plane KdV equation. 9.
MATCHING TO A NEAR-FIELD:
KdV.
The various KdV equations introduced here, together with the few solutions mentioned above, are all to be interpreted in terms of water waves. Specifically, we can examine the appropriate near-field of these equations and thereby derive the general form of the initial value problem for each KdV equation.
In the case of the similarity solutions it is also possible
to find the precise nature of a near-field which will match the far-field, although a neighbourhood of the origin must be excluded. The nearly plane (or plane) KdV equation can be examined as T -»• 0 by introducing the near-field variables X = (a^/6)x,
T = (a**/6)t,
Y = (a/6)y,
whence
n T T - n ^ = o, to leading order.
Thus for right-travelling waves, we have
n ~ f(x - T,Y), where f is an arbitrary function, and matching from the far-field is therefore possible if n + f (£,Y)
as
t+0.
(The plane wave corresponds to excluding the dependence on Y.) The concentric KdV equation can be discussed in a similar vein, where we now use the near-field variables ft = (a/6)2r,
T = (a/6)2t,
h = (6/a)n,
which gives rise to RR
R
R
to leading order as A -> 0 (see §6). The outwards moving wave is described by h ~ R
2
f (R - T)
as
R •* °°,
KdV EQUATION AND PROBLEMS IN WATER WAVE THEORY
where f is arbitrary.
37
Again, matching is possible if H ~ R-"1 f (£)
as
R -> 0.
The corresponding problem for the nearly concentric equation is less straightforward, and since it requires rather special initial data (which turn out to be barely realistic) we shall not pursue the analysis here: for more information, see Johnson (1980). Of rather more interest is the manner in which the similarity solutions can be matched to an appropriate near-field.
In particular, for both
A = 0,-8 in equation (8.2) (from the concentric KdV), we are able to match through a linear dispersive wave region to a linear concentric wave and finally recover the full equations (2.1) in a neighborhood of the origin (in r - t space).
The similar problem with A = 0, but for small amplitude
solutions of (8.3), has been considered by Miles (1978a).
The dispersive
wave region is most conveniently expressed in terms of a multiple-scale representation with
p = A- 3n R, R
(i
whence a + 9 3 ^ 2 „%, f°° 2^v i -* H ~ -f ( £ ) R 2 k cos(kx + 7- pkJ + 7r/4)dk, 6 7T JQ 6
for A = 0.
The constants a ,a
correspond to X
from matching to the P~2 solution.
>
0> X
<
0>
anc
*
are
obtained
In the case of A ~ -8, we obtain
2 2 ** f°° -± 1 3 H ~ ±j ( — ) k 2 cos(kx + 7 p k - 7T/4)dk, 77
A + 0,
JO
(x > 0) ,
'
with the same ordering as above, and this solution matches to the similarity solution described by (8.4). The above solutions can be matched, in turn, to a linear concentric wave solution.
It we introduce R = A" n R,
T = A~ n T
then we obtain | -j I sin(kT) JQ(kR)dk
(A = 0; R - T > 0)
and H -
~3 ( sin(kT) Y (kR)dk (A - -8;R - T <> 0), * '0 A close correspondence between our two similarity solutions is now very evident:
Jn
°
they each make use of one of the available solutions of Bessel's
38
R.S. JOHNSON
equation which is generated by the concentric wave equation. tion to — < n < —
The restric-
is necessary to obtain the correct ordering of the terms
as A •* 0; if n = -r , then the scalings used in defining R,T recover the full water-wave equations with all terms of comparable size.
The solution
of this latter set can not be determined and so the complete matching to an initial value problem (at T = 0) is not possible. If we adopt the same prescription for the similarity solution of the plane KdV equation (see (8.1)), then it is fairly easy to see that there is no regime where a linear problem is relevant.
It turns out that any
allowable scaling always recovers the original KdV equation or, when n = y as above, the full governing equations given in (2.1) but devoid of any parameters.
This full problem applies in an 0(a 2 ) neighbourhood of the
origin (in x - t space). 10.
SOME OTHER EQUATIONS OF KdV TYPE. To conclude this discussion of KdV equations, in the context of water
waves, we shall mention a few other versions which can be obtained by either incorporating new physical effects or extending still further some of our earlier work.
We start with the important property of dissipation by
the viscous stresses in the fluid.
For a relatively thin layer, where the
whole flow can be regarded as totally immersed in a boundary layer, the dominant gradients occur across the wave front giving rise to an equation of the form
This is the Korteweg-de Vries-Burgers equation, with properties reminiscent of those for the two underlying classical equations.
Equation (10.1) also
pertains for thicker layers when the surface is subjected to a wind-loading represented by Jeffreys sheltering model (Jeffreys 1925). An alternative approach to the role of the viscous terms is to assume that a thin boundary layer is formed below the wave-front and extending behind it.
If the basic flow is a fully developed viscous profile (as it
must be to avoid any inconsistences) then in a suitable parameter range, we obtain -2I Q n
3 T+
31, nnr + Jr\rcc = -o
4 5
5SS
Jo n5r (5
+ C O
V %r T
.
(10.2)
Here, as in (10.1), 0 represents the appropriate non-dimensional viscous coefficient.
In (10.2), we have used the notation introduced for equation
(3.4), but now the integrals are to be evaluated only for the Poiseuille
KdV EQUATION AND PROBLEMS IN WATER WAVE THEORY
profile.
39
Both equations (10.1) and (10.2) have very similar looking solu-
tions which adequately model the undular bore, the crucial point being that both include a dissipative term and it does not really matter the form that this takes.
More details are given in Byatt-Smith (1971a), Johnson
(1972a), Pfirsch and Sudan (1971). Higher order generalisations of the KdV equation can also be derived. For example, if surface tension effects are included then an equation of the fourth order is possible (10 3)
2nT + 3nn5 + 3 n K ? - ^^£>¥ in a special parameter range.
'
The constant W is a function of both the
surface tension coefficient and kinematic viscosity (see Byatt-Smith, 1971b), This same equation also describes the propagation of a wave on a surface which is composed of a material with density proportional to W.
(Such a
model is particularly relevant in cylindrical geometries for waves propagating through a liquid-filled flexible tube; equation (10.3) is again obtained.)
We can note that the inclusion of surface tension in the classi-
cal KdV problem merely alters the coefficient of the third order term: the original work of Korteweg and de Vries (1895).
see
Of course, by judicious
choice of the relative sizes of the various parameters, we can also produce equations wich combine the additional term in (10.3) with those in (10.1) or (10.2). Finally, we turn to two problems which are extensions of the work presented in §§3,6.
First we describe the nearly plane (2-D) KdV equation
over variable depth: from equations (2.1), with V = (3/8x, d/3y), we introduce x = (a*/5)x,
y - (a/6)y,
T - (a*/6)t,
* - (a*/6)<|>.
The bottom topography (introduced in equations (2.2)) is now written as z - b(X,Y);
X - C K , Y = ay,
whence with 5 = a
v
g(X,Y) - T;
g(X,Y) =
r D"
J
we obtain
2
dX,
D(X,Y) = 1 - b,
0 2
to leading order as a -> 0.
This can be expressed more conveniently as
(2D^ n x + \ v~h Dxn + 3D" 1 nn£ + \ D ng££)£ + D n ^ = o, (io
40
R.S. JOHNSON
where
y = y +f\
1 J
D^(
o
J
o
D"J/Z D y dX) dX . Y
Equation (10.4) may be compared with (3.6) and (6.1); the variables used in (10.4) agree with those required for general wave front propagation as given by Varley and Cumberbatch (1965). Our last example addresses the problem of how a shear flow distorts a ring wave:
this is therefore a combination of the ideas embodied in equa-
tions (3.4), (6.3) and (6.4).
If we use cylindrical polar coordinates, and
define C= (a/6)2[h(6)r - t], R = (a6/64)r, then the leading order (as a /6
2
2
[h + (h') ] f J
with 6 = 0(1),
-> 0) implies that
[(U - c){h cos 6 - h 1 sin 6} - I ] " 2 dz • 1,
(10.5)
0
where U(z) is the shear profile.
Here, c is a free parameter (by virtue of
the Galilean invariance) and the integral condition, which is a generalisation of Burns1 (1953) result, can be interpreted in terms of wave-front propagation in an inhomogeneous medium.
The solution, h(9), of (10.5),
describes the dominant distortion of the wave-front from circular (which itself corresponds to h E 1, U E c ) . The next order yields the equation for the surface profile which takes the form
a
+
A
T
H+
T He + a4 HHc + a5 HCCC = °'
(10 6)
'
where a. = a.(0) are complicated functions related to (10.5) (c.f. equations (3.3) and (3.4)).
The properties of (10.5) and (10.6) are currently under
investigation, but some novel features are already evident.
For example, if
U(z) > 1, then equation (10.5) predicts a critical layer but only for some 6 in a neighbourhood of the 'back1 of the ring wave.
The critical layer
moves off the bottom surface, rises to a maximum height and then drops down again, as 9 varies.
Furthermore, the usual singularity found in the trans-
verse velocity component for a wave moving obliquely over a shear is removed by the curvature effects.
The presence of the critical layer, for certain
6, is transmitted to the surface wave through the functions a.(0).
We may
note, however, that in the context of inverse scattering theory, there would seem to be very little immediate possibility of solving equation (10.6).
KdV EQUATION AND PROBLEMS IN WATER WAVE THEORY
41
REFERENCES ABLOWITZ, M.J. and SEGUR, H. (1977) Asymptotic solutions of the K dV equation, Stud. Appl. Math. 57_, 13-44. AIRAULT, H. (1979) Rational solutions of Painleve" equations, Stud. Appl. Math. 6^, 31-53. ANKER, D.A. and FREEMAN, N.C. (1978) Interpretation of three-soliton interactions in terms of resonant triads, J. Fluid Mech. 87, 17-31. BENJAMIN, T.B. (1962) The solitary wave on a stream with an arbitrary distribution of vorticity, J. Fluid Mech. 12, 97-116. BENJAMIN, T.B. and FEIR, J.E. (1967) The disintegration of wave trains on deep water, Part I: Theory, Jo Fluid Mech. 27, 417-430. BENNEY, D.J. and NEWELL, A.C. (1967) The propagation of nonlinear wave envelopes, J. Math. Phys. 46, 133-139. BENNEY, D.J. and ROSKES, G.J. (1969) Wave instabilities, Stud. Appl. Math. 48, 377-385. BEREZIN, Y.A. and KARPMAN, V.I. (1964) Theory of nonstationary finite-amplitude waves in a low-density plasma, Sov. Phys. JETP 19, 1265-1271. BURNS, J.C. (1953) Long waves in running water, Proc. Camb. Phil. Soc. 49, 695-706. BYATT-SMITH, J.G.B. (1971a) The effect of laminar viscosity on the solution of the undular bore, J. Fluid Mech. 48, 33-40. BYATT-SMITH, J.G.B. (1971b) Waves on a thin film of viscous liquid, AICHE J. 17, 557-561. CALOGERO, F. and DEGASPERIS, A. (1978) Solution by the spectral-transform method of a nonlinear evolution equation (including the cylindrical K dV equation), Lett. Nuovo Cim. 23, 150-153. CANDLER, S. and JOHNSON, R.S. (1981) On the asymptotic solution of the perturbed K dV equation, Phys. Lett. A86, 337-340. DAVEY, A. and STEWARTSON, K. (1974) On three-dimensional packets of surface waves, Proc. R. Soc. Lond. A338, 101-110. DRYUMA, V.S. (1974) Analytic solution of the two-dimensional K dV equation, Sov. Phys. JETP Lett. 1£, 387-388. FREEMAN, N.C. (1980) Soliton interactions in two dimensions, Adv. Appl. Mech. 2£, 1-37. FREEMAN, N.C. and DAVEY, A. (1975) On the evolution of packets of long surface waves, Proc. R. Soc. Lond. A344, 427-433. FREEMAN, N.C. and JOHNSON, R.S. (1970) Shallow water waves on shear flows, J. Fluid Mech. 42. 401-409.
42
R.S. JOHNSON
HASIMOTO, H. and ONO, H. (1972) Nonlinear modulation of gravity waves, J. Phys. Soc. Japan 33, 805-811. HIROTA, R. (1973) Exact N-soliton solutions of the wave equation of long waves in shallow water, J. Math. Phys. 14, 810-814. JEFFREYS, H. (1925) On the formation of water waves by wind, Proc. Roy. Soc. Lond. A107, 189-205. JOHNSON, R.S. (1972a) Shallow water waves on a viscous fluid - the undular bore, Phys. Fluids 11, 1693-1699. JOHNSON, R.S. (1972b) Some numerical solutions of a variable-coefficient Korteweg-de Vries equation, J. Fluid Mech. 54, 81-91. JOHNSON, R.S. (1973a) On the development of a solitary wave moving over an uneven bottom, Proc. Camb. Phil. Soc. 73, 183-203. JOHNSON, R.S. (1973b) On an asymptotic solution of the Korteweg-de Vries equation with slowly varying coefficients, J. Fluid Mech. 60, 813-824. JOHNSON, R.S. (1976) On the modulation of water waves on shear flows, Proc. R. Soc. Lond. A347, 537-546. JOHNSON, R.S. (1977) On the modulation of water waves in the neighborhood of kh « 1.363, Proc. R. Soc. Lond. A357, 131-141. JOHNSON, R.S. (1979) On the inverse scattering transform, the cylindrical Korteweg-de Vries equation and similarity solutions, Phys. Lett. A72, 197-199. JOHNSON, R.S. (1980) Water waves and Korteweg-de Vries equations, J. Fluid Mech. £7, 701-719. JOHNSON, R.S. and THOMPSON, S. (1978) A solution of the inverse scattering problem for the K-P equation, Phys. Lett. A66, 279-281. KADOMTSEV, B.B. and PETVIASHVILI, V.I. (1970) The stability of solitary waves in weakly dispersing media, Soviet Phys. Dokl. 15, 539-541. KAKUTANI, T. (1971) Effect of an uneven bottom on gravity waves, J. Phys. Soc. Japan 30, 272-276. KARPMAN, V.I. and MASLOV, E.M. (1978) Structure of tails produced under the action of perturbations on solitons, Sbv. Phys. JETP 48, 252-259. KAUP, D.J. and NEWELL, A.C. (1978) Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory, Proc. Roy. Soc. Lond. A361, 413-446. KNICKERBOCKER, C.J. and NEWELL, A.C. (1980) Shelves and the K dV equation, J. Fluid Mech. 9£, 803-818. K0RTEWEG, D.J. and de VRIES, G. (1895) On the change of form of long waves advancing in a rectangular canal, Phil. Mag. 39, 422-443.
KdV EQUATION AND PROBLEMS IN WATER WAVE THEORY
43
LAKE, B.M., YUEN, H.C., RUNDGALDIER, H. and FERGUSON, W.E. (1977) Nonlinear deep-water waves: theory and experiment, Part 2, J. Fluid Mech. 83, 49-74. MADSEN, O.S. and MEI, C.C. (1969) The transformation of a solitary wave over an uneven bottom, J. Fluid Mech. 39, 781-791. MAXON, S. and VIECELLI, J. (1974) Cylindrical solitons, Phys. Fluids 17, 1614-1616. MILES, J.W. (1977a) Obliquely interacting solitary waves, J. Fluid Mech. 79_, 157-170. MILES, J.W. (1977b) Resonantly interacting solitary waves, J. Fluid Mech. 79, 171-180. MILES, J.W. (1978a) An axisymmetric Boussinesq wave, J. Fluid Mech. 84, 181-192. MILES, J.W. (1978b) On the second Painleve transcendent, Proc. Roy. Soc. Lond. A361, 277-291. PFIRSCH, D. and SUDAN, R.N. (1971) Conditions for the existence of shocklike solutions of the K dV equation with dissipation, Phys. Fluids 14, 1033-1035. ROSALES, R. (1978) The similarity solution of the K dV equation and the related Painleve transcendent, Proc. Roy. Soc. Lond. A361, 265-275. SATSUMA, J. (1976) N-soliton solution of the two-dimensional K dV equation, J. Phys. Soc. Japan 40, 286-290. TAPPERT, F. and ZABUSKY, N.J. (1971) Gradient-induced fission of solitons, Phys. Rev. Lett. 27., 1774-1776. URSELL, F. (1953) The long-wave paradox in the theory of gravity waves, Proc. Camb. Phil. Soc. ^9, 685-694. VARLEY, E. and CUMBERBATCH, E. (1965) Non-linear theory of wave-front propagation, J. Irist. Math. AppljLc. 1_, 101-112. VARLEY, E., KAZAKIA, J.Y. and BLYTHE, P.A. (1977) The interaction of large amplitude barotropic waves with an ambient shear: critical flows, Phil, Trans. 287, 189-236. VELTHUIZEN, H.G.M. and VAN WUNGAARDEN, L. (1969) Gravity waves over a non~uniform flow, J. Fluid ttech. 39, 817-829.
CHAPTER 3 SOLITARY WAVES IN SLOWLY VARYING ENVIRONMENTS: LONG NONLINEAR WAVES R. GRIMSHAW Department of Mathematics University of Melbourne, Parkville Victoria 3052, AUSTRALIA
1.
INTRODUCTION. It is well known that for many different physical systems weakly non-
linear long waves are described by the Korteweg-de Vries equation when the wave propagation is uni-directional and the background medium is homogeneous (e.g. WhJtham, 1974, Chapter 17). In some circumstances, the canonical evolution equation is modified from the K-dV, either in the nonlinear term, or more commonly, in the dispersive term; an example of the latter case is the deep fluid equation (BDA) derived by Benjamin (1967) and Davis and Acrivos (1967)•
Most existing theories discuss these equations for
the uni-directional homogeneous case, when it is well known that they possess N-soliton solutions, and are exactly integrable through the inverse scattering transform technique (e.g. Ablowitz and Segur, 1981, Chapter 4 ) . On heuristic grounds we claim that the canonical equation to describe weakly nonlinear uni-directional long waves in an inhomogeneous medium is the variable-coefficient K-dV equation
Here the coefficients y and X are functions of T alone.
In (1.1) if £
2
is
a small parameter measuring the amplitude of the wave, then T is £ x, and r/*x -1 E, is the convected coordinate £i/ n c~
dx - t[, where x is distance, t is
time and cQ is the linear long wave phase speed; the medium is assumed to be inhomogeneous in the x-direction on a scale £ 3 ion of £ x.
, and so c n is a funct2
The wave amplitude A is chosen so that A
wave action flux in the x-direction.
is a measure of the
Equations of the type (1.1) were
first derived by Ostrovsky and Pelinovsky (1970) for the case of a surface gravity wave travelling over variable depth (see also Johnson (1973) and Shuto (1974)); in other physical contexts, equations of the type (1.1) arise in plasma physics (Nishikawa and Kaw (1975)) and for internal gravity waves (Grimshaw (1981a)).
SOLITARY WAVES IN SLOWLY VARYING ENVIRONMENTS
45
In Section 2 we present a generalization of (1.1) which allows for more general kinds of inhomogeneity, includes transverse variations and dissipative effects, and also replaces the dispersive term in (1.1) (i.e. the term whose coefficient is A) with a general linear operator.
It will
be shown how the linear long wave dispersion relation based on the linear long wave phase speed cft allows the introduction of a set of rays, which in turn provide the natural coordinates for the evolution equation.
In
the Appendix, we outline a general method for the derivation of the canonical evolution equation.
Our starting point here is a general system of
partial differential equations and boundary conditions, with the property that their linear part represents a slowly varying waveguide. In the remainder of this article we consider that class of asymptotic solutions of the canonical evolution equation which can be described as slowly varying solitary waves.
Unlike the case when the coefficients fi
and A. in (1.1) are constants, the variable-coefficient equation does not, in general, have exact solutions.
Hence instead we assume that the coef-
ficients A and // vary slowly with respect to a locally defined solitary wave.
The asymptotic solution is constructed using a multi-scale pertur-
bation method applied directly to (2.1), a technique first used in this context for the particular case of a surface solitary wave propagating over variable depth by Grimshaw (1970) and Johnson (1973).
A more general
account of this method is given by Grimshaw (1979) (see also Gorshkov and Ostrovsky (1981), or Kodama and Ablowitz (1981)).
We shall show that this
asymptotic solution consists of a solitary wave with a slowly varying amplitude and a trailing shelf.
An alternative procedure which leads to the
same result involves an adaptation of the inverse scattering transform technique (Karpman and Maslov (1977), or Kaup and Newell (1978)). In Sections 3 and 4, we consider the slowly varying solitary wave solution of the variable-coefficient K-dV equation (1.1), supplemented by the inclusion of a small dissipative term (see (3.1)). sists of two parts:
The expansion con-
an inner expansion, discussed in Section 3, for the
structure of the solitary wave, and an outer expansion, discussed in Section 4, for the trailing shelf.
Then in Section 5, the results of Sections
3 and 4 are applied and extended to discuss the decay of a solitary wave due to dissipation.
Finally in Section 6, we outline the analogous theory
for the BDA equation. 2.
CANONICAL FORM FOR THE EVOLUTION EQUATION. We shall consider waves propagating horizontally in a waveguide.
In
46
R. GRIMSHAW
general a waveguide can support an infinite set of vertical modes; here we select just one of these modes, whose linear long wave phase speed is c n , and seek an equation for the evolution of its amplitude.
Long waves of
small amplitude are characterized by two small parameters; one, a, is a measure of nonlinearity and is typically the ratio of the wave amplitude to the vertical scale of the waveguide; the other, £, is a measure of dispersion and is typically the ratio of the vertical scale of the waveguide to the horizontal wavelength. The evolution equation for the wave amplitude
A(T , £, rj) is
| p + l J A | | + A X ( | | ) + VT(A) + B = 0 2
iS. = 6 9 A 8?
an 2 '
(2.1a)
(2 ib)
Here T is a slow time variable, £ is a phase variable which describes slow spatial modulations in the wave direction relative to a frame which moves with the linear long wave phase speed c , and r) describes modulations transverse to the wave direction. X(A) is a linear operator describing dispersion, given by
w
= - i [ f(k)exp(ik£) ?(A)dk,
f
(2.2a)
exp(-ik£)Ad£.
(2.2b)
J-00
For instance, when f(k) = k 2 , X (A) - d2A/dE,2 and (2.1a) is an equation of K-dV type; the balance between nonlinearity and dispersion requires that 2 a = e . These equations typically occur when the waveguide has limited vertical extent, and there is an analogy with shallow water theory. By contrast, when the waveguide is abutted by a deep passive region, it is typically found that f(k) = |k|, and (2.1a) is a generalization of an equation derived by Benjamin (1967) and Davis and Acrivos (1967) for internal waves in deep fluids. requires that a = £.
The balance between nonlinearity and dispersion now T(A) is a linear operator describing dissipation,
given by T(A) = ^
I
g(k)exp(ikS)2(A)dk.
(2.3)
J 00
For instance, when g(k) - -k 2 , T(A) is d2A/d£2 and (1.1a) is a K-dV-Burgers equation.
When dissipation is principally due to frictional drag at a ri-
gid boundary, g(k) = (-ik) '
(Grimshaw, 1981).
pose that g(k) is equal to (-ik)
In general, we shall sup-
where m > 0; for a dissipative process,
we require Re{vg(k)} to be non-negative, and the sign of V i s chosen
SOLITARY WAVES IN SLOWLY VARYING ENVIRONMENTS
accordingly. here.
47
More general forms can be chosen, but will not be considered
Finally the coefficients y,A,v
and 8 are known functions of T, de-
termined from the structure of the vertical mode in the waveguide.
Equa-
tions of the type (2.1a,b) occur in a variety of physical situations; for internal waves in stratified shear flows a general derivation is described by Grimshaw (1981a).
An abstract rendering of this derivation is given in
the Appendix. In a homogeneous and horizontally isotropic medium x = £Ott, £=e(x-cot) where x is a coordinate in the wave direction and t is the time coordinate, while rj = ea
y where y is a coordinate normal to the wave direction.
coefficients y, A,V
The
and 6 are constants in this instance; in particular
note that the frequency u) for linear waves of wave number k in the x-direction, and I in the y-direction is given by 2
a) - kc Q + -5£- - Akf(k) - ivae 1-m g(k).
(2.4)
This dispersion relation provides an immediate interpretation of the coefficients A, V and 6.
Further it may be shown that 6 = y c Q .
Note that in
the dissipative term V is a non-dimensional measure of dissipation, and its relation to a dimensional quantity is such that the latter is 0(a£ m ) . In general, the coefficient y of the nonlinear term has no such simple interpretation. In an inhomogeneous medium the background environment varies on length and time scales of order $~ .
If the effect of this slow variation is to
be comparable with the effects of weak nonlinearity and dispersion, then 3 2 we must choose 3 = £(X (i.e. £ case).
for the K-dV case, or £
for the deep fluid
The main consequence of this is that the linear long wave phase
speed c_ will now vary on these long length and time scales, thus defining a set of rays whose trajectories in time and space define the wave direction and phase.
We introduce slow time and space variables T - 8t, X ± - 3x±,
(2.5)
where i = 1,2 corresponding to the number of horizontal space variables in the waveguide.
Then we introduce the phase variable £ - | S(T, X ± )
(2.6a)
and
0) - - |f, K ± - ||- .
(2.6b)
Now the linear long wave phase speed c^ will be a function of T, X. and if the medium is not isotropic, it will also be a function of the wave direct-
48
R. GRIMSHAW
ion £. which is the unit vector in the wave direction (£. = K . / K where K
2
i
= KiKi).
ii
The phase n then satisfies the dispersion relation 0) - Kc Q (T, X ± , £ ± ) - W(T, X ± , K ± ) .
Note that W is homogeneous in K with degree one.
(2.7)
Equation (2.7) is a par-
tial differential equation for the phase, whose solution is obtained by the method of characteristics or rays. long a ray.
Let T be a time-like parameter a-
Then the ray equations are dT
dX
-
do> _ 8W dT " dT '
i
.. dK
dz
8W
,
i _ 8W " " ax ± •
.
(2#8b)
Here V. is the long wave group velocity, and is related to c n by the expression
rv
(2 9)
'i-^+s^tj-vV-
-
Note that V.K. = c~ and so the component of group velocity in the wave direction is equal to the phase speed; in an isotropic medium the group velocity is equal to the phase velocity.
The ray equations (2.8a,b) are
solved subject to the initial conditions T, X ± = T Q (T i ),
X^C^),
H = H 0 (T ± ),
on T = 0,
(2.10a)
on T = 0.
(2.10b)
These equations determine an initial manifold on which T
are interior co-
ordinates; (2.10b) determines the initial phase on this manifold, and the for 0), K are are determined from (2.10b) and the dispersion initial values for relation (2.7).
The solution of (2.8a,b) is X
I
=
X (T
I J>>
K
=
I
K
<2-u>
I< T J>>
where we are adopting the convention that X Q = T, T Q = T, K = -a) and a capital letter index takes the values 0, 1, 2. The first set of equations in (2.11) determine T (i.e. T , T ) as functions of X , and if J is the J
Jacobian of this mapping, then
1
1 3J
i.
8V
i
,9 1 9 v
(2 12a)
j-fr = w:> where
-
. J - det[ -^
].
Next we observe that as a consequence of (2.8a) and (2.9),
(2.12b)
SOLITARY WAVES IN SLOWLY VARYING ENVIRONMENTS
^ -
49
-co + K ± V ± = 0,
(2.13)
and so H = - n ( T j) is an interior coordinate on the initial manifold. out loss of generality we may now identify!
With-
with H, and define r|=ot
T?.
It may then be shown that the evolution equation is (2.1a,b) provided that 2 the amplitude A is chosen so that A is a measure of the wave action flux along a ray tube (i.e. the quantity which is conserved in linear long wave theory in the absence of dissipation), and is proportional to J (wave action density).
The coefficients y, A, v and 6 generally differ from their
counterparts in a homogeneous medium by geometrical factors, and are functions of T ; however, the dependence on T-, T j
i
is passive, and they may be
z.
effectively regarded as functions of T only.
An outline sketch of the de-
rivation of (2.1a,b) is given in Appendix A (see also Grimshaw 1981a). As an illustration of the foregoing theory, suppose that the environment varies in just one space direction, and the wave is propagating in this direction.
Then we may assume that c~ = cn(X ) , and the solution of
the ray equation is
rxi « •
E=
T
" T> T 2 =2 V
2
(2 14)
'
The Jacobian J = c , the coefficient 6 = — c Q , and the coefficient \ differs from its counterpart in a homogeneous medium (i.e. when defined by 3 2 (2.4)) by the factor 1/c case.
in the K-dV case, and l/cQ
in the deep fluid
In the absence of transverse variations and dissipative effects, it
can be seen that for this uni-directional case,(2.1a,b) reduces to (1.1). 3.
SLOWLY VARYING SOLITARY WAVE (K-dV):
INNER EXPANSION.
Now let us consider the one-dimensional K-dV equation in an inhomogenous and dissipative medium.
Thus in (2.1a,b) we shall assume that A is in2 dependent of r\ and so B is zero; for the K-dV case, f(k) = k and (2.1a) becomes
-~+yA|f+X-^4 + av T(A) = 0. 3r
dE,
(3.1)
3?2
Here we shall further assume that the coefficients ]i, X and V are slowly varying functions of T , and so are functions of the slow variable s = ax.
(3.2)
Consistently with this hypothesis we have introduced a factor O into the dissipative term. variation.
Here a is a small parameter characterizing the slow
Because of the variable coefficients (3.1) does not in general
possess exact solutions analogous to the N-soliton solutions of the constant coefficient K-dV equation.
Instead we shall construct the slowly
50
R. GRIMSHAW
varying solitary wave asymptotic solution, using a multi-scale perturbation method applied directly to (3.1) (see Grimshaw (1979), Gorshkov and Ostrovsky (1981), or Kodama and Ablowitz (1981)).
An alternative procedure which
leads to the same result involves an adaptation of the inverse scattering transform technique (Karpman and Maslov (1977), or Kaup and Newell (1978)). To be specific we shall suppose that y and A are constant and v is zero for s < 0, so that in s < 0 equation (3.1) has a uniform solitary wave solution, Our aim is to determine the continuation of this solution into s > 0. We put A - A Q (0, s) + OA 1 (0, s) + ... ,
(3.3a)
1 fs 0 = £ - ± b(s f )ds f ,
(3.3b)
where
and b Q + ab
+ ... .
(3.3c)
Here 0 is a rapidly varying phase and b can be interpreted as the "speed" of the wave.
For the case when there is also a slow variable a£, see
Grimshaw (1979).
At leading order, we obtain the solitary wave solution
where
A Q - a seen2 36,
(3.4a)
ya = 3b Q - 12A$ 2 .
(3.4b)
Here a is the solitary wave amplitude, depends on s and is the principal quantity we wish to determine.
Note that it is permissible to center the
solitary wave at 0 = 0 , as the term in 0 arising from b- is equivalent to a phase shift. At the next order, we find that
-bo 9^ +
v
h
(A
iV
+ x
7 3 Ai = N r
(3 5a)
-
00 where
o. A
N
l
=
dA
~~os
~. b
tfA
l~a¥~ m
We note here that T(A ) with g(k) = (-ik)
V T(A
0)#
(3#5b)
can be expressed in either of
the alternative forms
_a3m r(m+l)sin™ f° sechV+u) ^ J 0 u a or,
m
a(- TTfr )
±f a i g
^
m
±nteger>
(3#6a)
2 sech 30,
if m is an integer,
(3.6b)
SOLITARY WAVES IN SLOWLY VARYING ENVIRONMENTS
T(An) = a3 0
r — cos(k36 - ±-TTm)dk. 2 Josinh(^)
51
(3.6c)
In (3,6a) the integral is divergent, and must be interpreted in the Hadamard sense of finite parts.
Integrating once, and using (3.4a,b), we find
that
2
2
^ Al
1
*
bn{-A- + (3 sech ee)A. + - ~ - — T T > - N- , U l X 43 36 -1N 1 = cQ + | N ^ e .
(3.7a) (3.7b)
Two independent solutions of the homogeneous equation (3.7a) are f - sech236 tanh 36, g = $6f + -~ cosh236 + ~ - sech236.
(3.8)
Note that f is proportional to 3A0/96. The method of variation of parameters then gives the general solution of (3.7a) as xf
+ C2g + ~ j Nxg d6 - I j Nxf d6,
(3.9a)
where 3W = - 3 j b Q .
(3.9b)
The general solution (3.7a) thus contains three undetermined functions of s, namely C , C , and C«. The solution (3.9a) contains terms which are exponentially large, proportional to exp(±236) as 6 -*• ±°°. These terms must be removed, and hence,
wc2
f N-fdS - - [ J0
L
J-oo
N.f d6, 1
(3.10a)
and so N f d6 = 0.
(3.10b)
Recalling that f is proportional to 3A0/36, (3.10b) may be integrated by parts, and hence we obtain
|-( f f Ajrae) + v f A T(A )d = o. J —00
(3.11)
J —00
Here we have used (3.5b) for N . Recalling (3.4a) we see that (3.12a) and I J _oo
where
A Q T(A Q )d6 = m o a 2 3 m " 1 ,
(3.12b)
52
R. GRIMSHAW
•r The constant m
T(sech2(())d(().
(3.12c)
can be evaluated using Parseval's theorem for Fourier trans-
forms; for instance, when m = y, we find that m~ = 0.72, and for m = 2, m n = - •=-=• .
Since k is determined in terms of a by (3.4b), (3.11) is the 1 2 Noting that r A is the wave
equation which determines the amplitude a.
action flux along a ray tube, we can identify (3.11) as the equation for wave action.
Next, it can be shown from (3.9a) that A± -> A*
as
where
6 + ±«, /•zoo
b A
0 l " ~C0 " J
Since C
(3.13a)
N d9#
l
(3.13b)
is still free, we now choose it so that A
is zero, and conditions
ahead of the solitary wave are undisturbed (Johnson, (1973).
It then fol-
lows that, using (3.5b),
b Q A~ = [ N d 8
(3.14a)
J —oo
b A
O l
=
" "55" (J
A d6)
O
*
(3.14b)
This latter equation shows that the solitary wave loses mass to a shelf trailing behind the wave whose amplitude at the rear of the solitary wave is A .
r
From (3.4a) we see that
J — oc
Proceeding to the second order, we find that
- b o "fe + v l e
+x
^ A2 " N2>
(3 16a
- >
where 8A N
2
Equation (3.16a) for A
3A
§1" *i-§T
3A + b
i-§e-
vT(A
i>-
(3 16b)
'
can be solved in a manner similar to that for A ;
the removal of exponentially large terms leads to the condition
[ J —oo
N 2 A Q d9 = 0.
(3.17)
On substitution of (3.16b), and some rearrangement using (3.5a) we obtain
|j ([ A^dS) + V [ {A2 T(A ) + AQ T(A1)}d6 + \ bQ(A~)2 = 0. « _OO
J ^00
(3.18)
SOLITARY WAVES IN SLOWLY VARYING ENVIRONMENTS
53
This can be identified as the second order wave action equation, recalling that (3.11) is the first order such equation.
Assuming, for simplicity,
that v is zero, (3.18) becomes ~
a b
aA~
?
and so serves to determine the phase speed correction b . zero, an analogous equation again determines b .
When v is not
At this point we have
determined the amplitude a by (3.11), the phase speed correction b (3.18) and the coefficients C remaining coefficient C
by
and C 2 in the expression (3.9a) for A .
The
and the second order phase speed correction b« are
determined by proceeding to the third order, although we note that it is permissible to put C- equal to zero as its effect is equivalent to the phase shift provided by b_.
Also note that
Ao ^ r - ~ ^ + 2 b Q 9s
A* 2
as 0 + ±°°.
(3.20)
Assuming that conditions ahead of the solitary wave are undisturbed, we have already put A- equal to zero, and may likewise put A_ equal to zero. However, A
is then determined (3.14b) and similarly we can find an expres-
sion for A~. 4.
SLOWLY VARYING SOLITARY WAVES (K-dV):
OUTER EXPANSION.
At this point, we observe that although the calculation of the first two terms in the expansion (3.3a,b,c) has been completed, certain non-uniformities have arisen.
The most severe of these are those due to the terms
A" and A~ (see (3.13a) and (3.20)), although there are also non-uniformities due to terms with algebraic decay as 0 •> -°° (arising from the dissipa2 tive term), and terms proportional to 0 exp(±2$0), and 0exp(±2$0) as 0-*±«> in A .
Our procedure for dealing with these is a generalization of that
used by Johnson (1973).
Since we have chosen A
and A_ equal to zero, let
us first consider the simpler case when 0 -> +°° (i.e. ahead of the solitary wave).
As 0 ->• °°, it may be shown from (3.4a) and (3.9a) that A ^ 4a exp(-230)(l + a(H*3 2 0 2 + H*30 + . . . ) + 0(a 2 )},
(4.1a)
where H* = C2bo3)~1( i -|^ ) , and H + = (2b 3)" 1 (- - — + — — 1 0 a 9s 28 9s
(4.1b)
54
R. GRIMSHAW
Thus we regard (3.3a,b,c) as an inner expansion, and seek an outer expansion of the form A ^ 4a exp{- ~ f(r,s; a )} + 0{exp(- ^j- )}, r - a£.
(4.2a)
To match with (4,1a), we require that f ^ 23© - H*3 2 © 2 + a(-H*3© + . . . ) + 0(a 2 ),
as 0 -> 0+,
(4.3a)
where 0 = ae «
b(s f )ds f .
-
r
(4.3b)
Jo The outer expansion is to hold in the region 0 > 0.
Since A is now a
function of r and s alone, equation (3.1) becomes |A + ^ |A + O2X d^ + Q m ds 9r 8r3 Substitution of (4.2a) then gives
If >3>+ ^H+ ^ f fi+ ^ < If >m 9r
Next we put
f = fQ + Of1 + ... ,
(4.6)
Substitution into (4.5) gives 8f
3f-
3fn
2 3f.
8f
0 .
0
3 9fn 3 2 f n
a
3fn
_
The initial conditions are obtained from (4.3a,b) and are f 0 - 0,
-97=2)8
on
0Q = r - j
f x - -23 [ b 1 (s t )ds t
b()(sl)dst=0,
on 0 Q = 0.
(4.8a) (4.8b)
The solution is obtained by the method of characteristics, and is f
3
= 163J(s )
fS s
X ds,
(4.9a)
0
where
fS0 2 fS U f f - I b o (s )ds = 123 (sQ) j X ds.
(4.9b)
SOLITARY WAVES IN SLOWLY VARYING ENVIRONMENTS
55
Here S-. is an initial condition on 0. = 0, and is related to s and r by (4.9b) which is the equation for the characteristics.
The solution remains
valid provided the characteristics do not intersect, which is the case provided that fs
3ft
3 | | (s0) j
s
Ads * X(so)B(so).
This is certainly satisfied initially (0 is negative.
(4.10)
0 = 0 ) , and is always true it 33/3s
However, if 33/3s is positive, (4.10) is violated at s = s.
say, and the solution cannot be continued beyond this point.
When the
characteristics do not intersect, f~ develops into a similarity solution as r,s •»• <»; it is obtained by fixing s Q and letting s -> °°,
f
9
r
o^I
\IO
{3
f
x ds}
— 1 /9
<4 n)
•
'
in agreement with the similarity solution found by Johnson (1973).
Turning
next to (4.7b) for f , we find that log{
[
+ (26(8Q))m [ Js
Vds.
(4.12)
o
As for f , the solution fails when (4.10) is violated and near s = s , A C 1/2 is proportional to (s
- s)
.
As 6 -*- -«, it may be shown from (3.4a), (3.9a) and (3.20) that the inner expansion is
O2( f ^3T+ A") 0(o3) b 2 + 2
4a exp(236){l + a(H^ZQZ + H^86 + ...) + 0(a2)
,
(4.13a)
where
H- = -(2b o 3)- 1 (|f ), H^ = (2bQ3)"1(- j | | + jg | | - V cos 7Tm(23)m + 2b;L3) - j A^ .
(4.13b) (4.13c)
Note here that the terms with algebraic decay arise due to the non-local nature of the dissipative operator T when m is not an integer; when m is an integer, none of these terms are present and instead, terms proportional to V appear in H".
We now seek an outer expansion in 0 < 0 of the form
56
R. GRIMSHAW
A = aB(r, s) + 4a exp{ ~ g(r, s; a)} + O(exp( ?f )), where r is again given by (4.2b),
(4.14)
To match with (4.13a), we require that
0 3A1 2 B * A + ~ - - ^ - + a(A + ...) + 0(a ) 1 bg os 2
g ^ 230 + H~3 2 © 2 + a(H~30 + . . . ) + 0(a 2 ),
as 9 + 0-,
(4.15b)
Substitution into (4.4) gives
|^ + ayB Ior5 + a2A ^34 + amv T(B) = 0, 9s
(4.16a)
ar
and 3
}
- + 12Aa
If
+ 4\& cos
TTm
l
+
2
4paB | ^
( 4I
m
0.
+ O(a)
(4.16b)
dr
Next we put
B + a B]
o
^
g = 80
n
+ ... >
(4.17a)
,,
and substitute into (4.16a,b) respectively.
(4.17b)
Using the matching condition
(4.15a), we readily find from (4.16a) that B Q - B Q (r),
where
b()(sl)dsf) = ^ ( s ) .
BQ( j
(4.18)
Recalling that r = oE, (4.2b), and the definition of 5 (2.6a), we see that the shelf, represented by B
to leading order, is a linear long wave tra-
velling with speed c n behind the solitary wave, and whose amplitude at the rear of the solitary wave is A-(s).
It extends from 0 = 0
(the solitary
wave) to r = 0, as for s < 0, A- is zero since in s < 0, a and 3 are constants.
B- is found in a similar way.
For B , some care is necessary when
evaluating the dissipative operator
since, for the case when m is not an
integer, it is not a local operator.
A composite expansion must be used
(compare Grimshaw, 1981b).
However, it may be shown that B can be determ
mined in the form
= y± - i m b e
o
2TWsm™ + m7r
||
_i
| p -
2
SOLITARY WAVES IN SLOWLY VARYING ENVIRONMENTS
57
Finally, the determination of g from (4.16b) is similar to the determination of f from (4,5).
Indeed gQ satisfies the same equation (4.7a) as f~,
with the same initial condition (4.8a).
Hence, g 0 is given by the right-
hand side of (4.9a) where now 0 < 0 and so s < s Q ; gQ is thus the continuation of f
into 0 < 0 along the same characteristics.
In particular, if
the characteristics intersect in 0 > 0 (< 0) they will not intersect in 0 < 0 (> 0 ) . f...
g- is determined by an equation similar in form to that for
However, it should be noted here that, unlike the region 0 > 0, the
exponentially small terms (i.e. 4a exp( •— g)) are dominated by all the terms in the expansion of B as a •*• 0. At this point we recapitulate that the slowly varying solitary wave is given by the inner expansion (3.3a,b,c) centered on the wave trajectory 0 = 0 , where A Q is given by (3.4a,b) and A a is determined from (3.11). expansion is (4.2a).
by (3.9a).
The wave amplitude
Ahead of the solitary wave (0 > 0) the outer
Behind the solitary wave (0 < 0) the outer expansion
is (4.14) where the dominant term OB- describes the trailing shelf and is a linear long wave of speed c n whose amplitude at the rear of the solitary wave is A- (2.14b).
Provided that the coefficients y, X and V are smooth
functions of s, the slowly varying solitary waves is an asymptotic solution of (3.1) as O -*• 0.
However, some care is needed if, for any particular
value of s, the coefficients y, X and V fail to be smooth functions of s. Suppose, for instance, that this is the case at s = 0.
It has already been
assumed that y, X are constant and V is zero in s < 0.
Let us now suppose
that y, X and V are smooth functions in s > 0, but that at s - 0, y and X have a discontinuous derivative, and v is discontinuous.
Then A_ and A,
remain well defined in s > 0 but B Q(r) is discontinuous at r = 0. unacceptable, and requires a boundary layer at r = 0.
This is
The boundary layer
coordinate is R
= a "2/3 r = a l/3 c .
(4.20)
Since B is now a function of s and R, (4.16a) becomes |2. + O 1 / 3 yB | | * A -5-1 + 0 m / 3 v ds 3R 3R3
T(B) - 0.
(4.21)
The matching conditions are, to leading order in cr, B + BQ(0+) = A~(0+)
as
R -> °°,
(4.22a)
and B -> 0
as
R •> -oo.
(4.22b)
58
R. GRIMSHAW
To leading order in a, the required solution is (Kodama and Ablowitz, 1981) Ai(p')dpf,
(4.23a)
p - R(3 f M s ^ d s 1 ) " 1 ' 3 .
(4.23b)
B = ^(0+) * —00
where
Equation (4.23a) describes a similarity solution, which for p > 0 is exponentially close to the constant one, but for p < 0 is a sequence of decaying oscillations.
An additional boundary layer region is required when
both r and s ->• 0; we shall not give details here as the solution in this initial region has been obtained by an adaptation of the inverse scattering transform technique (Karpman and Maslov (1977), or Kaup and Newell (1978)). If y, A and v have higher order discontinuities, so that A-(0+) is zero, but some derivative of A- has a discontinuity at s - 0, then a higher order boundary layer solution is needed. 5.
DECAYING SOLITARY WAVE (K-dV). In this section we shall apply and extend the results of the previous
section to discuss a solitary wave decaying due to dissipation.
We shall
suppose that y, A and v are constants, and that in the dissipative operator T (2.3), g(k) « (-ik)m, where 0 < m < 3 and m + 1 (for m = 1, T is nondissipative and simply gives an 0(aa) correction to the phase speed c Q , and for m = 3, T is again non-dissipative and gives an 0(a) correction to the coefficient A). We shall be particularly interested in m = 2 (the K-dVBurgers equation) and m = -r- (boundary-layer friction).
The equation for
the amplitude (3.11) then has the solution (see (3.12a,b) "1/m,
(5.1)
where 3 Q is the initial value of 3 at s = 0 and m Q is defined by (3.12c). The amplitude is determined from (3.4b) and so is proportional to 3 • Note that for a dissipative process Vnu > 0. (3.14b) and (3.15).
The shelf A, is found from
3vm
A" = - ^ 3m""1 .
(5.2)
o/ Hence as s •> °°, the amplitude decays as s" cays as s - m / m .
and the shelf amplitude de-
Since m < 3 the amplitude a decays faster than the shelf;
indeed, for m < 1, the shelf amplitude increases as s •* °°.
The solitary
wave position is given by 0 Q - 0 (4.8a); using (3.4b) we find that
1 +V
m
° ^
C
2 r
- [1 + T vmmn3nsj *
, if m * 2,
(5.3a)
SOLITARY WAVES IN SLOWLY VARYING ENVIRONMENTS
1r
59
?
f- = log{l + vmn3^s} if m = 2.
(5.3b)
Note that for 2 ^ m < 3, r -* °° as s -• °° and so the solitary wave travels to infinity (albeit with a decreasing speed).
However, for 0 < m < 2
r -*• r^ (a finite constant) as s •+ °°, and so the solitary wave stops in this frame of reference (i.e. travels with the linear long wave phase speed c ). The shelf is given by (4.18); using (5.2) and (5.3a,b), we find that
B
e
O-V r 3vm
{ i Vm+
3Vm BQ =
r }y = ~
vm n r -—- } eexpix { P
- 33 Q
Thus for 0 < m < 1, the shelf B decreases with r.
-v- c (m-2)
if
.»> 2
^
m = 2.
(5.4b)
increases with r, but for 1 < m < 3, B Q
Since V is a constant in s > 0, and by hypothesis is
zero in s < 0, the similarity solution (4.23a) is needed at r
0.
It is apparent from these expressions that the shelf amplitude, given by (5.2) at the rear of the solitary wave and by (5.4a,b) elsewhere will surpass the solitary wave amplitude (5.1).
This indicates a breakdown of
the asymptotic solution described in Sections 3 and 4. introduction of a longer scale for T than a then on a scale for s of 0(a
.
The remedy is the
First, suppose that K m < 3 ;
) , the solitary wave amplitude is compar-
able with the maximum shelf amplitude of 0(a) which occurs at r = 0. we put
Hence
m+2 g=
o
m/2
s = a
2
T,
(5.5)
and seek a solution of the form (compare (3.3a,b,c)) A = GA(8, s ) , where
1-m /\
e=
and
a 1 / 2 e = aj
a
f J
b = ob.
(5.6a) (5.6b)
0 (5.6c)
On substituting into (3.1), we find that
3s m-1 2 Using an expansion parameter a , the analysis of Section 3 may now be 39
86
dQ
repeated and we again obtain a slowly varying solitary wave with a trailing
60
R. GRIMSHAW
shelf.
Indeed the solution is precisely that obtained in Section 2 when it
is rewritten in the new variables s and 6. For the outer expansion corresponding to (5.6a,b,c) we first consider 2 <: m < 3 and put
m-2 9
m /0
r = a
r = a
£,
(5.8a)
and A = QA(s, r ) .
(5.8b)
The scale for r is determined from (5.3a,b) which locates the solitary wave position in r, s coordinates. *
A,
On substituting into (3.1) it follows that ^
1
m(m-l) 2
|l + PA $ + a"" i i + a 8s
8r
v T(5) = 0.
g£3
(5.9)
For the shelf this is to be solved in the region behind the solitary wave (0 = d~T~ § < 0) , with an initial condition as s -*• 0 determined by matching with the solution (5.4a,b).
By contrast with the solution obtained in Sec-
tion 3 where the shelf was shown to be a linear long wave, here (5.9) shows that the solution is a nonlinear long wave, and is given implicitly by 2-m A = B Q (a
2
(r - uAs) = BQ(r - aysA)
where B (r) is given by (5.4a,b).
(5.10)
It is readily shown that this matches
correctly with the inner expansion determined from (5.7); the details are similar to the corresponding discussion in Section 3.
Note from (5.10)
that the effect of nonlinearity is to increase the magnitude of the shelf as s increases.
Further it may be shown from,. (5.10) that A is 0(a m-z
z
) for
r of 0(1) for m > 2, but that when r is 0 ( a T ~ ) (i.e. r is 0(1)) the nonlinear effect increases to the extent that the solution (5.10) develops a singularity, when as is 0(1), which occurs first at r = 0.
When m = 2,
r = r and A is 0(a ' ) as r + °° (strictly when r is 0(|log a|)), but there is again a singularity which develops first at r = 0.
We resolve this
singularity by invoking a similarity solution in the vicinity of r = 0. Hence we put and
s
n
6s,
where 6 = on9
p = r(3a 2 As)" 1 / 3 ,
A = (a6) 2/3 A(s , p ) .
(5.11a) (5.11b)
n Here p is the similarity variable (4.23b), and for p of 0(1) r is O(cr
6
) when the evolution scale is given by s . n to (4.4), we find that
On substitution in-
SOLITARY WAVES IN SLOWLY VARYING ENVIRONMENTS
3s
n
3s
n
op
^
/o
A \ 1 / 3 op
(3s A)
*
3s
^ 3
r/o
n 3p
61
•vxni/3
6 (3s A)
2/3 1/3 When 6 = 1 , the boundary layer has width r of 0(0 ) , A is 0(a ) since A is 0(a), and the linear similarity solution (4.23a) pertains. However, 1/2 when 6 = 0 , A is 0(1) as A is 0(a), and the similarity solution is now nonlinear; it will have the structure F(p,s .•* r ) is a slow variable. power series in (3s
wnere
r = a
p(As . y
As s .„ -> 0, F may be constructed in the form of a
whose leading term is (4.23a). But as s- ,--*» ' -2/3 G p>s F is constructed in the form (3s-,/2^) ^ l / ? ; r^ w h e r e G i s a P o w e r series in (3s . A ) . As the scale for evolution is increased further to 1/3 6 = a, A is 0(1), but A is 0(a ' ) ; the boundary layer width is now r of 0(a ) . This solution is to be matched to (5.10) as p •><», and also to 1/2 the corresponding solution when 6 = 0 as s -* 0 (or equivalently s
/9
•* °°).
/0X)^'^,
The similarity solution will continue to dominate the structure
1/ ^
2
near r = 0 as the evolution scale is increased to 6 = O , when the boundary layer has width r of 0(1). Finally, when n = m(3 - m ) ~ (since 2 ^ m < 3, 2 ^ m(3 - m) < °°) , the dissipative term in (5.12) becomes of equal significance to the other terms, and simultaneously the similarity solution z—m has a width r of 0(<j3-m ) which is exactly comparable to the location of the solitary wave (r of 0(a
m
) ) . For longer scales the whole solution
is dominated by dissipation. For 1 < m < 2, r (5.8a) is a boundary layer variable, and the outer expansion is obtained from 0A(r, s) where a */2 |A + o y A |A + a 2 A ^ A + a m v os or •-, J
j
.
Q<
dr In this case, the shelf is a linear long wave given by (5.4a).
In the vi-
cinity of r = 0, the similarity solution described above (5.11a,b) again pertains and evolves in a manner analogous to that described above. however, when the critical scale n = m(3 - m) 1 -1 j < m(3 - m)
Now,
is reached (sinceol < m < 2 , m
< 2 ) , the width of the similarity zone is r of 0(a3-m ) 9
and
the similarity solution is still a boundary layer solution confined to the vicinity of r = 0,
It is linked by the shelf which occupies the zone
0 < r < r , to the solitary wave which is located at r . oo
— 0 (Q
oo
Next we consider the case 0 < m < 1; then on a scale for s of m 3-m ) , the solitary wave amplitude is comparable with the shelf ampli-
tude at the rear of the solitary wave, which, in contrast to the case 1 < m < 3, is now the location of the maximum shelf amplitude. Note also
62
R. GRIMSHAW
that this scale for s is now faster that a
. In the vicinity of r = 0,
the similarity solution described by (5.11a,b) again pertains, but for all values of the scale index n up to m(3 - m) er zone of width 0(a^"
m
).
is confined to a boundary lay-
In the vicinity of the solitary wave, there is
another boundary layer zone, within which we put S = a 3 ~ m s, r - r = O 3 ~ m R
(5.14a)
A - G 3 ~ m A(S, R)
(5.14b)
and On substitution into (4.4), we find that 3| | + yA | | + A ^ | + v T(A) = 0.
(5.15)
oR All terms in the equation are equally significant; the initial condition for (5.15) as s -> 0 is the slowly varying solitary wave (3.3a) where 3 and Ar are given by (5.1) and (5.2) respectively.
Between these two regions,
where 0 < r < r^, the shelf solution is given by the linear long wave (5.4a).
On longer scales for s the whole solution is dominated by dissi-
pation. We shall summarize this discussion by pointing to the contrast between the cases m = 2 (the K-dV-Burgers equation) and m = — (boundary layer friction) . When m = 2, the solitary wave travels to infinity, with its amplitude decaying as s
. Trailing behind the wave is the shelf (5.4b) which
is adjoined at r = 0 to a developing similarity solution. of 0(a
For the scale s
) , the similarity solution has absorbed the shelf zone, and over-
taken the solitary wave. When m == —, the solitary wave travels a finite -4 distance to r^, with its amplitude decaying as s . Trailing behind the wave is the shelf (5.4a) which grows as (r^ - r ) " r = 0 to the developing similarity solution.
, and is adjoined at
For the scale s of 0(a
),
the wave amplitude has fallen to the level of the shelf amplitude, while the similarity solution is still confined to the region r of 0(a
)„ In
both cases for longer scales, the solution is dominated by dissipation. 6.
SLOWLY VARYING SOLITARY WAVE (BDA). As a contrast to the theory developed for the K-dV equation we shall
give a brief outline of the corresponding theory for the deep fluid equation, or Benjamin-Davis-Acrivos (BDA) equation.
Thus, we shall assume that
in (2.1a,b) A is independent of T\ and so B is zero, and that f(k) = |k|. Then (1.1a) becomes
| ~ + y A - | | + A J C ( | | ) +ov T(A) = 0,
(6.1a)
SOLITARY WAVES IN SLOWLY VARYING ENVIRONMENTS
63
where - ±
\
|k|exp(lk5) 3(A)dk.
(6.1b)
Here the coefficients y, X and v are functions of s = err (3.2).
The con-
struction of the slowly varying solitary wave has been described by Grimshaw (1981b) and we shall only give the main results here.
We again seek
an inner expansion of the form (3.3a,b,c) where now A Q = a(l + (BB) 2 }" 1 ,
(6.2a)
where ya = 4b Q - 4A$.
(6.2b)
At the next order A_ satisfies an equation analogous to (3.5a). since
However,
«£ is not a local operator, the method of solution is more compli-
cated than that given in Section 3 for the K-dV equation. has been shown by Grimshaw (1981b) that equation
Nevertheless, it
(3.11) (the wave action
equation) again holds, and that there is again a shelf A- given by (3.14b). Here we find that f°° i
f
2
o
J—oo
AQ T(AQ)d6 = T T n ^ a 2 ^ 1 ,
(6.3b)
J _o
where
(6.3c)
fV
and
e=
T'
(6-3d)
J —oc
Hence the wave action equation for the determination of the amplitude is on using (6.2b) ^
(^
) + 4X,m()aem = 0 .
(6.4)
The amplitude of the shelf at the rear of the solitary wave is given by
A" = - i 2 | - ( £ ) . 1
b Q <3s
(6.5)
y
Note that the shelf amplitude is independent of the variation of the wave amplitude.
At the second order we again obtain (3.18) which serves to
determine b .
Finally for the outer expansion we refer to Grimshaw (1981b).
The main result is that the dominant term in the shelf behind the solitary wave is again the linear long wave given by (4.18).
Also, when y and A
have discontinuous derivatives at s = 0, a boundary layer and similarity solution is needed at r = 0.
64
R. GRIMSHAW
Next we shall suppose that y, X and V are constants and consider the decay of a solitary wave due to dissipation.
The solution of (6.4) is then
3 = 3Q{1 + 4vmm o $Qsr 1 / m .
(6.6)
Note that apart from a constant factor this is identical to the K-dV case (5.1).
However, the amplitude is now given by (6.2b) and so is proportion-
al to 3.
The significant difference from the K-dV case is that now A" = 0.
(6.7)
Thus here the decaying solitary wave has no shelf region, and there is consequently no need to consider the longer scales considered in Section 5. A different kind of decay which can occur for the BDA equation is that due to radiation damping.
This arises when the deep fluid region adjacent
to the waveguide can support outwardly propagating waves.
For instance,
suppose that the operator jl in (6.1a) is replaced by
=
" h f ^ " m2 ) 1/2ex P( ik ?> 3(A)dk,
(6.8a)
J —00
2
2 1/? - in ) x / ^ > 0, 2 2 1/2 Xk Im (k - in r > 0.
where either
Re(k
or
(6.8b)
The Fourier components with |k| < m correspond to radiating waves.
Assum-
ing that m is small it may be shown that (3.11) becomes (Grimshaw 1981b)
fc T
J
—oo
* —m
The shelf amplitude is again given by (3.14b) or (6.5).
Evaluating the
last term in (6.9), we find that (6.4) is replaced by
where g(x) = 1 + ^
{L2(X) - I 2 (X)}.
(6.10b)
Here L« is a modified Struve function, and I ? is a modified Bessel function. For x •> 0, g(x) » 1 and as x ->• °°, g(x) ~ 3x~2.
Thus when v is zero and X
and y are constants, (6.10a) gives a linear rate of decay for large amplitudes and an algebraic decay for small amplitudes. APPENDIX:
DERIVATION OF THE EVOLUTION EQUATION.
Suppose'first that the coordinate system consists of a vertical coordinate z which varies across the waveguide, and horizontal coordinates x-,
SOLITARY WAVES IN SLOWLY VARYING ENVIRONMENTS
x 2 which vary along the waveguide, while t is the time.
65
In order to de^
scribe long waves, we re-scale the horizontal coordinates and the time, and we put xj. = £x I , where x~ = t, etc.
(A.I)
Next we propose that the physical system is described
by L ( p r -j| ; X I , z; a, e)v + M = 0,
(A.2a)
where
Pl Here we recall that X
= -£r .
(A.2b)
= eax T (2.5), and describe the slow variation due
to the background environment.
In (A.2a), L is a linear operator, consist-
ing of a set of differential equations and boundary conditions, operating on the vector v, while M is a nonlinear term.
In (A.2a) the dependence on
a arises due to the slow variation of the environment, while the dependence on £ describes higher order dispersion. We introduce another set of slow 1/2 coordinates x" = £a x and note that the next member of this sequence is xf'f= X .
Then we seek a solution of the form v = v(?; x1^, Xz; z; a ) .
(A.3)
The equation to be satisfied is then
3L + a{
82L
8 V
3?: ~$t
+ eL1Qv + where
L = L(K
o
+
e2
3K
3 V
2 9pT3pT 9XT 3f} + L2ov +
M
+
•••
aL
=
01 v °>
i If 'fe; x r z;0>0)-
(A.4a)
(A 4b)
-
Here L ni is the first term in the expansion of L with respect to a (i.e. dL -5— ) , and L ., L o n are the first and second terms in the expansion of L
dot
1U ZU
with respect to e; note that in the K-dV case L..-. is zero. At the leading order, the solution is v = av Q = ar(K]. ||- ; Xj, z)A(5; x1^, XJ , where
L Q ( p r - ~ ; Xv
z)r( Pl ; X-j., z) = 0.
(A.5a) (A.5b)
66
R. GRIMSHAW
Thus r is the right null vector for L Q and (A.5b) is satisfied only when the dispersion relation p Q + W(X l5 is satisfied (compare (2,7)),
P±)
= 0,
Note that L
(A.6)
is an operator in z alone de-
scribing linear long waves and this is reflected in the homogeneity of the dispersion relation (A.6).
Next we introduce an inner product |w(-p x ), v(p t )},
(A.7)
which typically takes the form of an integral with respect to z (i.e. acrossthe waveguide), together with some boundary terms.
We shall assume that L^
is self-adjoint with respect to this inner product
|Lou, v } - { u , L o v(.
(A.8)
Consider now the inhomogeneous equation L Q v + f - 0.
(A.9)
A necessary and sufficient condition for a solution is
{r, f } - 0.
(A.10)
For all p_, we define D( P l ;
Xj.) - Jr, L Q r|.
(A.ll)
Clearly the dispersion relation (A.6) is equivalent to D = 0. Next, we put v - a(vQ + a 1 ' v± + av 2 + . . . ) , and substitute into (A.4a).
(A.12)
At the next order, we obtain
8L 8 v
o o
4
This is an equation of the form (A.9) and the compatibility equation (A. 10)
gives ^-r^T
-0
on D - 0,
(A.14a)
=
.
(A.14b)
or ^
^
0
Here we recall that V. is the long wave group velocity (2.8a), and (A.14b) shows that to leading order the amplitude A propagates with the group velocity.
At the next order, we obtain an equation for v« of the form (A.9)
and it may be shown that the compatibility condition (A.10) leads to (see Grimshaw, 1981a)
SOLITARY WAVES IN SLOWLY VARYING ENVIRONMENTS
3D
3A
,1
3
, 3D
57
3D_ + {r , M} - T(A)] • ;A -t- v-r— ) T^A;J + tr, n/ 2
x
2
2 3 P l 8 P j 3 x^3x Here n = 1, except in the K-dV case (L
J
0.
is zero) when n = 2.
The dissipa-
tive operator is given by
v |2- T(A) = {r, LQ1r}A - \ {r, -J- ( -^ )r}A.
(A.16)
At this point, we introduce the ray coordinates T , £ and r\9 where we recall 1/2 that T- = aE, and T = a n- !t may then be shown that the first group of terms in (A.15) is
Throughout these equations, p_ is the operator K T -r-p , and in particular 3D 3D -r— is an operator in £. However, since D is homogeneous in p , -r— is a dpQ
I
dpQ
homogeneous operator in £, and by a combination of adjusting the d e f i n i t i o n of A and integrating (A,15) with respect to £, we may replace (A.17) with dD d\
118
,
8D .
" ^ 3 7 " 2 J af( J ^ ) A "
dD
^
T.
T(A)
.
.
(A 18)
'
.
'
From this expression, we can identify |j T ~ | A as the wave action flux, and defining a new amplitude A as the square root of this expression, we see that (A.18) reduces to the first and fourth terms in (2.1a).
Next the
nonlinear term {r, M} will generally take the form of the second term in (2.1a), while the dispersion term {r, L rtr}A will become the third term in nU (2.1a); the details of the calculations involving these two terms requires a knowledge of the specific physical system being considered (for an application to stratified shear flows, see Grimshaw , 1981a). be shown that the last term in (A.15) takes the form 3D _ , 3B 1 32W 92A ^ - - B , where — = - ^ _ ^ - ^ . -1/2 Introducing the transverse coordinates ri = a T_(X ) , £f = a
T (X,) = a
Finally, it may ,A 1 Q N (A.19)
?, and noting that the homogeneity of W implies that
K^d W/Bi^dK. is zero, it follows that in (A.19) A contains only derivatives with respect to r). Hence 95
3n 2
68 where
R. GRIMSHAW . ~ 0 <S - I 9 W 2 2 6 " 2 8K i 9K j 9X ± 3Xj '
(A.20b)
and so the terms in (A. 19) correspond to the last term in (2.1a) and (2.1b). REFERENCES ABLOWITZ, M.J. and SEGUR, H. (1981) transform, SIAM, 440 pp.
Solitons and the inverse scattering
BENJAMIN, T.B. (1967) Internal waves of permanent form in fluids of great depth, J. Fluid Mech. 29^, 559-592. DAVIS, R. and ACRIVOS, A. (1967) J. Fluid Mech. 29^ 593-607.
Solitary internal waves in deep water,
GORSHKOV, K.A. and OSTROVSKY, L.A. (1981) Interaction of solitons in nonintegrable systems: direct perturbation method and applications, Physica 3D 112, 428-438. GRIMSHAW, R. (1979) Slowly varying solitary wave. equation, Proc. Roy. Soc. A368, 359-375.
I Korteweg-de Vries
GRIMSHAW, R. (1981a) Evolution equations for long, nonlinear internal waves in stratified shear flows, Studies Applied Math. 65, 159-188. GRIMSHAW, R. (1981b) Slowly varying solitary waves in deep fluids, Proc. Roy. Soc. A376, 319-332. GRIMSHAW, R. (1970) The solitary wave in water of variable depth, J_. Fluid Mech. 42^, 639-656. JOHNSON, R. (1973) On the asymptotic solution of the Korteweg-de Vries equation with slowly varying coefficient, J. Fluid Mech. 60, 313-324. KARPMAN, V.I. and MASLOV, E.M. (1977) Perturbation theory for solitons, Zh. Exsp. Teor. Fiz. 73, 537-559. KAUP, D.J. and NEWELL, A.C. (1978) and in slowly changing media: Roy. Soc. A361, 413-446.
Solitons as particles, oscillators a singular perturbation theory, Proc.
KODAMA, Y. and ABLOWITZ, M.J. (1981) Perturbations of solitons and solitary waves, Studies Applied Math. 64, 225-245. NISHIKAWA, K. and KAW, P.K. (1975) Propagation of solitary ion acoustic waves in inhomogeneous plasmas, Phys. Lett. 50A, 455-456. OSTROVSKY, L.A. and PELINOVSKY, E.N. (1970) Transformation of surface waves in fluids of variable depth, Izv. Atmospheric arid Oceanic Physics 6, 934-939. SHUTO, N. (1974) Nonlinear waves in a channel of variable section, Coastal Eng. in Japan 17, 1-12. WHITHAM, G.B. (1974) 636 pp.
Linear and Nonlinear Waves, Academic Press, New York,
CHAPTER 4 NONLINEAR WATER WAVES IN A CHANNEL M. C SHEN Department of Mathematics and Mathematics Research Center University of Wisconsin Madison, Wisconsin 53706
1.
INTRODUCTION. In this paper, we consider some approximate equations for the study
of nonlinear water waves in a channel of variable cross section.
For fi-
nite amplitude waves, a system of shallow water equations are given; for small amplitude waves, we present a K-dV equation with variable coefficients.
Some of their applications are discussed.
Some problems deserving
more study are mentioned at the ends of the following two paragraphs and in the conclusions to Sections 3 and 4. One of the interesting problems of water waves in a sloping channel concerns the breaking of a wave moving toward a shoreline, the development of a bore, and the movement of the shoreline after the bore reaches it. For the two dimensional case corresponding to a rectangular channel of variable depth, the bore run-up problem was studied by Keller et al. (1960), Ho and Meyer (1962), and Shen and Meyer (1963a,b) on the basis of shallow water equations (Stoker, 1957).
Later Gurtin (1975) derived a criterion
for the breaking of an acceleration wave in a two-dimensional channel, and his result was extended by Jeffrey and Mvungi (1980) to the case of a rectangular channel of variable width and depth.
We generalize Gurtin1s re-
sult to predict the breaking point of an acceleration wave in a channel of variable cross section and review some existent results regarding the bore run-up problem for a rectangular channel with a uniformly sloping bottom. Needless to say, the use of shallow water equations for the study of bore propagation may be open to criticism.
The issue would be settled if we
knew the precise conditions for the validity of shallow water equations. Up to date, the shallow water equations for a two-dimensional channel with analytical initial data have been justified by Kano and Nishida (1979), and for the three-dimensional case with a priori assumptions on the free surface by Berger (1976).
At present we may accept shallow water equations
as model equations, and the bore run-up problem for a general channel certainly deserves further investigation.
M. C. SHEN
70
Another application of our results deals with the development of a solitary wave in a channel of variable cross section.
Recently there have
been discussions on the so-called infinite mass dilemma, which arises from the formation of a shelf behind the solitary wave.
If the shelf were ex-
tended to infinity, then infinite mass would be created or annulled by a perturbation on the solitary wave.
A study of this problem may be found
in Miles (1979) and Knickerbocker and Newell (1980), based upon the K-dV equation for a rectangular channel of variable depth or width (Kakutani, 1971; Johnson, 1973; Shuto, 1974).
We shall establish a global existence
theorem for the solution of the K-dV equation for a general channel as a consequence of the existence results due to Kato (1975, 1980).
It follows
that the shelf, if formed behind the solitary wave in a general channel, can only be finite.
A rigorous justification of the validity of the K-dV
equation here should be an important contribution to the theory of water waves.
Work in this direction has been done by Berger (1974) and Nishida
(private communication). 2.
SHALLOW WATER EQUATIONS AND THE BREAKING OF A WAVE. We consider the irrotational motion of an inviscid, incompressible
fluid of constant density under gravity in a channel with a boundary defined by h*(x*, y*, z*) = 0, where z* is positive upward and x* is in the longitudinal direction (Figure 1 ) .
Fig. I.
ACROSS
SECTION OF THE CHANNEL
71
NONLINEAR WATER WAVES IN A CHANNEL The governing equations are V* • q* = 0,
(2.1)
V* xq* - 0,
(2.2)
p(q* + q* • Vq*) = - V*p* + g t* subject to the boundary conditions
(2.3)
n* + q* • V*£* = 0 t*
(2.4) at
C* = -£* + n*(t*,x*,y*) = 0,
p* = 0
(2.5)
q* . y*h* = 0
at
h* = 0.
(2.6)
Here V* - (3/3x*, 3/3y*, 3/3z*), q* » (u*, v*, w*) is the velocity, t* is the time, g = (0,0,-g) is the constant gravitational acceleration, p is the constant density, p* is the pressure, and z* = r\* is the equation of the free surface. assumptions.
To derive the shallow water equations, we make the following The channel boundary is convex, sufficiently smooth, and var-
ies slowly in the longitudinal direction; the magnitude of the transverse velocities is much smaller than that of the longitudinal velocity. As suggested by Friedrichs (1948), we introduce non-dimensional variables t = e~ 1 / 2 t*/(h/g) 1 / 2 ,
(x,y,z) = (3" 1/2 x*/H, y*/H, z*/H),
n = n*/H,
(u,v,w) = (u*/(gH) 1 / 2 , 3**v*/(gH)1/2, 3 * V ( g H ) 1 / 2 ) ,
h = h*/H ,
where 3^ = L/H and L and H are respectively the horizontal and transverse length scales.
In terms of them, (2.1) to (2.6) become u x
+ v + w =0, y z '
3 u = v , y x* u
(2.7)
u = w , v = w , z x' z y
t
+ uu + vu + wu + p = 0, x y z *x
t
+ uv + w + wv + 3p = 0 , x y z *y '
v
(2.9) v
w + u w + v w + w w + 3(p + 1) = 0, t x y z z at
z= n
P - 0, uh x
+ vh + wh = 0, y z
(2.8)
(2.10)
v
y
(2.11)
(2.12) (2.13)
at h = 0.
(2.14)
72
M. C. SHEN Assume that u,v,w,p and 3 possess an asymptotic expansion of the form
and substitute (2.15) in (2.7) to (2.14).
The equations for the zeroth
approximation are u
+ v
0x
0y
+ W
(2
0z = °»
V = u0z " °> u
0t
+ u u
0 0x + P0x + v0Uoy
V
(2 17
' >
+ W U
(2
0 0z = °'
p
0z " -1'
" °'
'18) (2 19)
'
n
0t + V o x + V o y "w 0 = °'
(2 20)
'
at
z-
Q
P o = 0,
U h +
+
0x W
'16>
(2.21)
at
V z " °'
h=0
'
(2 22)
-
As seen from (2.17), (2.19) and (2.21), u Q is a function of t,x only, and P 0 = -z + n Q .
(2.23)
This implies n is also a function of t,x only.
It follows from (2.17),
(2.18), and (2.23) that
V
+
Vox + %x - °'
(2 24)
'
Now we integrate (2.16) over a cross section D of the channel, apply the divergence theorem and make use of (2.20) and (2.22) to obtain
II
(v
0y
+ w
oz)dydz
=
-u0xA(t'x) =
" U 0 L h x (h y
+ h
z >
ds +
(n
0t
By rearranging the terms, we have n ot +u o n Ox +u Ox A(t,x)/B(t,x)-[u o
f
j
9
9-1/9
n o t +u o n O x +u O x A(t,x)/B(t,x)-[u o /B(t,x)]j h x (h y +h z Z )
where A(t,x) is the area.
i/Z
ds = 0,
(2.25)
B(t,x) is the width and L is the wetted bound-
ary of the cross section D (Figure 1 ) .
(2.24) and (2.25) form a system of
nonlinear equations, which may be used to model bore formation and its subsequent development in a channel of variable cross section. In the following we extend Gurtin's method to the case of a general channel.
The assumptions made are the following:
NONLINEAR WATER WAVES IN A CHANNEL
(1)
(2)
73
U Q J ^ Q are continuous
the first and second derivatives of u. and ru possess at most jump discontinuities,
(3)
u~ = ru = 0 ahead of the wave.
Denote the value of a function f immediately behind the wave front by f • Hereafter we also drop the subscript 0.
From assumptions (1), (2), we have
u" = r)~ = 0.
(2.26)
By total differentiation, u~ = -cu~ ,
n~ = -cn~ ,
where c is the speed of the wave front.
(2.27)
From (2.24), (2.25), and (2.26),
it follows that u" + n" = 0, t
vT + U~A"7B"~ = 0.
X
t
(2.28)
X
Comparing (2.27) and (2.28), we have -
-1 -
ut = c \
-1/9
,
c - (A /B )
± / Z
.
(2.29)
Now we differentiate (2.24) with respect to t and (2.25) with respect to x, and evaluate the equations behind the wave front. n
Then we eliminate
and make use of the expression 2 — — 2 — c u x t - ufct = c d(u x )/dx - cdu t /dx,
to obtain -2c d d ^ r V d x + ( D " ) " 1 [c f - r~(B~«] + 3c
x
= o,
where 2 2 —1/2 i v(h + h ) ds. x y z '
Hence, aQc
±/Z
[(3/2)aQ
c •'x o
D/Z
exp
T (2A ) •'x o rx -
L
dx1 dx + 1]
- _i
r (2A ) L dx,
exp
x
(2.30)
J
where a^ is the initial value of r) at x = x Q ,
We call x = £ a shoreline
if A"(£) = 0, but B~(£) ± 0, and let I(x) = (3/2)
fX
-5/2 fxl - -1 c D / Z exp r (2A ) dx1 dx.
X O
'X O
74
M. C. SHEN
Suppose a < 0. o
If I(&) • °°, then r\ = °° and the wave breaks before it x
reaches the shoreline.
If I(i) ^ °°, then either the wave breaks before it
reaches the shoreline or it breaks at the shoreline. If I(&) ^ °°, then the wave breaks at the shoreline.
Next suppose a
> 0,
Otherwise if I(&) = °°,
we evaluate the limit of r\~ given by (2.30) as x •+ I and obtain lim n x - (2/3)[-(d")f/4 + r"/2B"] x=: ^
Hence the wave will never break if (d~) f is finite at
where d~ - A"/B~. x = I.
(2.31)
However, for channels of variable cross section the equilibrium
water surface may converge to a point and this case is also of interest. Assume again a
> 0, I(£) - «.
If B""(Jl) - d~ (I) - 0 and (d~)' is finite
at x - i9 we assume h(x,y,z) - -z + g(x,y) = 0, and F
-
h (h
+ h
ds :
)
where y = -b, , b« are the endpoints of the width B (x). It follows from (2.31) that lim rf - (2/3)[-(d~)74 + g x /2] x=sjl , and the wave will never break. 3.
RUN-UP PROBLEM. We consider a bore propagating toward a shoreline in a rectangular
channel with a uniformly sloping bottom.
On the basis of the shallow water
equations, we can find a fairly complete solution of the bore run-up problem, which appears to be the only one available to date.
The bore path
from the point of breaking to the shoreline may be approximately determined by Whitham's rule (Whitham 1958) of which a justification was given by Ho and Meyer (1962).
Here we shall only consider the movement of the shore-
line after the bore reaches the shore. The shallow water equations for a rectangular channel of variable depth are obtained from (2.24) and (2.25) as u
+ uu
t n
+ri
=
0,
+ [u(n + d )] A.
(3.1)
xx O
- 0,
(3.2)
X
where we also drop the subscripts for u and ri and d
= -yx, y > 0.
sume t = 0 when the bore reaches the shoreline x = 0.
We as-
It was shown by Ho
and Meyer (1962) that u tends to a positive limit u° as the bore approaches the shore.
Let
NONLINEAR WATER WAVES IN A CHANNEL
75
c 2 = n + dQ, a = 2c + u + yt - u°,
(3.3)
3 = 2c - u - yt + u°.
(3.4)
In terms of a and 3, (3.1) and (3.2) can be expressed as x a = (u - c)ta,
Xg = (u + c)tg.
(3.5)
By cross differentiation of (3.5) and making use of (3.4), we have taf3 + 3(ta + t3)/[2(a + 3)] = 0.
(3.6)
If we introduce the canonical variables, a = (a + 3) 3 / 2 t a ,
b = (a + 3) 3 / 2 t 3 ,
(3.7)
(3.6) yields a system of equations (a + 3)a 3 = -3b/2,
(a + 3)ba = -3a/2.
(3.8)
Z = a - b.
(3.9)
Let Y = a + b, It follows from (3.8) that Y a 3 = 15Y/[4(a + 3) 2 ],
Z a 3 = 3Z/[4(a + g) 2 ].
(3.10)
In the a,3-plane, we prescribe sufficiently smooth data on 0c:a=0, 0 < e < 3 ^ 3*, and on CD:3 = 3*, 0 < a < a* (Figure 2 ) . However, the precise nature of the data is immaterial. to(0,3) < 0, and that as 3 + 0
We require only t (a,3*) > 0,
along a = 0,
p
lim a = a°
>
0,
lim x = lim t = 0,
lim y
=
u° > 0,
b(0,3) = 0 ( 3
where the existence of the positive limit a
9/2
(3.11)
),
and the behavior of b(0,3) for
small 3 were established by Ho and Meyer (1962).
Our discussion will be
based upon (3.10) instead of (3.6) so that the derivations will be simpler, but we have to assume smoother data on the boundary. Let G* be the region {(a,3), 0 < e < 3 ^ 3 * , 0 < a < show that the mapping from G be single-valued.
a*}. We first
in the a,3-plane to the x,t-plane ceases to
Since the Jacobian of transformation from the a,3 coor-
dinates to the x,t coordinates is 9(s,t)/9(a,3) = -2c"2ab, where c > 0 in G , we look for the lines a = 0 or b = 0 in G .
The Riemann
functions of (3.10) for Y and Z respectively are (Courant and Hilbert, 1962)
R(a,3; a',31) = F(-3/2, 5/2; 1, z) = F ^ z ) , F(-l/2, 3/2; 1, z) = F 2 (z),
(3.12)
M. C. SHEN
76
c N.
X
D(a?/3*) Ge
0 \
Fig.2.
Go IN THE
E
a , £ - PLANE
where z
= >(a' - a)(3f - 3)/(af + 3f)(a + 3),
and the Riemann representations for Y and Z in G
are found as
* ± (a,B) - * i (0,B*)F ± [a(e* - B)/(B*(ui + B) ]
J
a1 a '
i
t _ a)(3* - 3)/(a1 + 3*)(a + 3)]daf
* B ,(0,B l )F i [a(B l where $ 1 = Y, $
= Z,
To show that a and b change signs in G , we need
the following two lemmas. LEMMA 1.
(3.13)
As 3 + o+ along a • k3, 0 < k < K Q for arbitrary K Q , lim (a + b) - a°F1(k/(l + k)), llm (a - b) = a°F2(k/(l + k)) ,
uniformly for 0 < k < K .
NONLINEAR WATER WAVES IN A CHANNEL
77
LEMMA 2. lim (a°/2)[F (k/(l + k)) + F. (k/ (1 + k)) ] = -2a°/37r, lim (a°/2)[F (k/(l + k)) - F (k/(1 + k)) ] = a%JT. 1
Z
As seen from Lemma 1,2, a and b indeed change signs in their limiting values along a = k , and we have THEOREM 1.
Let ~G be the limiting region of G
as e -> 0.
Then there
exists both lines a = 0 and b = 0 in G, which terminate at the origin with finite slopes.
Along a line 3 = constant or a = constant, starting from
the boundary 0C or CD, the line a = 0 is encountered first.
The proof of
the Lemmas and Theorem 1 may be found in (Shen and Meyer, 1963). We choose a* so small that the line a = 0 starts from the boundary a = a*.
Denote by G
the line a = 0. x,t-plane.
the region delimited by a = o, 3 - 3*,
Our next step is to examine how G
a = a
* and
is mapped back to the
By taking limits along various directions towards the origin
in G , we find that the origin a = 3 = 0 is mapped to the parabola x = -Yt2/2 + u°t along which the water depth is zero.
t > 0,
(3.14)
As observed from (3.14), the shore-
line starts to move upwards after the bore reaches it.
At t = u /y, x
reaches the maximum value (u ) /2y and starts to move downwards and continue to recede.
However, since the line a = 0 approaches the origin with
a non-zero slope, its image in the x,t-plane can be shown to approach the shoreline eventually, and forms an upper bound of the physical admissible region.
Hence only an advancing bore which forms a part of the boundary
of the physically admissible region can catch up the shoreline.
We may
summarize our results as THEOREM 2.
The shoreline advances and recedes following the parabola x = -yt /2 + u°t,
t £ 0.
The solution breaks down again in the process of shoreline recession, and an advancing bore appears. From the above discussion, a rather complete solution for the bore run-up problem has been obtained for a two-dimensional sloping channel and one may marvel at the applicability of such simple equations to so complicated a problem.
It should be of great interest if the results could be
extended to general channels.
78 4.
M. C. SHEN K-dV EQUATION AND THE DEVELOPMENT OF A SOLITARY WAVE. We only sketch the derivation of the K-dV equation for a channel of
variable cross section; the details may be found in Shen and Zhong (1981). We introduce the non-dimensional variables -1/9
t - 3
J/
1 /9
-1/9
t*/(H/g) ± / Z ,
(x,y,z) - (3
r|, h, p and (u,v,w) are the same as before.
x*/H, y*/H, z*/H), The method used here is a
specialization of the procedure developed by Shen and Keller (1973).
We
assume that u,v,w,p and rj also depend explicitly upon a new variable 5 = 3S(t,x) where S is a function of t and x only, will be called a phase function. Then we assume that they possess an asymptotic expansion of the form
o + 3 ~ \ + 3~2<J>2 + ... And we assume that the zeroth approximation is given by
(u
=
o> V V
°' Po= "z« % = °'
The equation for the first approximation determines a Hamilton-Jacobi equation for S.
Let k = S , U) = ~St« 0) = kG(x),
Then
G(x) = ±[a(x)/b(x)] 1/Z ,
(4.1)
where a(x) is the area of a cross section D , and b(x) is the width of D , of water at rest (Figure 1 ) .
(4.1) may be solved by the method of charac-
teristics and the corresponding characteristic equations are dt/da = y,
dx/da = yG(x),
dk/da * -kyG'(x),
,. 2\
do)/da - dS/da - 0. where p is a proportionality factor.
We choose y = 1 so that O = t.
The
solutions of (4.2) determine a one-parameter family of bicharacteristics called rays, x = x(t,a 1 ), where O
is constant along a ray.
The equations for the second approxima-
tion determine a K-dV equation with variable coefficients
Vlt
+
Vlx
( 4 )
Here m Q = 2b(x),
(4.4)
NONLINEAR WATER WAVES IN A CHANNEL
n^ = 2 a ( x ) / G ( x ) ,
m9 = -[G(x)] 2
(4.5) 9
f
JLQ
9—1/9 1/Z
h (h yZ + h zZ ) x y z
—9
ds - G Z ( x ) G f ( x ) a ( x ) ,
(4.6)
m 3 = 3k[G(x)]"1b(x) - u T ^ y O : , x, y^ 0) - (|)y(t, x, y±, 0)],
(4.7)
m, — co-1
(4.8)
4
L
-1 L
79
ff JJ
2 dy dz, (V6)
Do
is the wetted boundary of D ; y = y , y are the endpoints of the width
of D ; and $ is the solution of the following Neumann problem V2<|> = k 2
in
2 = co
d)h+d)h =0 y y z z
D, o
at at
z -0 L. o
Since from (4.2) d/da = 3/9t + G(x) 9/9x,
dx/da - G(x),
along a ray, we may express (4.3) in terms of a and £
Vio
+ m n +
2 i -a V i e + V I K C
= 0)
or in terms of x and ^
To be definite, we choose G(x) S - -t +
fx J
0
-1 [G(z)] ^dz,
which i s a solution of ( 4 . 2 ) , and i t follows that OJ - 1,
k - G" 1 (x).
Note that other choices of S are also possible.
For rectangular and tri-
angular channels, the coefficients given in (4.4) to (4.8) can be explicitly evaluated (Shen and Zhong, 1981). 1)
Rectangular Channel Let d(x) and b(x) be the variable depth and width respectively.
m Q = 2b(x),
m± = 2b(x)d 1 / 2 (x),
m 3 = 3d~ 1 (x)b(x),
m,, = b f (x)d 1 / 2 (x) + d" 1 / 2 (x)d' (x)b(x)/2,
m 4 = (l/3)b(x)d(x).
80
2)
M. C. SHEN
Triangular Channel Let the two sides of a cross section D be defined by z = y, (x)y - d(x), o 1
z = -y2(x)y - d(x), where y ± (x) = d(x)/b i (x), i = 1,2. m o = 2[b1(x) + b 2 (x)] = 2b(x),
m1 =
2 d 1 / 2 ( x )b(x),
m2 = 2 d~1/2(x)[b'(x)d(x) + df(x)b(x)/2]/2, m 3 = 5d" 1 (x)b(x), 1 2 m 4 = [d" (x)/4][b(x)d (x) + (b^(x) +b 2 (x))/3]. In both cases, m- > 0, m
> 0, m, > 0 if d(x) > 0, b(x) > 0.
cases, as seen from (4,5), (4,8), m but the sign of m
For general
> 0, m, > 0 where we choose G(x) > 0,
given by (4.7) is not obvious.
We shall assume that
m_ ^ 0 is the case. Let T = [ m (xf)m " 1 (x f )dx f ,
(4.10)
A = m 3 (x)m 4 " 1 (x)n 1 .
(4.11)
In terms of T and A, (4.9) becomes
A
+
AA
•+-A
= H^T^ A
T > 0
—oo < £" < oo
(A 1 9^
subject to A(O,Q = A o ( Q ,
-«> < g < oo,
(4
.13)
where H(T) = -m o (x)m. z4
(x) - m1 (x)m, 14J
(x)m o (x) [m, (x)/m o (x) ] f . 4 3
A global existence theorem for (4.12) and (4.13) can be easily obtained by extending the existence result for the K-dV equation with constant coeffic c i e n t s due to Kato (1975,1980). Assume H(T) i s continuous and l e t H (-00,00) 2 denote the Sobolev space of order S of the L - t y p e . Since H(T)A s a t i s f i e s the conditions ( f 1 ) , (f2) in Kato (1975), we have the following
local
existence r e s u l t (Kato, 1980). THEOREM 3.
(4.12) subject to (4.13) with A e H S , S > 2 , has a unique
solution A e C[O,T f ; HS] n (^[O.T 1 ; H S ~ 3 ] ,
A(0,O -
AQ(O,
for some T* > 0; and A(t) depends upon A
continuously in the H -norm.
NONLINEAR WATER WAVES IN A CHANNEL
Hereafter, we shall denote an H -norm by || • || || • || .
81
and an L -norm by
To show that there exists a global solution in [0,T] for any T > 0 ,
we need a regularity result and an a priori estimate for | | A | L , which may be obtained by means of the first three conservation laws for the K-dV equation with constant coefficients (Lions, 1969). If A € C[0,T; H S ] is a solution to (4.12) with S > 2 and
THEOREM 4. if A(0) A
£ H
S
with S 1 > S, then A e C[0,T; H S ] with the same T.
This theorem is a simple extension of Katofs result (1980). o
To derive the H estimate for A ( T ) , T in [0,T], where T > 0, we first 4 2 assume A Q , A e H . Then we multiply (4.12) successively by A, A + 2A~£ and A + 3A^ + 6AA,. + (18/5)A^^^.^ and integrate to obtain estimates for
||A||, llAl^ and where $ T ( # ) depending upon T is a monotone increasing function with 1 2 $ (0) = 0.
For A , A e H , we may approximate A ,A by sequences of smooth-
er functions, make use of the results of Theorems 3 and 4 to complete the derivation. On the basis of the H
2
estimate and Theorem 4, we have the global
existence theorem: THEOREM 5.
For any T > 0, (4.12) possesses a unique solution
S
A c C[0,T; H ] n C^OjT; H S " 3 ] satisfying A(0,Q = A pends upon A
e H S , S > 2.
A de-
continuously in H -norm.
It is evident from the above theorem that if we prescribe A =a sech 3S at x = 0, then lim A = 0 for any T > 0.
If there is a shelf created behind
the solitary wave, it can never be extended to infinity for finite T.
We
also remark in passing that (4.3) has been used to study the fission of solitons in a channel of variable cross section (Johnson, 1973; Zhong and Shen, 1983) and a justification of the asymptotic method used should also be of interest. ACKNOWLEDGEMENTS.
The work reported here was supported in part by the
National Science Foundation under Grant MCS-800-1960 and the United States Army under Contract No. DAAG29-80-C-0041. REFERENCES BERGER, N. (1974) Estimates for the derivatives of the velocity and pressure in shallow water flow and approximate shallow water equations, SIAM J. Appl. Math. 27^ 256-280.
82
M.C. SHEN
BERGER, N. (1976) Derivation of approximate long wave equations in a nearby uniform channel of approximately rectangular cross section, SIAM J. Appl. Math. 31, 438-448. COURANT, R. and HUBERT, D. (1962) Methods of Mathematical Physics Vol. II, Interscience, New York. FRIEDRICHS, K.O. (1948) On the derivation of the shallow water theory, Comm. Pure Appl. Math. 1^, 81-85. GURTIN, M.E. (1975) On the breaking of water waves on a sloping beach of arbitrary shape, Quant. Appl. Math. 33, 187-189. HO, D.V. and MEYER, R.E. (1962) Climb of a base on a beach, Part 1, J_. Fluid Mech. 14, 305-318. JEFFREY, A. and MVUNGI, J. (1980) On the breaking of water waves in a channel of arbitrarily varying depth and width, J. Appl. Math. Phys. ZAMP 31, 758-761. JOHNSON, R.S. (1973) On the development of a solitary wave moving over an uneven bottom, Proc. Camb. Phil. Soc. 73, 183-203. KAKUTANI, T. (1971) Effects of an uneven bottom on gravity waves, J. Phys. Soc. Japan 30, 271-276. KANO, T. and NISHIDA, T. (1979) Sur les Ondes de Surface de l'Eau avec une Justification Mathematique des Equations des Ondes en Eau Peu Profonde, J. Math. Kyoto Univ. 1£, 335-370. KATO, T. (1975) Quasi-linear equations of evolution, with applications to partial differential equations, Lecture Notes in Mathematics 448, Springer-Verlag, New York, 25-70. KATO, T. (1980) The Cauchy Problem for the Korteweg-de Vries Equation, Research Notes in Mathematics 53, Pitnam, New York, 293-307. KELLER, H.G., LEVINE, D.A., and WHITHAM, G.B. (1960) Motion of a bore over a sloping beach, J. Fluid Mech. ]_, 302-316. KNICKERBOCKER, C.J. and NEWELL, A.C. (1980) Shelves and the Korteweg-de Vries Equation, J. Fluid Mech. 9£, 803-817. LIONS, J.L. (1969) Quelques Methodes de Resolution des Problems aux Limites Non Lineaires, Dunod, Paris. MILES, J.W. (1979) On the Korteweg-de Vries Equation for a gradually varying channel, J. Fluid Mech. 91^, 181-190. SHEN, M.C. and MEYER, R.E. (1963a) Climb of a bore on a beach, Part 2, J. Fluid Mech. 1(3, 108-112. SHEN, M.C. and ZHONG, X.C. (1981) Derivation of K-dV equations for water waves in a channel with variable cross section, J. de Mec. 20, 789-801. SHUTO, N. (1974) Nonlinear waves in a channel of variable cross section, Coastal Eng. in Japan 17, 1-12.
NONLINEAR WATER WAVES IN A CHANNEL
83
WHITHAM, G.B. (1958) On the propagation of shock waves through regions of non-uniform area of flow, J. Fluid Mech. 4_, 337-360. ZHONG, X.C. and SHEN, M.C. (1983) Fission of solitons in a symmetric triangular channel with variable cross section, Wave Motion, to appear.
CHAPTER 5 SOLITON BEHAVIOUR IN MODELS OF BAROCLINIC INSTABILITY IRENE M. MOROZ and JOHN BRINDLEY School of Mathematics The University Leeds LS2 9JT W.Yorks. England
1.
INTRODUCTION One of the major concerns of the meteorologist is the degree of
predictability of atmospheric motions.
The classic remarks made by
Lorenz (1963), in the now-celebrated paper in which deterministic equations were first shown to exhibit aperiodic and consequently unpredictable behaviour, that it may be impossible to predict the weather accurately beyond a few days, only too truly reflect the current state of affairs.
Although the availability of very fast computers and greater
accuracy of initial data have brought about some improvement in weather forecasts, there is still a disappointing limit on the length of time for which a weather prediction can be considered to be accurate.
Nevertheless
certain features of atmospheric motion are observed to persist for considerable lengths of time, usually associated with what are known as blocking situations (Berggren et al 1949); a notable persistent factor in another planetary atmosphere is Jupiter's Red Spot.
Among the models
proposed for these phenomena are modons (Fleierl et al 1981) and solitons (Maxworthy and Redekopp, 1976). We shall not be concerned in this article with direct modelling of atmospheric predictability; instead we shall concentrate on phenomena occurring in simple models exposing the essential physical behaviour, and demonstrate that under certain conditions, coherent persistent behaviour is possible. Cyclones, anticyclones and their associated frontal systems are a prominent feature of the mid-latitude westerlies of the Earth's lower atmosphere.
Their importance as weather-bearing systems and more gener-
ally their role in the general circulation of the atmosphere is wellknown if not yet well understood.
Their occurrence and rapidly changing
behaviour is strongly influenced by the existence of larger scale "longwaves" which are remarkable for their persistence and coherence over
SOLITON BEHAVIOUR IN MODELS OF BAROCLINIC INSTABILITY longer periods of time.
85
Both phenomena owe their existence to the avail-
ability of potential energy associated with the baroclinicity of the fluid, i.e. the non-coincidence of surfaces of constant gravitational potential and constant density, which is a possible equilibrium in a rotating system. Such an equilibrium is unstable and wave-like perturbations can develop at the expense of the potential energy if the trajectories of fluid particles are contained within the geopotentials and isopynals.
This
process is known as sloping convection or baroclinic instability, and the consequent waves as baroclinic waves. Laboratory studies of baroclinic instability have contributed much to our understanding of the properties of baroclinic waves.
Experimental
systems usually comprise two concentric cylinders containing the working fluid in the annular region between them.
The Earth's rotation is
simulated by co-rotating the cylinders and stable density stratification is achieved either by differentially heating the fluid (see Hide and Mason, 1975) or by imposing a differential shear between two immiscible fluids of different density (see Hart, 1979). An analysis of the thermally driven system reveals the existence of sixteen non-dimensional parameters which specify the behaviour of the system (Fowlis and Hide, 1965), although in practice only a small subset is of significance.
The behaviour of the flow is conventionally
summarised in a regime diagram of Thermal Rossby Number (a stratification parameter) against Taylor number (a viscosity parameter).
For a given
temperature contrast a variety of flows is observed ranging from axisymmetric zonal flow at low rotation speeds, through bands of periodic and modulated waves at moderate rotation speeds to aperiodic flow at high rotation speeds.
The review by Hide and Mason (1975) contains an
excellent account of the thermally driven annulus system;
similar
behaviours are encountered in the mechanically driven two layer model (see Hart, 1979 for a review). The development of mathematical models has proceeded in parallel with the development of laboratory models.
Mathematical models are
almost invariably infinite channel models (with a spatial periodicity imposed
to mimic an annular geometry);
the simplest are the
heterogeneous model due to Eady (1949), and the two layer model due to Phillips (1954).
The basic state for the Eady model is one of linear
vertical shear;
for the two layer model the zonal velocity is constant
(but different) in each layer.
Over the last decade a number of authors
86
I.M. MOROZ and J. BRINDLEY
have studied various aspects of the weakly nonlinear behaviour of wavelike perturbations to the basic states of both models, when the amplitude is permitted to vary slowly in both time and/or space.
Significant
contributions in this area have been made by Drazin (1970,1972) for the continuously stratified model and Pedlosky (1970,1971,1972) for the twolayer model. Depending on the orders of magnitude of various parameters which enter the formulation of the problem, it is possible to obtain a wide variety of nonlinear evolution equations, some of which are completely integrable equations with soliton solutions when friction is ignored, others admit aperiodic solutions and have strange attractors is present.
when friction
In this article was shall concentrate on the integrable
equations although it is of interest to indicate how friction modifies the form of the equations and the character of the solutions.
A fuller
treatment can be found in Moroz and Brindley (1982) . The motivation for considering baroclinic wave-packet behaviour lay partly in the experiments of Hide, et al (1977) and other experimenters who had long observed a modulation of the baroclinic wave and a recurrence property of data.
They interpreted the modulation in terms of a triad
interaction between the dominant wave, its sidebands, and the long wave. Subsequent numerical integrations performed by F a m e 11 and James (1977) failed to support this conjecture and a wave-packet model was thought to be an alternative and possibly better way Pedlosky (1972)
of viewing long-wave modulations.
had derived a coupled pair of equations describing
the evolution of a wave-packet in an inviscid two-layer model of baroclinic instability,
and Gibbon et al (1979)
were able to transform Pedlosky1s
wave-packet equations into the self-induced transparency (S.I.T.) and sineGordon equations, both of which admit soliton solutions.
Subsequently
Moroz (1981) and Moroz and Brindl'ey (1981) were able to show that the same equations arise in the three-layer model (and more generally the N-layer model) and the continuously stratified nonlinear Eady model. The material of this article draws heavily from the main results appearing in Gibbon et al (1979), Moroz (1981) and Moroz and Brindley (1981) .
§2 contains descriptions of both the continuously stratified
and the two-layer models as well as the linear stability theory for both. In §3 we indicate the nonlinear theory and show how the completely integrable equations arise.
§4 is a comparison of theoretical and
experimental results and finally in §5 we show how viscosity enters the problem.
88
I.M. MOROZ and J. BRINDLEY
y
= o
x lim _1_ f x-x» 2x J -x
and
at
y
yt
dx1
on y = + -
(2.2)
on y = + - 2
(2.3)
(2.4)
V'i'x
"(x,y)
where V2
=
B (_
3z 2 J
= x (f/9) f gr - f g , and (x,y) " xy y^x field. In the following we assume
is the non-dimensionalised pressure
,
s
1
= -s2 *
For a two
" l a Y e r model we
have ) + BY|
(x,y,
=
O
i = 1,2 (2.1)
subject to on
ix
y = 0,1
(2.2) '
-x where we have introduced the non-dimensional parameters B
Burger Number
(Stratification Parameter) (Dispersion Parameter)
, and F
Non-dimensional Boundary Slopes * Internal Rotational Froude Number
3
3-effect
(Dispersion Parameter)
s.
(i = 1,2)
(Stratification Parameter)
We seek wave-like perturbations to a simple zonal flow, writing ¥
=
if; =
-yz + \\)
for the continuous model
(Pcosh 2qz + Qsinh 2qz) e -U.y + ij/.
=
,1. ik(x-ct) (y) e
ft (-x—r )
for two-layer
. sin imry
model
(2.5) sin m7r(y+J5)
SOLITON BEHAVIOUR IN MODELS OF BAROCLINIC INSTABILITY
89
and linearise the resulting equations, we obtain characteristic relations: \, E
4q 2 c 2 + 2soq(tanhq + cothq) c + s 2 - (1 + q 2 ) Z £
=
+ q(tanhq. + cothq) 2 2 2 2 4q + B(k + m ^ ) j
for the continuous model where \
(2.6) and
a2 (a (a22 + 2P) c 2 - c [a 2 (a2 + 2 F ) (U. + U o ) - 26 (a2 + F)J
= Li
+ a 2 (a2 + 2F) u U_ + B 2 + F a 2 U 2 - $ (n 12 c
+ U j (a2 + F) 1 2 (2.6) '
for the two-layer model, where
a
= k
+ m T"
The dispersion relations are found by solving ^ The condition for marginal stability,
kc, = O ,
= O
for
c .
yields surfaces which
separate stable and unstable regions of parameter space.
These surfaces
often have a local form dominated by one particular parameter and can be conveniently represented in a two-dimensional plot of that (stability) parameter against zonal wave number stratified model,
s
(Fig.l) .
For the continuously
is the relevant stability parameter and the
condition for marginal stability is
s
2
_ -
4lq(tanhq + cothq) - (1 + q2)J
\z.n
(tanhq - cothq) for the two-layer model it is 9 XT
=
U
4B2F2 a^(4F 2 a' + )
=
U
- U
and
.
(2.7) •
The validity of a wave packet analysis rests on the existence of suitable behaviour in the linear problem.
This must yield a neutral
curve which, at least locally, takes the form shown schematically in figure 1. A ,
For small departures of order
A
of the stability parameter,
from critically, we see that a band of wave numbers of width
unstable, centred on the critical wave number, (1981) have shown that the stationary point for
k
.
A
is
Moroz and Brindley
in the "Eady" model occurs
s^ = 1 and at that point we have a coalescence of two modes with
90
IoM. MOROZ and Jo BRINDLEY
Figure 1 - Typical Neutral Stability Curve
SOLITON BEHAVIOUR IN MODELS OF BAROCLINIC INSTABILTY
identical phase speeds,
(c = -h) but different group speeds.
else on the neutral curve the phase speeds differ.
91
Everywhere
A similar result is
true for the two-layer model; Pedlosky (1972b) has shown that the stationary point is given by
U
= $/F
and at that point
c
c = U (twice). ^L
We shall see later that this is why we expect to obtain an evolution equation for the wave amplitude which is second order in space and time. The shape of the neutral curve near the stationary point therefore determines the bandwidth of excited waves;
the growth rate near the
marginal stability curve can also be computed. we have
For inviscid problems
kc. = 0(A ) .
3. NONLINEAR THEORY The results from linear theory suggest that we introduce new ables variables
X
and
T
resp , scaled respectively on the bandwidth and growth
rate of the most unstable wave, i.e. (3.1) We also require the variables amplitude
A(X.,T.)
and time.
(X ,T ) = |A|(x,t)
and assume that the
of the wave is a slowly varying function of space
The solution may then be developed as an expansion in a small
parameter, related to the departure from neutral stability.
This method
has been formalised by Newell (1972), Weissman (1979) and Gibbon and McGuinness (1981), and the result is a system of evolution equations which must be satisfied by the wave amplitude at each order of the expansion.
The coefficients of the linear terms in these equations are
identifiable as derivatives of the characteristic function, T respect to its various arguments;
, with
the nonlinear terms are specified by
the particular problem under consideration. For the inviscid baroclinic wave models we have •*A
+
=
0
(3.2)
\
-T\T(A)
(3.4)
92
I.M. MOROZ and J. BRINDLEY
A number of special cases arise as follows: (i) Marginally unstable wave packets with 3 or s For an inviscid model, ^ ^
= 0
X
and T
= 0
a
of 0(1)
everywhere on the neutral curve but
only at the critical point (Weissman, 1979).
The slow scales
are not required and the amplitude evolves according to the
second order equations
( +
° $( +C A> A 5'ANAB
where 52
is the square of the linear growth rate
N
is a positive constant
B(XJT )
is a second order correction to the basic state
and
_»1 T
g
ak
i
ft 2 _%
±
Qk Voo
l,2
1 }*2 oo
kk
are the group speeds. Gibbon et al (1979) have shown that the transformation S
=
± 1 - NB/|a| 2
R = /2 A
(3.6)
and the change of variable », ( x " c g i T )
i-i2
-"»
(X C
-
(3
'7)
results in the self-induced transparency equations R
with
S -*- ±1
#
£T
R-^0
as
=
RS
(3.8)
UI+00
.
If we assume, in addition, that R
=
4>
,
S
=
A
is real and write
icos*
,
(3.9)
then we obtain the sine-Gordon equation 4>£T = ± sin*
(3.10)
SOLITON BEHAVIOUR IN MODELS OF BAROCLINIC INSTABILITY
(ii) g (or s j
=
93
0
In the absence of sloping endwalls or 3 , the neutral curve is no longer as shown in figure 1; 1978 , figure 2) .
it is now the Eady stability curve (Drazin,
We may no longer consider a bandwidth of waves centred
at the critical point since this corresponds to choose a bandwidth centred about
k = O .
k = 0(1), which means that k-k = 0(A), P P
and the appropriate slow time and space scales are now still have
*^'a.= °
anc
Instead we
T
and
X
.
We
* obtain the nonlinear Schrodinger equation with
the roles of space and time reversed
*-\ \ (iii)
+
V- N 2 i A i 2 A
•
(3 11)
-
Neutral Waves
For neutral waves neither T 3' • T — ' by 3T i (iv)
=
7%o V i
nor *4
vanish and replacing
% 3 ' — TZ~~* again gives a nonlinear Schrodinger equation. 3x % i
Marginally unstable wave packets in the inviscid limit There are basic differences between the linear stability properties
for a model in which viscosity is initially set to zero and those for a model in which viscosity is retained and then allowed to approach zero. These differences are apparent only when dispersive effects are present. It is now no longer the case that ^
=0 o
and T* . = 0 k
on the stability
boundary, and the analysis produces an equation of nonlinear Schrodinger type. A full treatment of these cases is given in Moroz and Brindley (1982) . 4.
COMPARISON WITH EXPERIMENTS The quasi-geostrophic models of baroclinic instability behave
remarkably like the nonlinear optics problem of the interaction of a rapidly oscillating electric carrier wave with a two-level atomic medium, leading to self-induced transparency, where in various limits the NLS, S.I.T. and sine-Gordon equations all appear (see Lamb and McLaughlin, for a review).
It is also possible to compare the mechanisms operating
in the two systems. The baroclinic model can also be considered as a system with two energy levels, namely, the state of fully developed baroclinic waves and
94
I.M. MOROZ and J. BRINDLEY
the state of purely zonal flow, as determined by the function
S(£,T)
which is a measure of the available potential energy of the system. two extreme states correspond to a maximum (or upper state, when
The
S = + 1)
or a minimum (or lower state, when S = -1) of available potential energy. "2 For supercritical conditions,
G
> 0
and the upper state is zonal flow
(the lower state being fully-developed waves); for
o
< o .
the situation is reversed a2 > O
As in the S.I.T. case solutions associated with
are unstable, the asymptotic state being one of maximum available potential energy and we conclude that only the subcritical case is of importance in an infinite domain. Such results may well be appropriate for the oceans where typical wavelengths of disturbances are
0(50-100)km
in a general circulation of
5000km, and it is therefore a good approximation to permit a continuous spectrum of waves to be excited.
For laboratory experiments and the
atmosphere, typical wavelengths are of an order of magnitude larger and spatially periodic boundary conditions are more appropriate.
Exact
analytical solutions are known for the sine-Gordon equation in a periodic domain and they take the form of Jacobi elliptic functions (Matveev, 1976). Costabile et al (1978) recognised the limited value of such general expressions and instead derived nonlinear standing wave solutions arising from the nonlinear interaction of two periodic standing waves.
For a
finite Josephson transmission line three fundamentally distinct types of solution exist:
the plasma, breather and fluxon oscillation.
They
represent respectively an oscillation about zero mean, a bound state oscillation of a vortex-anti-vortex pair and the repeated reflexion at the ends of the transmission line of a fluxon which emerges alternatively as fluxon or anti-fluxon after reflexion. To ascertain which of these is appropriate to the baroclinic wave problem, Moroz and Brindley (1981) considered each solution in turn and concluded that the plasma oscillation may be the most relevant because ofits validity in the small amplitude limit when it can be expressed as the sum of two travelling modulations. Preliminary comparisons between the small amplitude approximation of the plasma oscillation and the observations reported by Hide et al (1977) give reasonable agreement.
Laboratory experimenters have long recognised
the tendency for the regular regime of baroclinic waves to persist for considerable ranges of parameter values and as Hide (1979) remarks, this
SOLITON BEHAVIOUR IN MODELS OF BAROCLINIC INSTABILITY
95
coherent behaviour is due largely to nonlinear effects rather than viscosity.
Such behaviour is certainly consistent with soliton
solutions described here. 5.
EFFECTS OF FRICTION It is customary when discussing the consequences of introducing
friction to exactly integrable equations to treat friction as an inhomogeneity and simply add an additional constant term.
For the
baroclinic wave models described above this is not a mathematically consistent approach and, depending on the amount of friction present and the relative magnitudes of all the non-dimensional parameters present in the problem, dramatically different evolution equations can arise:
an
indication of this was apparent in case (iv) above. The addition of small frictional effects in the baroclinic wave models give rise to the equations related to the Lorenz Strange Attractor equations modified by the presence of a spatial derivative (Alexander et al, 1982);
exactly the same behaviour occurs in the nonlinear optics
problem (Haken, 1975).
When strong friction is present, an equation
arises which resembles the nonlinear Schrodinger equation but which has complex coefficients, usually called the Ginzburg-Landau equation, (Moroz and Brindley, 1982).
Both of these equations are known to
possess chaotic solutions and their occurrence in this context provides a very interesting field of study in view of the recent interest in the appearance of aperiodic solutions when exactly integrable equations are perturbed (Holmes, 1981).
We summarise the evolution equations in Table
1 . We have indicated that the models of baroclinic instability described here are capable of supporting a wide variety of very different types of solution depending very sensitively on parameter values.
In
several value ranges the infinite domain solutions are of soliton type, and we are at present examining numerically the development, as the friction parameter increases, of soliton solutions to the N.L.S. equation (case (i) above) into the chaotic solution of the Ginzburg-Landau equation.
It is clear that the assumption of an infinite domain is not
a good one for direct comparison with the behaviour of baroclinic waves in the atmosphere or in rotating annular experiments, in which typical baroclinic wavelengths are more than 10% of the channel "length". Related periodic solutions exemplified by the fluxon solution of the
I.M. MOROZ and J. BRINDLEY
96
r
N.
or
6 1 s
i
1= o
0(A) "
•,'
or
3 }
| = 0(1)
=
(S.I.T.-SINE-GORDON)
!"
SOLITON SOLUTIONS BEHAVIOU R UNKNOWN
FOR INFINITE DOMAIN; DOUBLY PERIODIC
(EQUATIONS AR E N.L.S.,WITH
JACOBI ELLIPTIC
USUAL SPAC E AND TIME DEPENDENCES INTERCHANGED)
SOLUTIONS FOR 1
BOUNDED DOMAINS
_ j
r = 0 (N.L.S.) SOLITON SOLUTIONS FOR INFINITE DOMAIN F.P.U. RECURRENCE IN BOUNDED DOMAINS
' SPATIAL LORINZ EQUATIONS UNDER 11IVESTIGATION ;
r = 0(A)
i
BEHAVIO UR UNDER
j
INVESTI GATION.
r = 0(1)
EQUATIONS DELATED TO
;N.L .S., SOME COMP LEX COEFFIECIENTS
Table 1
PERIODIC AND APERIODIC SOLUTIONS ACCORDING TO PARAMETER VALUES (T.D.G.L. EQUATION)
Summary of Wave Packet Behaviour
SOLITON BEHAVIOUR IN MODELS OF BAROCLINIC INSTABILITY
97
sine-Gordon equation (Moroz and Brindley, 1981) are probably more relevant. However, the scales of baroclinic waves in the oceans and in some other planetary atmospheres certainly justify the infinite domain assumption and motivate further examination of the models described. In summary then, the baroclinic wave models provide an excellent example of a new area of physics in which solitons can arise, showing how viscosity alters the form of the exactly integrable equations occurring for an inviscid model.
Moreover the natural development of the N.L.S.
into the Ginzburg-Landau equation as viscosity is introduced, implying a change from predictable to unpredictable behaviour, suggests a new way of approaching soliton perturbation problems and a new direction of interest in the study of Hamiltonian systems under perturbation, an area of considerable interest and excitement in dynamical systems theory. REFERENCES ALEXANDER, M.E., BRINDLEY, J. and MOROZ, I.M. (1982) Baroclinic Wave Packets described by Spatially Varying Lorenz Equations, Phys.Lett. 87A 240-244. BERGGREN, R., BOLIN, B. and ROSSBY, C-G. (1949) An Aerological Study of Zonal Motion, its Perturbations and Breakdown, Tellus 1 14-37. BRINDLEY, J. and MOROZ, I.M. (1981) Aperiodic Behaviour of Baroclinic Wave Packets. Phys.Lett. 83A 259-262. COSTABILE, G. PARMENTIER, R.D., SAVO, B. MCLAUGHLIN, D.W. and SCOTT, A.C. (.1978) Exact solutions of the sine-Gordon equation describing oscillations in a long (but finite) Josephson function, Appl.Phys.Lett 32_ 587-589. DRAZIN, P.G. (1970) Nonlinear baroclinic instability of a continuous zonal flow Q.J.R.Met.Soc. 96 667-676 DRAZIN, P.G. (1972) Nonlinear baroclinic instability of a continuous zonal flow of a viscous fluid. J.Fluid Mech. 55 577-588. DRAZIN, P.G. (1978) Variations on a theme of Eady in Rotating Fluids in Geophysics (Ed. P.H.Roberts and A.M.Soward) Chap.Ill 139-169, Academic Press, London. EADY, E.T. (1949) Long Waves and Cyclone Waves, Tellus 1, 33-52. FARNELL, L. and JAMES, I.N. (1977) Where have all the sidebands gone? Meteorological Office Internal Report. Unpublished. FLEIERL,G.R.,LARICHEV, V.D., McWILLIAMS, J.C. and REZNICK, G.M. (1981) The dynamics of baroclinic and barotropic solitary eddies. Dyn.Atmos.Oceans 5 1-41.
98
I.M. MOROZ and J. BRINDLEY
FOWLIS, W.W. and HIDE, R. (1965) Thermal convection in a rotating fluid annulus: effect of viscosity on the transition between axisymmetric and non-axisymmetric flow regimes. J.Atmos.Sci. 22^ 541-558. GIBBON, J.D., James, I.N. and Moroz, I.M. (1979) An example of soliton behaviour in a rotating baroclinic fluid. Proc.R.Soc.Lond.A 367 219-237. GIBBON, J.D. and McGuinness, M.J. (1981) Amplitude equations at the critical point of unstable physical systems. Proc.R.Soc.Lond. A 377 185-219. HART, J.E. (1979) Finite amplitude baroclinic instability. Mech. 11 147-172. HIDE, R. (1980) Regular baroclinic waves: some recent work. Summer School Report 217-221. Woods Hole, M.A. HIDE, R. and MASON, P.J. (1975) Adv.Phys. 2!4 47-100.
Ann.Rev.Fluid Woods Hole
Sloping convection in a rotating fluid.
HIDE, R., MASON, P.J. and PLUMB, R.A. (1977) Thermal convection in a rotating fluid subject to a horizontal temperature gradient: spatial and temperal characteristics of fully developed baroclinic waves. J.Atmos.Sci. 3£ 930-960. HOLMES, P.J. (1981) Space- and time-periodic perturbations of the sineGordon equation in Lecture Notes in Mathematics. (Ed. D.A.Rand and L-S Young) 898 164-191 Springer-Verlag. LAMB, G . L . and MCLAUGHLIN, D.W. (1980) Aspects of soliton physics, in Solitons (Ed. R.K.Bullough and P.J.Caudrey) Topics in Current Physics, vol.r^, Chap.II, 65-106. Springer Verlag, Berlin. LORENZ, E.N. (1963) Deterministic non-periodic flow. J.Atmos.Sci. 20, 130-141. MATVEEV, V.B. (1976) Abelian functions and solitons. Preprint no.373, Institute of Theoretical Physics, University of Warsaw (unpublished). MAXWORTHY, T. and REDEKOPP, L.G. (1970) A Solitary Wave Theory of the Great Red Spot and other Observed Features in the Jovian Atmosphere. Icanus 2!9 261-271. MOROZ, I.M. (1981) Slowly modulated Baroclinic Waves in a Three-layer Model. J.Atmos.Sci. 3£ 600-608. MOROZ, I.M. and BRINDLEY, J. (1981) Evolution of baroclinic wave packets in a flow with continous shear and stratification. Proc.R.Soc. London, A 377, 379-404. MOROZ, I.M. and BRINDLEY, J. 1982. Nonlinear amplitude evolution of baroclinic wave trains and wave packets. Preprint. NEWELL, A.C. (1972) The post bifurcation stage of baroclinic instability J.Atmost.Sci. 29 64-76.
SOLITON BEHAVIOUR IN MODELS OF BAROCLINIC INSTABILITY
PEDLOSKY, J. (1970) 15-30.
Finite amplitude baroclinic waves.
99
J.Atmos.Sci. 27
PEDLOSKY, J. (1971) Finite amplitude baroclinic waves with small dissipation. J.Atmos.Sci. 28 587-597. PEDLOSKY, J. (1972) Finite amplitude baroclinic wave packets. J.Atmos.Sci. 29^ 680-686. PHILLIPS, N.A. (1954) Energy transformations and meridional circulations associated with simple baroclinic waves in a two-level quasi-geostrophic model. Tellus 6^ 273-280. WEISSMAN , M.A. (1979) Nonlinear wave packets in the Kelvin-Helmholtz Instability. Phil.Trans.R.Soc. Lond. A 290. 639-681
CHAPTER 6 WAVES AND WAVE GROUPS IN DEEP WATER PETER J.BRYANT Department of Mathematics, University of Canterbury, Christchurch, New Zealand.
1.
INTRODUCTION. Analytical methods of modelling water waves of small but finite
height are based on the linear theory and improved with weakly nonlinear theories (West, 1981).
An alternative is to develop, with computer
assistance, water wave models which are nonlinear in their lowest approximation and are valid for a range of heights up to the onset of wave breaking (Schwarz & Fenton, 1982).
The present approach falls into
the latter category, and is concerned with investigating wave geometries which occur locally in deep water. Water waves propagating from a surface disturbance are subject to dispersion modified by nonlinear wave interactions.
This property
suggests that the numerical resolution into Fourier components of the nonlinear equations governing the evolution of a water wave system models the dispersion and its modification, and is therefore a natural method for investigating water wave properties.
Fornberg & Whitham (1978) used
this approach in studying certain nonlinear model equations for wave phanomena.
It is applied here to Laplace's equation with the nonlinear
boundary conditions appropriate to irrotational gravity wave propagation in deep water. Analytical solutions in the form of perturbation expansions exist for two dimensional water waves of permanent shape in deep water (Stokes waves) for which the dispersion and nonlinear modification are in balance.
A number of computer-based methods have been used (Schwartz
& Fenton, 1982, §2) to extend the calculations up to the highest waves of permanent form.
The present method is demonstrated first (§2) for
the calculation of two dimensional permanent waves.
Three dimensional
permanent waves have been found recently as perturbations to two dimensional permanent waves.
The present method allows calculations of
three dimensional waves independently of two dimensional waves, and one
WAVES AND WAVE GROUPS IN DEEP WATER
101
such example is presented below (§3). Waves on the ocean surface often occur locally as a wave group with an envelope that changes slowly as the waves propagate.
Analytical
solutions exist for weakly nonlinear wave groups of permanent envelope in two and three dimensions.
The present method is applied to the
calculation of wave groups of permanent envelope in two dimensions (§4) and in three dimensions (§5), in both cases without the restrictions on wave height which are needed for the analytical solutions. Specific wave geometries which occur in deep water are calculated by a numerical method based on Fourier analysis.
Examples are presented
for wave parameters outside the range of validity of analytical models. Wave properties, such as the form of permanent waves of finite crest length, and the approach to wave breaking, are demonstrated.
Although
the method is applied here only to gravity waves in deep water, it may be generalized to further forms of nonlinear wave motion. 2.
METHOD OF CALCULATION. The set of equations governing gravity waves in inviscid
irrotational motion on the surface of deep water is ^xx
+
^yy
+
*zz
=
z < eT
°'
l(x,y,t),
(2.1a)
V V ^ z "* °' z "*" °°' n
n
t " *z +
*t
+
+
^x^x
\Z
+ en
((f) 2 +
x
=
*
*y 2
+
°'
(2#lb) Z =
er
* z 2 ) = °'
l(x,y,t), Z =
^(X'y't)-
(2.1c) (2.Id)
The dimensional variables are the surface displacement an, the velocity potential (g&) a((), and £x, £y, Zz, (Vg) t, where a is a measure of water wave amplitude, 2TT£ is a typical wavelength, and £ = a/& is a measure of wave steepness.
The origin of coordinates lies in the mean
water surface with the z-axis vertically upwards. Symmetric two dimensional permanent wave solutions of the set of equations (2.1) exist for which N Tl = T a cos{k(x-ct)}, k=l k N 4> = I b, e sin{k(x-ct)}. k=l k
(2.2a)
(2.2b)
The permanent waves propagate with a velocity c(g&) , where c is an
102
P.J. BRYANT
unknown function of £, and 2TT£ is the wavelength.
The number of
harmonics N is determined numerically by trial and error so that the set of amplitudes a, , b, includes all those amplitudes greater in magnitude than some small prescribed value. Although the series (2.2b) for (j) has a zero mean at constant z, it should be noted that in satisfying the boundary condition (2.Id) <j) does not have a zero mean on z = eri(x,y,t). When the series (2.2a,b) are substituted into the boundary conditions (2.1c,d), with c, denoting the cosine in equation (2.2a) and s the sine in equation (2.2b), the resulting expressions may be written
- £(2ka k s k ) x (2kb k e EkT1 c k ) = 0 ,
(2.3a)
G = jE (akck - k c b k e £ k \ ) = 0, &
\*
r*.
is.
£ ] ^ J \ .
(2.3b)
J\.
where e
= exp(£k2a c ) . If the measure of amplitude, a, is taken to P 1 be half the height of the wave crest above the wave trough, -r-{r|(0) — r| (TT) }, then H =
2 a. - 1 = 0. kodd k
(2.3c)
Equations (2.3a,b) are transformed numerically to F = S F S =0, (2.4a) m m m G = S GC =0, (2.4b) m m m from which F = G = 0 , all m. (2.5) m m The Fourier components F , G are nonlinear functions of a, , b, , and c mm K. K. for given £. Equations (2.5) may be solved numerically by Newton's method, which for F is described by 3b, 3F'
F Fm<
m
for all m. Each coefficient on the left of equation (2.6) is an m Fourier coefficient of a partial derivative of equation (2.3a), and the prime denotes the new value of each variable.
The coefficients and the
right of equations (2.6) are evaluated at the old values of the variables. There is a similar set of equations derived from G and a single equation derived from H. The complete set of linear equations is solved
WAVES AND WAVE GROUPS IN DEEP WATER numerically for the differences
a
k~
a
^./ ^ k ~ ^ k ' c ~ c ' '
tne new
103 values of
the variables are calculated, and the procedure is repeated until the differences are less than some small arbitrary number (usually 10~ 8 ). The range of permanent wave solutions may be explored as the amplitude ratio e is changed step by step.
No difficulties were experienced with
the convergence of the numerical method, and the solutions agreed with those calculated elsewhere.
Since the number of harmonics increases
rapidly as the limiting wave is approached, and the wave properties near the limiting wave are fully described already (for references, see Schwartz & Fenton, 1982, §2), the present calculations have been continued only up to about 95% of the limiting wave height.
Rienecker
& Fenton (1981) have developed a method similar to that described here except that Fourier transforms are not used and equations (2.3) are solved directly by evaluating them at a number of points spaced along the wave profile.
The Fourier transform method is demonstrated now for
more general wave geometries. 3. THREE DIMENSIONAL PERMANENT WAVES Three dimensional permanent wave solutions of the set of equations (2.1) exist for which J k2(j) n = I I a cos{k(x-ct) + jy/r}, (3.1a) j=0 k=kx(j) D J k2(j) <(> = I I b. k exp{(k2 + j 2 /r 2 ) z}sin{k(x-ct) + jy/r}, j=ok=k 1 (j) J {31h) having wavelengths 2TT£ in the x-direction and 27Tr& in the y-direction, whose profile is steady relative to a frame of reference moving with velocity c(g£)
in the x-direction.
The bounds of the summations are
determined numerically by trial and error so that the set of amplitudes a., includes all those amplitudes greater in magnitude than some small prescribed value.
Since r| is chosen to have a zero mean and the
argument is symmetric in k when j = 0, the lower bound ki(0) may be set equal to 1 without loss of generality.
Other lower bounds ki(j), j > 0,
may be negative. Recent investigations summarised by Yuen & Lake (1982, ppl32-146) have considered three dimensional permanent waves which are perturbations to uniform two dimensional permanent waves, and which are steady relative to the two dimensional waves. Less restrictive assumptions are implicit in the analysis of Roberts (1983) and Roberts & Peregrine (1983), where
104
P.J. BRYANT
perturbation expansions in wave steepness are developed.
No such constraints
are required for the three dimensional permanent wave description above (equations 3.1). When the series (3.1a,b) are substituted into the boundary conditions (2.1c,d), with c., denoting the cosine in equation (2.2a) and s ., the sine in equation (2.2b), the resulting expressions may be written F = 2 2
kca . s ., - K ., b . e
3 K v.
s
(
2 2 kb. i e j k 3k
£K.,ri D c. 3k
£K f -ikn - £(2 2 (j/r)a.ks ) x 2 2 (j/r)b^ e D K c^J = 0, 3 k J J lj k DK
G = 2 2 a . . c.^-kcb.ue j k [ ]k ]k Dk £K
2
e 2 2kb.,e lj k 3k
n -ik D
£K
where
£Kt
K.. = (k2 + j 2 /r 2 ) 3k
c .. jk I2
i f
cc, 3kJ
-ikn
and e
s
k
(3.2a)
£K
+ + \z \z 2 22 2 (j/ ( j /r )b., r ) b ee 2 2 llj k ^k ^2
J
n ik D D
=0
I2
c .J J 3k
(3 2b)
-
ri
= exp{£K.. 2 2 a c }. 3k p q pq pq
If the measure of amplitude, a, is taken to be the central surface displacement r)(0,0,0), then H = 2 2 a . , - 1 = 0. j k 3k Equations (3.2a,b) are transformed numerically to F = 2 2 F S =0, m nroftron G = 2 2 G ^ c ^ = 0,
(3.2c)
(3.3a) (3.3b)
from which F = G = 0 , all m,n. (3.4) mn mn The Fourier coefficients F , G are nonlinear functions of a.. , b.. , mn mn 3k' 3k and c for given £ and r, and these harmonics and the wave velocity may be calculated by Newton's method as in §2. A three dimensional permanent wave example is presented now which lies outside the range of analytical solutions described previously. This example, drawn in perspective in figure 1, has parameter values £ = 0-25, r = 10, with c = 1-0262.
The dominant harmonics have
amplitudes aii = 0-65 and ao1 = 0«44, with all other harmonics having smaller amplitudes because they are the result of non-resonant nonlinear
WAVES AND WAVE GROUPS IN DEEP WATER
105
Figure 1 Perspective view of long-crested permanent waves showing 8 wavelengths in the x-direction and 2 wavelengths in the y~direction, with vertical magnification 5, when £ = 0-25 and r = 10.
Figure 2 Detailed view of the end structure of the long-crested permanent waves in figure 1, with vertical magnification 5.
106
P.J. BRYANT
interactions between the dominant pair.
The harmonic j = 0, k = 1
propagates in the x-direction with a velocity c(g£) , while the harmonic j = 1, k = 1 propagates at an angle tan velocity component c(g£)
(0*1) to the x-direction with a
in the x-direction.
The net result is a long-
crested three dimensional permanent wave propagating at an angle 0 = 0*06 to the x-direction, whose wave height (trough to crest) to wavelength ratio is 0*071.
The wave speed of the permanent wave is c(g£) cos6,
and its wavelength is 27r£cos6.
The wave structure at the ends of the
crests, drawn in detail in figure 2, propagates in the x-direction with a velocity c(g£) , and reduces the wave height to wavelength ratio to 0 021 at its lowest point.
Relative to a frame of reference moving with
the long-crested permanent wave, the end structure propagates along the wave crests with a velocity c(g£) sin6, producing a three dimensional wave which has a steady profile relative to a frame of reference moving with velocity c(gil)
in the x-direction.
The end structure illustrated
in figure 2 is of the same form as that calculated analytically by Roberts & Peregrine (1983, figure 4) for semi-infinite long-crested permanent waves. The data for the present solution is as follows.
Equation (3.1a)
contains 173 harmonics in 17 wavebands 0 < j ^ 1 6 , the wavenumber range being - 6 ^ k ^ 9.
(Most calculations were made with the order of
summation reversed, that is, 10 wavebands 0 < k ^ 9 with - 13 ^ j ^ 1 6 , because faster numerical convergence was obtained).
The maximum
Fourier coefficient F , G not included in the calculation has mn mn magnitude 3*2 x io~" . The maximum magnitude of F and G over the 64 x 32 -4
points used in the final calculation is 7*6x 10 — 4-
with a root mean square
deviation of F and G from zero of 1-2x10 Yuen & Lake (1982) describe two forms of three dimensional permanent waves, either symmetric or skewed about the direction of propagation.
Both forms can be expected to generalize to the range of
waves described by equations (3.1), and investigations of these and related forms of three dimensional permanent waves are continuing. 4.
TWO DIMENSIONAL WAVE GROUPS Two dimensional wave groups with envelopes of permanent shape,
which are periodic in the x-direction with a group length 21TL, are composed of harmonics with wavenumbers k/L where k has integer values only.
If 2TT£ is the wavelength of a typical wave in the group, with
k 0 = L/£ not necessarily an integer, then to a first approximation the
WAVES AND WAVE GROUPS IN DEEP WATER
107
spectrum of the wave group in nondimensional wavenumber space is centred on k 0 .
The velocity of the group, to a first approximation, is ^(gJl)
in deep water.
The nondimensional wave frequency, w(k), expanded about
the central wavenumber, k 0 , is w(k) = w(ko) + (k~ko) r^r, ^dkjko , 1
k-k0
2k0
+—(k-ko) 2.
2
(4.1)
k0
for waves in deep water with w = (k/ko) .
The wave group is described
then by
kx
1
£==• - wwt| t ,, n = 2 a cos|£==k * = 2 a. c o s ] k ~ k ° k K { k0
(4.2a)
(x-frt) + x - ( l + 3)t>, J
(4.2b)
where 3 is an unknown nondimensional frequency correction.
This form for
a periodic wave group satisfies the nonlinear Schrodinger equation, which in the present nondimensional notation is i(A t + ^A x )
where
- J £ 2 | A | 2 A = 0,
- g-A
(4.3a)
n =
(4.3b)
The nonlinear Schrodinger equation is valid for waves of small but finite height, £ «
1, whose spectrum lies in a narrow waveband
|k-k o |/k o ~ e . A generalization of equations (4.2) which satisfies the nonlinear free surface conditions (2.1c,d) is
=
£ H(j) I
with
(() = I
where
a = — + 3 •
I
\
j=0 k=kx(j)
a b
jK
fk
l
(4.4a)
cos]^- ix-ht) - jat , exp - ^ sini-p (x-ht) - jat} , k
°
^ ko
*
(4.4b)
The index j = 0, for which kx (0) = 1 since rj has
a zero mean, describes the surface displacement and water motion which is steady relative to the group structure.
The index j = 1 refers to
the harmonics in equations (4.2) for the dominant waveband with k near kg.
Each higher value of j describes a set of harmonics in a waveband
about jko, where k ^ j ) , j > 0, may be negative.
When the series (4.4a,b)
are substituted into the boundary conditions (2.1c,d), equations similar in form to equations (3.2) are obtained, which are solved by the same
108
P.J. BRYANT
method as is described previously for a., , b., , a as functions of £ and 3K 3K k0. It was found that the calculated spectral peak in the dominant waveband, j = 1, moved outside the neighbourhood of ko as £ increased to larger values.
Such solutions need rescaling with a more appropriate
wavelength 2TT£' S O that kj = L/£' corresponds to the actual centre of the dominant waveband.
The nondimensional group velocity is then changed
from — to v according to j(gi)h = v{qV)h
,
(4.5)
and equations (4.4) have the same form in the new nondimensional variables except that — is replaced by v.
If the primes are dropped
from the new variables, equations (4.4) become J kz(j) rk
n =
>|
I I a cos-(x-vt)-jat , j=ok=k1(j) D k tko J J
k 2 (j)
• = I I b j=0 k=k1(j) D
|,i
r,
(4.6a) ^
e x p ^ s i n f (x-vt)-jat , k J ° ^k°
(4.6b)
where for any particular solution the choice of ko determines v and conversely. Two examples are presented of two dimensional wave groups with envelopes of permanent shape, both of which lie outside the range of validity of the nonlinear Schrodinger equation.
The first is of a wave
group containing two wavelengths per group length, k 0 = 2, in the range of larger wave heights for such groups.
Its parameter values are
£ = 0-470, v = 0-6, with a = 0*4308, and the wave height (trough to crest) to wavelength ratio for a wave at the centre of the group is 0-100.
One
group length is sketched in figure 3, which shows the water surface displacement at an instant, and the envelope of permanent shape.
The
upper envelope is a height 0-470£ above the mean level at the centre of the group, while the lower envelope is a depth 0-308& below the mean level there.
This is one obvious departure from the permanent envelope
analytical solutions of the nonlinear Schrodinger equation (4.3) because they are symmetric about the mean level. A wave at the centre of the group of permanent envelope is compared in figure 4 with the wave of permanent shape having the same wave height and wavelength.
The wave in the group is more peaked, with the height
of its crest above the mean level being 0-470£ compared with 0-37l£ for the permanent wave.
The horizontal particle velocity at the crest is
WAVES AND GROUPS IN DEEP WATER
109
Figure 3 The wave group of permanent envelope with e = 0-470 and v = 0-6, showing the water surface displacement at an instant (the solid curve) and the envelope (the dashed curves), drawn with the horizontal and vertical scales equal.
Figure 4 A wave at the centre of the group of figure 3 (the solid curve) compared with the permanent wave of the same height and length (the dashed curve), both drawn with the horizontal and vertical scales equal.
110
P.J. BRYANT
0-70(g£)
compared with 0-45(g£)
for the permanent wave.
The horizontal
component of particle acceleration in front of the crest, and deceleration behind the crest, is of maximum magnitude 0-46g compared with 0-31g for the permanent wave.
This comparison indicates that a wave passing through
the centre of the group is closer to the point of wave breaking than a permanent wave of the same wave height and wavelength.
The calculations
could not be extended to wave groups of greater height because of lack of computer capacity. The wave group solution contains 302 harmonics (605 variables) in 14 wavebands 0 < j < 13, the wavenumber range being - 17 < k < 40. More harmonics would have been added if more computer capacity had been available.
Although the inability to add more harmonics meant that the
kinematic boundary condition F = 0 (equation 2.1c) was not satisfied adequately at some isolated points, the properties described above had converged to within the precision stated.
The maximum magnitude of F
over the 128X32 points used in the final calculation was 0*186, but the root mean square deviation of F from zero was 0-018.
The maximum
magnitude of G over the same points was 0-022 with a root mean square deviation from zero of 0-002. The second example is of a wave group containing one wavelength per group length, k 0 = 1.
As the wave train propagates through the
group structure, each wave shape oscillates with an angular frequency a(g/£)
about a symmetric shape.
Since the dominant harmonics in
equations (4-6) are now those for which j = k, k = 1,2,..., the shape oscillation is modelled better by changing the summation in equations (4.6) to keep k ^ 0/ with c = v + a
and
m = k - j.
The series then
become ni2
r| =
k2 (m)
I I m=-mx k=0
A
k2(m)
m2
=0
B
cos{k(x-ct) + mat},
(4.7a)
e
(4.7b)
sin{k(x-ct) + mat}.
ITlK
The dominant waveband, m = 0, describes a steady wave propagating with a velocity c(g£) .
Other wavebands, m ^ 0, describe the cyclic
oscillation of this wave shape as it propagates in the x-direction, as though it were a periodic wave train set into an oscillation about a permanent shape.
For this reason, these waves have been named cyclic
waves and have been described in more detail elsewhere (Bryant, 1983a). The third example from this previous investigation is sketched in
WAVES AND WAVE GROUPS IN DEEP WATER
111
figure 5, showing the change in shape of the wave profile over one wavelength and one cycle of shape.
It was shown that when a cyclic wave
rigure 5 The profiles of the cyclic wave with € = 0-05, c = 1*0672 and a = 0*3972 over one wavelength and one cycle of shape, at times (from the bottom) at = 0, TT/2 , IT, 3TT/2, 2TT, relative to a frame of reference moving with the wave velocity. Each profile is drawn with the horizontal and vertical scales equal.
passes through points of maximum wave height (at the centre of the wave group), it is closer to wave breaking than is a steady permanent wave of the same height and wavelength, the comparison being similar to that for the wave group example above.
Since ocean waves of large height are
never completely steady in shape, it may be more realistic to model them with cyclic waves rather than with steady waves. 5.
THREE DIMENSIONAL WAVE GROUPS The simplest of the three dimensional wave groups are the oblique
wave groups for which the straight parallel lines of constant wave phase are oblique to the straight parallel lines of constant group phase. One well-known example is that of the waves along the boundary wedge of the Kelvin ship-wave pattern.
A previous investigation of oblique wave
groups (Bryant, 1983b) is summarised here.
Further forms of three
dimensional wave group structures, such as that resulting from the interaction of two two-dimensional wave groups, are yet to be investigated. The horizontal axes Ox, Oy are chosen so that the wave train of typical wavelength 2TT£ propagates at an angle 0 to the x-direction while
112
P.J. BRYANT
the group structure propagates with a velocity h(g$L) cose in the x-direction.
Equation (4.2b) then generalizes to • x cose + y sine - (l + $)ti, (5.1)
where the group length in the x-direction is 2TTL, the wavenumber components in this direction are k/L with k taking integer values only, and k 0 = Lcos6/£
(5.2)
is the number of wavelengths per group length in the x-direction.
This
form for a periodic oblique wave group satisfies the nonlinear Schrodinger equation 1
\
I V.
+ U
7
A y £. A I J
" £" A Y*
+
AA
O
JAvv " 7 £ 2 | A ! 2 A ft
X X
£
=
° '
(5.3a)
where
n =
(5.3b)
and
X = x cose + y sin6 , Y = - x sin6 + y cose .
(5.3c)
A generalization of equation (5.1) which satisfies the nonlinear free surface conditions (2.1c,d) is r I
n =
i with
(j) =
k
£(j) I a
?k5#>3
fk 1 cosKr— (x - ^coset) cose + j (y sine - at) K
h J, I -v j=0 k=kx(j) D k
ex
ff°k
V
P1 U ~ c o s 0 ^k° J
+
(jsin9) 2
(5 4a)
H J
'
z x
r^ (x- h coset)cose + j(y sine - at) \ , ko J where
a = 1 - -r-cos2e + 3-
(5.4b)
When the series (5.4a,b) are substituted
into the boundary conditions (2.1c,d), equations similar in form to equations (3.2) are obtained, which are solved by the same method as is described there for a., , b., , a as functions of £, ko , and Q . Two interesting properties of oblique wave groups and their relationship with the nonlinear Schrodinger equation (5.3) are found from the numerical calculations.
The nonlinear Schrodinger equation requires that
the spectrum of the water surface displacement contains a single central peak in wavenumber space.
Resonance occurs near wavenumbers for which
the forced wave harmonics in the dominant waveband of the oblique wave group (equation 5.4a) coincide with free wave harmonics. criterion for deep water gravity waves is
This resonance
WAVES AND WAVE GROUPS IN DEEP WATER
113
ja + j (k/k0)cos26 = { (k cos6/ka)2 + (j sinG) 2 } l / l + .
(5.5)
When a significant second peak occurs in wavenumber space near the location described by equation (5.5) , the nonlinear Schrodinger equation was found not to be a valid model equation for oblique wave groups. The nonlinear Schrodinger equation (5.3) has apparent wave group solutions for which the complex amplitude A passes through zero, causing the envelope of the group, ]A|, to be non-differentiable at such a point. No numerical solutions could be found corresponding to these solutions of the nonlinear Schrodinger equation.
The numerical evidence is that
such solutions do not exist for the full equations (2.1). The nonlinear Schrodinger equation (5.3) does not have an oblique wave group solution at the critical angle 0 = tan second derivatives in the equation cancel.
(1//2) at which the
However, an oblique wave
group solution of equations (2.1) does exist for this angle because the resonant second spectral peak described by equation (5.5) is of comparable magnitude to the central spectral peak.
A sketch of an
oblique wave group at the critical angle is shown in figure 6.
The
boundary wedge of the Kelvin ship-wave pattern, illustrated for example by Lighthill (1978, figures 70, 71), has the same structure inside the group as is seen in figure 6, but does not have the wave structure outside the group which accompanies the periodic wave group structure in figure 6. 6.
DISCUSSION The particular wave geometries described here are ohly a small
selection of those which may be calculated by the Fourier transform method applied to the full nonlinear governing equations.
Generalizations
to the examples above include standing wave geometries, waves in finite depth, and short waves influenced by surface tension.
Most analytical
methods and model equations are valid only for linear or weakly nonlinear waves, that is, for waves of small but finite height.
The
present method, as is shown above, is set up for nonlinear waves without restriction on wave height.
Also, the calculation of the irrotational
velocity field simultaneously with the water surface displacement provides insight into the physical properties of the water wave motion.
114
P.J. BRYANT
Figure 6 Perspective view of one group length in the x and y directions with e = 0-05, k 0 = 10, 6 = tan and vertical magnification 2-5TT.
(1//2),
WAVES AND WAVE GROUPS IN DEEP WATER REFERENCES BRYANT, P.J. (1983a) Cyclic gravity waves in deep water, J. Aust. Math. Soc. B25 (in the press). BRYANT, P.J. (1983b) (sub judice).
Oblique wave groups in deep water, J. Fluid Mech.
FORNBERG, B. & WHITHAM, G.B. (1978) A numerical and theoretical study of certain nonlinear wave phenomena, Phil. Trans. Roy. Soc. A289, 373-404. LIGHTHILL, James (1978)
Waves in Fluids, Cambridge University Press.
RIENECKER, M.M. & FENTON, J.D. (1981) A Fourier approximation method for steady water waves, J. Fluid Mech. 104, 119-137. ROBERTS, A.J. (1983) (preprint).
Highly nonlinear short-crested water waves,
ROBERTS, A.J. & PEREGRINE, D.H. (1983) waves, (preprint).
Notes on long-crested water
SCHWARTZ, L.W. & FENTON, J.D. (1982) Strongly nonlinear waves, Ann. Rev. Fluid Mech. 14, 39-60. WEST, Bruce J. (1981) On the simpler aspects of nonlinear fluctuating deep water gravity waves (Weak interaction theory), SpringerVerlag Berlin Heidelberg New York. YUEN, Henry C. & LAKE, Bruce M. (1982) Nonlinear dynamics of deep-water gravity waves, Adv. Appl. Mech. 22, 67-229.
115
CHAPTER 7 TWO-AND THREE-WAVE RESONANCE ALEX D. D. CRAIK
Department of Applied Mathematics University of St. Andrews St. Andrews, Fife, Scotland.
1.
INTRODUCTION Resonance of linear and weakly nonlinear waves gives rise to a wealth of physical phenomena and many interesting mathematical problems. Resonant interactions are responsible for exchange of energy between existing wave modes, for preferential amplification from infinitesimal levels of previously undetected modes, for the disintegration of initially uniform wavetrains into more complex motion and for enhanced extraction of energy from primary shear flows. While simple linear models may provide acceptable theoretical descriptions of some aspects of these phenomena, others require more sophisticated theories such as that of inverse scattering and of strange attractors, or the brute force of modern computers. Sometimes, too, one is fortunate enough to find simple particular solutions of the nonlinear evolution equations. Linear, or direct, resonance of two waves is considered in Section 2, and Sections 3-5 concern weakly nonlinear three-wave resonance. The aim has been to provide an overview of the range of phenomena and a review of the available mathematical solutions. Particular attention is paid in Section 5 to non-conservative systems, for which the mathematical theory is least developed. Special cases such as subharmonic resonance and further complications involving quadratic interaction of more than three waves are mentioned only briefly. Cubic and higher-order nonlinearities are not discussed, owing to lack of space. Likewise, the more general problem of weak interactions among Fourier components of a continuous wave spectrum is not confronted. 2. LINEAR RESONANCE It is convenient, first, to consider the three-layer fluid flow of Figure 1. Incompressible inviscid fluid has density p(z) and unidirectional primary velocity [U(z),0,0], relative to Cartesian axes x,y,z, given by the piecewise-constant functions
TWO- AND THREE-WAVE RESONANCE U
Pi
with pi < p£ < P3-
l
P2
U = U2
P3
"3
117
(z > h) (|z 1 < h) (z < -h)
(2.1)
Here, z is measured upwards, gravitational
acceleration is (0,0,-g) and there may be interfacial surface tensions Y, y' at the respective interfaces z = h and -h. If Ui = U2 = U3 = 0, linear, irrotational, capillary-gravity waves may exist at either interface and the properties of waves on one interface are influenced by the proximity of the other.
Waves with
periodicity exp[i(kx-cat) ] are normally only weakly affected by the other interface when kh > 1.
Then, the dispersion relations for waves at the
upper and lower interfaces remain rather close to those for waves on a single interface with the other replaced by a plane rigid wall.
In this
case, OJ
* ±00 (k) U
or
±OJ
(k),
36
where subscripts u and I respectively designate the upper and lower interface.
Such approximate dispersion curves are sketched in Figure
2 (a). Significant departures from this approximation arise if the particular constants p. (j = 1,2,3), Y and y' admit curves which intersect (as shown at points A and A') at some wavenumber kQ.
In the
vicinity of such points, the exact dispersion relationship oj(k) has solutions which behave as shown in Figure 2(b): curves labelled u and I do not, in fact, intersect, but either curve changes its identity in the vicinity of the points where OJ = w.. With a primary flow U, rather different behaviour may result.
For
instance, if U 2 and U3 remain zero, there is a range of values of Ui for which waves on the upper interface have frequencies OJ * OJ ,OJ roughly as shown in Figure 3(a), while those on the lower interface remain virtually unchanged from ±OJ of Figure 2(a) .
It is found that, for
ki < k < k 2 , where the frequency of the lower branch OJU becomes positive, the energy associated with the wave OJ = OJU is negative, whereas the energy is positive at all other wavenumbers.
If the portion
ki < k < k 2 of OJ_ approaches or intersects the curve oj^(k) at some k =k 0 , then the local character of the exact solution OJ resembles Figure 3(b), not 2(b), with complex conjugate roots OJ appearing for |k-k(j| < A.
A.D.D. CRAIK
118
U 2 ,P 2
U 3 =0,p 3
Figure 1. Three-layer flow configuration.
Figure 2 (a) Approximate dispersion curves for three-layer model when U =0. (b) Nature of exact dispersion curves near linear resonance of waves with like energy sign. (b)
2
k
ih
I
L ko-A
ko+A
k
Figure 3 (a) Approximate dispersion curves for upper-interface modes when Ux > 0. (b) Nature of exact dispersion curves near linear resonance of waves of opposite energy sign.
TWO- AND THREE-WAVE RESONANCE
119
The complex roots denote the occurrence of linear instability due to a direct, linear resonance of the modes on either interface.
Such
instability arises in conservative systems only when the energies associated with the participating modes have opposite signs:
these modes
coalesce, having zero energy, at ko ± A and so may grow while total energy remains conserved. When Ui is sufficiently large, the lower branch w U approaches the upper one w
and the resulting interaction of positive- and negative-
energy modes on the same interface also gives rise to instability - the well-known Kelvin-Helmholtz one.
Though such instabilities have long
been known, the physical explanation given here is comparatively recent and first arose in analogous work in plasma physics.
Discussion of the
energetics of such waves and detailed dispersion curves for the threelayer flow (2.1) are given by Cairns (1979) and Craik £ Adam (1979).
An
equivalent interpretation in terms of 'wave action1 may also be given. The general initial-value problem for linear evolution of arbitrary localized disturbances was recently considered by Akylas § Benney (1982). Even for 'stable1 interactions among waves of like energy sign, the contribution from Fourier modes near points of direct resonance was found to dominate the disturbance at sufficiently large times. Such linear, or direct, resonance arises in many areas of physics. Within fluid mechanics, a similar interpretation in terms of energy applies to flows with continuous density stratification.
Waves can then
propagate vertically, as well as horizontally, and consideration of the energy of internal waves on either side of a velocity discontinuity provides an insight into the phenomenon of over-reflexion (see, e.g. Acheson 1976). Extension of such ideas to flows with continuous, curved velocity profiles is not straightforward, because of the presence of a critical layer where U = u)/k.
Nevertheless, computed results for shear-layer
profiles (e.g. Drazin et al, 1979) reveal qualitative similarities with those discussed above. Recently, Cairns § Lashmore-Davies (1983 a,b) have proposed an interesting model for cases where inhomogeneity of the medium causes slow spatial variations of the dispersion relation.
If (^(kjX), o)2(k,x)
are the frequencies of two uncoupled modes, which vary slowly with x, then the local dispersion relation near a 'mode-crossing point1 may be approximated by
X=XQ
120
A.D.D. CRAIK (u) - a)i) (a) - 0)2)
= 5
where 6 is a small number, taken as real, and o)i(ko) = w 2(ko)
E
^0
sa
X'
The disturbance is regarded as having constant frequency UQ and so is composed of two slowly-varying wavenumbers k = k^(x), k2(x) which locally coalesce into kg near X = X Q .
AS the waves pass through the
resonance region, their interaction may cause substantial changes in the amplitudes, which require calculation.
Regarding either mode as having
periodicity exp i(kox - u)Qt) near x o and slowly-varying complex amplitude A.(x), where j=1 or 2, Cairns § Lashmore-Davies were led to postulate the mode-coupling equations (d/dC + iaiOAi
= iXA2, (2.2)
(d/dc + where c E x - x 0 , a.. = (awj/ao/Ow^/ak), A E [6(8u)1/3k)"1(aa)2/ak)"1]*. These satisfy the conservation law
£(| A l |2 • |A2|2) = 0 which, with suitable normalization, expresses conservation of wave energy or wave action when the two waves have like energy signs.
Elimination
of Aj (or A2) from (2.2) and a change of variable yields Weber's equation, with a parabolic cylinder function as solution.
The asymptotic
representation of this function for large |c| readily reveals the net changes in Aj and A2 as the waves propagate through the resonance region. For waves of differing energy signs, the analysis is similar. This remarkably simple model is known to give results in agreement with detailed analyses of particular problems in plasma physics. Essentially, it is an extension of Whithamfs theory of slowly-varying waves to encompass direct, linear resonance.
A rigorous justification
of the model equations (2.2), perhaps via an averaged Lagrangian, has not yet been given but may reasonably be expected. 3.
THREE-WAVE RESONANCE IN FLUIDS. When finite-amplitude disturbances are sufficiently small, linear
theory usually provides a good first approximation.
Successively
improved approximations may be constructed by expansion of the disturbance in ascending powers of (dimensionless) wave amplitude.
Some
TWO- AND THREE-WAVE RESONANCE
121
such expansions have been developed with full mathematical rigour to yield firm results in particular limiting cases.
More often, a more
casual approach is adopted to questions of convergence of the amplitude expansion and the validity of truncating the expansion at some chosen, typically low, order.
The justification for these unrigorous analyses
rests upon their frequent success in modelling interesting situations less amenable to exact description. The study of a single dominant wave mode, with slowly-varying amplitude and phase, is central to theories of nonlinear wave motion and hydrodynamic stability.
The interaction of a primary wave, of amplitude
0(a) say, with its own 0(a2) second harmonic and with any 0(a2) mean flow correction produces 0(a3) forcing terms in phase with the original wave.
These cubic nonlinearities describe the strongest nonlinear
effects felt by the wave. one mode is dominant:
However, there are many situations where no
then the dominant nonlinear effects may be of
quadratic order, and therefore stronger than those of 0(a 3 ).
The most
important such case is three-wave resonance. The quadratic interaction of two waves with horizontal wavenumbers kx,k2 and frequencies u)i,u>2 gives rise to forcing terms with periodicities exp{±i[(k!.x - o>it) ± (k2.x - cu2t)]}. If there exists a linear wavemode with wavenumber k3 and frequency CJ3 such that (ki,u>i) ± (k2,u)2) ± (k3,o)3) = (0,0)
(3.1)
then some of the forcing terms are in phase with this linear mode and resonance occurs.
If all three waves are of comparable amplitude 0(a),
and no others are of importance, the governing equations for their complex amplitudes A.(x,t) (j = 1,2,3) take the form
jt
+
Yl' V) Al
= a
iAl
+
M
2
A*
+ 0(a3)
A2A^A* + 0(a3)
(3.2)
= a 3 A 3 + A3A*A* + 0(a3) where, for definiteness, we suppose that ki + k 2 + k 3 = 0 and 0)1 + u)2 + o)3 = 0.
Here, the v. denote (constant) linear group velocities
(3u>/3k). of each mode, a. are the linear growth rates (a. being negative
122
A.D.D. CRAIK
for linear damping) and X. are the quadratic coupling coefficients. The superscript * denotes complex conjugate. The X. are usually evaluated only by considerable effort: for conservative systems, they are most readily found by a variational formulation (Simmons 1969) and, even for dissipative flows, a variational approach may be advantageous (Usher § Craik 1974). In conservative systems, the a. are normally zero in cases of interest and each X. has the form iKsgn(3D/3u))3 .
X. j
I 3D 3D
where K is a real constant, D(u),k) = 0 is the linear dispersion relation in a suitably-chosen form and ( ) . denotes evaluation at (k.,a3.) (j = 1,2,3). Further, the energy E. of each wave mode per unit horizontal area may be expressed as (see Cairns 1979)
j
=
3 ' 3l
*V
The equations (3.2) may then be renormalized and truncated at 0(a3) to give (^*YI^)QI-TIQK
" t
(3-3)
(A * »•')
Q3 ~ Y3Q1Q2
where y. = sgn[u). E . ] . If the nonlinear interaction is not energy conserving, even when the a. are zero the scaled equations similar to (3.3) have complex y. of the form y. = exp(i<J>.) with constant phases .. 3 3 3 3 General solutions for conservative cases are now well-known and are briefly described below. In contrast, only a few particular or approximate solutions have been found for the non-conservative equations. Before discussing these solutions, it is appropriate to describe some of the contexts within fluid mechanics in which such equations have arisen. The first influential study of nonlinear resonant wave interactions in fluids was that of Phillips (1960), who examined irrotational surface gravity waves in inviscid fluid otherwise at rest. Ironically, such waves do not exhibit three-wave resonance and Phillips boldly
TWO- AND THREE-WAVE RESONANCE
123
proceeded to consider four-wave resonance driven by cubic nonlinearities. But capillary-gravity waves do_ exhibit three-wave resonance (McGoldrick 1965, 1970a; Simmons 1969; Case SChiu 1977).
So also do surface and
interfacial gravity waves on a uniform shear flow (Craik 1968) and in the three-layer flow of Figure 1 (Craik § Adam 1979).
Likewise, resonant
interactions occur among three internal waves in continuously stratified fluid and among two surface waves and one internal wave (Ball, 1964; Thorpe, 1966; Davis § Acrivos, 1967).
These and other examples are
described in the books of Phillips (1977) and Leblond $ Mysak (1978). Resonance occurs in the presence of many primary shear flows.
Kelly
(1968) gives two inviscid examples, while Raetz (1959) and Craik (1971) drew attention to the possible importance of three-wave resonance among viscous Tollmein-Schlichting waves in laminar boundary layers.
Such
resonance is normally manifested by an initially-dominant two-dimensional plane wave 'driving up1 a pair of waves of lower frequency from originally infinitesimal levels.
Sometimes, as in Kelly's examples, the
latter waves may be two-dimensional, with wavenumbers ^z>^-3 collinear with that of the original wave, k\. ^2
=
^3
=
%^i>
W
2
=
^3
=
w
% i i
s a
The particular, degenerate, case
subharmonic resonance involving just
two wave modes (ki,a)i), (%ki,%cai).
Experimental evidence of such
subharmonic resonance has been found in unstable shear layers (e.g. Miksad, 1972, 1973) but not in symmetric wakes (Sato 1970).
A similar
subharmonic resonance exists among capillary waves (McGoldrick, 1970b). In boundary layers, three-wave resonance among two-dimensional collinear wave trains is not normally possible but a two-dimensional wave ki may interact resonantly with a particular pair of oblique plane waves %ki ± I where kj . I = 0.
Craik (1971) (see also Usher § Craik,
1975) argued that this interaction should be particularly powerful because all three waves then have the same critical layer at which the downstream phase speed equals the primary flow velocity.
The famous
experiments of Klebanoff et al (1962) did not reveal evidence of these particular resonances; but recent observations by Kozlov & Levchenko (1983) and Thomas $ Saric (1981) have confirmed their existence, the frequencies of preferentially-amplified oblique waves being found to centre on half that of the two-dimensional wave excited by a vibrating ribbon.
In contrast, the early development of three-dimensionality in
the experiments of Klebanoff et al seems to have been due to a resonance (at higher order) of oblique wavenumbers kx ± Jl with the
124
A.D.D. CRAIK
second harmonic of the fundamental wave (Craik, 1980; Nayfeh $ Bozatli, 1979). The interaction among resonant triads in primary shear flows is generally non-conservative : there may be net energy exchange between waves and mean flow, both through linear growth or damping a. and through the nonlinear coupling coefficients X. which are usually complex.
Such
cases are of very real interest and there is a need for further studies of the non-conservative equations (3.2) which govern them. The influence of higher-order nonlinearities has received some attention.
Finite-amplitude two-dimensional waves may be described
either by higher-order amplitude expansions (Herbert 1980) or by direct computation from the Navier-Stokes equations (Orszag $ Kells 1980) and such equilibrium states are typically unstable to three-dimensional disturbances, in the form of oblique wave modes or of spanwise-periodic longitudinal vortices.
It is now recognized that onset of three-
dimensionality leads to regions of enhanced shear which exhibit local and rapid secondary instability.
Such secondary instability is an important
precursor of transition to turbulence in boundary-layer and channel flows. 4.
CONSERVATIVE THREE-WAVE RESONANCE Here we consider solutions of the equation (3.3) characteristic of
three-wave resonance in conservative systems.
When the normalized
complex amplitudes Q. depend on just a single variable, say £ = a.x + 3t for some constant a and 3, exact solutions are readily found.
These
were obtained independently by Jurkus § Robson (1960), Armstrong et al (1962) and Bretherton (1964) - the importance of three-wave resonance in many areas of physics not being matched by rapid communication between them! When all three waves have the same energy sign, the solutions are periodic Jacobi elliptic functions; in certain limiting cases, these yield
f
shock-like1 hyperbolic tangents and fpulse-like1 hyperbolic
secants.
The corresponding solutions, for waves of unlike energy signs,
exhibit a singularity at a particular value of £: 1
evolution, this normally signifies an 'explosive finite time.
for purely temporal breakdown after a
Note that energy conservation is not violated by the
appearance of this singularity: three waves grow.
Ex + E 2 + E 3 remains constant while all
Details of these, and many of the following, solutions
are given in the valuable review article by Kaup et al (1979) (but note
TWO- AND THREE-WAVE RESONANCE
125
that the original published version contains many misprints, corrected in the offprints).
Wieland § Wilhelmsson (1977) describe further numerical
and analytical solutions of related equations which include both linear damping terms and cubic nonlinearities. The development of a singularity signals the breakdown of the approximations underlying equations (3.3) and might be thought to correspond to a dramatic and rapid physical event.
Surprisingly, there
is still no totally convincing experimental demonstration of this:
the
growth of oblique waves in unstable boundary-layers (see above) is most striking, but this occurs in a non-conservative system.
Experimentalists
may care to note that the three-layer flow of Figure 1 supports resonant triads of the required type (Craik § Adam, 1979). When the amplitudes Q. depend on more than one independent variable, exact solutions are also known, but they are much more complex.
Their
most general form is described by an inverse-scattering transform.
Such
solutions with dependence on one space dimension and time, or on two space dimensions, were first given by Zakharov $ Manakov (1973,1975) and those depending on two space dimensions and time by Zakharov (1976). These and subsequent work are succinctly reviewed by Kaup et al (1979) and Kaup (1981). The wave envelope Q. of each mode may be associated with a discrete set of n-solitons (n = 1,2,...) and a continuous spectrum.
The mode
interactions may be described as interactions among three such solitons. Typically, a soliton from one envelope is 'converted' into two solitons belonging to the other two envelopes.
These 'soliton exchange inter-
actions' depend crucially upon the relative frequencies and group velocities of the modes.
For instance, with dependence on x and t only,
and wave modes of like energy sign, a mode having the middle group velocity and highest frequency always loses its solitons to the two 'outer' modes.
In contrast, solitons of the 'outer' modes are converted
to solitons of the 'middle' mode only in certain exceptional cases.
A
fuller discussion and illustrative numerical examples are given by Kaup et al (1979). In the potentially 'explosive' case of waves with differing energy signs, initially-localized wave envelopes may or may not separate, and so decouple, before a singularity can develop.
For instance, when the
highest-frequency mode has energy of opposite sign to the other two, and travels with
the middle group velocity, singularities develop after a
126
A.D.D. CRAIK
finite time only if the initial amplitudes and extents of the wave envelopes are large enough (see Kaup et al). An especially simple exact solution, found by Craik (1978), varies with two space coordinates x, y and time t, and is actually a special case of Zakharov's (1976) general solution.
Rescaled amplitude equations,
with complex amplitudes b. (j = 1,2,3), are satisfied by the relations
b
l- V
b
' = V b3
where £ = x - y - t , n = x + y - t , <j> = x are characteristic coordinates and subscripts denote partial derivatives.
Here, 3 < 0 for waves of like
energy sign and 3 > 0 in potentially 'explosive1 cases.
A class of
functions satisfying (4.1) is F(5,n,4O = sgn(3) log [f(O + g(n) for these, Craik gives a precise criterion for the appearance, or not, of singularities when 3 > 0. 5.
NON-CONSERVATIVE THREE-WAVE RESONANCE. When the waves have non-zero linear growth rates a. and/or their
nonlinear interaction is not energy conserving, only a few exact analytical solutions of (3.2), truncated at 0(a 3 ), are known. Wieland § Wilhelmsson (1977) give several examples, all with amplitudes depending on a single variable.
One, originally due to Fuchs
§ Beaudry (1975), has just one wave linearly damped and the nonlinear interaction is conservative.
Their coupled equations reduce to
duo/dt = -U1U2,
dui/dt =
where u Q + Uj is a constant which may be normalized to unity.
On
setting UQ = sin(jY), uj = cos(£Y), where Y = Y(t), these equations yield d 2 Y/dt 2 + pdY/dt + sin Y = 0,
u 2 = -JdY/dt.
This is identical to the equation governing finite oscillations of a simple pendulum with linear damping.
Approximate solutions for weak and
strong damping are readily constructed. McEwan et al (1972) have examined, both theoretically and experimentally, the possible steady states of resonantly-interacting,
TWO- AND THREE-WAVE RESONANCE
127
linearly-damped standing internal waves in stratified fluid, with one wave forced by an oscillating boundary.
With weak forcing, there is
only a single forced wave mode; but, as the forcing is progressively increased, a threshold amplitude is reached at which this single-mode solution is unstable and the other two resonating waves appear.
These
two new waves increase in amplitude, as the forcing is further increased, while the forced wave retains its constant threshold amplitude. and experiment are in agreement.
Theory
Possible steady states of a five-wave
interaction, comprising two linearly-damped resonant triads with one wave in common, are also considered.
When the common wave is externally
forced and has greatest absolute frequency, the single-mode solution is stable at low forcing; with greater forcing, the only stable steady state is that consisting of the triad with the lower threshold amplitude. Particular solutions for temporal evolution of a five-wave system are given by Wieland $ Wilhelmsson (1977, Ch.13). Another interesting example is that of Wersinger et al (1980), who consider temporal evolution of a nearly-resonant triad with wavenumbers k. satisfying kj = k 2 +k3 and small frequency mismatch 6 = 0)2+^3-0)1. They suppose that one wave is linearly unstable, the other two linearly damped, and the nonlinear coupling coefficients are as for conservative interactions.
Their equations scale to dbi/dt = bi + b2b3 exp (i6t) b
exp (-i exp (-i
where Q2, a3 are positive.
This system has a remarkably rich structure:
in various parameter ranges, several bifurcating periodic solutions and the chaotic behaviour of a strange attractor have been found. When one wave is heavily damped, linear damping and nonlinear forcing may be nearly in balance.
If, for instance, A 3 is such a wave
in (3.2), it approximately satisfies A3 = (A3/a3)A*A*. If this is so, and if waves Ai and A2 are undamped, the latter satisfy the scaled and truncated equations (3/at + Vi-Vjlx
= -Iil
(a/at + y 2 .v)i 2 =
2
128
A.D.D. CRAIK
with I, (k = 1,2) proportional to |A,|2. Solutions which depend on x and t are (c.f. Wilhelmsson et al 1977) i ! = -9F/3T,
I 2 = 3F/35,
F = log[Z(O
- T(T)],
where
Here, ci, c 2 are the x-components of vi, v 2 and Z(O > T ( T ) a r e differentiable functions which may be chosen to satisfy arbitrary initial conditions for Ij and I 2 . Corresponding solutions depending x, y and t are given by Chu § Karney (1977). When the interaction coefficients X. represent a non-conservative process and when no wave is heavily damped, the author knows of no solutions which depend on more than one independent variable. However, approximate solutions, valid when one wave is much larger than the other two, are available, the governing equations then being linear (see, e.g. Craik § Adam, 1978). This fpump-wave approximation1 has been widely used in plasma physics to describe parametric instabilities. Craik (1971) gives two particular nonlinear solutions with dependence on t only. One is a periodic solution for waves which, according to linear theory, would be damped or amplified. The other has no linear growth or damping and satisfies the coupling equations b3b2,
db2/dt = b 3 b*, db3/dt = e
For these, the interaction is conservative only when 6 = 0 (waves of differing energy signs) and 6 = TT (waves of same energy sign), since only then is there a conserved quantity | b x | 2 + | b 2 | * ± |b 3 | 2 . Nevertheless, a particular solution for any fixed 0 is Be 1 X l ,
b 2 = Be 1 X 2 , b 3 = B 3 e 1 X 3 ,
D _ f c°sXn 1 nB3 B n
"[cos(6-xo)J [ J
B3
' '
R
where K i s an a r b i t r a r y p o s i t i v e c o n s t a n t , are
" 1-tKcosxo ' the phases x « ( t ) (j = 1,2,3)
X k = X k ( 0 ) - t a n X 0 l o g ( l - tK c o s
Xo)
(k = 1 , 2 )
TWO- AND THREE-WAVE RESONANCE
129
X3 = Xo + Xi + X2 and the constant xo must satisfy tan(6 - x o ) c o t Xo = 2. Roots xo o f tnis last relationship have cos xo negative or positive: the negative roots give algebraic decay with increasing (positive) t and the positive an 'explosive1 growth with singularity at t = (Kcosxo) Provided exp(iS) ^ -1, initial phases may always be found which lead to this singularity. Since interactions among waves in shear flows are not normally conservative, such rapid nonlinear growth may be the rule rather than the exception! There is a real need for further theoretical work on non-conservative interactions. 6. HIGHER-ORDER INTERACTIONS Lack of space prevents consideration of higher-order interactions. Cubic nonlinearities admit resonance among four wave modes. Such resonance is particularly important among gravity waves, where no threewave resonance occurs. Also, many features of nonlinear instability of shear flows arise from cubic and higher-order nonlinearities. These and other aspects of wave interactions in fluids are discussed by the author in a monograph which is in preparation. Other recent accounts are Gollub § Swinney (1981), Meyer (1981), Yuen § Lake (1982) and West (1981). REFERENCES ACHESON, D.J. 1976 J. Fluid Mech. 77^ 433-472, On over-reflexion. AKYLAS, T.R. $ BENNEY, D.J. 1982 Stud. Appl. Math. 67, 107-123, The evolution of waves near direct-resonance conditions. ARMSTRONG, J.A., BLOEMBERGEN, N., DUCUING, J. $ PERSHAN, P.S. 1962 Phys. Rev. 127, 1918-1939, Interactions between light waves in a nonlinear dielectric. BALL, F.K. 1964 J. Fluid Mech. j^9, 465-478, Energy transfer between external and internal gravity waves. BRETHERTON, F.P. 1964 J. Fluid Mech. 2£, 457-479, Resonant interactions between waves. The case of discrete oscillations. CAIRNS, R.A. 1979 J. Fluid Mech. 92, 1-14, The role of negative energy waves in some instabilities of parallel flows. CAIRNS, R.A. $ LASHMORE-DAVIES, C.N. 1983a Proc. 3rd Joint VarennaGrenoble Internat. Symp. on Heating in Toroidal Plasmas, Grenoble 1982, vol.2, 755-760, A mode conversion interpretation of the absorption of electron cyclotron radiation for perpendicular propagation. CAIRNS, R.A. § LASHMORE-DAVIES, C.N. 1983b Phys. Fluids (to appear). A unified theory of a class of mode conversion problems.
130
A.D.D. CRAIK
CASE, K.M. & CHIU, S.C. 1977 Phys. Fluids 20, 742-745, Three-wave resonant interactions of gravity-capillary waves. CHU, F.Y.F. § KARNEY, C.F.F. 1977 Phys. Fluids 20, 1728-1732, Solution of the three-wave resonant equations with one wave heavily damped. CRAIK, A.D.D. 1968 J. Fluid Mech. 34, 531-549, Resonant gravity-wave interactions in a shear flow. CRAIK, A.D.D. 1971 J. Fluid Mech. 5£, 393-413, Nonlinear resonant instability in boundary layers. CRAIK, A.D.D. 1978 Proc. Roy. Soc. Lond. A363, 257-269, Evolution in space and time of resonant wave triads. II A class of exact solutions. CRAIK, A.D.D. 1980 J. Fluid Mech. 9£, 247-265, Nonlinear evolution and breakdown in unstable boundary layers. CRAIK, A.D.D. & ADAM, J.A. 1978 Proc. Roy. Soc. Lond. A363, 243-255, Evolution in space and time of resonant wave triads. I The 'pumpwave approximation1. CRAIK, A.D.D. $ ADAM, J.A. 1979 J. Fluid Mech. £2, 15-33, 'Explosive1 resonant wave interactions in a three-layer fluid flow. DAVIS, R.E. § ACRIVOS, A. 1967 J. Fluid Mech. .30* 723-736, The stability of oscillatory internal waves. DRAZIN, P.G., ZATURSKA, M.B. § BANKS, W.H.H. 1979 J. Fluid Mech. £5, 681-705, On the normal modes of parallel flow of inviscid stratified fluid. Part 2 : Unbounded flow with propagation at infinity. FUCHS, V. $ BEAUDRY, G. 1975 J. Math. Phys. 16, 616-619, Effects of damping on nonlinear three-wave interaction. HERBERT, T. 1980 AIAA Jour. liB, 243-248, Nonlinear stability of parallel flows by high-order amplitude expansions. JURKUS, A. § ROBSON, P.N. 1960 Proc. I.E.E. 107b, 119-122, Saturation effects in a travelling-wave parametric amplifier. KAUP, D.J. 1981 Nonlinear phenomena in physics and biology. Proc. NATO, Advanced Study Institute, 1980, Banff, Canada, Plenum, pp 95-123. The linearity of nonlinear soliton equations and the three wave resonance interaction. KAUP, D.J., RIEMAN, A. § BERS, A. 1979 Rev. Mod. Phys. 51^, 275-310 [Errata, in Rev. Mod. Phys. 51, 915, are corrected in reprints]. Space-time evolution of nonlinear three-wave interactions. I Interactions in a homogeneous medium. KELLY, R.E. 1968 J. Fluid Mech. 31, 789-799, On the resonant interaction of neutral disturbances in two inviscid shear flows. KLEBANOFF, P.S., TIDSTROM, K.D. § SARGENT, L.M. 1962 J. Fluid Mech. j^, 1-34, The three-dimensional nature of boundary-layer instability. KOZLOV, V.V. & LEVCHENKO, V.Y. 1983 J. Fluid Mech. (to appear). LEBLOND, P.H. § MYSAK, L.A. 1978 Waves in the ocean.Elsevier; Amsterdam, Oxford, New York. McEWAN, A.D., MANDER, D.W. § SMITH, R.K. 1972 J. Fluid Mech. S5, 589-608. Forced resonant second-order interactions between damped internal waves.
TWO- AND THREE-WAVE RESONANCE
131
McGOLDRICK, L.F. 1965 J. Fluid Mech. 2J^, 305-331, Resonant interactions among capillary-gravity waves. McGOLDRICK, L.F. 1970a J. Fluid Mech. £0, 251-271, An experiment on second-order capillary-gravity resonant wave interactions. McGOLDRICK, L.F. 1970b J. Fluid Mech. 4£, 193-200, On Wilton's ripples: a special case of resonant interactions. MEYER, R.E. (ed) 1981 Transition and turbulence. Academic. MIKSAD, R.W. 1972 J. Fluid Mech. 5£, 695-719, Experiments on the nonlinear stages of free-shear-layer transition. MIKSAD, R.W. 1973 J. Fluid Mech. 59, 1-21, Experiments on nonlinear interactions in the transition of a free shear layer. NAYFEH, A.H. $ BOZATLI, A.N. 1979 Proc. AIAA 12th Fluid $ Plasma Dyn. Conf. Williamsburg, Va.: AIAA Paper 79-1456, Nonlinear wave interactions in boundary layers. ORSZAG, S.A. & KELLS, L.C. 1980 J. Fluid Mech. 9£, 159-205, Transition to turbulence in plane Poiseuille and plane Couette flow. PHILLIPS, O.M. 1960 J. Fluid Mech. £, 193-217, On the dynamics of unsteady gravity waves of finite amplitude, Part 1. The elementary interactions. PHILLIPS, O.M. 1977 The Dynamics of the Upper Ocean, 2nd ed. Cambridge University Press. RAETZ, G.S. 1959 Norair Rep. NOR-59-383. Hawthorne, Calif. A new theory of the cause of transition in fluid flows. SATO, H. 1970 J. Fluid Mech. 44, 741-765. An experimental study of nonlinear interaction of velocity fluctuations in the transition region of a two-dimensional wake. SIMMONS, W.F. 1969 Proc. Roy. Soc. Lond. A309, 551-575, A variational method for weak resonant wave interactions. SWINNEY, H.L. § GOLLUB, J.P. (eds) 1981 Hydrodynamic instabilities and the transition to turbulence. Springer. THOMAS, A.S.W. $ SARIC, W.S. 1981 Bull. Amer. Phys. Soc. 26, 152. Harmonic and subharmonic waves during boundary layer transition. THORPE, S.A. 1966 J. Fluid Mech. 24, 737-751, On wave interactions in a stratified fluIcT USHER, J.R. $ CRAIK, A.D.D. 1974 J. Fluid Mech. 6£, 209-221, Nonlinear wave interactions in shear flows. Part 1. A variational formulation. USHER, J.R. $ CRAIK, A.D.D. 1975 J. Fluid Mech. 70, 437-461, Nonlinear wave interactions in shear flows. Part 2. "Third-order theory. WERSINGER, J-M., FINN, J.M. § OTT, E. 1980 Phys. Rev. Lett. 44, 453-456, Bifurcations and strange behaviour in instability saturation by nonlinear mode coupling. WEST, B.J. 1981 Deep water gravity waves. Lecture Notes in Physics 146, Springer. WIELAND, J. § WILHELMSSON, H. 1977 Coherent nonlinear interaction of waves in plasmas. Oxford : Pergamon.
132
A.D.D. CRAIK
WILHELMSSON, H., WATANABE, M. $ NISHIKAWA, K. 1977 Phys. Lett. 60A, 311313, Theory for space-time evolution of explosive-type instability. YUEN, H.C. § LAKE, B.M. 1982 Adv. in Appl. Mech. 2£, 68-229, Nonlinear dynamics of deep-water gravity waves. ZAKHAROV, V.E. 1976 Dokl. Akad. Nauk. S.S.S.R. 228, 1314-1316 [Soviet Phys. Dokl. 21, 322-323]. Exact solutions to the problem of the parametric interaction of three-dimensional wave packets. ZAKHAROV, V.E. § MANAKOV, S.V. 1973 Zh. Eksp. Teor. Fiz. Pis'ma Red. 1J3, 413-417 [Soviet Phys. - J.E.T.P. Lett. 18, 243-247]. Resonant Interaction of wave packets in nonlinear media. ZAKHAROV, V.E. § MANAKOV, S.V. 1975 Zh. Eksp. Teor. Fiz. 6£, 1654-1673 [Soviet Phys. - J.E.T.P. £2, 842-850 (1976)]. The theory of resonance interaction of wave packets in nonlinear media.
CHAPTER 8 NONLINEAR ELECTROMAGNETIC WAVES IN FLOWING PLASMA N. E. ANDREEV, V. P. SILIN and P. V. SILIN Academy of Sciences of the USSR P.N. Lebedev Physical Institute Moscow, U.S.S.R.
1.
INTRODUCTION. In the nonlinear plasma electrodynamics the wide range of phenomena is
connected with the ponderomotive Miller force [1-3].
Already the beginning
of the theoretical research in this direction has predicted the original self-focussed states of the electromagnetic field in plasmas [4-8].
In
paper [9], it was shown that the hysteresis nonlinear picture of strong electromagnetic field increases in plasma.
A special interest connected
with the ponderomotive Miller force influence onto the plasma density changes has grown in connection with the problem of the strong p-polarized radiation transformation in space nonuniform plasmas [10-14].
Here it is
necessary to underline that a number of authors (see, for example [15-25]) have investigated numerically the nonlinear dynamics of plasmas that are subjected to strong radiation interaction.
Alongside the numerical results
let us note the description of the strikingly nonstationary picture of fields in space nonuniform flowing plasmas.
This picture is connected with
the creation and with the disappearance of the cavitons—the density caves into which is created the high intensity field that accounts for the anomalous growth of the field absorption rate.
Recently, this picture was
corroborated once again in paper [26], For the purpose of our paper, it is important to note the results of paper [27] based on numerical simulation. In £27] the effect of the suppression of hot electrons in the transition from subsonic to supersonic plasma flow was found.
It turned out that
such suppression was connected with caviton suppression and therefore, it was connected with the strong plasma field suppression that gave us the reduction of the reflected harmonics of the pumping radiation [25,28]. Meanwhile, it is necessary to underline that similar phenomena have been predicted only for the case of the p-polarized field irradiation interaction with plasmas.
The experimental results [29,30] are of interest.
They indicate the suppression of harmonics generation due to the transition
134
N.E. ANDREEV, V.P. SILIN and P.V. SILIN
from subsonic to supersonic regime, not only in the case of p-polarized fields, but also in the case of s-polarization.
To explain the last result
in paper [29], a thesis about the field self-restriction phenomena was suggested.
The theory of this phenomenon is formulated in our paper [31]
(see also the review talks [32,33]).
The field self-restriction originates
in a distinctive feature of the stationary supersonic plasma flow when the ponderomotive Miller force does not push away the plasma particle as usual in subsonic flow.
Instead, the ponderomotive force shovels the plasma
particles into the region of the maxima of the electric field intensity of the electromagnetic wave.
Theoretical prediction of this feature was gi-
ven in 1977 in papers [34-36] and the results of the experimental investigation were reported in [37]. In connection with this feature of shoveling plasma in the high field region, it is easy to understand that the plasma density increase created by the electromagnetic field growth must produce such a condition when the plasma becomes nearly opaque.
Therefore,
we see that in supersonic plasma flow, it is possible to realize the field self-restriction phenomenon.
Later in this paper we shall give the results
of today's theory of such phenomena. The main part of our paper is devoted to the stationary theory of the interaction of an electromagnetic field with moving plasmas.
So, the first
paragraph expounds an analysis of the hydrodynamics equations and the material equations that reduces the whole problem to simply investigating the nonlinear equations of the electromagnetic field.
The second section
deals with nonlinear s-polarized waves in plasma which is moving with the speed near the velocity of sound.
This section lets us see the simple
analytical regularities that describe the transitions inside the plasma from subsonic to supersonic velocity, including the regularities that determine the structure of the distinctive cavitons.
The third section deals
with the principal results of the theory of the s-polarized field self-restriction, and the fourth section concerns the corresponding p-polarized theory in case of supersonic flow in a plasma that is in average uniform. For space nonuniform plasmas, the subject of the fifth section, a theory of the s-polarized radiation self-restriction phenomenon is given based on some features of the Painleve equation.
The sixth section presents the
field self-restriction phenomenon in the nonstationary dynamics of a plasma expansion that is determined by the ponderomotive force, so the absorption of the plasma heated radiation.
All such nonstationary phenomena on
the level of global hydrodynamics, heat transfer, and so on are described
NONLINEAR ELECTROMAGNETIC WAVES IN FLOWING PLASMAS
135
with the help of code "MEDUSA11 [38], but on the level of "microdescription11 of the ponderomotive force influence we use the code "LAST" [39]. The main conclusions of our paper are presented in Section 7. 2.
NONLINEAR MATERIAL EQUATIONS IN A MOVING PLASMA. The one-dimensional stationary flow of a nonisothermal plasma acted
on by a ponderomotive force due to a high frequency field is described by the hydrodynamic equations
i|fl0, V
8v _ V s 3x n
(2.1)
2 8n 3x ~
Ze 4M.m 0) i e o
9 ,-g ,2 2 3i IEOI •
,
(2
'2)
where n is the density of ions, v is their hydrodynamic velocity, v s = V ^ T /M.
is the speed of sound, z|e| and M. are the charge and mass
of the ions, e and m
are the charge and mass of the electrons, 0) is the
frequency, and E (x) is the coordinate-dependent electric field intensity. E(x,z,t) = Re{E (x)exp[-io) t + iz (a) /c)sin 0]} o o o where 0 is the angle of incidence.
(2.3)
In (2.2), we assume the temperature,
and hence the speed of sound, to be independent of the coordinates, as is approximately the case in experiment in the laser plasma corona. Equations of motion (2.1), (2.2) have the obvious integrals n(x)v(x) = NV = const, 2 2 1 9 9 Ze E (x) Z Z y v (x) + v in n(x) + 2 _ 2 S z 4M.m i e o
, 9 L vz +
2
(2-4)
v
Z s
9
In N.
(2.5)
Eliminating v(x), we can write down the following equation, which gives nonlinear deformation of the plasma density as a function of the electric field intensity: 2 V 2 ss Ze 2 | o 1 NN2 1 s y -y + - V An n(x) + ° r- = i + -\ z Z n (x) Vz 4V M4m o> VZ
£n N,
(2.6)
i e o
where N and V are the density and velocity of the plasma flow at the points, and electric field intensity is zero. In case of the spherical symmetrical stationary plasma flow, instead of Equation (2.1) we have -^ (nvr2) = 0 ,
(2.7)
136
N.E. ANDREEV, V.P. SILIN and P.V. SILIN
where v is the only nonzero radial velocity component.
As this takes
place, the momentum equation takes the form ( .2) where by x is meant radial coordinate r.
Upon integrating (2.7) we obtain n(r)v(r) = NV/(r/R) 2 .
(2.8)
Taking into account the smallness of the electric field scale length in comparison with the characteristic radius of the plasma flow, let us introduce the coordinate x along with nonlinear changing of the field (2.3), so that r/R = 1 + x/R where |x/R| «
1.
The integral (2.8) then can be written down as n(x)v(x) = NV • (1 - 2x/R)
(2.9)
which can be used for relating the plasma density n(x) and the electric field intensity E (x):
°N2 v2 Ze 2 |E| 2 . v2 i(l-4f) -f + -^-fei n(x) + — 5 *—z- i + -^-£n N. 1 R n (x) V2 4V2M.mu L 2 V2
(2.10)
i e o It is obvious that this relationship turns into Equation (2.6) as R -• °°. When the hydrodynamic equations (2.2), (2.1) (or (2.7)) are valid, the electroneutrality condition holds, meaning that the characteristic distance over which the field changes is large compared with the Debye radius.
Un-
der these conditions we can use the following expression for the high-frequency dielectric constant:
where n
e = i - ~^- , c 2 2 = m a) /(4TTe Z) is the critical density of the ions.
sipative effects are neglected in formula (2.11).
(2.11) Small dis-
We disregard these ef-
fects now, all the more since the general scheme for taking such processes into account in nonlinear electrodynamics was developed earlier in a paper
17]. Equations (2.6) (or (2.10)) and (2.11) constitute the material equations of the nonlinear electrodynamics of a stationary one-dimemsional plasma flow in case of the plane or spherically symmetric dispersion of a plasma.
It must be emphasized that the qualitative difference between the
nonlinear electrodynamics of supersonic and subsonic flow.
This difference
can be seen from the following relation that results from equations (2.1), (2.2):
_ vV 2O. J
s 'n 3x
167m M. dx c 1
NONLINEAR ELECTROMAGNETIC WAVES IN FLOWING PLASMAS We see therefore that whereas in subsonic flow maxima of |E (X)|
137 corre-
spond to minima of the density, in supersonic plasma flow they correspond to density maxima.
In other words, in the subsonic flow the strong field
expels the plasma and thereby decreases the nonlinear reaction of the plasma on the field, thus precluding limitations on the field strength. On the contrary, in a supersonic flow ponderomotive force sweeps the plasma into the region of the strong field, thus increasing the nonlinear reaction of the plasma on the electromagnetic field.
Inasmuch as in a dense plasma
the electromagnetic field is subject to the skin effect, it should be clear that an upper bound on the electromagnetic wave field intensity should exist in supersonic flow. In the case of interest to us, that of fields whose pressure is low compared with the thermal pressure, it is possible to demonstrate distinctly the peculiarity of the nonlinear electrodynamic properties of a supersonic plasma stream. We shall also present analytical results for a unique regime in which the stream velocity oscillates in the region of field propagation between the sonic and supersonic values [36]. It should be noted that for appearance of this regime the plasma flow in the critical density region with subsonic velocity. Bearing;in mind that electric field intensity is comparatively small, we can write down the plasma density in the form n(x)/N = 1/(1 + 6(x)) where |6| « 1.
(2.13)
In doing so for plane plasma flow, we obtain from (2.6) the
following equation for the quantity 6(x) which determines the density deformation under the influence of a high-frequency electric field: v
2
2
v2
2
6 [1 + - V ] + 26[1 - -~- ] + 2 5L ||(x) |2 = o, V V V °
(2.14)
where a = l/(16Tm M . ) . Inasmuch as 6(x) is small the addendum in (2.14), which is quadratic in 6, can be obviously neglected except when the stream velocity V is closed to the sound speed, v ; when S v 2
is small (|v| « 1). ->•
^—j
2
V = 7 ( Az-- 1) (2.15) V As this takes place, we have from (2.13), (2.14) 2
l - v[l± Vl - a 2 |E o | 2 /(Vv) 2 ]
(2.I6)
138
N.E. ANDREEV, V.P. SILIN and P.V. SILIN
It is easy to see using Equation (2.14) that the two branches (±) of the solution (2.16) correspond to the subsonic and supersonic plasma flow respectively.
The continuous transition under the influence of the electro-
magnetic field from subsonic to supersonic flow is possible in those points of the space only, where |E | has maximum and the radicand of the expression (2.16) goes to zero. Once the velocity of unperturbed plasma flow V is not close to the speed of sound, the change in the nature of the flow (transition under radiation, with the pressure low compared with the thermal pressure, from subsonic to supersonic flow) is impossible. from (2.14)
n(|Ij 2 )
As this takes place, we have
a2|I |2
(217)
T — ^ 7 77 7 * s
Analogously to (2.17) in the case of inhomogeneous plasma flow which corresponds to the spherically symmetrical expansion, we have from (2.10)
n(H o | 2 , x)
2
2
a2\lf V
- v
2
where a (x) = 1 + x(2v - l)/(vR ). Relations (2.11), (2.16)-(2.18) constitute the concrete material equations, which we shall use in the analysis of nonlinear electromagnetic waves in a moving plasma. 3.
NONLINEAR S-POLARIZED ELECTROMAGNETIC WAVES IN A PLASMA MOVING WITH NEAR-SONIC VELOCITY. We consider first the case of s-polarization, when E (x) is oriented
along the y axis.
Assuming E Qy (x) - iE(x) expt-icKx)],
(3-D
we can write (2.3) in the form E (x,z,t) - E(x)sin[o) t - Z(CD /c)sin 6 + 4>(x)], y o o where E(x) and (f)(x) obey Maxwell equations 2 GO ^ - ^ E(x) + - — - ( £ - sin2 9)E(x) - (^)E(x) = 0, dX dx c
(3.2)
(3.3)
»§£•>« 4
dx Taking into account (2.11) and introducing dimensionless coordinate X = x(o) /c)cos 0 0, we can write down the following two integrals of these field equations:
NONLINEAR ELECTROMAGNETIC WAVES IN FLOWING PLASMAS -E2(X)(t)f(X) = M
139
= const,
(3.5)
s (E f (X)) 2 + M 2 / E 2 + U(E 2 ) = E s where
= const,
(3.6)
E2
(3.7) 2 n cos 0 c In this section, we consider the nonlinear solutions, corresponding 0
to the material Equations (2.16).
In doing so it is convenient to intro-
duce dimensionless electric field strength
and dimensionless coordinate y
= ^~|^
X E^|v
x -^ cos 0.
(3.9)
In the case of standing waves when there is no energy flux along the x axis (i.e. M g = 0 ) , we derive from (3.6) and (3.7):
0J;')2 + | d -f )* 2 ± (1 - * V / 2 - 1 - E±
(3.10)
Here A defines the deviation of unperturbed plasma density from the value n c cos 2 0 (|A| « 1 ) : N = n
cos 2 0(1 + A)
(3.11)
and signs (±) correspond to two branches of the solution (2.16)
M*2)
/
2
—-= ^— = 1 + A - V ± v\l - * n cos 0 c We discuss now boundary conditions for the equation (3.10).
(3.12) The
plasma is thought to flow from the X •* +°° with the velocity V being small 2 in comparison with the speed of sound (v > 0) and density N > n cos 0, giveng the case of linear nontransparency of the unperturbed plasma flow. For solutions decreasing (i|/ = \p' = 0 ) as X -*• +°°, it follows from (3.10) that E = 0. It is clear from (3.12) that continuous transition to the 2 branch n (ij; ) corresponds to the supersonic plasma flow at those space points y where the extremum of the electric field strength takes place: m ~ ^ |i|Ky ) | = i|> ~ 1- F o r this reason it follows from (3.10) that E_=E+=0 for solutions describing the transition under radiation from subsonic to supersonic flow.
Besides, the following relation between density and ve-
locity of the unperturbed plasma flow should hold (cf. Eq. (10) , Ref. [36]):
140
N.E. ANDREEV, V.P. SILIN and P.V. SILIN
That is n c cos 2 G
6
The Equation (3.10) then takes on the form
( |J ) 2 = (1 - t2)(l ±Vl - * 2 )
(3.14)
This equation can be integrated with the help of elementary functions.
By
virtue of the fact that it does not depend on any parameters, all dependency of its solutions on physical parameters (plasma density or velocity at upstream infinity) is determined by relations (3,8) and (3,9) subject to (3.13). The function Y Q (y) = 2 th(y/v>2)/ch(y/v£)
(3.15)
represents the solution of Equation (3.14), which describes the diminishing 2 electromagnetic field for subsonic plasma flow (n = n (\p )) in the regions y > y , y < -y 1 as well as the branch of the solution passing through the zero point (-1 < ij; < 1) for supersonic flow in the interval -y.. < y < y (see Figure 1 ) .
At the points y =
y- the plasma stream velocity is equal
to the sound speed and electric field intensity has extrema: o (*y 1 )
=
1,
Y1 = V 2
n(V2 + l)
The solutions of considered type (which are decreasing as X
(3.16) +
and have
at least one extremum \il/\ = 1 ) all may be constructed from a different number of discussed parts of the function M7 (y) (3.15).
For instance, the
simple soliton ^ ( y ) has one maximum of the field and consists of two diminishing parts of the function (3.15): (3.17)
The function ¥ (y) itself describes a "bisoliton" with two extrema. o 11
"Tri-
soliton , which has three extrema of the field, consist of two "supersonic" parts of the function ¥ (y) and two diminishing "subsonic":
NONLINEAR ELECTROMAGNETIC WAVES IN FLOWING PLASMAS
141
(a)
-5
-3 -r
?^4 FIG 1.
The spatial structure of the dimensionless magnitude of the electric field (a), (c) and perturbations of the velocity density
(b), (d).
142
N.E. ANDREEV, V.P. SILIN and P.V. SILIN
In general, the N-soliton solution with N extrema consists of N - l supersonic parts of the function ¥ (y) in the region of the nonlinear transpar2 °2 ency, when n = n_0[> ) < n cos 0, as well as two diminishing "tails" in c 2 the subsonic plasma flow, when n = n (ip ) at y > y. and y < -(2N - 3)y- : \ ( y ) = (-Dm * o (y + 2m yx),
(3.19)
where m consecutively takes on whole values from 0 to N - 2 in the different space regions: m = 0,
-y < y < 4<»
m = l,2,...,N-3,
-(2m + l)y 1 < y < -(2m - l)y 1
m=N-2,
(3.20)
- o o < y < -(2 m - l)y .
When N •* °°, the left subsonic region is absent and the solution ^ ( y ) (3.19), (3.20) describes the nonlinear standing wave in the supersonic plasma flow in a half-space y < y.. , and is subject to the skin effect in the region of subsonic flow at y > y . This solution was discussed in a paper [36] in order to describe the interaction of the laser light (incident from X -»- -00) with an expanding plasma corona. It should be stressed that the maximum of the electric field intensity for considered solutions (3.19) as it followed from (3.8) is unambiguously related to the velocity V (or density N, see (3.13)) of the unperturbed plasma flow: 2
E
9
V
42 4
!S
5
2
\
(
4
"- T \-1>2-6<1-4M
^ - - D (3.21)
4irn M. y Vx v V v n cos 0 x c l s e s s c where V_ = eE/(m aj ) is the electron vibration velocity in the electromagE eo netic wave field and V T Te 2
2
the region of the nonlinear transparency (n = n (ty ) < n cos 0) the plasc 2 ma density therewith oscillates in space between the values n . = n (\h =0) 2 mm and n = n_(\\) =1), which are equal in accordance with (3.12), (3.13), max — to and (2.15) 2 n A - 2V v = 1), 2 n cos 0 c 2 n 1 v (3.2 + A - V = l - : •v = 1 - 3 1). n cos2 0 v2 in the solution ¥m_ (3. 19) is connected (following the paper [36]) with the incident electric field E by If the maximum value of the field E
NONLINEAR ELECTROMAGNETIC WAVES IN FLOWING PLASMAS
143
WKB approximation E
L • 4E o 2 /Vl-i(n mln + nmax)/nc cos2 0
(3.23)
we obtain from (3.21) and (3.23) the following relationship between the incident electromagnetic field intensity at X •* -00 and the plasma flow velocity V at X -> +»: E
?
2
E
o _ ~o ~o 4Trn T " 2 ce e V xT e
K
16
V2 v s
2
C
1}
lf2
V
*
Since we have the single value of the incident electric field E
(3-24)
(3.24)
for the predetermined plasma flow velocity V < v , it may be thought that s the realization of such a stationary regime of the laser beam interaction with an expanding plasma is difficult.
It should be born in mind at the
same time that the velocity of the plasma stream flowing in the region of the electromagnetic field propagation can be itself changed under radiation. For the velocity of the unperturbed plasma flow close to the speed of sounc^ we obtain in accordance with (3.24) the following estimate of this change 6V = V - v : s
6V
v8
s
~M,n
,4/5 T ' ' ce e
o
< 3 ' 25 >
It is clear from this relation that the increase of the incident field intensity leads to an extension of the range of plasma flow velocities in 2 the vicinity of the reflection point (n(X) = n tionary regime described by ^
cos 0) for which the sta-
(3.19) can be reached.
However, in the case
of interest to us, that of fields with pressure low compared with the ther2 mal pressure ( E Q < 4TT n T ) , it is evident from (3.25) that the solution y^ can be realized for flow velocity in the vicinity of the critical density, which is quite near the speed of sound. In the laser plasma corona, the substance flow velocity in the^ critical region is determined by the global features of the plasma expansion (such as geometry of the flow, conditions on the ablation surface, processes of laser light absorption and thermal transport) and can be rather far from the local speed of sound.
When the plasma flow velocity in the
vicinity of the critical density is well above the speed of sound, the 2 comparatively weak laser field (E « 4ir n T ) cannot hinder the supero ce e sonic flow and the regime of the laser plasma interaction described by the solution V
(3.19) is not realized.
144
N.E. ANDREEV, V.P. SILIN and P.V. SILIN The nonlinear solutions for electromagnetic waves in supersonic plas-
ma flow are discussed in the next sections of the paper. 4.
PROPAGATION OF THE S-POLARIZED RADIATION IN SUPERSONIC PLASMA FLOW. In the case of supersonic plasma motion, when the velocity of the un-
perturbed flow is not too close to the speed of sound, the nonlinear deformation of the plasma density is determined by the formulae (2.17). 2
Then
the value U(E ) (3.27) governing the integral (3.6) of the Maxwell equations for an s-polarized field (3.1) takes the form U(E 2 ) = A • E 2 + \ E 4 / E 2 V ,
(4.1)
where ^ ~ 2 — = (e - sin 2 0)/cos 2 0, cos 0 £ C o o n cos TT2 M ± (V Z -v g Z )n c cosZ0/N E 1 6 ^ T £ ( - ^ - j - 1) - £ — ^ 1
n
^
c
v
.
(4.2)
s
The left hand side of (3.6) can be regarded as the sum of the effective kinetic, centrifugal, and potential energies, and the corresponding problem of finding the intensity of the electric field becomes analogous to the problem of the particle motion in a central field.
Figure 2 shows the
dependence of the sum of the centrifugal and potential energy $(E ) = M
S
/E
+ U(E ) in the case of linear transparency of the plasma 2 2
flow, when A = (e
- sin
0)/cos
0 > 0.
It is seen from this figure that
in this effective potential a finite motion, which corresponds to the wave solutions of field equations, is possible at those values of E
which do s
not exceed a certain maximum.
The maximum possible value of the electric
field strength for such solutions has than an upper bound determining the location of the effective potential maximum (see Figure 2 ) . In the case when there is no energy flux (standing waves), and M E field value s,max occurs at E = E_ SE s,max V ~ The space structure of the field at M g = 0 and E
= 0 , the maximum (4.3) (4.4)
2 2 < (A /2)E^ in accordance
with (3.6), (4.1) takes on the following form
E(X) = Ey/A - e1 shaV-y- 1 >V A T T ^ ' where
(4#5)
145
NONLINEAR ELECTROMAGNETIC WAVES IN FLOWING PLASMAS
FIG 2.
The dependence of the sum of the centrifugal and potential energy 4>(E2) = M 2 / E 2 + U ( E 2 ) on the electric field intensity.
£
1 -
2i
and sn(u,k) is the elliptic function.
s/
E
V2>1/2'
(4.6)
We note that when the equality (4.4)
is satisfied, the oscillatory dependence (4.5) turns into E(X) = E y v^ th(xA72).
(4.7)
This solution corresponds to the following plasma density and velocity distribution n(X) - N = n
cos 2 0 A th 2 (xA?2"),
v(X) - V = -V(n
cos20/N) A th 2 (x/K/2).
The foregoing formulas allow us to call the solution (4.7) a supersonic solution.
In contrast to the caviton, in which the density well is filled
with an intense high-frequency electric field, in our case (4.7) the bottom of the density well corresponds to a zero electric-field intensity. On the contrary, for the magnetic field we have
146
N.E. ANDREEV, V.P. SILIN and P.V. SILIN
Bx(X,z,t) = -sinG
E y (X,z,t),
B (X,z,t) = -%. E__ COS[OJ t - z(o> /c)sinei/ch (X/A/2) • z vT V o o
(4.8)
In particular, at 0 - 0 the magnetic field inside the supersonic soliton has a maximum and decreases with increasing distance from the density well. We must point out in this connection the possibility of propagation in the plasma of peculiar magnetoacoustic nonlinear waves constituting travelling density wells filled with a high-frequency magnetic field, similar in some respects to electroacoustic nonlinear waves [40]. For a travelling wave, when M
^ 0, the maximum electric field inten-
sity turns out to be less than given by formula (4.3) and decreases somewhat with increasing energy flux density in the travelling wave.
When the
energy flux density reaches a value close to that determined by the righthand side of (4.3), wave propagation becomes impossible.
This can be di-
rectly seen from the following solution obtained for equations (3.6), (4,1) at M g + 0 (cf. Ref. [7]): E2(X) = E
2
+ (E
2
•
2 2 _ 2 2 where E- ^ E = E - E Q are the three real roots of the equation 1 z max j M g 2 / E 2 + E2A - i E 4 /E v 2 = E g . We note that at E
(4.10)
» E« formula (4.9) describes a nonlinear plane wave
with constant E(x). A solution in the form of a plane wave E(x) = E exists also at E^ - E~ = E
, when formula (4.9) describes a soliton
E2(X) = E, 2 + ( that goes over as M
-*- 0 (E
•> 0) into soliton (4.7).
According to (4.10), propagation of the travelling waves becomes impossible when M M M
«
S
exceeds the value 2 32TT
n
T M
s,max"7^nc Te(1 3/3 e
N
n
c
5T > cos 0
3/2
,
(
V2
~2 " v s
n
1}
c
~ N
COS
°
f
(4
'12)
The maximum electric field intensity corresponding to the maximum value of the energy flux density is given by
NONLINEAR ELECTROMAGNETIC WAVES IN FLOWING PLASMAS
147
Thus, the foregoing analysis allows us to state that in a plasma expanding with supersonic velocity, the electric field intensity of the s-polarized stationary nonlinear waves is bounded by the condition E
2
41T n T ce e
2
n
cos 2 0
<4(fTl)(-^v ^ ' N v
- 1).
v
(4.14)
It follows from this, in particular, that the maximum possible field amplitude decreases both when the velocity of the unperturbed plasma flow V approaches the speed of sound and when the density N approaches the value 2 n cos 0 corresponding to the boundary of linear transparency of the plasma. To conclude this section, we emphasize that the analytic formulas obtained in this section for the field are valid under conditions of weak 2 2 (cubic) nonlinearity, which call for smallness of A = (e - sin 0)/cos 0 2 2 ° in comparison with (1 - v 5.
/V ) •
SELF-LIMITING OF THE P-POLARIZED WAVE FIELD IN SUPERSONIC PLASMA FLOW. We proceed now to consider p-polarized radiation.
Bearing in mind
the application of the results to a plasma with infrequent collisions and with a density higher than critical in the interior of the plasma, we confine ourselves here to the case of standing waves.
The principles of non-
linear electrodynamics for p-polarized waves are treated in a paper [41]. Assuming in accordance with that reference E x = E^.(x) and E
= iE (x) ,
we have Ex(x,z,t) = E x ( x ) cos[o) t - z(u) /c)sin 0 ] , E (x,z,t) = E (x) sin[u) t - z(u) /c)sin 0 ] .
(5.1)
The system of field equations reduces then to the following: 2 2 d E dEL. 0) a) ,2 dx c 2 z' dx c dE z ^o 2 ^o — sin 0 + E v — sin 0 = — e E v , dx X c c X'
£ (£Ex) = f sin 9 eEz,
(5.2)
where, in accordance with (2.17) e = e - (E__ + E o X ; 2
e =1 N/ " v o
h
= 167T n c
V2
V —2 -
e
n
1}
it •
(5 3)
-
148
N.E. ANDREEV, V.P. SILIN and P.V. SILIN
The last of these three equations (5.2) is a consequence of the first two. It will be useful, however, for the analysis that follows.
In accordance
with reference [41], the integral of the field equations takes on the form dE
2
< ~d >
+
0) 2
-j
o o {£ (E + E z O *
>-
9 9 sin 0
h -
(EY2 + E 2)
i
} =E P
= c o n s t - (5-4)
With the aid of this relation and the first-order equations of the system (5.2), we can both investigate the solutions qualitatively and solve the field equations. To make the exposition more compact, we introduce the notation (5<5)
2 a) e E_. ~oo ~
(5.6)
we write down, in accordance with (5.2), (5.4), the following system of equations: ( h 1 ) 2 - a 2 g 2 - |-(1 - g 2 - h 2 ) 2 = - | i
2
,
(5.7)
-ah 1 = g[l - a 2 - g 2 - h 2 ] , g 1 [l - 3g 2 - h 2 ] - 2ghh 1 = ah[l - g 2 - h 2 ] . 2 Equations (5.7) and (5.8) enable us to express h
(5.8) (5.9) 2
in terms of g :
h 2 = f ± (g 2 )E{-2g 4 H-2g 2 -a 2 (l+g 2 )±a[4a 2 g 4 -E 2 (2g 2 -a 2 )] 1/2 }(2g 2 - a 2 ) " 1 (5.10) Next, with the aid of this formula and Equation (5.8), we ultimately find rg f±W) rg 2 2 _ a2 ^^ # (5.11) x _ d g ______ f + (g 2 ) a * [ 4 a 8 " E ( 2 g
°
This formula determines the dependence of E__(x) on the coordinates, and in accord with (5.10) also the dependence of E (x). In the particular case z 2 E = 0, we have from (5.10), (5.11) E x (x) = E-Zr" cos[x(03o/c)sin 0 ] , E z (x) = Evv/E^" sin[x(o)o/c)sin 0 ] .
(5.12)
The solution (5.12) corresponds, according to (5.3), to a zero nonlinear dielectric constant.
This means that in this state, the plasma density
NONLINEAR ELECTROMAGNETIC WAVES IN FLOWING PLASMAS has become equal to critical.
149
The peculiarity of the solution (5.12) can
be understood by recalling the singular behavior of the solutions of linear electrodynamics near e = 0.
It is important that the limiting state (5.12)
is realized at a finite electric field intensity, when P.max _ , , _V ceTe v2 s
iwi
47rn
_ JL A nc} N •
f5 ( 5
-m '13)
Just as in the case of (4.14), the left-hand side decreases here when V approaches the speed of sound, and also when N approaches the critical density.
It should be noted that the solution (5.12) can take place both in 2 the classical region of the linear transparency, when N < n cos 0 and in c 2 the region of linear opacity n
cos 0 < N < n .
The general discussion of the maximum field of a p-polarized wave in a plasma with a dielectric constant (5.3) is best carried out on the basis of Equations (5.7)-(5.9). tremum of the field E tremal points X
We note first that according to (5.8) the ex-
can be realized at points of two types.
At the ex-
of the first type E^ vanishes: g(X t ) = 0,
h 2 (X t ) = 1 ± |E|.
(5.14)
At the extremal points of the second type, we have g 2 (X t ) + h 2 (X t ) = 1 - a2 and
g 2 (X ) = - i - (E2 - a 4 ) , t 2a 2
h 2 (X ) - _ L . [i - (i t 2a
The largest extremal point, corresponding to E
2
z
the plus sign.
(5.15) a
2
)
2
- E2]
(5.16)
, is given by (5.14) with 2
This point corresponds to the minimum value of h
on the
segment (+00, 1 + | E | ) , which does not correspond to spatially oscillating solutions.
The remaining two possible extremal points, as seen from (5.14)
and (5.15) are such that in them g 2 (X t ) + h 2 (X t ) < 1.
(5.17)
This means that the intensity of the electric field at these points cannot exceed the value determined by (5.13). We note next that from the fact that the left-hand-side of (5.15) is positive it follows that the extremal points of the second type are possible only under conditions corresponding to the classical region of linear 2 2 transparency a
< 1 or, equivalently, N < n
cos
.
In addition, it fol-
lows from (5.16) that the following inequalities should hold:
150
N.E. ANDREEV, V.P. SILIN and P.V. SILIN 1 > 1 - (1 - a 2 ) > E 2 > a 4
(5.18)
Therefore, we have according to (5.16) and (5.18) h 2 (X ) < 1 - a 2 , t
g2(X ) < - i j (1 - a 4 ) . c 2
(5.19)
We see thus that the z component of the electric field intensity cannot exceed the value determined by (5.13).
We note that the smaller extremal
point (5.14) can be realized , just like the points of the second type, only under the condition E
< 1.
The largest of all possible values of the
field h, corresponding to solutions that oscillate is space, will take ~ 2 place at E = 0, when h
= 1, i.e., in the case of the solution (5.12).
We turn now to a discussion of the limitation imposed on the longitudinal field Kj..
At the extremal points Xo, corresponding to this field,
when g (X«) = 0, we have in accordance with (5.9) -2g(X £ )h(X £ )h 1 (X £ ) = ah(X£)[l - g 2 (X £ ) - h 2 (X^)].
(5.20)
If we assume that h(X 0 ) ^ 0, it can be easily verified that a simultaneous 2 solution of Equations (5.8) and (5.2) leads only to values g (X«) < 0. It remains therefore to consider h(X p ) = 0 .
It follows then from (5.7) and
(5.8) that g2(X£)[l - g 2 ( X £ ) ] 2 = a 2 [g 2 (X £ ) + i(l - E 2 ) - | g 4 (X A )].
(5.21)
In order for the right-hand-side of this equation to be negative, it is necessary to satisfy the inequality g 2 (X £ ) < ^[1 +
1 + 3(1 - E 2 ) ] < 1.
Account is taken here of the fact that 0 < E 2 < 1.
(5.22)
It follows from (5.22)
that the intensity of the longitudinal field is also limited to the value ~2 (5.13), which is realized only at E = 0 . Thus, when p-polarized radiation interacts with a supersonic plasma stream, both electric-field components are bounded, and the maximum possible field values of both components decrease when the unperturbed density approaches the critical value (see Equation (5.13)). 6.
NONLINEAR REFLECTION OF S-POLARIZED RADIATION FROM SUPERSONIC FLOW OF A NONUNIFORM PLASMA. The foregoing analysis allows us to state that when radiation propa-
gates in a supersonic stream of a nonuniform plasma, the amplitude of the possible stationary solutions is also limited, at least in the case of sufficiently gently sloping profiles of the unperturbed density.
To make
this conclusion even more obvious, er present here results pertaining to
NONLINEAR ELECTROMAGNETIC WAVES IN FLOWING PLASMAS
151
the case of a linear space dependency of the unperturbed plasma density, when the linear dielectric constant e O
= sin 2 0 - c o s 2 0 ~ . L
(6.1)
For spherically symmetric stationary expansion of a plasma the characteristic scale length of the nonuniform density (L) in the vicinity of the reflection point equal in accord with (2,18) cos20/N.
L - Ivl R n 1
•
(6.2)
c
Equation (3.3) for s-polarized standing (d<j>/dx = 0) wave (3.2) with regard to (2.11), (2.18) can be written in the following form 2 ^ = X u(X) + U 3 (X) sign(V2 - v *) Z
(6.3)
S
We have used here dimensionless variables X = x(a> cos0/c) (Leo cos0/c)" 1 / 3 , o o u(X) - ^ -
(LU) cos0/c) 1 / 3 ,
where E-. is determined by the formula (4.2).
(6.4)
When obtaining this field
equation, we neglect in (2.18) the dependence of the coefficient a(x) on the coordinate (a(X) = 1 ) . This is permissible near the critical density (reflection point), when the parameter (La) cos0/c)
is small compared
with unity. The Equation (6.3) in the case of supersonic plasma motion (sign(V
- v
s
) - + 1) is Painleve equation of the second type, which has
solutions with movable singular points [42-44].
We consider for this equa-
tion solutions that decrease at X -*- +°°, corresponding to the same asymptotic 2 2 form of the solutions for both possible values of sign(V - v ) :
u(X) -
^rjr exp(- 4 X 3 / 2 ) ,
For small values of the amplitude A «
X -> +».
(6.5)
1, Equation (6.3) corresponds to the
linear Airy equation, and its solutions using the asymptotic form (6.5) go 2 2 over into the Airy function regardless of the sign of V - v , as seen in Figure 3a, which shows the solution of Equation (6.3) for A = 0.1 in the asymptotic form (6.3). In the case of subsonic flow (V
< v
) , the solution of Equation s
(6.3) retains the essential properties of the Airy function with increasing amplitude A, although the deformation of the density profile does cause an
152
FIG 3.
N.E. ANDREEV, V.P. SILIN and P.V. SILIN
The spatial structure of the electric field in the linear plasma layer in case of subsonic (a), (b) and supersonic (c) flow.
obvious shift of the reflection point and a shift of the maximum of the field towards the denser layers of the plasma.
This is clearly demonstrated
by comparing Figures 3a and 3b, the latter showing the solution of Equation (6.3) for A = 10 and sign (V
- v
) = - 1 . In this case, naturally, there s is no restriction on the amplitude of the field. The situation is qualitatively different in the case of supersonic 2 2 plasma flow, when V > v . Above all, in this case the deformation of the s density profile causes the maximum of the field to shift toward the more tenuous plasma layers, as seen from Figure 3c, which shows the solution of 2 2 Equation (6.3) for sign (V - v ) = +1 and A = 1.414 using the asymptotic s form (6.5).
The most important property of the supersonic plasma flow in
our analysis, however, is the absence of solutions without singularities of the Painleve equation (6.3) for a field with decreasing asymptotic form (6.5) at an amplitude A >
2 [43-44]; this obviously means a limitation on
the possible value of the electric field intensity of the electromagnetic wave propagating in an inhomogeneous supersonic plasma stream.
We empha-
size that according to (6.4) the maximum possible electric field intensity
NONLINEAR ELECTROMAGNETIC WAVES IN FLOWING PLASMAS decreases like (La) cos 0/c)
153
with increasing characteristic dimension
of the plasma inhomogeneity. Summarizing all the foregoing from Sections 3-5, we can draw the following conclusion.
In a supersonic plasma stream, the plasma striction
nonlinearity that leads to an increase of the particle density in the region with stronger fields, a limitation is imposed on the possible field intensity, inasmuch as the plasma transparency decreases with increasing field.
This conclusion allows us to state that when radiation propagates
in an inhomogeneous supersonic stream under the condition that the plasma be subject to cubic nonlinearity (i.e., when the pressure of the field is insufficient to slow down the supersonic flow), the stationary value of the electric field in the vicinity of the reflection point turns out to be limited and decreases with increasing characteristic dimension of the inhomogeneity, in contrast to the situation in linear electrodynamics.
This
conclusion distinguishes qualitatively the foregoing results from those obtained earlier in the theory of a quiscent plasma [ 7] and in the theory devoted to the case of subsonic influx of plasma into the critical-density region [36] (see Section 2 of this paper). 7.
THE EFFECT OF WAVE FIELD self-limiting IN THE DYNAMIC INTERACTION OF POWERFUL RADIATION WITH PLASMA. In this section we shall present the numerical calculations and re-
sults for the interaction of powerful laser radiation with plasma and demonstrate how field self-restriction effect shows in nonstationary dymanics of plasma corona flyaway, due to ponderomotive force as well as to absorption of plasma heating radiation.
Laser energy absorption was determined
by detailed consideration of electromagnetic, small-scale hydrodynamic and kinetic processes in the vicinity of the critical density with code "LAST" [39] use; and hydrodynamic boundary conditions, ion and electron temperature for small absorption region were determined by "large-scale" hydrodynamics of expansion being calculated with code "MEDUSA" [38]. The calculation results for short laser pulses with time < 100 ps at normal incidence of s-polarized radiation [45] demonstrate that self-consistent absorption of Nd laser radiation at 5.10^ order (for the target with Z
f
w/cm^ is a value of 30%
- 6) and fluctuates with time in accordance
with evolution of electric field and plasma density profiles.
The radia-
tion energy, relatively rapidly contributed into plasma, therewith leads to rapid heating of the target corona, and supersonic plasma movement in the vicinity of critical density arises already at early stages of corona
154
N.E. ANDREEV, V.P. SILIN and P.V. SILIN
expansion.
When the relative speed of plasma flow through the critical
density region exceeds the local speed of sound, the effect of field-limiting is clearly seen (see Figure 4 ) . It should be emphasized that the de-
-2
(c)
o.s <0 FIG 4.
15
20
Typical "LAST" code simulation results for the interaction of a plane electromagnetic wave at normal incidence with supersonic plasma flow. (a), (b) and (c) show, respecitvely, the density profile, the flow velocity, and field structure.
crease of electromagnetic field intensity when approaching the critical density value, leading to absorption decrease and increase of reflection of laser radiation, is the consequence of the supersonic flow response to ponderomotive force action.
In order to distinguish the determining role of
ponderomotive force in the absorption process the calculation with the exception of Miller force was made.
Shortly after (time ~ 5 ps) switching
off ponderomotive force, smoothing of the density profile occurs and fieldseljrliiniting effect disappeared.
The field intensity in the critical den-
sity region therewith increases (see Figure 5) which leads to increase of absorption up to the value ~ 60%, weakly changing with time. Numerical modelling for non-linear interaction of powerful p-polarized radiation with plasma at ns time of laser pulse was performed for parame-
NONLINEAR ELECTROMAGNETIC WAVES IN FLOWING PLASMAS
155
LU FIG 5.
The same as on Figure 4, 5 ps after switching of the Miller force.
ters close to ones realized in experiments, made in the Physical Institute of the Academy of Sciences of the USSR, under the action of Nd - laser at ~ 5.101
w/cm2 on a flat aluminum target [46]. The calculations [47], made
with three different limiting values of coefficients of plasma heat conductivity f = 1; 0.1; 0.03 from free streaming flux n
V-p T e , demonstra-
ted that the plasma expansion speed in the critical point vicinity exceeds local sound speed approximately in 2ns after laser pulse start (depending on f value and target thickness).
The typical scale length of plasma den-v
sity nonuniformity in the critical point vicinity at t ~ 2ns, when transition to supersonic regime of expansion occurs, is of order 10-20 pm. Plasma parameters (nonuniformity dimension and the speed of substance flowing into) obtained with code MEDUSA, characterizing the vicinity of the critical plasma density during time intervals close to laser pulse middle, were used as initial and boundary conditions for the detailed analysis of nonlinear interaction of radiation with plasma by code LAST at subsonic and supersonic regimes of expansion.
156
N.E. ANDREEV, V.P. SILIN and P.V. SILIN The angle of p-polarized laser radiation incidence
in these calcu-
lations was chosen to be close to the optimum ((La) /c)
sin 0 = 0,7) with
initial scale length of plasma density nonuniformity at critical point L = 10A - 10]im. o In the case of subsonic plasma flow, the interaction is characterized by intensive excitation of Langmuir oscillations in space region occupied by plasma with density n - ( 0 . 7 U ) n .
Then the absorption coefficient of
laser radiation is close to 80%, and about one third of absorbed energy is transferred to hot electrons, the distribution of which may be characterized by temperature approximately one order higher than the temperature of the main body of electrons.
The intensity of the second and the third harmon-
ics generation in subsonic regime reaches, respectively, K. - 10"
and
K~ - 10""^ of the intensity of laser radiation incident onto plasma.
With
the increase of plasma flowing speed through the region with critical density, the field self-limiting effect in supersonic plasma stream is clearly seen, leading to decrease of laser radiation absorption.
In Figure 6,
the distributions of density, speed, and of two electric field components (obtained in calculations with plasma speed flowing into the critical density region 1.5 times the speed of sound), are presented.
Note that as
with s-polarized radiation, the field decrease with approach to the critical point is the consequence of nonlinear deformation of plasma density under ponderomotive force action in supersonic flow, when maxima (averaged over plasma oscillation scales) correspond to density maxima. The absorption coefficient of the incident radiation in this case is about 40%, and the proportion of incident to plasma energy transferred to hot electrons does not exceed 10%.
Besides, "tails11 of the electron dis-
tribution function are characterized by significantly lower energy than in the case of the subsonic regime of plasma flowing expansion.
The sharp
decrease of coefficients of transformation into higher harmonics up to -4 —7 values K« - 10
and K
- 10
is the result of decrease of Langmuir wave
generation in the critical density vicinity (see Figure 6 ) . The above consideration shows that in the case of short laser pulses as well as targets heating by laser radiation with ns pulse duration, aroused supersonic plasma movement in the vicinity of critical density leads to suppression of hot electron generation and higher harmonics of heating radiation due to manifestation of the field self-limiting effect leading to the increase of laser radiation reflection.
This conclusion
qualitatively agrees with the experiments [29,30] indicating the field
157
NONLINEAR ELECTROMAGNETIC WAVES IN FLOWING PLASMAS
10 0.5
2 FIG 6.
3
H
S (,
Typical "LAST" code simulation results near critical density region for the interaction of a plane p-polarized electromagnetic wave with supersonic plasma flowo
decrease in plasma with transition from subsonic to supersonic plasma flow*. 8.
CONCLUSION. The previous sections contents show that nonlinear plasma properties
in the limit of collisions neglecting as well as under conditions when plasma heating is due to inverse Bremsstrahlung or small scale collective effects, are essentially determined by hydrodynamic plasma flowing.
This,
first of all, is related to qualitative change of plasma particles re-distribution under Miller's force depending on whether stationary plasma flow is subsonic or supersonic.
The effect arising here of wave self-limiting
in plasma, apart from its manifestation in suppression of harmonics generation [29,30], may be seen, also, in simultaneous suppression of fast electron generation and in accompanying phenomena.
It is supposed to be
158
N.E. ANDREEV, V.P. SILIN and P.V. SILIN
expedient to investigate experimentally, in detail, the effect of plasma flowing on its nonlinear electrodynamics under laboratory conditions (compare [37]), where—unlike experiments on interaction of laser radiation with plasma [29,30]—it is possible to obtain significantly more detailed information about occurring phenomena. After account of achievements in certain domains of plasma electrodynamics we shall dwell on some outlooks for future development.
Just in
this connection, in conclusion, it should be emphasized that the demonstration of relatively simple theoretical laws appears to be possible applying to stationary plasma flow.
On the other hand, in paper [31], we already
indicated that in the case of non-stationary interaction of radiation with plasma, for instance, the conditions of changing of the sign of the effective force, pushing apart or shovelling plasma, may differ from that in stationary conditions.
In fact this means that non-stationary pattern of
nonlinear interaction of electromagnetic radiation with nonstationary plasma flow may give us many unexpected phenomena.
Besides, it may be sugges-
ted that one of the simplest and, at the same time, one of the most interesting non-stationary phenomena may be the phenomenon, currently called stimulated Mandelstam-Brilluen scattering and which, in nonlinear regimes requires for its description as well as in the material, given above, application of essentially nonlinear notions of plasma hydrodynamics (see [48]). REFERENCES 1.
GAPONOV, A.V. and MILLER, M.A. On the Potential Wells for Charged Particles in the High Frequency Electromagnetic Field, Zh. Eksp. Teor. Fiz. 34_ (1958) 242-243.
2.
PITAEVSKY, L.P. Electric Forces in Transparent Dispersive Media, Zh. Eksp. Teor. Fiz. 39^ (1960) 1450-1456.
3.
HORA, H., PFIRCH, D. and SCHLUTER, A. Beschleunigung von in Homogenen Plasmen durch Laserlicht, Z. Naturforschung 22A (1967) 278-280.
4.
VOLKOV, T.F. Influence of High Frequency Electromagnetic Field on Plasma Oscillations, Sbornik Fiz. i Problema Upravljaemih Termojadernih Reaktsij (Plasma Physics and Problem of Controlled Thermo-Nuclear Reactions) <4 (1958) 98-108.
5.
WEIBEL, E.S. Confinement of a Plasma Column by Radiation Pressure, In: Landshoff, R.K.M. (edit.), The Plasma in a Magnetic Field, (Stanford Univ. Press, 1958).
6.
ASKARJAN, G.A. Effect of the Gradient of a Strong Electromagnetic Ray on Electrons and Atoms, Zh. Eksp. Teor. Fiz. 42^ (1962) 1567-1570.
7.
TALANOV, V.I. On the Self-Focusing of Electromagnetic Waves Nonlinear Media, Izv. Vusov, Radiofizika, 7 (1964) 564-565.
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159
8.
LITVAK, A.G. On the Problem of .Electromagnetic Waves Self-Focusing in Nonlinear Media, Izv. Vusov, Radiofizika 9 (1966) 629-631.
9.
SILIN, V.P. Nonlinear Theory of Penetration of a High Frequency Field into a Conductor, Zh. Eksp. Teor. Fiz, 53. (1967) 1662-1677.
10.
GILDENBURG, V.B. Nonlinear Effects in an Inhomogeneous Eksp. Teor. Fiz. 4£ (1964) 2156-2164.
Plasma, Zh.
11.
GILDENBURG, V.B. and FRAJMAN, C M . Deformation of Plasma Resonance Region in a Strong High Frequency Field, Zh. Eksp. Teor. Fiz. §9_ (1975) 1601-1606.
12.
CHEN, H.H. and LIU, C.S. 37 (1976) 693-697.
13.
SHUKLA, P.K. and SPATSCHEK, K.H. Profile Modifications at Critical Density of Laser Produced Plasmas, J. Plasma Phys. A$_ (1978) 387-403.
14.
GILDENBURG, V.B., LITVAK, A.G. and FRAJMAN, G.M. Deformation of Density Profile and Efficiency of Resonant Absorption of Laser Radiation in Non-Uniform Plasma, Pis'ma Zh. Eksp. Teor.Fiz. 28. (1978) 433-436.
15.
VALEO, E.J. and KRUER, W.L. Solitons and Resonant Absorption, Phys. Rev. Lett. 33^ (1974) 750-753.
16.
FORSLUND, D.W., KINDEL, J.M., LEE, K., LINDMAN, E.L. and MORSE, R.L. Theory and Simulation of Resonant Absorption in a Hot Plasma, Phys. Rev. A _U (1975) 679-683.
17.
ESTABROOK, K.G., VALEO, E.J. and KRUER, W.L. Two-Dimensional Relativistic Simulations of Resonance Absorption, Phys. Fluids 1£ (1975) 1151-1159.
18.
ESTABROOK, K.G. Critical Surface Bubbles and Corrugations and their Implications to Laser Fusion, Phys. Fluids 19 (1977) 1733-1739.
19.
MORALES, G.J. Plasma, Density Plasma,
20.
ANDREEV, N.E. and SAUER, K. Jumps in the Nonlinear Reflectivity of an Overdense Plasma Layer, Phys. Lett. A 61 (1977) 165-166.
21.
SAUER, K., ANDREEV, N.E. and BAUMGRATEL, K. Wave Absorption and SuperReflectivity of Laser Plasma due to Electromagnetic Structure Resonances, "Plasma Physics and Controlled Nuclear Fusion Research, 1978". VII Intern. Conf., Innsbruck, Austria, 23-30 August, 1978, Est. Syn., p. 69; IAEA-CN-37/V-2-2, Vol. Ill, p. 187-199.
22.
ANDREEV, N.E. and SILIN, V.P. Dynamics of Nonlinear Absorption of Intensive Radiation by Moving Plasma, Fiz. Plasmy 4_ (1978) 908913.
23.
ANDREEV, N.E., SILIN, V.P. and STENCHIKOV, G.L. Nonlinear Interaction between Radiation and an Expanding Plasma, Zh. Eksp. Teor. Fiz. 78^ (1980) 1396-1407.
Solitons ii\ Non-Uniform Media, Phys. Rev. Lett.
and LEE, Y.C. Ponderomotive-Force Effects in Non-Uniform Phys. Rev. Lett. 33 (1974) 1016-1019; Generation of Cavities and Localized Electric Fields in a Non-Uniform Phys. Fluids 20 (1977) 1135-1147.
160
N.E. ANDREEV, V.P. SILIN and P.V. SILIN
24.
KOVRIZHNYH, L.M. and SAHAROV, A.S. Cavitons Generation in the Plasma Resonance Region, Fiz. Plasmy 6^ (1980) 150-158; Dynamics of Interaction HF Field with Non-Uniform Plasma, Sbornik, Vzaimodejstvie sil'nyh elektromagnetnyh voln s besstolknovitelfnoj plasmoj (Interaction of strong electromagnetic waves with collisionless plasma), p. 117-155, Gorkij, IPF AN SSSR, 1980.
25.
ANDREEV, N.E., SILIN, V.P. and STENCHIKOV, G.L. Nonlinear Interaction of the Radiation with Ponderomotive-Force Deformated Plasmas, Physica D2 (1981) 146-157.
26.
ADAM. J.C., SERVENIER, A.G. and LAVAL, G. Efficiency of Resonant Absorption of Electromagnetic Waves in an Inhomogeneous Plasma, Phys. Fluids 25^ (1982) 376-383.
27.
ANDREEV, N.E., SILIN, V.P. and STENCHIKOV, G.L. Effect of Suppression of the Possible Generation of Fast Electrons in a Plasma, Pis'ma Zh. Eksp. Teor. Fiz. _28 (1978) 533-537.
28.
ANDREEV, N.E., SILIN, V.P. and STENCHIKOV, G.L. Dynamics of Harmonics Generation in Laser Plasma, Fiz. Plasmy 8 (1982) 600-606.
29.
ANDREEV, N.E., et al. Effect of Suppression of Second Harmonic Generation in an Expanding Plasma, Pis'ma Zh. Eksp. Teor. Fiz. 31^ (1980) 639-642.
30.
ANDREEV, N.E., et al. Spectral-Temporal Investigations of the Backscattered Radiation from Laser Plasma, Phys. Lett. A82 (1981) 177-179.
31.
ANDREEV, N.E., SILIN, V.P. and SILIN, P.V. Self-Limiting of a Wave Field Following Supersonic Dispersal of a Plasma, Zh. Eksp. Teor. Fiz. 79^ (1980) 1293-1302.
32.
SILIN, V.P. Nonlinear Laser Plasma Interaction and Heating,, The Physics of Ionized Gases, Invited Lectures and Progress Reports of SPIG-80, Dubrovnik, 25-30 August, 1980, edited by Matic, M. Beograd, 1980, p. 575-635.
33.
SILIN, V.P. Physical Processes in a Laser-Produced Plasma, Proc. XV Int. Conf. on Phenomena in Ionized Gases, Invited Papers, Minsk, 14-18 July, 1981, USSR, p. 357-366.
34.
TSINTSADZE, N.L. and TSHAKAJA, D.D. On the Theory of Electrosound Waves in a Plasma, Zh. Eksp. Teor. Fiz. 72 (1977) 480-487.
35.
MULSER, P. and KESSEL, VAN C. Profile Modifications and Plateau Formation due to Light Pressure in Laser-Irradiated Targets, Phys. Rev. Lett. 3^ (1977) 902-905.
36.
LEE, K., FORSLUND, D.W., KINDNEL, J.M. and LINDMAN, E.t. Theoretical Derivation of Laser Induced Plasma Profiles, Phys. Fluids 20. (1977) 51-54.
37.
AKIYAMA, H., MATSUMOTO, 0. and TAKEDA, S. Proc. Int. Conf. on Plasma Physics, 1980, v.l, p. -11 -08, p. 393, Nagoya, Japan.
38.
CHRISTIANSEN, J.P., et al. MEDUSA, a One-Dimensional Laser Fusion Code, Computer. Phys. Commun. 7 (1974) 271-287.
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161
39.
ANDREEV, N.E., SILIN, V.P. and STENCHIKOV, G.L. Dynamics of Parametric Turbulence in a Plasma, Sbornik, Vzaimodejstvie sil-nyh elektromagnitnyh voln s besstolknovitel1noj plasmoj (interaction of strong electromagnetic waves with collisionless plasma), p. 156-185, Gorkij, IPF AN SSSR, 1980.
40.
KARPMAN, V.J. Nelinejnye volny v dispergiruyushchin sredah (Nonlinear Waves in Dispersive Media), Nauka, Moskwa, 1973.
41.
ELEONSKY, V.M. and SILIN, V.P. Nonlinear Theory of Penetration of p-Polarized Waves into a Conductor, Zh. Eksp. Teor. Fiz. 60 (1971) 1927-1937.
42.
DAVIS, H.T. Introduction to Nonlinear Differential and Integral Equation, Dover, New York, 1962.
43.
ABLOWITZ, M.J., RAMANI, A. and SEGUR, H. A Connection between Nonlinear Evolution Equations and Ordinary Differential Equations of p-Type, J. Math. Phys. 2J, (1980) 715-721; 1006-1015.
44.
ALBOWITZ, M.J. Remarks on Nonlinear Evolution Equations and Ordinary Differential Equations of Painleve1 Type, Physica D 3 (1981) 129-143.
45.
ANDREEV, N.E., et al. Computer Simulation of Laser-Plasma Interaction by LAST and MEDUSA Codes, Kratkie Soobshcheniya po Fiz. FIAN N 9 (1981) 57-62.
46.
ARTSIMOVICH, V.L., et al. Investigation of Scattering Processes in a Laser-Produced Plasma, Zh. Eksp. Teor. Fiz. J30 (1981) 1859-1867.
47.
ANDREEV, N.E., et al. Nonlinear Absorption of Laser Radiation by Expanding Plasma Corona, Kratkie Soobshchenya po Fiz. FIAN N 3 (1982) 26-32.
48.
SILIN, V.P. and TIKHONCHUK, V.T. Theory of Nonlinear Saturation of Stimulated Mandelstam-Brillouin Scattering in a Plasma, Zh. Eksp. Teor. Fiz. 83 (1982) 1332-1345.
CHAPTER 9 SUPERLUMINOUS WAVES IN PLASMAS P. C. CLEMMOW Dept. of Applied Mathematics & Theoretical Physics, Silver St., Cambridge CB3 9EW, England
1.
INTRODUCTION The basis of the theory of the development of
wave motion in
classical physics is to treat the wave as a perturbation on an equilibrium or steady state, and to linearize the governing equations in the perturbation variables.
This familiar technique has been fully exploited in
the theory of the propagation of electromagnetic waves in plasmas, where it has produced a wealth of detailed analysis which has application to a wide range of physical phenomena. On the other hand, waves in plasmas are subject to nonlinear effects which manifest themselves quite readily.
Typical is the 'Luxembourg effect1,
observed in the early years of broadcasting, in which the field of a 'disturbing1 wave so influences the ionospheric environment through which a 'wanted1 wave is propagated that the latter is noticeably modulated. Many other kinds of 'wave-interaction' are possible, and a variety of studies of the interaction between two or more waves probably forms the bulk of the literature on nonlinear plasma waves.
The emphasis in such
work is mainly on obtaining the first order corrections to linear theory, and investigating the conditions most favourable for interaction. Another corner of nonlinear theory, which may perhaps have more attraction for the mathematician, owes its stimulus to comparatively recent additions to our physical understanding.
It is recognized that
both the laser and the pulsar are capable of generating an electromagnetic wave that, at its specific frequency, is so strong that its passage through a plasma constitutes a finite-amplitude, maybe even a largeamplitude, wave, whose description is quite outside the scope of linear theory.
This poses the problem of finding solutions of the exact (non-
linear) governing equations.
Specifically, it is of interest to seek
periodic solutions, with the reasonable expectation that there will be those that recover the familiar waves of linear theory in the small-
SUPERLUMINOUS WAVES IN PLASMAS amplitude limit.
163
The earliest work of this kind, Akhiezer & Polovin
(1956), in fact predated the appreciation of the applications mentioned, and did not apparently attract much attention until comparatively recently. A comprehensive review of developments up to a few years ago is available in Decoster (1978), and some more recent work is covered in the present paper. The wide range of application of linear theory has thrown up a bewildering variety of monochromatic waves whose detailed nature depends on a host of plasma parameters.
To explore nonlinear theory it is desir-
able, initially at least, to concentrate on the simplest possible physical model which contains the basic elements of interest, and on the simplest class of wave.
The discussion here is therefore restricted to a homo-
geneous, cold, magnetized plasma in which electrons are the single mobile constituent; and to plane waves of constant velocity.
A further help to
clarity is to distinguish between superluminous and subluminous waves; that is, between waves whose speed is, respectively, greater than or less than the vacuum speed of light.
The former seem in some ways easier to
handle, and they alone are considered in the present account. For a treatment of subluminous waves an obvious device is to work in a Lorentz frame of reference that moves with the wave; in such a frame the wave is stationary, and hence depends, not on time, but solely on one cartesian coordinate, z, say.
The governing equations are then ordinary
differential equations, the derivatives being with respect to z.
For
superluminous waves an equally useful device is to work in a Lorentz frame in which all quantities are independent of position, so that the governing equations are ordinary differential equations, the derivatives being with respect to time, t.
These are displayed in §2, where they are shown to
lead to a second order vector differential equation for the reduced velocity, ja, of electrons.
This is the key equation of the theory, and the
remainder of the paper is a description of progress towards elucidating the nature of its periodic solutions. To discuss the results of the theory in the context of plasma physics it would be necessary to make the transformation back to the laboratory frame, and to relate the character of the waves to the physical parameters involved.
Here, however, the objective is to illuminate the basic mathe-
matical content of the theory, and the analysis is therefore presented solely in terms of dimensionless quantitites referred to the spaceindependent frame.
164 2.
P.C. CLEMMOW THE VECTOR DIFFERENTIAL EQUATION For the cold electron plasma model the variables are E and B^ the
electric and magnetic vectors of the wave; and N and j7, the number density and (hydrodynamic) velocity of the electrons.
The equations are Maxwell1s
equations VAE A-v*
V.E -
- -B , ~
VAB
=
A**
'
i E + y e(N.v. - Nv) , lMm.
P
o
« - (N. - N) , eo
1<~1
**
'
V.B = 0 ,
i
-
and the electron equation of motion
with Y
=
(1 - v 2 / c 2 ) ~ * .
The constants here are -e and m, the electron charge and rest mass;
N.
and v., the number density and velocity of positive ions (of charge e, and assumed infinitely massive so that they are not subject to acceleration); JB , the ambient magnetic field vector; and e , y
and c, the
vacuum permittivity, permeability and speed of light. The object is to look for solutions in which the variables depend on space and time only through the combination t - nz/c , where n is a constant, taken to be non-negative without loss of generality. Such a solution represents a plane wave travelling in the z-direction with speed c/n, and the discussion is confined to superluminous waves by taking 0 _< n £ 1 . The simplification that results from restricting the class of solutions in this way is much enhanced by referring the analysis to the frame of reference, S, that moves in the z-direction with speed nc.
For the
time variable pertaining to S is itself T(t - nz/c) , where
Y
=
(1 - n )
;
so that in S the wave depends on the time variable only. The equations referred to S are thus obtained by dropping all the spatial derivatives in the set displayed above.
Two of these equations
then imply that B and N (= N.) are constants, and the remaining two may therefore be written
SUPERLUMINOUS WAVES IN PLASMAS
165
The constant v^y being the velocity of the positive ions in S, is one of the parameters characterizing the plasma.
Its retention as an
arbitrary parameter only keeps explicit the application to streaming plasmas, which is not essential to the main objective.
The natural speci-
fic case to consider is that in which the positive ions are at rest in the laboratory frame, giving v£
=
-ncz ,
where z^ is the unit vector in the z-direction. It is now convenient to introduce the dimensionless parameter where 03 = (Ne2/e m ) * and ft = eB/m are plasma and (vector) gyro frequencies, together with the dimensionless variables T
s o) t ,
jj^ =
yv/c .
The elimination of £ then gives
u? - X (u/Y)f
+ u/y = - n£ ,
(2.1)
where the prime denotes differentiation with respect to T, and Y
(1 + u 2 ) * .
=
Equation (2.1), given in Clemmow (1977), may be regarded as the definitive equation of the theory.
It should be stressed that, for the
chosen model, it has been reached without making any approximations. Nonlinearity, present in several terms in the laboratory frame formulation, appears only through the y factor, and thus in a different guise from the types of nonlinearity familiar in other branches of wave theory* For ease of description an analogy may be drawn with Newtonian particle dynamics by regarding ^u as the position vector of a particle of unit mass under the action of three fields of force, namely a centrifugal force
- ja/y, a uniform force
- nj^,
and a 'magnetic1 force 2^,^/y)1 •
Solutions of (2.1) are represented by paths of the particle, and a first integral of (2.1), |fuf - X (u/y + nz)l 2 + y + nu shows that each path is bounded. particle orbits.
=
constant ,
(2.2)
Periodic solutions correspond to closed
The particle has a position of stable equilibrium at
166 ju =
P.C. CLEMMOW - nf£,
and there are two normal modes of small oscillation in the
neighbourhood of the equilibrium point which transform in the laboratory frame to the familiar monochromatic waves of the linear approximation (magneto-ionic theory).
It is reasonably straightforward to obtain cor-
rections to linear theory by expanding in powers of suitable amplitude parameters, but work on these lines is not pursued here. Linear theory lays particular emphasis on waves propagated either along the ambient magnetostatic field or across (perpendicular to) it. The nature of these extreme cases goes some way towards indicating what happens when the direction of propagation makes a general angle with the magnetic field.
In the exact theory, it is easy to find solutions of
(2.1) representing periodic waves of arbitrary amplitude propagated along the field, that reproduce the waves of linear theory in the small amplitude limit: the analysis is outlined in §3.
For propagation across the field,
though, results are harder to come by: most attention has been given to a treatment of highly superluminous waves using n as a small parameter, as described in §§4, 5 and 6.
This latter work throws up a lot of as yet
unanswered questions, and the paper closes in §7 with some tentative suggestions for further exploration. 3.
PROPAGATION ALONG THE MAGNETIC FIELD For the type of nonlinear, periodic wave under consideration the
ambient static fields are defined by the averages (mean values) of the oscillating fields.
It is easy to see that, in the laboratory frame in
which the positive ions are stationary, the average electric field is zero, and the average magnetic field differs from the constant field JB^ of frame S (appearing in (2.1)) only through its x and y components having the extra factor T.
The case of propagation along the field is therefore
specified by taking X. =
(0,0,X) ,
so that, writing
u. = (£>n,C; , the components of (2.1) are ?" + X(n/Y)' + ?/Y n"
CV with
- A(5/Y)'
+
n/Y
+ S/Y
-
0 ,
(3.3)
-
o
,
(3.4)
-
- n ,
(3.5)
SUPERLUMINOUS WAVES IN PLASMAS
Y
-
167
(1 + ? 2 + n 2 + C 2 ) * .
Two classes of periodic solution can be established by elementary analysis.
The first has both £ and y constant, implying also, of course,
the constancy of 5 2 + r)2 . 5
«
Then (3.5) is
- ny,
(3.6)
and (3.3), (3.4) become in effect linear equations, and hence trivially soluble. £
Self-consistency is obtained by choosing the solutions for which
+ TI is constant. £
=
With suitable time origin these may be written
a sin (COT) ,
n
=
a cos (COT) ,
(3.7)
where a is an arbitrary constant, and CO has either of the values determined by u)2 - Xco/y - 1/Y
=
0 .
(3.8)
Evidently (3.6) and (3.7) give Y
-
rU+a2)*
.
(3.9)
This class of solution represents purely transverse, circularly polarized waves, of arbitrary amplitude, with a mathematically simple dependence of velocity on amplitude and frequency. merely by dropping the a
2
The small-amplitude limit is given
term in (3.9), and the familiar waves of
magneto-ionic theory are then recovered. The other readily recognized class of periodic solution is that in which
C = H - 0,
and j satisfies
5" + C/Y
=
- n
(3.10)
with Y
=
(1 + C 2 ) * •
That the solutions of (3.10) are periodic, of arbitrary amplitude, may be deduced in the standard way from the first integral V1
+ Y + n£
=
constant ,
(3.11)
and their investigation is in principle straightforward.
If desired,
expressions in closed form can be obtained in terms of elliptic integrals. The waves thus represented are purely longitudinal, and the ambient magnetostatic field has, of course, no effect on them.
Their small-amplitude
limit degenerates to oscillation at the plasma frequency, reflected in the fact that the amplitude dependent frequency of the finite-amplitude wave is independent of the wave velocity.
168
4.
P.C. CLEMMOW
PROPAGATION ACROSS THE MAGNETIC FIELD This case is covered by taking
JL - (^,0,0) . The first component of (2.1) is then 5" + 5/Y
=
0 ,
and the remaining two components involve £ only through the y factor.
It
seems likely, then, that there is a class of solutions for which £ is identically zero.
This implies that the electric field lies in the plane
perpendicular to the ambient magnetostatic field, which is indeed (for propagation across the field) the polarization of one of the characteristic waves of linear theory.
With the adoption of the ansatz £ = 0 the
equations to be considered are
n" + A(e/Y)' + n/Y
» o
(4.2)
C" " Mn/Y) 1
= - n
(4.3)
+
Zh
with y
=
( i + n2
+
c 2 )* •
As yet, not much is known about the periodic solutions of these equations for general values of n between 0 and 1.
Most of the work on
them has been concerned with highly superluminous waves, for which n is treated as a small parameter, and solutions are sought as perturbations on solutions for the case n = 0.
An exception is the case when there is
no average magnetic field; that is,
X = 0.
It is natural to envisage solutions over the positive octant of the three-dimensional (n,X>A) parameter space, where A is some convenient (dimensionless) measure of wave amplitude. results are for the domain n «
Since the only extensive
1, it is henceforth assumed that this
condition holds until stated to the contrary.
It is then important to
recognize that what classes of periodic solution exist depends on the magnitudes of X and A.
Particularly noteworthy is the distinction between
the domains A «= 0(1) and A = 0(n), since only the latter can yield the results of linear theory: these domains are considered in §5 and §6, respectively, due attention being paid to the order of magnitude of X.
SUPERLUMINOUS WAVES IN PLASMAS 5.
PROPAGATION ACROSS THE MAGNETIC FIELD: n «
169
1, A = 0(1)
When n = 0 it is evident that (4.2), (4.3) have solutions n
=
a sin(VT),
C
-
a COS(VT),
(5.1)
where a is an arbitrary constant, and V takes either of the two values given by V2 +
r V 2
(1+a )*
—r
=
0 .
(5.2)
2
(1+a )*
These solutions, represented by circular orbits in the r|,£ plane,are analogous to the solutions (3.7) of (3.3), (3.4). By introducing 'polar* variables p, through T) =
p sin <(> ,
£
=
p cos (j)
(4.2), (4.3) can be expressed as differential equations for p and $' as functions of , and a perturbation on the n=0 solution p = a ((j)f = v) sought in the form 00
p/a = 1 + I
m=l
Pm(4>)nm .
By taking the analysis to order n 2 Clemmow (1982a) found closed orbits p/a
=
1 + np
cos <|> + n 2 p
cos(2<)>) ,
(5.3)
where p x and p 2 are constants specified in terms of a and V; which orbits are ellipses with centre at r| = 0,
£ = nap x , and eccentricity of order n 2 .
The validity of the perturbation analysis requires n «
1, but other
conditions are also involved.
It appears that, in terms of the dynamical
analogy, the perturbing force
- nz^ must be much weaker than the centri-
fugal force; so, for a < 1, the more stringent condition n « over.
a takes
This is reflected in the fact that, although apx remains finite as
a -> 0, the dominant term in ap
is proportional to I/a.
As regards the
'magnetic1 force, X can be taken to zero without affecting the validity of the analysis, but must keep clear of certain 'resonance' values that depend on the value of a.
These'resonances' arise by virtue of the co-
efficients in the expansion of p becoming infinite. (5.3), p
In particular, in
is always finite, but p 2 is infinite if ! (1+a 2 )*
(1+a 2 ) 2
=
0 .
(5.4)
This condition on V is inconsistent with the negative root of (5.2), but is consistent with the positive root for a particular value of X, dependent on a.
Specifically, (5.2) and (5.4) together imply that X behaves
170
P.C. CLEMMOW
like {a2 as a -> 0, is 0.54 when a = 2
2
, and approaches (a/6)
as a •* ».
Although, as stated, the solutions just described apply equally to the unmagnetized plasma (A. = 0 ) , there exist also other classes of solution for this case, which have indeed been longer recognized and more fully investigated.
The equations for A = 0, being
n" + n/y = o ,
c" + c/y = - n ,
(5.5)
evidently admit the purely longitudinal periodic solutions, with n identically zero, already described at the end of §3; and these, in fact, hold for all values of n between 0 and 1. solutions which for n « roughly
They also admit another class of
1 may be described as quasi-transverse in that,
speaking, the amplitude of the £ oscillation is smaller than the
(arbitrary) amplitude of the r\ oscillation by a factor of order n.
The
general character of such solutions is readily revealed: for if £ is of order n then y is approximately (1 + r|2) , and so Ti satisfies
n11 + n/d+n 2 )*
- o;
(5.6)
this, of course, gives periodic solutions which, without loss of generality can be taken to be anti-symmetric functions; with y then a known function having half the period of r\, the equation for £ can be shown to yield solutions also with half the period. A first integral of (5.5) is (cf. (2.2))
i(nf2 + c 1 2 ) * + y + nc - w ,
(5.7)
so that any orbit associated with a specific value of the constant W ( > 1/F) lies within the domain of the n»C plane bounded by y + n£ = W, which is the ellipse
n 2 + (e + nr 2 w) 2 /r 2 - r 2 w 2 - 1 .
(5.8)
The closed orbit of a longitudinal solution is degenerate, being simply the back and forth traversal of the major axis of the ellipse (5.8).
The
closed orbit of a quasi-transverse solution is also degenerate, being the back and forth traversal of a simple arc that is symmetric about the £ axis and terminates at the ellipse, meeting it along the (inward) normal. Chian & Clemmow (1975) have shown that, for each value of W ( > I/I1), there is just one solution with such an orbit; moreover, this solution, also, exists for all values of n between 0 and 1.
The main change as n
approaches unity is that the/oscillation becomes more nearly sinusoidal (about an increasingly large negative mean value) with a relatively greater
SUPERLUMINOUS WAVES IN PLASMAS 2
amplitude that approaches 2
171
times that of the ^ oscillation asn-? 1.
It seems plausible that the two classes of solution just described for the case X = 0, namely longitudinal and quasi-transverse, continue to exist if n «
1 and X is non-zero but sufficiently small.
example, that X were 0(n 2 ).
Suppose, for
Then the form of (4.2) would allow r\ to be
0(n 2 ), so that, from (4.3), £ would differ negligibly from the n = 0 case; the waves would be quasi-longitudinal.
Alternatively, (4.3) would allow £
to be 0(n), and then, from (4.2) ri would differ little from the quasitransverse X = 0 case. This surmise, that, in addition to the quasi-circular waves described by (5.3), there exist quasi-longitudinal and quasi-transverse waves for X sufficiently small, is complemented by the suspicion that the latter two types of wave cease to be present if X is too large.
Some insight into
the possibilities is afforded by the analysis of the next section, where r| and C are themselves taken to be 0(n) (that is, A = 0(n)).
This case for
the unmagnetized plasma (X = 0) was examined by Clemmow (1982b) with the aim of discovering the fate of the quasi-circular solution in the approach to the small-smplitude limit, where linear theory is known to admit only the quasi-longitudinal and quasi-transverse solutions.
In §6 the method
is extended to the magnetized plasma (X £ 0 ) , with results which help to explain the variation in the nature of the solutions over the \, A parameter space when n « 6.
1.
PROPAGATION ACROSS THE MAGNETIC FIELD:
n «
1, A = 0(n)
It is convenient first to recapitulate very briefly the analysis in Clemmow (1982b) for the case X = 0. *
-
With
VT
(6.1)
(4.1) and (4.2) can be written V 2 n/n + n/ny
-
v2 £/n + C/ny
0,
-
-1,
(6.2)
where the dot denotes differentiation with respect to \p. The introduction of the expansions V2 =
1 + V x n 2 + \>2nk + ...
n/n - n 0 + n ^ 2 C/n
=
+
n 2 nif
+
,
(6.3)
••• >
C o + S x n 2 + ^ n * + ...
,
(6 4)
-
(6.5)
then provides a pair of equations from each set of terms of a particular order in n . The first two pairs are, from terms independent of n,
P.C. CLEMMOW
and from the n 2 terms,
The solutions r\Q and £ Q of (6.6), linear in sin ip and cos ip, when substituted into the right-hand sides of (6.7) give linear combinations of sin(m\JO, cos(mip), m = 0, 1, 2, 3. £
The condition for the solutions n
to be periodic is, then, that the m = o terms are absent.
and
In this way
it is found that, with appropriate choice of time origin, ri
=
a sin
ty,
£
-
-1 + b costyy
(6.8)
where a^2vi + 1 + J(3a2 + b 2 ) ] =
0 ,
(6.9)
and
+3 • J(a2 + 3b 2 )] = 0 . The elimination of v
(6.10)
from (6.9) and (6.10) gives
ab{l - J(a2 - b 2 )] = 0 ,
(6.11)
so that there are three possible cases (I)
a = 0,
(II)
b - 0,
(III)
a2 -b2 = 4
These represent, respectively, the longitudinal, quasi-transverse and quasi-circular solutions discussed in §5.
In particular, (III) corresponds
to an elliptic orbit which is indeed nearly circular when a and b greatly exceed unity, but which becomes increasingly eccentric as a and b decrease, and finally merges into (II) when a = 2,
b = 0.
Thus in the small-
amplitude limit only cases (I) and (II) are present, in agreement with conventional linear theory. The above analysis is carried through more thoroughly, and to the next stage, in Clemmow (1982b).
The present aim, however, is not to
pursue the details, but rather to see broadly the effect of introducing a magnetic field by retaining the X terms in (4.2) and (4.3). particularly instructive to take X to be 0(n 2 ).
It appears
With (6.3), (6.4) and
(6.5) retained, and also X
=
Xxn2
,
(6.12)
it is evident that (6.6) still holds, and that (6.7) is modified only by having on the right-hand sides further terms - X £ , X ri , respectively. From (6.8) the conditions for periodic solutions, modifying (6.9) and
SUPERLIMINOUS WAVES IN PLASMAS
173
( 6 . 1 0 ) , are then seen to be a | 2 v + 1 + l ( 3 a 2 + b 2 )} +2X b
= 0 ,
(6.13)
b [ 2 v + 3 + J ( a 2 + 3b 2 )} +2Xxa
= 0 ,
(6.14)
from which the e l i m i n a t i o n of v
gives (cf.
ab[l - J(a2 - b 2 ) ] + X ^ a 2 - b2) The relation (6.15) is a key result.
(6.11)) = 0 .
(6.15)
It is conveniently displayed
as a curve in the a, b plane, and figs. 1 and 2 show (diagrammatically) the two cases X
«
1,
(6.15), the dashed line
X
»
1,
respectively.
a2 - b 2 = 4 ,
The full line represents
and the dotted line
ab = 4X . As l
explained more fully in §7, fig. 1 can be interpreted entirely in terms of solutions that have already been introduced.
When (a2 + b 2 )
is appreciably
greater than 2 the quasi-longitudinal (q-1, a-0), quasi-transverse (q-t, b-0) and quasi-circular (q-c, a 2 -b 2 ) solutions are all present; but when 2 2 i (a + b ) is appreciably less than 2, only the q-1 and q-t solutions remain.
In fig. 2, on the other hand, the distinction is between solutions
for which (a2 + b 2 ) * is appreciably greater or less than 2/2X
.
In the
former case the solutions are again q-1, q-t and q-c, but in the latter only q-c. 7.
DISCUSSION The analysis represented by figs. 1 and 2 is based on the assumption
that n «
1, and that r\ and £ are 0(n). However, it is reasonable to
suppose that solutions in the domain a 2 + b 2 »
1 should link up with
solutions for which r\ and y are 0(1). This supposition has, indeed, been confirmed explicitly for the case X = 0 (Clemmow 1982b).
It may also be
confidently applied to fig. 1, because it seems quite acceptable, as a 2 + b 2 increases beyond 4, to identify the branches asymptotic to a 2 - b 2 = 4 with the q-c solutions (5.3), and those aymptotic to a = 0 and b = 0 with the q-1 and q-t solutions mooted in the penultimate paragraph of §5. No question of resonance in the q-c solution arises here, because X
a2 + b 2 .
«
The interpretation of the case a 2 + b 2 »
1 is less straightforward
in fig. 2, because the distinction between the cases a 2 + b 2 < 8X a
2
+ b
2
> 8X
now comes in.
and
In each of these latter cases there are q-c
l
solutions, presumably to be identified with (5.3); but whereas the q-c solution with b < 0 is represented by a single branch, that with b > 0 has two branches separated by a domain in the vicinity of the point
174
P . C . CLEMMOW
FIGTOE 1
SUPERLUMINOUS WAVES IN PLASMAS
FIGU2E 2
175
176
P.C. CLEMMOW
a = b = 2\ .
This point, as might be expected, is the location of the
resonance described by (5.4) (at A = Ja2, in the notation of §5). Moreover, although only the q-c solutions are present when a 2 + b 2 is appreciably less than 8A , they are supplemented by q-1 and q-t solutions when a 2 + b 2 appreciably exceeds 8A .
It might be conjectured, therefore, that, when
ri and £ are 0(1), periodic q-1 and q-t solutions exist in addition to the q-c solutions provided the amplitude is sufficiently large compared with the magnetic field. At the other extreme in figs. 1 and 2, it appears that they correctly reflect the approach to the small-amplitude limit. q-t solutions for A
«
1 to q-c solutions when A
The swing from q-1 and »
1 is readily confirmed
in conventional linear theory. In conclusion, it may be noted that the case of propagation at an arbitrary angle to the magnetic field is likely to be amenable to the sort of treatment outlined in §§5 and 6.
However, it should be emphasized how
much of the available analysis is restricted to the assumption n «
1.
Direct computation from the differential equations for the case A = 0 has checked the results of the analysis quantitatively for values of n up to about 0.2, and qualitatively for somewhat larger values; but there seems to be a transition stage around n = 0.4 where the situation has yet to be clarified.
A little work has been done on the case n - 1, and this might
be worth pursuing, particularly with a view to investigating the transition to subluminous waves (n > 1). REFERENCES AKHIEZER, A.I. & POLOVIN, R.V. (1956) Theory of wave motion of an electron plasma, Soviet Phys. JETP, 3_, 696. CHIAN, A.C.-L. & CLEMMOW, P.C. (1975) Nonlinear, periodic waves in a cold plasma: a quantitative analysis, J. Plasma Phys., 14_(3), 505-527. CLEMMOW, P.C. (1977) Nonlinear, superluminous, periodic waves in a plasma with magnetic field, J. Plasma Phys., .H (2) , 301-316. CLEMMOW, P.C. (1982a) Further analysis of nonlinear, periodic, highly superluminous waves in a magnetized plasma, J. Plasma Phys., 27(1), 177-187. CiLEMMOW, P.C. (1982b) Periodic, highly superluminous, nonlinear waves in a cold, unmagnetized plasma, J. Plasma Phys., 27^(2), 267-276. DECOSTER, A. (1978) Nonlinear travelling waves in a homogeneous cold collisionless plasma, Physics Reports, 47.(5), 285-422.
CHAPTER 10 ELECTROSTATIC ION CYCLOTRON WAVES AND ION HEATING IN A MAGNETIC FIELD H. OKUDA Plasma Physics Laboratory Princeton University Princeton, New Jersey 08544 U.S.A.
1.
INTRODUCTION. It is well known that the electrostatic ion cyclotron (EIC) waves may
be destabilized by drifting electrons (current) through the stationary ions or by ion beams along magnetic field (Drummond and Rosenbluth, 1962; Kindel and Kennel, 1971; Perkins, 1976; Okuda et al., 1981).
EIC waves are par-
ticularly important in an isothermal plasma (T » T.) where the ion acoustic waves may be stable due to Landau damping.
The threshold drift speed
is a fraction of electron thermal speed so that EIC waves may easily be excited under a variety of conditions. Laboratory experiments as well as space craft measurements have been reported recently on the observation of EIC waves in tokamaks (TFR Group, 1978), linear devices (Yamada and Hendel, 1978), and space plasmas (Kintner et al., 1979).
Large amplitude density fluctuations, heating of ions
across magnetic field and anomalous cross-field particle diffusion have been observed in these measurements. There are a number of theoretical considerations on the linear as well as nonlinear behavior of the current-driven EIC instabilities (Drummond and Rosenbluth, 1962; Kindel and Kennel, 1971; Okuda and Ashour-Abdalla, 1982). Linear theory predicts that the unstable modes satisfy 0) x n ft., k p, ^ n where p_^ is the ion thermal gyro radius, ft. is the ion gyro frequency, and n is an integer representing a cyclotron harmonic. When the electron drift speed is above the threshold, EIC waves grow to large amplitude until limited by nonlinear effects.
In the absence of
electron source, plateau formation on the electron distribution due to quasilinear diffusion gives rise to the nonlinear saturation of the EIC waves resulting in a modest ion heating (Drummond and Rosenbluth, 1962). There are situations, however, in which a flux of fresh electrons constantly
178
H. OKUDA
replenish the distribution function of electrons so that complete stabilization due to plateau formation cannot take place.
Plasma heating experi-
ments by injection of electron beams (Yamada and Hendel, 1978), ion beams (Eubank et al., 1979) and field-aligned auroral currents where ionosphere acts as a reservoir of fresh electrons are such examples.
For these cases,
the duration of beams is much longer than the characteristic time scale of the EIC waves so that a presence of beam source play an important role on the nonlinear behavior of the EIC waves, ion heating and plasma transport. Here we would like to present results from analytic theory as well as from numerical simulations on EIC waves.
In Section 2, linear theory is
reviewed for the current-driven EIC waves with detailed numerical solutions to the dispersion relation.
Plasma heating via injection of electron beam
into a plasma is analyzed theoretically in Section 3.
Numerical results
obtained from the initial value simulations without a plasma source are given in Section 4.
In Section 5, we present simulation results in which a
plasma is subject to electron beam injection at one end of the system. Conclusions of the work and possible applications to laboratory and space experiments will be given in Section 6. 2.
LINEAR THEORY. Let us consider a low $ plasma in a strong external magnetic field.
The electrostatic dispersion relation for low frequency (a) « ft ) and long wavelengths (k p
«
1) oscillations may be given by (Okuda and Ashour-
Abdalla, 1981)
1+
V^1+ ..
0) - k,, v,
v v
K
te
rf £
v, I I de) )] ]
- k
(JO
*(( *
ll
W
v
te
T z)
'
0) - nO (1 - T /T
where the ion distribution function is assumed a bi-Maxwellian with parallel and perpendicular temperatures denoted by T... and T.
and the electron
distribution along magnetic field is a drifting Maxwellian with the drift 1/2 velocity given by v, . Q is the ion gyrofrequency^ v ... = (2T.. /m.) , v
= (2T /m.) tiLJ. i-L i
, v
te
= (2T /m )1'2 are the ion parallel, perpendicular, e e 9 9 9 9 9 9
k and electron thermal speeds, r (y,,) = exp(-y_,) InCyJ, ^ = k n
i
i
i
l
l
f= vv . /2ftf ti±
1
2
2 2.1/2 k p i where In i s the modified Bessel function of order n, p =(T / m Q y i s the ion thermal gyroradius, X - v. lyf~2~u> and A. = v..l\/~2u> . are. the e te P^ -*ti P-*Debye lengths of electrons and ions and Z is the plasma dispersion function.
ELECTROSTATIC ION CYCLOTRON WAVES
179
Near the fundamental frequency, a) « ft., and T ../T. not too small, the real part of Equation (1.1) is reduced to
h
[T
iii/Te
which approaches 0) - ft. T.M
T
indicating that the wave frequency shifts to ft. as T.
> T....
Since the
EIC waves heat ions preferentially perpendicularly to magnetic field, ion heating can be one of the nonlinear saturation mechanisms by shifting the wave frequency downward thereby enhancing the ion cyclotron damping.
As
will be shown in Section 5, simulation results indicate that the ion heating is the nonlinear saturation mechanism in the absence of electron plateau formation.
As the ions are heated perpendicularly as a result of EIC
waves, the critical electron drift velocity increases as a result of enhanced ion cyclotron damping.
Just exactly how much ion heating is required
for total stabilization can be found from Equation (1.1) for marginal stability (Okuda and Ashour-Abdalla, 1982). Before calculating the maximum ion heating from marginal stability analysis, let us first show numerical solutions to the dispersion relation, Equation (1.1), in order to study the growth rate and the wave frequency for a given electron drift. Figures 1,2,3 show such examples for hydrogen ions (m./m ft /a)
= 5, T
= T.,| = T
= 1837) with
. Figure 1 shows the frequency and growth rate
for several hydrogen cyclotron harmonics as a function of drift speed. all cases, the angle of propagation is kept constant at k,,/k generally gives the most unstable waves.
For
= 0 . 1 which
Figure 1 (a) corresponds to
k p. = 1.2 in which the fundamental harmonic U) ~ ft. is expected to be most unstable.
The growth rate is a fraction of gyrofrequency when the drift
speed is of the order of thermal speed. fundamental harmonic is about v, /v
The critical drift speed for the
= 0.3 for hydrogen ions in agreement
with earlier work (Drummond and Rosenbluth, 1962).
For shorter wavelength,
k p. = 2.4 shown in Figure 1 (b), the second harmonic is the most unstable mode although the first and the third harmonics remain unstable.
Figure 1
(c) shows the frequency and the growth rate for k p. = 3.6 where the third harmonic has the maximum growth rate.
H. OKUDA
180
1
•
1
1 —•
/ftj
tUg / f t j
<,,/ftj
_ ^"
1.0 _-
w 3/ft: ^
/-
(a)
1
/ 1
111
.0
-
1 -
y /ft. 1 2
1/ /
—
/
-
/
/A\i / Vr\r —
I 1 1 l|
10-
1 1 1 11
o"
w 4 /ftj /
H
-
CD,/ft j =
i
I
1
ayftj
i
/
-
5
-
1
0.5
11
1
1
I
1
1.0
1.5
1
/
0.5
2.0
/l
1.0 v
1.5
q
'•^1
\i
y
in"
10
2.0
10
i 0.5
de/vte
/
/ I / 1.0 v
I \\ 1.5
2.0
de/vte
Fig. 1. Numerical solutions of the dispersion relation given by Eq. (1.1) for'a hydrogen plasma. T e /T ± = 1, k,^^ = 0.1 and k ± p i = 1.2 (a), 2.4 (b), and 3.6 (c).
!On P) P i \O PE t-1 (D rtz=
•-« —
1
1
1
1 1
CD 0Q T) • (-{ ho
"1
i
i
1
1 11 1 1 1
co •
e Oi
t-J CO . rt
rt" I— 1
5- h-
1
bi
*~
(D
CO — Hi o O h-1
H«
<
CO rt rt fD
<
rt CD
s-
b
—
'
||
p
rt O
1 1111 f
o
Hi o
CO
->>
o
II
rt
M
fD
—
en
1
1
l
l i
III
1
1
1 1 11 1 1
\
-
Figure 2 shows the dependence on k p. of the first, second and third harmonics for k,/k
= 0 . 1 and v, /v t e = 1.
For these parameters, the wave
frequency is above the corresponding cyclotron harmonic. an unstable band in k p. as seen in Figure 2.
Each harmonic has
ELECTROSTATIC ION CYCLOTRON WAVES
181
Figure 3 shows the dependence of growth rate and the frequency with respect to the angle of propagation.
Here
v, /v = Ti±and de te ill k± p ± - 1 (a) and 2 (b). It is clear that there is an optimum angle of propagation where the growth rate takes its maximum at around k.Jk
=0.1.
Decreas-
ing or increasing the angle reduces the growth rate.
This
characteristic is the same as
Fig. 3. Numerical solutions of the dispersion relation with respect to V ± f o r k±P± - 1 in (a) and k p. = 2 in (b). T e /T ± • V de / v te = X a n d 5. e' pe
seen in Figure 2 and happens because either the condition for instability (v de > 0)/k|() is violated if kMis too small or
ion cyclotron damping is enhanced for large k||. As we have seen in Figures 1-3, EIC waves can be strongly unstable when the electron drift speed is comparable to the thermal speed.
The non-
linear consequence on the ions is the heating in the perpendicular direction (Okuda and Abdalla, 1981).
Assuming the plateau formation of the
electron distribution is inhibited, one can calculate the maximum ion perpendicular heating from the marginal stability analysis of Equation (1.1) (Okuda and Ashour-Abdalla, 1982). Near the fundamental ion cyclotron frequency, a) « ft. and k p. * 1, Equation (1.1) is reduced to
1 + k,,v
v
de
te
a) - n .
Oi-kv Z(
a.
V te
V ill T
T
1 -
F
J
=
0.
(2.1)
i For a small electron drift, v, /v tion (2.1) may be approximated by ginal stability (real w ) , we find
< 1, the electron contribution to Equa- k.v )/k..v so that for marII te II te
182
H. OKUDA
r
P r 1 z R ( P )(T e /T i
i
T
)+ i-(V iii)[(1-ri)(1-Tiii/Tiiiiiiioi (2.2)
for the real part, and
for the imaginary part.
Here p and A are defined by 03 F — -
v, /v
ft 03 - ft. and A' g-i .
(2.4ab)
can be minimized numerically with respect to A and p or y. and
p using Equations (2.2) and (2.3) for a given value of T /T.,,.
For a given
set of ( y ^ p ) , Equation (2.2) is first used to find A which is then used to calculate v, /v For v, /v
from Equation (2.3). >, 1, the small argument of the electron Z function breaks
down and one must retain the full Z function for the electrons.
One then
finds, for the real part, "-"i
. ft
r
i <
T
i | |
/ T
l l >
p Z
R
( p ) ( T
e
/ T
l | | > , ((2.5) 25)
" p I ^ p H y T ^ ) + [l+(^ ) j v d e )/k 1 1 v t e Z R (a)-k 1 | v d e /k ( | v t e )] -A
where A = ( V T i l l ) [ ( 1 " T i | | / T i i ) ( 1 " and for the imaginary part, k v
u ^o~(j°
k v -0) 2
T
F } + T
l
i||/Tii(1 " ro)/yi "
co-ft, (1-T
/T )
1]>
03-
.
k v
te
|| te
Noting that for v, /v
(2#5a)
(2.6)
,> 1, the exponential electron term is the dominant
term, the critical drift speed is given by V
A
te
' v te
[
(T e /T i | | )p(A
where A and p are defined as before.
+
T1||/Til)riexp(-p^) 1
(2'7>
Equations (2.5) and (2.7) can be
solved numerically to find the minimum drift v, /v
for a given X... /T-
which in turn should give the marginal stability for a given drift speed. v
de / v teFigure 4 shows the values of T. ,/T.II for a given v, /v
of T /T.|. for hydrogen ions.
as a function
Since the ion parallel temperature changes
little compared with the perpendicular temperature, T. / T.,, maybe considered to represent the
ELECTROSTATIC ION CYCLOTRON WAVES
• 82T0I07
maximum ion perpendicular heating for a given electron drift.
183
As
the electron temperature increases relative to the ion parallel temperature, T /T^ > 1, Figure 4 indicates the maximum ion perpendicular heating increases.
This
is seen from Equation (1.2) or Equation (2.2) which indicates that the frequency of the EIC waves increases with T
e
so that
more perpendicular heating is required for the marginal stability. It is seen from Figure 4, a large temperature anisotropy, T /T. >10 may be obtained for a drift speed v,
comparable to thermal speed
v . For a sufficiently large electron drift speed, Figure 4 indicates there is no marginal stability for any temperature anisotropy.
Fig. 4. Maximum attainable ion temperature anisotropy as a function of electron drift speed. Note for a sufficiently large drift, there is no temperature anisotropy for marginal stability.
For such a large electron drift speed, perpendicular heating alone
cannot stabilize the EIC instability, and therefore, other effects such as electron velocity space diffusion as well as electron heating which reduces the ratio v, /v must be taken into account for the stabilization of the de te EIC instabilities. 3.
ELECTRON BEAM PROPAGATION AND ION HEATING. When a plasma is unstable with respect to an EIC instability, small
amplitude perturbations grow in time until limited by nonlinear effects. EIC waves, once they reach large amplitude, can in turn react hack on Ion and electron distribution functions.
For the case of EIC waves, the elec-
tron velocity space diffusion will lead to saturation of instability (Drummond and Rosenbluth, 1962). Let us calculate the energy released from the drifting Maxwellian electrons to the waves by forming a plateau on the electron distribution function.
The amount of kinetic energy released from the electrons is cal-
culated from
AE = (l/2)men f(f* -- f*) vj dv IS
(3.1)
184
H. OKUDA
where f Q e is the initial drifting Maxwellian and f®
is the final electron
distribution with a plateau, S is the cross-sectional area of the electron beam, I is the length of the beam and n is the number density. drift, v
/v
«
1, we find m
and for v
/v
For a small
»
e nJlSv de
1, m n&S v , AE •
e
3
de
In the absence of energy loss for a plasma such as due to wave convection and heat conduction, most of the energy lost from the electrons goes to ion perpendicular kinetic energy.
The increase of the ion perpendicular tem-
perature would be
V
= For vj /v
te
m v % j
2 for v, /v » de te
1 .
(3.2b)
< 1, only a modest increase of the ion perpendicular tempera-
ture can therfore be expected for T « T..
This is because the energy lost
from the electrons by plateau formation is a small fraction of thermal energy for v d e / v t e < 1. Let us now consider a driven system where a plasma is subject to a constant injection of electron beam along magnetic field.
Such a system
may be sketched as shown in Figure 5 where a plasma located at x > 0 is subject to electron beam coming from the left.
As the electrons stream
through the ions, electron velocity distribution develops a plateau at x ?= H shown in Figure 5 by exciting the ion cyclotron waves.
For x > &,
there is no instability and therefore no ion heating takes place initially. Since the source region is sending the electrons continuously, the ion perpendicular temperature for 0 < x < & keeps on increasing until it reaches a point where the ion cyclotron waves are stable for a given electron drift speed as shown in Figure 4.
Beyond that time, the source electrons can
reach x = £- without forming a plateau since the ion cyclotron waves are now stable for 0 <, x < A.
If there is no loss of energy from a plasma, the
ELECTROSTATIC ION CYCLOTRON WAVES
185 #82TO23I
Fig. 5. Model for theoretical analysis and numerical simulations in which a plasma located at x > 0 is subject to electron injection from x < 0. same process must repeat for x > & as the time goes on. The length & on which the electrons develop a plateau formation can be estimated from
(3.3) where T
is the electron diffusion time and Y is the growth rate of the ion
cyclotron waves given by Y/^. * T (T /T.)(v, /v 1962).
) (Drummond and Rosenbluth,
The time for the ions heated to the maximum temperature determined
from Figure 4 is therefore given by T S
where T.
T.f/AT. ,
= T D
(3.4)
lJL
ll
is the maximum temperature determined from Figure 4 for a given
electron drift speed.
The speed at which the ion heat pulse propagates a-
long magnetic field is therefore given by
AT.ii de T
il
(3.5)
186
H. OKUDA
Note that this speed is equal to the propagation speed of the EIC waves and also electron beam along magnetic field since the beam propagation is possible only after the EIC waves saturate so that the anomalous resistivity drops to zero. So long as the above one-dimensional model is valid, ion cyclotron turbulence and ion heating propagate without limit along magnetic field lines.
One cannot, however, neglect the ion perpendicular heat loss at the
distance where the ion heat loss across magnetic field dominates the ion heating due to ion cyclotron waves.
In this case, the energy in the elec-
tron beam is effectively scattered by the ions so that there is a maximum length, £ , of the electron beam to propagate through a plasma. pendicular heat loss during time T D over the length of I
Ion per-
is given by
q\ = V K x AT ii T D (3 6)
'
where a is the size of the electron beam and K heat conductivity.
is the ion perpendicular
Equating the loss, Equation (3.6), to the heating from
the electrons given by Equation (3.1), we find the length £
will be given
by, for v d e /v t e >, 1,
a
m v, 2 >* / e d e
2
2 "p.2 where K
= nD
2 and S = a .
D
x
v
te
T
T
i
i
(
'
Note D. is the cross-field particle diffusion
coefficient and D_ = p. ft is the Bohm diffusion coefficient. x>
v
~ v
and T
1
1
Assuming
~ T , we find 1/2 m e
a
2
2 p i
D
B D
In the presence of ion cyclotron turbulence, D comparable to D
}
may be smaller than or
for v, ~ v (Drummond and Rosenbluth, 1962; Okuda et aL, B de te If we assume a/p. * 5 and D w /D = 10, we find I /p. * 10^ for a l B l t i hydrogen plasma. When the electron drift speed is less than the thermal speed, beam 1980).
ELECTROSTATIC ION CYCLOTRON WAVES
propagation may be localized over a distance of &.
187
In this case, the ion
heat loss along and across magnetic field can take the energy away before ion temperature reaches the point given by the marginal stability.
Under
such conditions, EIC waves are confined in a narrow region of the order of £ and the ion heating beyond x = £ is due to heat conduction along magnetic field. 4.
RESULTS OF SIMULATIONS WITHOUT SOURCE. In this Section, we shall present results obtained from one- and two-
dimensional numerical simulations on the EIC instabilities.
The simulation
model used is an electrostatic model in which full dynamics is retained for the ion motion while guiding center drift approximation is used for the electrons.
This approximation for the electrons is valid for low frequency,
0) « ft , and long wavelength, k p
<< 1, oscillations where electrons are
treated as guiding center particles.
Electron motion along magnetic field
is solved exactly (Lee and Okuda, 1978). dv m. ^—- = I dt
(E (x.) + - v. x B ) v ~ v~i c ~i ~
e
dV
In essence, a set of equations
He
dt
"
e E
| |
eg v±e
xB
= B2 dt
cj) =
-
>JL 4iieKn
- ne
E - - V$ have been solved on a spatial grid using a finite difference integration in time (Lee and Okuda, 1978). The initial conditions for particles are a stationary Maxwellian for the ions and a drifting Maxwellian for the electrons along magnetic field. Initially, spatial distribution is uniformly loaded using random numbers. Then parameters of the simulations are m./m
= 1837, U) /ft = 0 . 2 and
T e = T ± initially. Let us first study results obtained from a one-dimensional simulation in which a uniform magnetic field is taken in the y - z plane with B /B =0.1. Perturbations are allowed only in the y-direction (one-dimensional model). v, /v
= 1.4, L = 1024 A where L is the system length and A is the mesh
188
H. OKUDA
size and A = A. The allowed wavelengths on the grid points in this system is given by
27rm (m = 0, ±1, ±2m L so that k^p. varies 0.05 to 10 with the spacing 8(k^p.) = 0.05.
70,000 ions
and electrons are used wllich gives about 70 particles per grid for each species.
0.4
Figures 6(a) and (b) indicate the time history of the real and imaginary parts of the electrosta- ^ 01 tic potential, e<J>,(t)/T , for the -0.2 K. e third mode, (a), and 23rd mode and -0.4 their frequency spectrum, (b).
-9
For the third mode, k p = 0.16, (b)
only the fundamental cyclotron
I0
23rd mode, k p. = 1.2, several
o3
amplitude for these modes reaches
-0.2
e<j) /T = 0 . 2 before saturation.
-0.4
Nonlinear saturation takes place when the electron velocity shown in Figure 7. Atft.t= 10, a plateau is formed for the region < v
de:
40
80
J
Fig. 6. Time history of the real and imaginary parts of the potential, e
distribution develops a plateau as
of velocity space, v
o2 20
'* j
4
harmonic is excited while for the harmonics are clearly seen. The
-6 ^-3
which
then creates a positive slope afe/9v
> 0 for v
in Figure 7.
> v, as seen de This new region of
the positive slope is flattened
^
later as the long wavelength modes j> corresponding to larger phase velocity are destabilized. Heating of ions perpendicular to magnetic field and the loss of electron kinetic energy along magnetic field is shown in Figure 8. The relative change for both ions and electrons remains small as
'en'Me Fig. 7. Electron velocity distribution along magnetic field at different times. Note the development of a plateau.
ELECTROSTATIC ION CYCLOTRON WAVES discussed in Section 3 because the plateau formation on the electron
189
1.5
distribution releases only a fraction of electron kinetic energy. The rate at which the electron drift speed is dissipated is shown in Figure 8 for both
For both cases, a rapid initial decay is followed by the slower relaxation.
F i g .8 . Heating ener and
The anomalous resis-
ft
of ion perpendicular gy cooling of electron kinetic energy along magnetic field,
9
/
tivity defined by n = -(m /n ) (dv, / d t ) / v , i s found T]/r\ » 1.9 x io" e e de de o
for
the initial rapid decay and 3.6 x 10" 5 for the later slower decay for v
/v
= 1.4.
Here n
er for smaller drift.
= 4rr/o3
. The anomalous resistivity becomes small-
Note here r)/n = v ./GO holds where v . is the efo ei pe ei
fective collision frequency of the electrons with the ions.
The numerical
values observed in the numerical simulations are in good quantitative agreement with the theoretical predictions form a quasilinear theory (Okuda et al., 1981). Simulations are extended to two-dimensions in which spatial variations are allowed in the x-y plane with the external magnetic field oriented as before.
Results of two-dimensional simulations is expected to reveal pro-
cesses not allowed in one-dimensional results such as mode-coupling among modes at different angles of propagation and cross-field particle diffusion across magnetic field. Figure 9 shows a plot of test particle positions at an instant of time,ft.t= 30, for electrons (top) and ions (bottom) which were initially located at a narrow strip centered at x = 20.
The spread in the x-direction
suggests the presence of cross-field particle diffusion due to the ion cy2 clotron instabilities. The measured diffusion coefficient, D =<(Ax ) >/t« -3 2 . x e 5 x 10
A 0)
suggesting a presence of large anomaly for the transport
coefficients associated with ion cyclotron waves. From the two-dimensional simulations, it was found that the modes which propagate strictly across a magnetic field, k = k , are excited by nonlinear mode-coupling process.
These two-dimensional modes (k.. = 0) are
long-lived since thermal motion of particles along magnetic field cannot short out such density perturbations and may be responsible for determining the size of the thin auroral arc bands along auroral field lines (Kan and Okuda, 1982).
190
H. OKUDA
Ion Density
2.0
I ' ' '
i
•
• •
. . .
(a)
64
E
E
E
E
1.5
1
1
>
•
'
'
'
t = 110
E
I
1
i
• I - -
:1 ii 1
"f
1
i
•
1
1
I.
B
i
1.0
• 1 1
I E
.5
1
6
1
A
,
E 1
.•.'
I
e i •
i * ft m
i 1 .
I
'
i
s> *-
12//
€?
$
E E
0 2.0
• E g I *
a
1.5\-
!1 32
I
i I
j±»
I I
! I
if,
I
r
i |U i i
i II
v
« \
I i
1
I
1 I
it
i
I I
i1
Ix
l
i i
fi/f = 2 2 0
I
I
1.5
i1 1
ii
I
1.0
!
i
i
I ^ I '
20
^
f
It I
I
i
II
/{ • ! /
* I
I
i
I1
1
•5h
I
iT 1
1
i
1 1 .!>
I
-/V
n.
M
64
I
i
i ,
If
;
i
X
I
32 Fig. 9. Plot of the test particle positions at &j_t = 30 for electrons (top) and ions (bottom) which were initially located at x = 20.
.5 0
,, ,
. . .
.
,
.
!
. . . 1
200 400 600 800 X
1024
Fig. 10. Ion density profile with a source located at x < 0. Note the large ion density perturbations propagate from left to right.
191
ELECTROSTATIC ION CYCLOTRON WAVES
5.
RESULTS OF SIMULATIONS WITH A SOURCE. We have seen earlier in Section 4 that the ion cyclotron instabilities
saturate at a low level giving rise to a modest heating of ions in the absence of electron source.
Here we present results of simulations obtained
using a model in which a plasma is subject to a source which constantly injects electrons into a plasma system with the initial drifting Maxwellian velocity distribution as shown in Figure 5.
Such a model may represent an
electron beam injection experiment in a plasma or an auroral current along magnetic field driven by the ionsphere-magnetosphere coupling (Okuda and Ashour-Abdalla, 1982).
As discussed theoretically in Section 3, ion heat-
ing is expected to be much stronger in the presence of an electron source due to the inhibition of plateau formation on the electron distribution function.
= 320
ne/no to
0 2000
100
50 1
|
1
. .
1
|
150 200 ^ 0 Sift 1
1
1
1
|
1
50
1500 (b)
(b)
1500
1000
-
1000
500
500 0 Fig. 11.
100 150 200
Vi
-1
Ji
J
-
-
-1
Electron density fluctuation with time at x = 320A and 416A and their frequency spectra.
192
H. OKUDA Figure 10 shows the profiles of ion density normalized by the average
density for the entire system length plotted at three different times.
The
parameters of the simulations are the same to the one-dimensional simulam./me 1837, 1.4 de The dashed vertical lines denotes the boundary
tions reported in Section 4 with L = 1024A, v
0 . 1 a n d ft /co = 5 . e pe lines for different bins for diagnostic purposes.
At time ft^ = 110 (upper
panel), the larger density perturbations, 6n/n * 0.25 are mostly confined to the first two bins.
At later times shown in the middle and lower panels,
the ion density perturbations are seen to propagate and extend further right, confirming the physical picture described in Section 3.
From Figure
10, one can estimate the propagation speed along magnetic field to be 0.007 v
.
This speed is much smaller than the electron drift speed and is
in good agreement with the self-induced propagation speed calculated to be 0.01 v, from Section 3. de The fact that the density modulation is associated with the ion cyclotron waves is confirmed by measuring the frequency at several different locations along field lines.
This is shown in Figure 11 where the time his-
lon Density
2.0r
n{t =175 j
1.0
t\
f\f\/\
0
A
f\f
200 400 600 800 X
1024
Perpendicular Ion Distribution Functions .
Bin 2
Bin 1
Bin 3
1000
500
-10
-5
0
5
10 -10
-5
0
5
10 -10
-5
0
5
10
vy/vti Fig. 12.
Ion density profile and perpendicular velocity distribution in three different bins atf].t = 175.
ELECTROSTATIC ION CYCLOTRON WAVES
193
tory of electron density at two different locations, x = 320A and x = 416A are shown in the upper panel (a), while in the lower panel, (b), frequency spectra of the density fluctuations are shown.
The frequency analysis con-
firms the presence of coherent peaks above ft.. In addition, there are much smaller, but clearly coherent peaks near 0) = k C indicating oblique ion II s acoustic waves propagating nearly perpendicular to magnetic field. The effects of ion cyclotron waves on the ions are shown in Figure 12 where the perpendicular ion velocity distributions are shown for three bins at ^ i t = 1 7 5 .
The large amplitude density perturbations associated with
ion cyclotron waves have penetrated up to the third bin by this time.
We
therefore expect that the temperature in the first and second bins is more or less heated up to the maximum value determined from the marginal stability analysis given in Section 3.
It is very interesting to realize the presence of high energy tail in Figure 12 where the total distribution is separated into bulk and tail parts.
This can be seen more clearly in Figure 13 where the logarithm of
the distribution in bin 1 at Q.t = 175 is plotted as a function of energy
H. OKUDA
194
Electron Density
2.0
= 175 '-
1.5 *****
ne/n0 to 0.5b r, . .
200 400 600 800 1024 X
0
Parallel Electron Distribution Functions
1400
T
|
I
I
I
I—|
1
Bin 3
Bin 1
1
1
I""p T
Bin 4
~~t*0
1000
500
-5
-5
-5
v
v
e,/ te
Fig. 14.
Electron density profile and the parallel velocity distribution at bins 1, 3 and 4. #82T0232
100
200 300 400 500 600 700 800 900 X
Fig. 15. Ion perpendicular temperature profile at different times.
ELECTROSTATIC ION CYCLOTRON WAVES v
2
2 /v . .
195
The initial Maxwellian distribution is also shown for comparison
in the figure.
Clearly the heated distribution has a high energy tail ex-
tending almost as much as 100 times of the initial thermal energy.
The
temperature of the bulk distribution is about 5-10 times of the initial Maxwellian while that of the tail distribution 50-100 times of the initial temperature. Figure 14 shows similar plots for the electron distribution at ft.t=175 for three different bins.
It is seen that the electron distribution gen-
erally has a plateau at far right such as in bin 4, whereas the distribution tends to have a positive slope for v
< v
in bins 1 and 2.
This
suggests that the electron distribution is determined by a delicate balance between wave-induced diffusion and input from the source.
Note the plateau
formation on the electrons is a subtle process which does not require much energy at all. Figure 15 shows the macroscopic ion perpendicular temperature along x at different time steps.
We note that at an earlier time the temperature
gradient is confined to the left with no heating to the right.
At late
times the temperature gradient moves to the right with the propagation of electron beam in a plasma.
In the absence of energy loss across magnetic
field such as in the present simulations, the ion heating pulse propagates to the right all the way if one waits sufficiently long.
In the presence
of energy loss across magnetic field associated with cross-field particle diffusion and heat conduction, we would expect that the beam propagation is limited to a finite length as discussed in Section 3. 6.
DISCUSSIONS. Linear and nonlinear properties of the electrostatic ion cyclotron
waves have been studied in detail both analytically and by numerical simulations.
It is shown that, when the electron drift speed along magnetic
field is comparable to the thermal speed, the instability can grow to large amplitude causing ion heating and anomalous transport process. sence of a source, ion heating remains modest.
In the ab-
In a driven system in which
the electron velocity distribution is maintained by the injection of a beam from a boundary, ion heating can be very large generating a high energy tail.
Beam propagation is limited by the dissipation processes such as ion
hea,t conduction associated with the ion cyclotron waves.
These results may
be used in a variety of laboratory experiments and auroral plasma physics where beam propagation into a plasma manifests the dynamics of the system.
196 ACKNOWLEDGEMENT.
H. OKUDA This work was supported by the National Science Founda-
tion grant ATM81-15257 and the United States Department of Energy Contract No. DE-AC02-76-CH03073. REFERENCES DRUMMOND, W.E. and ROSENBLUTH, M.N. (1962) Anomalous Diffusion Arising from Microinstabilities in a Plasma, Phys. Fluids 5, 1507-1531. EUBANK, H., GOLDSTON, R., ARUNASALAM, V., BITTER, M., BOL, K., BOYD, D., BRETZ, N., BUSSAC, J.-P., COHEN, S., COLESTOCK, P., DAVIS, S., DIMOCK, D., DYLLA, H., EFTHIMION, P., GRISHAM, L., HAWRYLUK , R., HILL, K., HINNOV, E., HOSEA, J., HSUAN, H., JOHNSON, D., MARTIN, G., MEDLEY, S., MESERVEY, E., SAUTHOFF, N., SCHILLING, G., SCHINELL, J., SCHMIDT, G., STAUFFER, F., STEWART, L., STODIEK, W., STOOKSBERRY, R., STRACHAN, J., SUCKEWER, S., TAKAHASHI, H., TAIT, G., ULRICKSON, M., VON GOELER, S., YAMADA, M. , T$AI, C , STIRLING, W., DAGENHART, W., GARDNER, W. , MENON, M., and HASELTbN, H. (1979) Neutral-Beam-Heating Results from the Princeton Large Torus, Phys. Rev. Lett. 43, 270-274. KAN, J.R. and OKUDA, H. (1982) Generation of Thin Auroral Arcs Banded in the Inverted V Precipitation Structure, Geophys. Res. Lett, (to be published). KINDEL, J.M. and KENNEL, C.F. (1971) Top Side Current Instabilities, ,J. Geophys. Res. 76, 3055-3078. KINTNER, P.M., KELLEY, M.C., SHARP, R.D., GHIELMETTI, A.G., TEMERIN, M., CATTELL, C , MIZERA, P.F., and FENNELL, J.F. (1979) Simultaneous Observations of Energetic (keV) Upstreaming Ions and Electrostatic Hydrogen Cyclotron Waves, J. Geophys. Res., 7201-7212. LEE, W.W. and OKUDA, H. (1978) A Simulation Model for Studying Low Frequency Microstabilities, J. Compt. Phys. 26, 139-152. OKUDA, H., CHENG, C.Z., and LEE, W.W. (1980) Anomalous Diffusion and Ion Heating in the Presence of Electrostatic Ion Cyclotron Instabilities, Phys. Rev. Lett. 46^ 427-430. OKUDA, H., LEE, W.W., and CHENG, C.Z. (1981) Numerical Simulation of Electrostatic Hydrogen Cyclotron Instabilities, Phys. Fluids 24, 1060-1068. OKUDA, H. and ASHOUR-ABDALLA, M. (1981) Formation of a Conical Distribution and Intense Ion Heating in the Presence of Hydrogen Cyclotron Waves, Geophys. Res. Lett. 8_, 811-814. OKUDA, H. and ASHOUR-ABDALLA, M. (1982) Acceleration of Hydrogen Ions and Conic Formation on Auroral Field Lines, J. Geophys. Res., (to be published) . PERKINS, F.W. (1976) Ion Streaming Instabilities: tromagnetic, Phys. Fluids 19, 1012-1020.
Electrostatic and Elec-
TFR GROUP (1978) Ion-Cyclotron Instabilities in the TFR Tokamak, Phys. Rev. Lett. 41_, 113-116. YAMADA, M. and HENDEL, H. (1978) Current-Driven Instabilities and Resultant Anomalous Effects in Isothermal Inhomogeneous Plasmas, Phys. Fluids 21, 1555-1568.
CHAPTER 11 SOLITONS IN PLASMA PHYSICS
P. K. SHUKLA Institut fiir Theoretische Physik Ruhr - Universitat Bochum 4630 Bochum Federal Republic of Germany
1.
INTRODUCTION. In an unmagnetized plasma, three kinds of waves can propagate.
are the ion-acoustic, electron plasma, and light waves.
These
The linear disper-
sion relations of these waves are, respectively, given by
where c
03 « k cg(l - k 2 A 2 / 2 ) ,
(1.1)
U)2 = o)2e(l + 3 k 2 A 2 ) ,
(1.2)
O)2 = u)2 + c 2 k 2 , pe
(1.3)
is the sound speed, A
is the electron Debye length, (A)
is the
electron plasma frequency, and c is the velocity of light. The most important role of nonlinear effects is to cause steepening of the leading edge of a wave.
However, it is frequently found that the
dispersion effects become significant as the steepness of the front increases.
The competition between nonlinearity and dispersion leads to lo-
calized waves often called "solitons". In the present article, we review theories of solitons in an unmagnetized plasma.
To understand the physics, we have chosen three simple waves
(as given above) and have worked out nonlinear theories for them.
As is
well-known (Sagdeev, 1966), supersonic compressional ion-acoustic solitons have a maximum potential ~ 1.3 T /e, and the corresponding speed is about 1.6 c . On the other hand, finite amplitude envelope Langmuir and light s wave solitons are obtained by incorporating a fully nonlinear analysis of the low-frequency plasma motion.
It is found that under certain circum-
stances, equations governing the stationary solitons are exactly integrable and can be written in terms of the energy integral of a classical particle. By analogy with the motion of an oscillator in a potential well, we analyze the conditions under which localized solutions exist.
Our theory of Lang-
muir wave envelope solitons compares favorably with laboratory experiments.
198
p.K. SHUKLA
In the small amplitude limit, a perturbation theory has been introduced which enables us to derive a set of nonlinear evolution equations describing Langmuir soliton turbulence.
A new kind of electromagnetic soliton
emerges when the relativistic corrections to the nonlinear current density and the ponderomotive force are taken into account.
Here, the forced elec-
tron density perturbation contains a depression at the center, together with shoulders of density excess on the sides.
Such kinds of nonlinear en-
tities can effectively accelerate electrons in plasma. In particular, Wong (1982) suggests that two adjacent one dimensional envelope solitons represent electrostatic potential barriers for ions.
The
latter are accelerated by the ambipolar potential peaks into the valleys between these peaks.
This process of inducing fusion by a nonlinear state
is appealing. The manuscript is organized as follows:
In Section 2, we discuss the
well-celebrated non-envelope ion-acoustic solitons.
In Section 3, a finite
amplitude theory of Langmuir envelope solitons is presented.
Depending on
the space and time scales and the amplitude of the nonlinear perturbations, a great variety of nonlinear evolution equations can be generated from the two warm fluid equations.
Section 4 contains the description of a new kind
of circularly polarized electromagnetic solitons.
The last section is de-
voted to the discussion of the significant progress made in the area of soliton physics as well as a few problems which should be solved in the future. 2.
ION-ACOUSTIC
SOLITONS.
For a preliminary discussion of the propagation properties of nonlinear ion-acoustic waves, we consider a homogeneous unmagnetized plasma with cold ions but warm electrons (T
»
T.)«
When the phase velocity of the
waves lies between the electron and ion thermal velocities, then in one space dimension, the dynamics of nonlinear ion-acoustic waves is governed by 9 n + 8
t
xnv
=
°*
9 v + v3 v = -8 (J), t
X 82 ( | ) = n
(2#1)
(2.2)
X - n,
n e = exp((J)),
(2.3) (2.4)
where the ion (electron) density n(n ) , ion fluid velocity v, and the am-
SOLITONS IN PLASMA PHYSICS
199
bipolar potential are normalized by the unperturbed plasma density n n , the 1/2 ion sound speed c = (T /m,) , and T /e, respectively. The space and s e l e 2 2 time are normalized by the electron Debye length X
(A^
= T / 4 m u e ) and
ion plasma period U) . . It should be noted that a linearization of (2.1)-(2.4) about the uniform equilibrium n = l, v = 0 , = 0 reproduces the usual ion acoustic wave dispersion relation quoted in the introduction. In the dimensionless vari9 ^—2 1 ables used here, the dispersion relation is 0) = (1 + k ) for perturbation with x and t variation of the form exp(ikx - iu)t). We look for stationary nonlinear solutions to (2.1)-(2.4) that dependent on x and t through the variable £ = x - Mt (M = Mach number = V n /c = u s pulse-speed/c ) . The appropriate boundary conditions for an isolated s pulselike solution are v
1
(j) - 0, (j)1 + 0,
"° v + 0, 1
n
n
1
*1} - 0
+ , m
as
|5| -co
(2.5)
at £ = 0,
(2.$)
where the primes denote differentiation with respect to the variables £ a n d (j> is the maximum height of the localized solution at the center, m Integrating (2.1) and (2.2), and using the boundary conditions (2.5), one finds n = M/(M - v ) ,
(2.7)
(M - v ) 2 = M 2 - 2(j).
(2.8)
Combining the last two equations, we get n =
(1 - 2/M2)"^ .
(2.9) f
Substituting (2.4) and (2.9) into (2.3), multiplying by (j) , and integrating once, we obtain (Sagdeev, 1966) i(<j)f)2 + iK<|>) = 0, where (2.5) has been used again.
(2.10)
The Sagdeev potential \\) is given by
(Chen, 1974) H) - 1 - exp() + M 2 \l - (l Equation (2.10) is the energy integral of a classical particle with unit mass.
If the potential is a real well, a particle entering from the
left (say (j) = 0) will go to the right-hand side of the well ((J) > 0 ) ,
200
P.K. SHUKLA
reflects from the place where (j) = <j> and returns to <|> = 0 making a single transit.
It does not go back again since i|;(0) = di^(0)/d(() .
= 0. This
orbit corresponds to a very special solution in (J); a single pulse solution called a soliton. < 0
for
Thus for soliton solutions to exist, one must have
° < • - * > dxp/d<|) ^ = A >(<) 0 for <|> > 0(<)0.
It turns out that
only compressive ((j) > 0) solitons are possible, provided that exp(cf>m)
(2.12)
The other condition for the existence of soliton is that U»(<|>) should have double roots near <|> = 0. Expanding Equation (2.11) for (f> « 1 gives 2 4 M
R It follows that ^() < 0 for M
MZ
1
2
3
"6 ?JT 3
(2.13)
> 1. The speed of the exact soliton can be
written in terms of its peak amplitude m (2.11) one obtains a nonlinear dispersion ! [exp((j> ) 9 2 exp((f>m) -
(where ())f = 0 ) . Hence from relation - 1] 1-
(2.14)
Now it is of interest to see whether there is an upper limit to the soliton speed.
To determine this we note from (2.9) that for the ion den-
sity to be everywhere real we must have
M2
2
V
We also note that $ must lie to the left of the critical value (j) = M /2 Y m cr in order for the ion velocity to remain real throughout the soliton.
An
equivalent condition for the function \p is
cr) > o so that the condition for physical solution to exist becomes 1 + M 2 - exp(M2/2) > 0, that is, M < M - 1.6. The condition (J) tion (2.14)) leads to (J) « 1.3. m
= M 2 /2 (with M
For M close to unity (i.e., SM = M - 1 «
(2.15) given by Equa-
1 ) , and finite but small
amplitudes, one obtains from (2.13)
iK4>,6M) - f *2(4> - 3(SM >-
(2.16)
Substituting (2.16) into (2.10) and integrating once, we obtain the result
SOLITONS IN PLASMA PHYSICS
(J> = 3 6M sech2
201
(-y) (x - Mt) .
(2.17)
We see from (2.17) that the maximum soliton amplitude (j) = 36M « 1 is re/6)"~22. . Of Of interest interest here here lated with the soliton width which is given by ((j) /6) is the scaling of the x and t variables in (2.17).
Introducing £ = 6M « 1, Introducinj me which is a measure of the pulse amplitude, the argument of sech 2 can be expressed as 2"^[e^(x - t) - e 3 / 2 t].
(2.18)
This gives the appropriate scaling (in a frame moving with M = 1) of the space and time variable that is required to construct a weakly nonlinear theory of ion-acoustic waves which describes the evolution of small but inite amplitude localized perturbations.
Thus, one can introduce the fol-
lowing "stretched" variables (Washimi and Taniuti, 1966)
5 = A x - t), T = e 3/2 t
(2.19)
in carrying out a nonlinear expansion of (2.1)-(2.4), We assume that n, (f>, and v have power series expansions in £ about a homogeneous field-free equilibrium, i.e.,
v
= ^ ( D + e 2 v ( 2 ) + ... ,
(2.20)
From (2.19), we have 3
= e% 3 r , 3 = e 3 / 2 3 - e^ 3 r .
X
c,
t
T
(2.21)
c,
Inserting (2.20) and (2.21) into Equations (2.1)-(2.4), we get to lowest order
n (D = for localized perturbations.
„,(!) = V (D
(2>22)
To next order in £, (2.1)-(2.4) become
(Davidson, 1972)
- 3 c n ( 2 ) + 3 T n< X ) + ^(n^
v(1)) + ^
_a?v(2) J g ^ a ) + V (D g^a) = _^ ^2)^ 3|*(2) = * ( 2 ) + | ( * ( 1 ) ) 2 - n ( 2 ) .
= 0,
(2.23)
(2#24) (2.25)
Adding (2.23) and (2.24) eliminates v^2\ Differentiating (2.23) with respect to ? once, substituting (3>- (j)
- 3r"
) in terms of first order
quantities, and further using (2.21), one (Washimi and Taniuti, 1966) finds
202
P-K. SHUKLA
that n
evolves according to dTna)
+ n ( 1 ) ^ n ( 1 ) + | 3^n (1) = 0,
which is just the Korteweg-de Vries (K-dV) equation.
(2.26)
It serves to model
weakly dispersive and weakly nonlinear ion-acoustic waves in an unmagnetized plasma. The origins of the various terms in Equation (2.26) are clear.
The
nonlinear term arises f r ^ the combination of ion convection, ion flux, and the expansion of the Boltzmann distribution (Equation 2.4) for (j) « dispersive term -r- d n in Equation (2.3).
1.
The
comes from deviation from exact charge neutrality
Nonlinearities tend to steepen the wave whereas disper-
sion tends to spread it out.
Therefore, it is possible to obtain a sta-
tionary solution if these two exactly balance each other.
Such a solution
is in fact the "solitary" wave solution which is given by Equation (2.17). One of the remarkable properties of these solitary waves is that they are quite stable and preserve their identity even through a collision process. In other words, if two solitons are set on a collision course they will emerge from it retaining their initial sizes and speeds.
The K-dV equation
is exactly integrable with the aid of the inverse scattering technique (Gardner et al., 1967). We close this section with two remarks.
First, departure from the
Boltzmann distribution function for the electrons (e.g., due to trapped particles) leads to a modified K-dV equation (Schamel, 1973) or the nonlinear Schrodinger equation (Shukla, 1977a) governing the dynamics of nonlinear ion-acoustic waves.
Secondly, exact ion-acoustic solitons also exist
in the presence of an external magnetic field (Shukla and Yu, 1978; Yu et al., 1980).
The evolution of the small amplitude ion-acoustic wave in a
magnetoplasma is governed by a multidimensional K-dV equation (Zakharov and Kuznetsov, 1974). 3.
FINITE AMPLITUDE ENVELOPE SOLITONS. It has been shown by Shukla and Tagare (1977) that nonlinear electron
plasma wave in an electron gas can have the saw-tooth shaped profile.
On
the other hand, in a two component (electron-ion) plasma, nonlinear interaction of a large amplitude Langmuir wave with slow plasma motion (consisting of enhanced ion wave fluctuations) gives rise to a self-consistent density cavities which, in turn, trap the localized Langmuir waves.
Such en-
tities are referred to as 'envelope solitons'. Physically, the envelope solitons originate in the following manner:
SOLITONS IN PLASMA PHYSICS
203
Suppose a small amplitude modulation is introduced to the uniform Langmuir wave.
Then some of the plasma particles are expelled from the large ampli-
tude region by the ponderomotive force (Chen, 1974),
As a result, the lo-
cal plasma density and hence the local plasma frequency is decreased in the large amplitude region.
A decrease of the plasma frequency implies a de-
crease of phase velocity and therefore there is an increase of the wavenumber.
It follows that the local wave number becomes larger behind the crest
than in the front.
Subsequently, the group velocity becomes greater behind
the crest than in the front due to the positive group dispersion. sult, the wave energy is piled up in the crest region.
As a re-
In this way the
small amplitude modulation is amplified, giving rise to a solition-like structure when finally nonlinearity balances exactly the dispersive effects. Consider the propagation of a finite amplitude Langmuir wave packet E~ = E(x,t)exp(-io)
t) + c.c. .
The slowly varying complex amplitude
E(x,t) arising from interaction with low-frequency plasma motion is given by the equation (Schamel et al., 1977) 2 i y 3 % where y = (m /m.) and n tion.
e
l
E + 3 3^ E = (n e - 1)E,
(3.1)
e is the slowly varying electron density perturba-
We have nondimensionalized x, t, n , E by the electron Debye length _2 e A^, the inverse ion plasma frequency 0) ., the unperturbed number density D ^ pi n , and (4ir n Q (T
+ T ^ ) 2 , respectively.
Equation (3.1) is valid for small
|E|, S O that higher harmonic generation is precluded.
Furthermore, we have
assumed that no significant increase in the electron temperature occurs, so that the latter can be treated as constant.
The neglect of electron Landau
damping shall be justified later. We average the slow electron equation of motion over one plasma period (2TT/O3 ) of Langmuir wave oscillation and obtain po T d
I
i2
~T~ V-n. dx ' 0'
e
= — m
e
d
•
-r—Y <S> dx m
e
e
d
_
where in view of the low frequency perturbations (o)/k « inertia has been neglected.
Here, v
/o
~T~ In n , dx e'
Ov
(3.2)
v
) the electron
is the electron thermal velocity and
<J) is the ambipolar potential associated with slow plasma motion.
Further-
more, V Q is the amplitude of the rapidly oscillating electron velocity (v = v n exp(-ia) t) + c.c.) and is related to E by v n = e En/im co . e u po u u e po writing Equation (3.2), we have also assumed an isothermal equation of state for the slowly varying pressure and accordingly let p
= n s
T
= const.
e
In
T , where e
204
P.K. SHUKLA
From Equation (3.2), it follows that (3.3) i 12 2 where $ = ed)/T , and <j> = e E /m a) is the vponderomotive rpotential of the e' p ' ' e po electrons. When the phase velocity (u)/k) of the slow plasma oscillation is much smaller than the ion thermal velocity v . (adiabatic response), then the slowly varying ion density perturbation is given be ~
= exp(-a^),
(3.4)
o where O = T /T.. 6
1
The ponderomotive potential of ions is smaller by the 2
electron to ion mass ratio (y ) and is therefore neglected.
The Poisson
equation (2.3) completes the set of basic equations. On the other hand, for w/k >> v ., the slow ion motion (e.g., enhanced ion fluctuations) is described by Equations (2.1)-(2.3), and (3.3). First, we investigate the adiabatic (quasi-static) case in the quasineutral approximation (n
= n ) . Thus from (3.3) and (3.4), we find $ = e <(.p T±/Te(Te + T ± ) , ^ = o
exp(-|E| 2 ).
(3.5) (3.6)
Combining (3.1) and (3.6), we find
2 i y 9 E + 3 3 2 E + E [ 1 - e x p ( - | E | 2 ) ] = 0. t x
(3.7)
We look for a stationary solution of (3.7) in the form 2 where r| is a constant.
E(x,t) = w(x)exp(i D 2 t ) ,
(3.8)
Thus, (3.7) becomes
3 92w + (1 - A)w - w exp(-|w|2) = 0,
(3.9)
2 where A = 2 y T) . Multiplying (3.9) by 9 w and integrating, we have 3(3 w ) 2 + (1 - A)w 2 - 1 + exp(-w2) = 0,
(3.10)
where the boundary conditions w = 9 w = 0 at Ixl -*°° have been used. '' Combining (3.6) and (3.10), wexget \ where
(3n)
V(n) = 0,
(3.11)
SOLITONS IN PLASMA PHYSICS
205
_2 V(n) = with n = n /n .
2n
3
ln n
[(1 - A)In n + 1 - n ) ] ,
(3.12)
Analysis of (3.12) reveals that V < 0 and V 1 (1) = V(l) = 0,
V(N) = 0, V f (N) < 0, where N = n
at |x| = 0.
Thus, there exists local-
max ized solutions consisting of density cavity and Langmuir electric field envelopes.
The nonlinear frequency shift A is obtained by using the bound-
ary conditions w(0) = W and 3 w = 0 at |x| = 0. We find from (3.10) A = 1 + [-1 + exp(-W2)]/W2 > 0, (3.13) where W is the maximum value of w at the center. yields w
On the other hand, (3.6)
= -In N for a standing soliton.
We now consider the small amplitude limit.
Writing n = 1 + n, where
n << 1, we have from (3.13) A = W / 2 , and (3.12) becomes ~2 9 - ^ - (WZ + n ) .
(3.14)
An exact analytical solution for the density variation can be written as
n = -|nQ| sech [ |-~ -w where n =
W
2
(3.15) • ) •
is the maximum of the density depletion.
Secondly, we consider the hydrodynamic limit.
Here, for a quasi-neu-
tral plasma our basic system of equations are 2i y 3tE + 3 32E = (n - 1)E, 3tn + 3x(nv) = 0, 3 v + v 3 v = -3 (In n +
t
(3.16) (3.17)
|E|2).
(3.18)
xx
We now look for stationary solutions of (3.16)-(3.18) in the form E(x,t) = w(x - Mt) exp (i[6(t) + T(x)]} ,
(3.19)
n(x,t) = n(x - M t ) ,
(3.20a)
v(x,t) = v(x - M t ) ,
(3.20b)
where M is the Mach number. Substituting (3.19) into Equation (3.16), we obtain from the imaginary part T(x) = yMx/3.
(3.21)
The real part gives 3 32w = (n + A - l)w,
(3.22)
where the nonlinear frequency shift A is defined by A = 2 y 3 t 6 + y 2 M 2 /3.
(3.23)
206
P«K. SHUKLA
The ion equations can be integrated to give n U = -M, 2
U /2 + In n + w
(3.24) 2
2
= M /2,
(3.25)
where U = v - M, and the plasma is assumed to be unperturbed (n = 1, v = 0) at infinity.
From the last two equations we obtain w 2 = M 2 (l - n"2)/2 - In n.
(3.26)
Using this relation, Equation (3,22) can be integrated once to yield 3 32w +
(1 - A)w 2 + n(l + M 2 /n 2 ) = 1 + M 2 ,
(3.27)
where we have again used the boundary conditions at infinity. We seek two-parameter localized solutions of Equations (3.24), (3.25), and (3.27) with a bell shape for w(x) and an inverted bell shape for n. Accordingly, we let W = w(0) by the maximum of the electric field w, and N = n(0) be the minimum of the density n.
The Mach number, being uniquely
determined by Equation (3.24) evaluated at x - Mt = 0, is M 2 = 2(W2 + In N)/(l - N " 2 ) .
(3.28)
Similarly, the frequency shift A is obtained from Equation (3.24) and reads A - 1 + (N - 1)(1 - M 2 /N)/W 2 .
(3.29)
Making use of (3.26), we can write (3.27) in the form \
(9 n ) 2 + V(n;N,W) - 0,
z
(3.30)
x
which is analogous to the energy integral of the classical particle. potential energy V is given by 9 2 2 9 9 Zn W Z Z V(n;N,W) 9 9 9 [(1 - A)w + (n - 1) (1 - M /n) ],
The
(3.31)
3(1 - M V n ) where w is related to n by Equation (3.26). We now discuss the conditions under which (3.30) leads to soliton solutions.
Consider the behavior of V near n = N and n = 1 (note that it
2 We obtain (3.31) holds for N < n < 1 ) . I", . 1 - -~ N from /, 1 ( l -M— \ J\ ,(n - N) N 1 -N V
>
L
- 2A(1 - n)2/3
NW \
/J
for n^N,
(3.32)
for n Z 1.
Thus, V behaves appropriately at the boundaries n = 1 and n = N.
Another
condition for the existence of soliton solutions is that V should be negative.
This requirement is satisfied as long as N
follows.
The frequency shift 6d) = - -r- a)
> M , from which A > 0
of the high-frequency waves is
SOLITONS IN PLASMA PHYSICS always negative. overdense
207
The region outside the density trough is, therefore,
for the waves which are localized in the underdense cavity re-
gion. In Figure 1 we have indicated the region of existence of soliton solutions.
Lines of constant soliton velocity are also plotted for several
Mach numbers.
Clearly, no soliton can exist for M
can draw the following general conclusions.
< 0.
From Figure 1 we
First, for a fixed density de-
pression, the field intensity varies opposite to the velocity.
This can be
seen from Equation (3.28), which corresponds to an energy conservation law 2 2 J2 if (1 - N )/N
is interpreted as an effective mass, w
ergy, and -In N as the total energy. imum field intensity.
as the potential en-
A standing soliton therefore has max-
Second, Figure 1 indicates that high speed (M < 1)
solitons necessarily have small amplitudes.
Finally, we point out that the
usual small amplitude theory (Rudakov, 1973; Karpman, 1975) is valid to the left of the dotted line shown in Figure 1.
Our assumed profile for W
6N does not allow soliton solutions below the line M = N.
and
As we shall see
shortly that near this line the assumption of quasi-neutrality for slow plasma motion is violated.
Inclusion of Poisson's equation leads to a new
type of soliton (Nishikawa et al., 1974; Makhankov, 1974; Spatschek et al., 1981; Rao and Varma, 1982).
In particular, Rao and Varma (1982) in their
elegant paper solved the problem of finite amplitude envelope solitons by relaxing the assumption of quasi-neutrality. consists of a double hump at the center.
The electric field profile
This occurs at M > 0.6.
We now derive the small amplitude limit by letting (6N) 2 « W 2 <> 6N « 1. We obtain, by a cumbersome Taylor expansion,
I)
(3.33)
(3 34>
-
and the classical potential energy V becomes V(6n;6N,W2) = -(6n/3) (6N - 6n), where 6n = 1 - n.
and
(3.36)
It follows that 6n = 6N sech2[(6N/6) x ] ,
(3.37)
w 2 = W 2 sech2[(6N/6) x ] ,
(3.38)
P.K.
208
SHUKLA
Fig. 1. The existence region of solitons. Here, W =| 6N = 1 - N, where N = n m ^ n /n o . Solitons exist between the lines M = 0 and M = N. Note that ion nonlinearity becomes important to the right of the dotted line. 6(t) = ( J S ) ( A - y 2 M 2 /3)t.
(3.39)
Therefore, the solution for the real electric field E becomes v (16TT
/ X -
^ — 5 - - W sech I n T )* \
V t \
r - S - I cos(kx L /
(3.40)
where the soliton width L, the group velocity, the wave number k, and the frequency of the internal oscillation are given by v g = Mc s ,
(3.41)
and 3(kA D ) 2 - 6N/2 + 0(6N 2 )].
(3.42)
SOLITONS IN PLASMA PHYSICS
209
The frequency spectrum (3.42) shows that the high-frequency wave packet is 2 2 composed of Langmuir waves (o> = 4TTn e /m ) whose frequency is downshifted due to the presence of the density cavity. waves is n (1 - S N / 2 ) . cavity.
The effective density for these
The wave envelopes, therefore, exist only in the
Envelopes are evanescent outside and are expressed by the sech
profile. The solutions (3.37) and (3.38) satisfy (Karpman, 1971, 1975; Zakharov, 1972; Rudakov, 1973) 2i y E + 3 E = 6nE, t xx 6n
where 6n = - 6n.
tt "
6n
xx
=
(|E
(3.43)
!2)xx'
(3
'44)
We would like the reader to note that in the small ampli-
tude limit, ion equations (3.19)-(3.20) reduce to Equation (3.44).
The
latter represents the wave equation for the ion-acoustic fluctuations driven by the low-frequency ponderomotive force of the high-frequency waves. 2 Depending upon the relative strength of W /<$N, the soliton solutions expressed by (3.37) and (3.38) represent (i) the subsonic soliton of Rudakov if SN 2 « W 2 <, 6N « 1, and (ii) the sonic soliton of Karpman if 2 2 6N « W « 6N « 1. The soliton of Karpman, for example, prevails if W 2 = 0(6N 3 / 2 ). M
2
= N
2
Its velocity M 2 = 1 - 0 ( 6 N ^ ) lies above the critical line For smaller values of W 2 , so that W 2 =0(6N 2 ),
= 1 - 6N in Figure 1.
we pass the line M
= 1 - 6 N , and the above theory breaks down.
The reason
is that the terms involving biquadratic nonlinearity and the dispersive effects in the ion wave equation become of the same order as the one retained in (3.44).
Inclusion of these two terms leads to an equation of the form:
6n
- 6n = | |E|2 + 6n2 + 6n 1 . xx tt xx L1 ' -lxx
(3.45)
Soliton solutions of the system (3.43) and (3.45) were investigated by Hakhankov (1974) and Nishikawa et al. (1974).
We have
6n = - 6N sech2 z, E = - ^r 32
sech z tanh z exp[i(G(t) + T(x))],
(3.46) (3.47)
where z = (SN/18)^(x - Mt), M = 1 - 5 6N/9, 6(t) = 6Nt/12y, and T(x) = yMx/3. This solution is a one^parametrie family of soliton whose electric field has a node at the center. W
= 1 - (10/9)6N < M
The soliton speed is given by
= N « 1 - 6N.
In the language of quantum mechanics,
this soliton represents the first excited state whereas the former solitons
210
P.K. SHUKLA
can be considered as the ground state. Three-parametric soliton solution appear if deviations of the isothermal equation of state for the electrons are taken into account (Schamel and Shukla, 1976).
Ion equation now takes the form:
6ntt-
!2
where b is the electron trapping parameter and is given by b = (1 - 3)n"" Here, 3 = T /T e , and T e
2
is the temperature associated with the trapped-
electron distribution function.
For example, 3 ~ 0 corresponds to a flat
trapped-electron distribution (Schamel, 1972, 1973, 1979).
On the other
hand, a hole in the trapped region, corresponding to an under population of trapped electrons, is represented by a negative value of 3. It can be shown that soliton solutions of the system (3.43) and (3.48) exist provided that (3.49) This condition sets an upper bound for the trapping parameter b.
The Mach-
number is given by M = 1
W2
" I6N (1 " P
-1
)-
(3 50)
"
Explicit analytical soliton solutions exist only for special cases.
For
p" 1 = 2/3, we find (Schamel and Shukla, 1976) 6n = - 6 N [ 1 - 4 sinh2 z exp(-2z)] , |E| = W exp(-z) |(2 - exp(-2z))(l - | sinh2 z)J^,
(3.51) (3.52)
where z = 0.32 6N2|x - Mt |. In the isothermal limit p
"*" 0, our three-paramtric solution reduces
to that obtained by Karpman (1975). We now compare our analytical results with existing experiments.
In
order for envelope solitons to exist, two basic requirements have to be fulfilled.
Firstly, there must exist a source of Langmuir waves.
This can
presumably be an electron beam which excites Langmuir waves due to the twostream instability, or the process of linear mode conversion at the resonance layer of an inhomogeneous plasma can lead to enhanced Langmuir waves, or by installing an electric field at resonance in a capacitor experiment. Secondly, the processes which may destroy soliton formation should be weak or must be balanced by additional arragements.
Damping, linear and nonlin-
ear instabilities of a soliton, multidimensionality and plasma nonunifornrity
SOLITONS IN PLASMA PHYSICS
211
causing wave convection and acceleration are examples.
Finally, there also
enters the problem of the accessibility and the development of the soliton leading to a stationary state. Wong and Quon (1975) performed a series of experiments in a double plasma device at the UCLA.
They distinguished three different cases for
the development of the system depending upon the beam strength with typical values n b /n
~ 0.05 - 0.1 and v,/v
~ 5 - 10.
For a weak beam, the maxi-
mum amplitude of the beam mode was below the threshold of OTSI and consequently, no modulation was observed.
For values just above the threshold,
the parametric process sets in and ion wave with k = 2k
grew.
The system
evolved and after several ion plasma periods, a chain of soliton-like structure emerged.
For a stronger beam, the development was basically the
same except that the localized structures had finally larger amplitude and their formation time was shorter.
Three cases of stationary solitons were
observed with properties i) they were not propagating in space, and ii) the normalized maximum field in each of them was nearly equal to the normalized depth of the cavity.
Typical results for each regime are indicated in the
Figure 1 by I, II, III, respectively. 2 to W
For example, soliton II corresponds
- 0.1, 6N = 1 - N - 0.1 and lies exactly on the curve for a non-pro-
pagating, i.e., M = 0, soliton.
The latter is true for all three solitons,
which therefore have the maximum field intensity for their given 6N.
The
good agreement between our theory and this experiment leads us to believe that the observed spikes, sometimes called cavitons or spikons, are nothing else but the quasi-stationary solitons discussed here. On the other hand, in the experiments of Ikezi et al (1976) an ionacoustic test wave was launched and the density modification was studied by switching on the beam. <$N * 0,2, W 2 »
Ikezi et al observed a chain of solitons with
0.1, and M * 0.6.
This result, which is characterized by a
weaker field intensity and a larger density dip, also agrees with our calculations (point IV in the Figure 1 ) . Furthermore, if the beam is switched off, they observed that the high-frequency field damps away rapidly, whereas the density dip persists and attains a higher speed. sponds to following the line of constant
N downward.
This feature correNote that there ex-
ists an upper limit for the speed of such a soliton, given by M = N « 0.8 for 6N - 0,2,
The observed value of M * 1 indicates a transition from the
chain of envelope solitons to a nonlinear ion-acoustic wave packet, in the form of cnoidal waves or a chain of solitons.
212
P.K. SHUKLA Our comparison with the beam experiments assumes that the beam no
longer feeds energy into the solitons, thus allowing the exclusion of the source term in the basic equations.
Numerical simulations (Schamel et al.,
1976) in which the beam is taken into account indicate that decoupling between the beam and the beam excited mode occurs when the density fluctuations have reached a certain value (typically, of the order of 6N
W
0.1).
The reason for this decoupling is the frequency shift of the beam mode due to the parametric process, which detunes the beam-plasma resonance.
At the
same time as the wave parametric coupling becomes fully operative, the beam, which is still present in the plasma, attains a finite thermal speed. can still create new modes from noise, but on a slower time scale.
It
In
addition, these new modes are ncDt phase-locked and therefore, are not expected to interfere with the solitons. Finally, we mention that the decoupling of the beam, together with the assumption of one-dimensionality, do not seem to be fully realized in the experiment of Ikezi et al.
Namely if the beam is switched off, they ob-
served a rapid diminution of the field intensity, which appears as particle energy both parallel and perpendicular to the beam direction.
The observed
stationarity therefore seems to be due more to a balance of energy fed in by the beam and energy loss by dissipation and wave convection.
Thus, al-
though the assumptions of the present theory do not exactly correspond to the conditions in this experiment, nevertheless the results are comparative. The leakage of energy in the perpendicular direction allows the formation of the one-dimensional, stationary soliton in the direction parallel to the beam.
However, if one prevents (Wong, 1977) the flow of energy in the per-
pendicular direction, say by applying an external magnetic field, a strong overshooting of the electric field occurs and no stationary solitary structures will result.
In such a strongly driven regime, a new theory
CSchamel and Els&sser, 1978) is called for to explain the transient features. Let us estimate the effect of Landau damping of the high-frequency waryes.
For completeness, we shall also account for the effects of the non-
linear frequency shift which enhances the rate of dissipation.
As noted
earlier, our results indicate that the -validity of the expression for the small amplitude soliton extends well into the nonlinear regime, so that in this range the soliton can be represented by (3.37) and (3.38).
Thus, in
Fourier space, we have ^
sech[tfrk/2)(.SN/6) 2 ] .
(3.53)
SOLITONS IN PLASMA PHYSICS
213
The Landau damping rate is given by
Y = -cope(TT/8)V3 exp - | - (1 - 6N/4)/2k2J.
(3.54)
The rate of energy dissipation due to the electron Landau damping is therefore
-j
Y l \ | 2 dk/ j
|Ek|2 dk,
(3.55)
which depends only on the density depth 6N. In units of co ± , we find T = 2 x 10~5(m./m ) T = 5 x io" (m /m )\ for 6N = 0.2.
2
for 6N = 0.1 and
Thus, for argon plasma, where T = 0.1
for 6N = 0.2, Landau damping might become important for 6N > 0.2. 4.
LOCALIZED ELECTROMAGNETIC PULSES. The interaction of electromagnetic radiation with plasmas is of pri-
mary interest for laser fusion and plasma heating.
For weak pulses, the
self-filamentation (Kaw et al., 1973) occurs and the laser light is trapped into the large scale density cavities (Shukla et al., 1976a).
When the
laser intensity is very high, besides the density modulation, the mass of the electrons is also modulated as it oscillates relativistically in the laser field (Shukla et al., 1977).
In this case, stationary solutions such
as shock waves as well as solitons with density humps (Tsintsadze and Tskhakaya, 1977; Yu et al., 1978) or cusps (Kotetishvili et al., 1982) can also appear. Here, we consider the Raman interaction in which circularly polarized electromagnetic waves are modulated by forced Langmuir oscillations.
The
role of relativistic corrections to the nonlinear current density and the ponderoniotive force (Yu et al. , 1978) in laser plasma interaction is examined.
These effects are important because the laser-induced electron os-
cillation velocity can approach 0.6 times the speed of light (e.g., CO^ laser at 1 0 1 6 W/cm 2 ). Yu et al (1982) have shown that a powerful circularly polarized electromagnetic wave interacting with an electron plasma can give rise to solitary wave structures of which the forced density profile contains a depression at the center, together with shoulders of density excess on the sides. The density shoulders are due to charge separation effects, since thermal dispersion has been neglected. shape.
The vector potential assumes the usual bell
It should be noted that such profiles have been observed (Kozlov
et al. , 1979) in laser plasma interaction simulations and may play very important role to the particle acceleration.
214
P.K. SHUKLA We start with the wave equation for the high-frequency (circularly
polarized electromagnetic waves) motion 32A - c 2 V 2 A = 4TT cj",
(4.1)
where A is the vector potential, and j = -nev is the electron current density.
Here, n is the electron density which is modulated by Langmuir (low-
frequency) oscillations, e is the electron charge, and v is the electron velocity. The equations of motion including both high- and low-frequency motions are 3 n + V • (nv) = 0,
(4.2)
3 p + v • Vp = -e(E + v x B/c) - TV In n,
(4.3)
where p = m v/(1 - v /c ) 2, E = - 3 A/c - V(J), m
and T are the electron
rest mass and temperature, and c|> is the electrostatic potential of the Langmuir oscillations. Combining (4.2), (4.3), and using the definition for p, we obtain (Yu et al., 1978) 9 p + Tp • Vp/mQ = e 3 A/c - e Tp x (V x X)/m Q c + eV<() - TV In n, where T = (1 + p /c m o ) 2.
(4.4)
We have also used the relation v = Fp/m .
Assuming A = A(x) (x cos 0 - y sin 9 ) , where 0 = kz - 0)t, we obtain from (4.4) p = m where ]p = eA/m c , o
cif,
and 3(1 + ^ 2 ) ^ - 3 = * - In n,
where 3 = c m /T. X
(4.6)
We have normalized (j), n, and V by T/e, n Q , and
= (4irn e /T)~~^, respectively.
$ = 0, i j ; = 0 , n = l a t sent problem.
(4.5)
We have also used the boundary conditions
|x| -*• °°, and neglected electron inertia for the pre-
The last assumption means that the time variation of the law-
frequency electrostatic oscillation is less than the electron plasma frequency.
Equations (4.5) and (4.6) describe the high- and low-frequency mo-
tions, respectively.
We note that the ponderomotive force term appearing
in (4.6) is exact. Using the Poisson equation V2(() = n - 1,
(4.7)
SOLITONS IN PLASMA PHYSICS
215
one obtains n - 1 + 3V 2 (1 +
ty2)h>
(4.8)
where the pressure term (proportional to In n ) , which describes the dispersive effects of the electrostatic oscillations, has been neglected, assuming long wavelength disturbance.
The time scale of the interaction is
taken to be much shorter than those of the ion motion, the ions, therefore, form a fixed neutralizing background. From Equations (4.7) and (4.8), which describe the slow motion, it is clear that the modulation under consideration is not due to any (linear) normal mode, rather it is due to a driven mode, in which the relativistic ponderomotive force on the electrons is balanced by the space charge electric field. It is convenient to introduce the complex modulational representation % = -| i|/(t,z)(x - iy)exp(- id)t + ikz) 4- c . c , where 9 * «
(#, d \p «
(4.9)
k*.
The wave equation (4.1) becomes 2 i a 9 • + i a 3 i|) + 3 3 2 * + ^ t z z
= n^(l + | * | 2 ) " ^ ,
(4.10)
2 2 2 2 2 2 where a = 0)/a) , a = 2c k/a) X , and A = (a) - c k )/w . We have normalized t by 0)" , Stationary solutions of (4.10) can be found by introducing * = *(£) exp[i 0(t) + i <|>(z)], where £ = z - Mt, and M is a constant.
We obtain
3 a|* + 6* = n*(l + 1*1 V ^ , where 6 = A - 2a 0
(4.11)
2 2 2 + 2kc <|> /a) + a M /$ is the frequency shift, and
aMz/£. Multiplying (4.11) by 3>-i|;, using (4.8), we obtain, after some manipulation,
3 O ? ^ ) 2 + vojo = 0,
(4.12)
where
VOW = [6*2 + 2 - 2(1 + *2)^](1 + * 2 ) , and we have used the boundary condition \p -*• 0 for |^| -> °°.
(4.13) The effective
potential V(*) differs from the plane polarized case (Kotetishvili et al., 1982) by not containing a singularity.
216
P.K. SHUKLA
To verify that localized solutions of (4.13) exist, let us examine the behavior of V(ip).
For if; -* 0, we have
V0I0 = (6 - l)ty + —ty,
(4.14)
which leads to the condition 6 < 1, so that V(ip) < 0. On the other hand, at the maximum ty , we have from V(I/J ) = 0
6=
0,
(4.15)
and
3 Vl
= \p (1 + if;2) [6 - (1 + i|;2)~ \ > 0,
* h(7
m
m
m
(4.16)
since 6 > (1 + i ^ 2 ) ^ . m Thus, V(ij;) indeed has the profile necessary for the existence of localized solutions for ty. In fact, Equation (4.12) is readily integrable numerically.
A typical soliton profile is given schematically in Figure 2.
The density profile has been obtained from (4.8).
The low-frequency elec-
trostatic potential (j) given by (4.7) has a profile similar to that of -ty.
Fig, 2. Schematic profiles of the normalized vector potential envelope i[», wave magnetic field |B|, and electron density perturbation n - 1. In the small amplitude limit, Equation (4.12) can be integrated directly, using the approximate form of V(ip) given by (4.14).
i/; = ty sech x>
The result is
(4.17)
SOLITONS IN PLASMA PHYSICS
217
—2 — V where T = 4(1 - 6) « 1, and X = * £/23 2. m m The electron density, which can be obtained from (4,8), is n = 1 + j ij^(2 sech2x - 3 sech 4 x).
(4.18)
—4 We note that, here, the density variation scales like \f> ; it is small compared with those of other interactions (Yu et al., 1978). By means of numerical calculation, our investigation can easily be extended to include thermal effects as well as ion dynamics.
In this case,
the slow motion involves Raman as well as Brillouin effects, the latter being due to natural or forced ion-acoustic motion.
Although one expects
that these effects are physically separated, recent numerical investigations (Kozlov et al., 1979) seem to indicate that there may be cases in which both effects are simultaneously important. 5.
DISCUSSION. In this article, we have reviewed the present state of art of solitons
in plasma physics.
We have presented simple examples for constructing
exact solitons in an unmagnetized plasma.
There have also been substantial
efforts to find exact planar solitons in the presence of an external magnetic field.
Non-envelope solitons include the ion-acoustic (Shukla and Yu,
1978; Yu et al., 1980) and the Alfven solitons (e.g., Shukla et al., 1982) in a low-beta plasma.
In a magnetized plasma, numerous kinds of high- and
low-frequency oscillations exist and their nonlinear interaction can give rise to a great variety of envelope solitons.
A few examples of envelope
solitons in a magnetoplasma are the localized upper-hybrid (Kaufman and Stenflo, 1975; Porkolab and Goldman, 1976; Yu and Shukla, 1977; Shukla, 1977b; Dysthe et al., 1978; Shukla and Yu, 1982), the lower-hybrid (Spatschek et al., 1977), and the whistler (Shukla et al., 1976b; Spatschek et al., 1979; Laedke et al., 1982) wave packets.
These nonlinear entities are
actually observed in the laboratory (Cho and Tanaka, 1980) as well as in space plasmas. In our opinion, there still remains the problem of the finite amplitude envelope soliton when the quasi-neutrality condition is abandoned.
It
seems highly unlikely that a single equation representing the energy integral of a classical particle can be obtained.
It is, therefore, suggestive
that numerical work should be performed in order to understand the fine structure of the solitons.
Furthermore, care has to be exercised for those
solitons whose width is of the order of the Debye length. that Landau damping Becomes quite significant. done in this area.
The reason is
Clearly, much remains to be
P.K. SHUKLA
218
We have not discussed the dynamics of solitons in an inhomogeneous medium.
However, there exists many investigations demonstrating that a
soliton can be treated as a Newtonian particle.
For example, a soliton is
accelerated down the density gradient in a non-uniform medium (Chen and Lin, 1977; Shukla and Spatschek, 1978).
Inclusion of ion inertia (Chukbar and
Yankov, 1977) in the dynamics of slow plasma motion essentially drags the soliton and the sound waves are emitted (Bondeson, 1980).
Finally, to our
knowledge, there do not exist methods which solved the initial value problem of the Zakharov and the coupled Schrodinger-KdV equations. ACKNOWLEDGEMENTS.
The author is grateful to Hans Schamel and Ming Yu for
their fruitful collaboration, the results of which are the basis for part of the present review. kal is acknowledged.
The benefit of useful discussions with Martin KrusThe author thanks Mrs. H. Kurzbein for her commenda-
ble typing of this manuscript.
This work was sponsored by the Deutsche
Forschungsgemeinschaft through the Sonderforschungsbereich 162 Plasmaphysik Bo chum/Jill ich. REFERENCES BONDESON, A. (1980) Perturbation analysis of single Langmuir solitons, Phys. Fluids 23, 746-754. CHEN, F.F. (1974) Introduction to Plasma Physics, Chapter 8, Plenum Press, New York. CHEN, H.H. and LIU, C.S. (1977) Soliton generation at resonance and density modification in laser-irradiated plasmas, Phys. Rev. Lett. 39, 1147-1151. CHO, T. and TANAKA, S. (1980) Observation of an upper-hybrid soliton, Phys. Rev. Lett. 45, 1403-1406. CHUKBAR, K.V. and YANKOV, K.V. (1977) Langmuir solitons in an inhomogeneous plasma, Sov. J. Plasma Phys. _3> 780-782. DAVIDSON, R.C. (1972) Methods in Nonlinear Plasma Theory, Chapter II, Academic Press, New York. DYSTHE, K.B., MJ0LHUS, E., PECSELI, H.L. and STENFLO, L. (1978) Langmuir solitons in magnetized plasmas, Plasma Phys. 20, 1087-1099. GARDNER, C.S., GREENE, J.M., KRUSKAL, M.D., and MIURA, R.M. (1967) Method for solving the Korteweg~de Vries equation, Phys. Rev. Lett. 19, 1095-1097. IKEZI, H., CHANG, R.P.H., and STERN, R.A. (1976) Nonlinear evolution of the electron beam instability, Phys. Rev. Lett. 36, 1047-1051. KARPMAN, V.I. (1971) High-frequency electromagnetic field in plasma with negative dielectric constant, Plasma Phys. 13, 477-490. KARPMAN, V.I. (1975) On the dynamics of sonic-Langmuir solitons, Phys. Scr. JU, 263-265.
SOLITONS IN PLASMA PHYSICS
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KAUFMAN, A.N. and STENFLO, L. (1975) Upper-hybrid solitons, Phys. Scr. 11,, 269-270. KAW, P.K., SCHMIDT, G., and WILCOX, T. (1973) Filamentation and trapping of electromagnetic radiation in plasmas, Phys. Fluids 16, 1522-1525. KOTETISHVILI, K.V., KAW, P.K., and TSINTSADZE, N.L. (1982) Some nonlinear phenomena in transparent electron plasma, Sov. J. Plasma Phys. 8^, in press. KOZLOV, V.A., LITVAK, A.G., and SUVOROV, E.V. (1979) Envelope solitions of relativistic strong electromagnetic waves, Sov. Phys. JETP 49, 75-80. LAEDKE, E.W., SHUKLA, P.K., SPATSCHEK, K.H., and YU, M.Y. (1982) The lifetime of nonlinear whistlers, Phys. Fluids 25, 1693-1695. MAKHANKOV, V.G. (1974) On stationary solitons of the Schrodinger equation with a self-consistent potential satisfying Boussinesq's equation, Phys. Lett. A 5£, 42-44. NISHIKAWA, K., HOJO, H., MIMA, K., and IKEZI, H. (1974) Coupled nonlinear electron plasma and ion-acoustic waves, Phys. Rev. Lett. 33, 148-151. PORKOLAB, M.V. and GOLDMAN, M.V. (1976) Upper-hybrid solitons and oscillating two-stream instabilities, Phys. Fluids 19, 872-881. RAO, N.N. and VARMA, R.K. (1982) A theory of sonic Langmuir solitons, J_. Plasma Phys. ,27, 95-120. RUDAKOV, L.I. (1973) Deceleration of electron beams in a plasma with a high level of Langmuir turbulence, Sov. Phys. Doklady 17, 1166-1167. SAGDEEV, R.Z. (1966) Cooperative phenomena and shock waves in collisionless plasmas, In Reviews of Plasma Physics (Ed. M.A. Leontovich) Vol. IY, pp. 23^91, Consultants Bureau, New York. SCHAMEL, H. (1972) Stationary solitary, snoidal, and sinusoidal ion acoustic waves, Plasma Phys. 14, 905-924. SCHAMEL, H. (1973) A modified Korteweg-de Vries equation for ion acoustic wave due to resonant electrons, J. Plasma Phys. 9_, 377-387. SCHAMEL, H. and SHUKLA, P.K. (1976) Envelope solitons in the presence of nonisothermal electrons, Phys. Rev. Lett. 36, 968-971. SCHAMEL, H., LEE, Y.C., and MORALES, G.J. (1976) Parametric excitation of ion density perturbations in the relativistic beam-plasma interaction, Phys. Fluids 19, 849-856. SCHAMEL, H., YU, M.Y., and SHUKLA, P.K. (1977) Finite amplitude envelope solitons, Phys. Fluids 20, 1286-1288. SCHAMEL, H. and ELSASSER, K. (1978) The dynamics of cavitons induced by the soliton flash, Plasma Phys. _20, 837-847. SCHAMEL, H. (1979) Role of trapped particles and waves in plasma solitonstheory and application, Phys. Scr. 20, 306-316. SHUKLA, P.K., YU, M.Y., and TAGARE, S.G. (1976a) Solitary filaments of coupled electromagnetic wave and finite amplitude ion fluctuations, Z. Naturforsch. 31A, 1516-1518. SHUKLA, P.K., YU, M.Y., and TAGARE, S.G. (1976b) Evolution of finite amplitude whistler wave, Phys. Fluids 20, 702-703. SHUKLA, P.K. and TAGARE, S.G. (1977) Nonlinear electron plasma waves, J_. Phys. A; Math. Gen. 1£, L267-L268.
220
P.K. SHUKLA
SHUKLA, P.K. (1977a) Stability of a finite amplitude ion acoustic wave in plasmas, Phys. Fluids 20, 1579-1580. SHUKLA, P.K. (1977b) Nonlinear propagation of high-frequency plasma waves in a magnetized plasma, J. Plasma Phys. 18, 249-256. SHUKLA, P.K., YU, M.Y., and SPATSCHEK, K.H. (1977) Intense circularly polarized waves in plasmas, Phys. Lett. A 62, 332-334. SHUKLA, P.K. and SPATSCHEK, K.H. (1978) Profile modifications at critical density of laser produced plasmas, J. Plasma Phys. 19, 387-403. SHUKLA, P.K. and YU, M.Y. (1978) Exact solitary ion acoustic waves in a magnetoplasma, J. Math. Phys. 19, 2506-2508. SHUKLA, P.K. and YU, M.Y. (1982) Comment on "Upper-hybrid wave collapse", Phys. Rev. Lett. 49, 696. SHUKLA, P.K., RAHMAN, H.U., and SHARMA, R.P. (1982) Alfven soliton in a low-beta plasma, J. Plasma Phys. 28, 125-132. SPATSCHEK, K.H., SHUKLA, P.K., and YU, M.Y. (1977) Filamentation of lowerhybrid cones, Nucl. Fus. 18, 290-294. SPATSCHEK, K.H., SHUKLA, P.K., YU, M.Y., and KARPMAN, V.I. (1979) Finite amplitude localized whistler wave, Phys. Fluids 23, 576-582. SPATSCHEK, K.H., LAEDKE, E.W., YU, M.Y., and SHUKLA, P.K. (1981) Theoretical aspects of profile steepening in laser-produced plasmas, In Laser Interaction and Related Phenomena (Eds. H. Schwarz, H. Hora, M. Lubin, and B. Yakubi) Vol. 5, p. 755, Plenum, New York. TSINTSADZE, N.L. and TSKHAKAYA, D.D. (1977) On the theory of electrosound waves in a plasma, Sov. Phys. JETP 45, 252-256. WASHIMI, H. and TANIUTI, T. (1966) Propagation of ion-acoustic solitary waves of small amplitude, Phys. Rev. Lett. 17, 996-998. WONG, A.Y. and QUON, B.H. (1975) Spatial collapse of beam driven plasma wave, Phys. Rev. Lett. 34, 1499-1502. WONG, A.Y. (1977) Electromagnetic wave interactions with inhomogeneous plasmas, In Laser Interaction and Related Phenomena (Eds. H. Schwarz and H. Hora) Vol. 4B, p. 783, Plenum, New York. WONG, A.Y. (1982) Caviton induced fusion, In PPG 644, Center for Plasma Physics and Fusion Engineering, UCLA. YU, M.Y. and SHUKLA, P.K. (1977) On the formation of upper-hybrid solitons, Plasma Phys. 1£, 889-894. YU, M.Y., SHUKLA, P.K., and SPATSCHEK, K.H. (1978) Localization of highpower laser pulses, Phys. Rev. A 18, 1591-1596. YU, M.Y., SHUKLA, P.K., and BUJARBARUA, S. (1980) Fully nonlinear ionacoustic solitary waves in a magnetized plasma, Phys. Fluids 23, 2146-2147. YU, M.Y., SHUKLA, P.K., and TSINTSADZE, N.L. (1982) Nonlinear interaction of a powerful laser with an electron plasma, Phys. Fluids 25, 1049-1050. ZAKHAROV, V.E. (1972) Collapse of Langmuir waves, Sov. Phys. JETP 35, 908-914. ZAKHAROV, V.E. and KUZNETSOV, E.A. (1974) Three dimensional solitons, Sov. Phys. JETP 39, 285-286.
CHAPTER 12 A THEORY FOR THE PROPAGATION OF SLOWLY VARYING NONLINEAR WAVES IN A NON-UNIFORM PLASMA R. J. GRIBBEN Department of Mathematics University of Strathclyde Livingstone Tower, 26 Richmond Street, Glasgow Gl 1XH.
1.
INTRODUCTION This article is concerned with the uniformly-valid perturbation of
those solutions of the Boltzmann and Maxwell equations which represent the propagation of periodic waves in a uniform plasma.
No restriction on the
amplitude of the wave is required but it is assumed that the basic monochromatic wave train is slowly-varying in the sense that, for example, such parameters of the wave as its amplitude or wavelength may change significantly only over a large number of periods.
The work is thus
related to the well-known Whitham theory (Whitham 1965,1967,1974) for treating similar wave phenomena in other fields, especially those of water waves, but no averaged Lagrangian technique has been obtained in general for the present problem although that approach will be discussed for the special cases of a cold and warm plasma in Section 3.3.
On the other hand
here the perturbation method is applied directly to the basic equations. Although in the original presentation (Butler and Gribben 1968) the latter were expressed in a relativistically-invariant form leading to a rather neat mathematical appearance for the general results in the present article we consider from the outset the simplest form of the equations of interest. In particular the plasma is taken to vary only in the direction of propagation of the one-dimensional wave considered; neglected and the magnetic field is zero.
furthermore, collisions are
In these circumstances the basic
equations reduce to the Vlasov and Poisson equations in one space variable and time and the direct interpretation of the equations which result from application of the theory is simpler. We should perhaps emphasize that we are not considering here the effects of wave-wave interactions but concentrating on the interaction of particles with the wave.
The general theory covers in detail the
behaviour of particles in the trapped regions (see Section 3.1) but as
222
R.J. GRIBBEN
yet no problems have been solved for such cases. The main purpose of the paper is to review the current status of this theory of slowly varying waves and, in so doing, advantage has been taken of the opportunity to summarize in Section 2 some appropriate uniform wave solutions available.
These mainly consist of the classical,
so-called BGK type (after Bernstein et al 1957), especially those introduced recently by Schamel (1972a,b; 1975).
In Section 3 the theory
of slowly varying waves is developed using a two-scale method and some of its general implications are described as well as its application to simple special cases.
The results of the theory emerge as the formulation
of a set of integro-differential equations.
Consideration of the type of
these equations has important consequences with regard to the question of stability of the waves (see Whitham 1974).
Again the identification
has only been obtained, so far, in simple circumstances but current developments and future areas of research, including open questions, are indicated in the concluding Section. 2.
UNIFORM WAVE SOLUTIONS
2.1 BGK SOLUTIONS The plasma is assumed to consist of ions and electrons of different types.
The basic equation satisfied by f (x,t,v), the one-dimensional
distribution function of the sth-species of particles, is, with no magnetic field, 3f
3f
e E 3f s
where e and m are charge and mass of a particle respectively and E(x,t) is the electric field intensity.
Further E satisfies Maxwell's equations
which here take the form
(2
where e
is the permittivity of free space.
equations for f
s
This system of three
and E is self-consistent because, from (2.1),
JL ( |
f dv
| + J L (j
V
fsdv|=o.
-3)
NONLINEAR WAVES IN A NON-UNIFORM PLASMA
223
Uniform periodic wave solutions are sought in which the variables =
depend on x and t in the combination x
^ x " <*>t where k and OJ are the
wave number and frequency and oo/k = V, the phase velocity.
Then (2.1),
(2.2) and (2.3) become, on introducing the electrostatic potential cj> defined by E = - 3<|>/3x, 3f u
k2
e k 9X ' " m '
dfi + -_L Z i
dX 2
s
3f
¥r^T'O 3^ 9 e
S
^
,00
Z e
uf
s
s
L
f
(2.4)
du = 0,
du = 0,
(2.5)
(2.6)
where u = v - V is the velocity in the frame moving with the wave. Equations (2.4) and (2.5) are simultaneous equations for and f
and the
latter must satisfy (2.6), the zero current condition in the wave frame. The characteristics of (2.4) are given by , e k ,. du _ _ s dc|) dx " m g u dx and hence the general solution can be written as f u as = u2 +
2eS
s
= F (a ) where s s
s
and is proportional to the energy of a particle in the wave frame. is not a single-valued function of a F+ s F
s< a s)
and hence we define, s , a :> 6 s > o,
F (a ) s s
(«„),
F" ( a ) :, a - 8 > u < o, s * s s s F°s (a.),' 6 S <' as < e s '
(2.7)
where 3 = max(2e <j>/m ) , 3 = min(2e cj>/m ) . For the periodic potentials s s s s s s considered F (a ) corresponds to those particles trapped between the wave s s crests and such particles have closed trajectories in the (u,x)-plane. On the other hand the untrapped particles moving faster (F ) and more slowly s (F ) than the wave have open trajectories and simply speed up and slow down on travelling over the crests and troughs of the wave respectively. The remaining equations (2.5) and (2.6) become,
224
R.J. GRIBBEN
Tc2
d
,o. $ +
r
T
-1
y
Le
77 F dx
o
s
os
F
da
S
S
(a -=~ \ s
* S
S e s S
F
L
s d a s = °-
(2 9)
'
s
The contour L
is the map of -°° < u < «> in the complex a -plane cut along s s the real axis from the branch point at 2e <^/m to +». We suppose that s s
the period of is 2TT and this defines k.
Further, from (2.8), the
maxima of $ bisect its minima and is symmetric about its maxima and minima. For any given F (a ) the solution of (2.8), a highly nonlinear s s differentio-integral equation, is difficult and the first general method of tackling the problem was made by Bernstein et al (1957) by a somewhat inverse approach.
For simplicity they take a plasma in which only one
species of ions and electrons are present (denoted by s = i,e, say) and suppose that the potential profile
anc
*
tne
distributions for the
untrapped particles F (a ) and F (a ) are known.
Then, depending on the
particular region of the <j> profile, the distribution of one type of trapped particles, ions or electrons, can be found analytically from (2.8) if the distribution of the remaining type of trapped particles is also prescribed.
The solution is obtained from solving an Abel integral
equation and the manner in which the solution can be systematically advanced between successive maxima and minima of (j) is described in the original paper and elsewhere (e.g. Neunzert
1965).
In the latter
reference also an alternative method of deriving this solution is given which is of some interest and which may have application in obtaining solutions in more space dimensions. In his method Neunzert takes the Fourier transform in u-space of the unknown distribution and then the problem reduces to an initial value problem for a rather simple hyperbolic equation,
— I where f(x>p) =
—ivp e f(x,v)dv has known values on x
=
* X > say, and
J —oo
p = 0 and f(x,v) is the unknown trapped particle distribution.
Neunzert
NONLINEAR WAVES IN A NON-UNIFORM PLASMA
225
then obtains the analytic solution of this problem in terms of its Riemann 2e function G(x,p;x;pT) = I Q i/^
(4> (x)-
zeroth order Bessel function of the first kind.
Finally, the inverse
Fourier transform of the function f(x,p) can be found explicitly and yields the solution obtained by Bernstein et al (1957). A key feature of posing the Vlasov-Poisson problem in the manner indicated in this section is that it is thereby expressed as a linear problem for the unknown trapped distribution.
This of course is a great
simplification but there remains the shortcoming that there is no guarantee that a positive solution will emerge as required to represent a real distribution.
An alternative strategy, which avoids this
difficulty and involves a direct attack on the problem of solving the nonlinear Poisson equation, will be described below.
In neither of these
approaches is it necessary to restrict the solution to periodic waves. 2.2.
SCHAMEL SOLUTIONS In a series of papers Schamel (e.g. 1972a,b; 1975) has proposed a
method for obtaining uniform wave solutions of equations (2.1), (2.2) and (2.3).
In essence, the method consists of prescribing all distributions
for ions and electrons, trapped and untrapped, as functions of a. or a respectively and then solving equation (2.8) for . Although any relevant choice could be made for f
and f., in practice Schamel assumes
on physical grounds that in the absence of the wave the untrapped distribution functions should reduce to fshifted1 Maxwellians in the wave frame, e.g.
(cf. (2.7)) since it is assumed that min(4>) = 0.
The trapped electron
distribution is assumed to be Maxwellian and continuous with the corresponding untrapped distribution at the trapping boundary a
=0.
Again, for the electrons,
0, (2.U)
226
R.J. GRIBBEN
wherety= max(<J>) and y electrons.
If y
is a measure of temperature of the trapped
< 0 the trapped electron distribution has a depression
and if y > 0 it has an elevation. Similar distributions are assumed for e the ions, with suffix i replacing e and a. - 2£$L replacing a ; F.' (a.) i are then defined for a. > — ^ and F?(a.) for 0 < a. < —^- •
l
parameters N
., T
m.
., V
I
., y
l
l m.
Not all the
. i n these expressions are independent;
equation (2.9) becomes N V e e
= N.V.,
(2.12)
l I
and there are further conditions to be satisfied as will emerge below. Next the Poisson equation (2.8) can be integrated once to give
(2.13)
VQX/ where in the present case,
/
2^
f
z
2^
^
\T(<|>,...) = V - T T - Jm I F (a + - ^ H da +m. I F./a. - - M da. V ° 2 e o | / J L e V m e / 6 lJL " V 1 m i / X and \J
is arbitrary and independent of $.
(2.14)
For periodic waves Schamel
chooses the $ profile to have its maximum value, \p, at x s 0 and its minimum value, 0, at x = ^-
It then follows from (2.13) that
1^(0,...) = \/0ls...) - 0.
The first condition determines y
and the
second involves, in general, the parameters in the distributions as well as \|; itself.
Integration of (2.13) then yields the solution for the
potential
since <J>(TT) = 0.
Finally, of course, § (0) = ip so that d
^
— -*••
(2.16)
The relations (2.12), (2.16) and V (^,...) = 0 all involve the parameters appearing in the distribution functions as well as i|; and k. Now the wave velocity corresponding to the BGK solutions is
NONLINEAR WAVES IN A NON-UNIFORM PLASMA
227
arbitrary because the analysis is carried out in the wave frame.
Hence in
practice it can be chosen for convenience and, for example, in problems involving streaming cold ions the laboratory frame is often chosen to be that in which the ions are at rest so that the wave velocity is the negative of the ion velocity in the wave frame.
In more general cases the
choice is less clear and Schamel (1975), for example, chooses the laboratory frame as that in which the unperturbed plasma (i.e. for ty = 0) would have zero mean current and from the consequent definition of V derives a dispersion relation. Schamel goes on to discuss several cases of these solutions corresponding to both periodic and solitary waves; include ion-acoustic waves and Langmuir waves.
special cases treated
More recently the same
ideas have been extended to discuss the theory of electron holes and double layers (Schamel 1979, 1983). In order to simplify the presentation Schamel uses a dimensionless notation.
We point out that in using these or any other uniform solutions
as a basis for treating non-uniform waves it is necessary to use dimensional variables because the quantities used in the non-dimensionalisation are allowed to vary in the more general problem. 2.3
OTHER SOLUTIONS In this section we mention some other solutions of BGK type which
have been obtained. Neel and Flynn (1973) have examined the finite amplitude wave problem associated with distribution functions for trapped and untrapped particles modified for this purpose by a prescription given by Allis (1969).
Such modes are known as Allis modes and they have the property
of consistency with physically realistic distributions whilst reducing to linear waves in the small amplitude limit and to nonlinear waves described by the fluid equations in the limit of zero temperatures.
For
distributions which are Maxwellian in the unperturbed plasma the prescription yields a special case of the Schamel distributions described already.
In order to obtain analytic solutions as far as possible Neel
and Flynn study nonlinear periodic waves using the Allis method applied to the Cauchy distribution and obtain propagation properties of the waves. A somewhat degenerate BGK solution has been obtained by Rowlands (1969).
Although he considered two counter-streaming cold electron beams
oscillating against a uniform stationary ion background the same solution
228
R.J. GRIBBEN
applies to the case of single streams of cold electrons and ions.
Trapped
particles are excluded here and the nonlinear solution, with the perturbed motion of the ions neglected, can be written in the indirect form a) x j 1 y = a sin i^~ + (a 2 -y 2 ) 5 - af
L o
(2.17)
J
where the potential (j) is related to y by mV2 e o and the electron density n and velocity u are n = Nd+y)"1,
u = V(l+y).
(2.19)
In these expressions N and V respectively, w
are the values at x = 0 of n and u °i = (e2N/e m ) 5 and a is the wave amplitude parameter
restricted to be less than one for no particle trapping.
In fact the solu-
tion corresponds to a BGK solution in which the electron distribution is a delta function in energy space, viz. f (a ) = 2NV 6(a - V 2 ) . A non-trivial generalization of the above solution for single streams of electrons and ions has been obtained by Coffey (1971) where effects of plasma temperature are taken into account by including a pressure term in the momentum equation for the electrons.
The pressure is taken to be
proportional to n 3 and the model assumes that the electron heat flow relative to the mean motion is negligible implying that the wavelength of the wave is large compared with the Debye length. T
The equations are then 1
identical, to those describing the single water bag model of a plasma (Davidson 1972).
In this case a full analytic solution is not available
in closed form but a solution in powers of amplitude is sin x +
^ ; 3(l-3) 2
(cos x -cos 2 X ) + 0(a 3 ),
(2.20)
+ f9n+l\ (3-4cos x + cos 2 X ) + 0(a 3 ) , (2.21) m V^ 12(1-3) e o where 3 = 3V2 /V2 < 1 and V , (aN) is the thermal velocity of the electrons th o tn in the absence of a disturbance. In addition the condition of no trapping becomes now a 2 < 1 - 3/3 - 8 3 V 3 + 23 5 ,
(2.22)
NONLINEAR WAVES IN A NON-UNIFORM PLASMA
229
which becomes increasingly restrictive on a as 3 increases from its cold plasma value of zero. The solutions given in the last two paragraphs for cold and warm plasmas will be discussed again in the next section in the context of slowly varying plasmas.
But to conclude this section we mention that many
other authors have obtained particular examples of periodic BGK solutions (e.g. Ozawa et al 1964, Thompson 1971).
In addition there has been
experimental (Wharton et al 1968, Franklin et al 1972, Ikezi et al 1972) and numerical (Berk and Roberts 1967, Armstrong and Montgomery 1967) evidence for the existence of such waves.
Except for very special
circumstances the question of stability of these waves remains largely unresolved. 3.
SLOWLY VARYING WAVES
3.1 GENERAL THEORY In this section we consider the general question of determining uniformly-valid perturbations of solutions of the type described in Section 2.
The plasma is assumed to be varying on time and length scales
that are large compared with the plasma period and Debye length respectively and this fact is recognised explicitly by introducing the additional independent variables X = ex,
T = et,
where e is a small parameter, into the solutions of the basic equations (2.1) and (2.2).
The definition of the phase variable x is also
generalized so that in terms of it the slowly-varying frequency and wave number are given by u)(X,T) = - | * , k(X,T) = | *
.
Hence 9o)
¥x
+
9k
9
"9T " 9T
3k
_
9X
/I^TTN
_
n
c\
i\
the 'conservation of waves' equation. The distribution functions and potential are assumed to take the forms f (a ,x,X,T) and cf>(x,X,T) respectively. Substitution into (2.1), s s regarding the variables x> x and T as independent, yields an equation involving f
corresponding to untrapped particles the s equation is integrated over a period in x when the periodicity condition s
and $,
For f
230
R.J. GRIBBEN
on f
and cj> yields an equation that must be satisfied as a necessary
condition for non-secular solutions;
for the trapped particles the
corresponding integration is over one complete orbit of a particle and the result divided by two.
The condition can be summarized as
"3f
r
9T
[+ Vs
/
2e
re Z
m g / J3X
,
Lm s \^T
( (a
s
s
' (3-2)
m s
where r is the integration contour described above for untrapped and trapped particles. Before applying the same analysis to the Poisson equation (2.2) it is convenient to combine it with (2.1) and write the result in the equivalent form, a / f°° \ * I f°° 2 e E2\ (E m vf dv + 7 - E m v f dv - - 2 — = 0 . ~9 t \s s j _ M s / 3x \8 s J ^ s 2 / We then obtain for the condition for non-secular solutions of this equation,
2egcj) J m ^s
s s
m
s (3.3)
2e (J) 5 2 - —?-) 1 f da
[v+(a
s
s
m
Equations (3.2) and (3.3) are correct to all orders in e but to make progress it is convenient to expand f
and cf> in series in e in the form, s f = F (a ,X,T) + e G (a ,X,T) + ... , s s s s s $(X,X,T)
NONLINEAR WAVES IN A NON-UNIFORM PLASMA
231
Therefore, in the leading order theory the equations to be satisfied are (3.2) and (3.3) with f
s
and $ replaced by F
and $ respectively and the
s
last term (containing e) omitted in (3.3), and (3.1).
In addition the
Poisson equation in the form (2.8) is assumed to remain locally valid, contains X and T as parameters and must be satisfied with replaced by $ along with the current condition (2.9). The conditions linking the different branches of F
in phase space s
must also be considered and are of two types depending on the manner in which the uniform waves are perturbed.
In general the perturbed motion
is not periodic and the dividing trajectory through one saddle point does not pass through the next.
This allows particles to transfer from one
region (+, - or 0) to another and there are two fundamental types of transfer (three of each type) that can occur, viz. particles from one region enter the other two or particles from two regions enter the remaining one.
The former case is called ^on-mixing1 and the latter
1
'mixing , since there alternate layers (of thickness of 0(e)) of particles from different regions are present.
An example of each type is sketched
in Figure 1. Non-mixing
Mixing
Figure 1. Perturbed dividing trajectories; A,A1 are saddle points of uniform motion. (a) Configuration in which particles from f o f region enter f + T and f-f regions, (b) configuration in which particles from f + f and f-? regions enter f o f region. The boundary conditions corresponding to these cases can be derived on physical grounds (Butler and Gribben 1968) or on mathematical grounds (Gribben 1972).
The results show that for non-mixing cases F
continuous at the trapping boundary, i.e.F
=F s
=F s
=
on a s
is
s
3 • On s
the other hand for mixing cases this is not in general true and then just a single condition,
R . J . GRIBBEN
232
(P +R ) F + + (P -R y)F s s s s s s
must be satisfied, where >2e , P
S
- 2P F° = 0 , ss '
a
s
•*
(3.A)
\ / 33
2* \m = lim a +3 J o s s
2TT
o It seems clear that the distribution functions for the mixing cases cannot be strictly defined and a reformulation of the problem in terms of weak functions for the distributions can then be made (see Butler and Gribben 1968).
Such a step builds irreversibility into the system and
weakly convergent distributions may well have physical relevance in serving as a basis for the study of collisionless shock waves, for example, where a balance might be established between the turbulence of the mixing and the nonlinearity.
An interpretation of the boundary condition (3.4)
for these cases in terms of the probabilities of particles entering one region from the other two is possible. Finally in this section we point out that the non secular condition (3.2) derived from the Vlasov equation can be regarded as a statement of the fact that particles of the distribution F are conserved along their s time-average paths. This follows since the coefficients of 8F /8T, dF /8X s s and 8F /9a can be identified as the change in time, distance and energy s s incurred when a particle traverses a complete period of the wave or completes half a cycle if it is trapped. 3.2 HIGHER ORDER THEORY We briefly mention the use of the non-secular conditions (3.2) and (3.3) to discuss the problem correct to terms of 0(e) in the expansions. The analysis has been presented in detail in the original paper (Gribben 1972). The main point of interest lies in the form of (3.2) which can be written as
NONLINEAR WAVES IN A NON-UNIFORM PLASMA
3F
_
9)T 8^~ s 3F +
(p
8F
83T iiS" V s
_
s" e K sp ) " 3ST 3T s
_
233
(p
s""£Ksp)
8F £K
sh> - ^s 3X 'V^sh* = ° ( £ 2 ) '
where p
and H are phase averages of the momentum and energy of a particle, s s respectively, and both they and F are now correct to 0(e). The quantities
K
sp
and K , = VK are functions of X, T and a and whilst their physical sh sp s
significance is unclear their appearance at this level of approximation is, perhaps, not unexpected since they contain relative values of momentum and energy with respect to the corresponding phase averages.
In other
f
words they involve a quantity of standard deviation* type whereas leading order theory involves only
f
averages1.
More important, the Debye
length scale, defined within the basic Vlasov-Poisson system of equations but not in the equations of the leading order theory, reappears in the first order theory.
At this stage, therefore, a possible scale of shock
wave structure emerges and this result could be taken in conjunction with the discussion of Section 3.1 to provide a possible description of the formation and structure of such phenomena.
Of course, in a shock the
assumption of slowly-varying solutions breaks down but qualitative features might appear from the theory as in the use of the Navier-Stokes equations for such purposes in ordinary gas dynamics.
The fleading order
theory1 in the latter case yields the Euler equations which contain no length scale for discussing shock wave structures.
On this basis it might
be worth investigating a weak shock transition from one state to a neighbouring one using the equations of first order theory.
Although the
equations are complicated progress might be possible if the shock strength were sufficiently weak. 3.3 SPECIAL CASES
(a) COLD PLASMA STREAM
Here we consider the perturbations of the solution given by (2.17), (2.18) and (2.19) due to a slowly-varying background plasma (see also Gribben and Parkes 1977).
The governing equations for the cold electron
motion are
|f • £ (nv) = 0,
(3.5)
234
R . J . GRIBBEN
»•«> of which (2.17), (2.18) and (2.19) represent the solutions for uniform periodic waves satisfying the boundary conditions given in Section 2.3. When the theory described in Section 3.1 is applied to these equations there results the following set of equations for the faveraged1 quantities,
f + -k t N(v+ Vi - °>
(3 8)
-
0,
(3.9)
NONLINEAR WAVES IN A NON-UNIFORM PLASMA
way.
235
The result may be written as,
I. {*fcr^J* * K * ^TT^J r2Tr
r
p
L
o
NV
r
-i
r
NV V
v
n
e
o m e
/
j
whence, on using the uniform solution (2.19), (3.8) is obtained.
The
same process is used in deriving (3.9) but now we take the first moment of (3.2) with respect to the average particle velocity relative to the wave i.e. we multiply (3.2) by (1/2TT) result from a
=3
Jo
to °°.
(a e +
)
dx and integrate the
p
Equation (3.10) is most easily recovered from the general theory by observing that the first moment of the Vlasov equation (2.1) for the electrons can be combined with the Poisson equation to form
f f vf J —o
e
dv + i- ( f v^f dv - !2_ (|±\1 = £ |i r f. dv 2 I3j dx S I J
e
2m I3xj f
00
g
\
/
m- 8x I 1
l
J —00
Application of the slowly-varying theory and substitution of the 6-distribution for f
then yields (3.10) if, as we have assumed throughout,
the perturbation motion of the ions is neglected. As discussed further in Section 4 one of the key questions to decide in the study of the general evolutionary equations is the type of system under consideration i.e. whether elliptic, hyperbolic etc..
In the case
of the cold plasma stream it is readily seen that the equations form a 1
degenerate1 hyperbolic system having a single real characteristic
velocity, V + V , with multiplicity four.
This implies that there is no
splitting of the characteristics in going from the linear to the nonlinear regimes and moreover the uniform waves are stable to disturbances which satisfy the assumptions of the theory (i.e. are slowly-varying) (see Whitham 1967). For this cold plasma case an exact solution of (3.8) - (3.11) for a nonlinear initial value problem can be found. values of V, V
It is assumed that initial
and N are all constant and A « X(L-X) ; 0 < X < L, T = 0,
and A = 0 elsewhere.
The solution proceeds by transforming X to the
236
R.J. GRIBBEN
characteristic coordinate and distorting the time variable in a suitable way.
The results show, for a particular choice of parameters, that the
initial parabolic profile for A, essentially the square of the amplitude, grows and narrows before the solution breaks down after a finite time T . However, before T
is reached the model itself fails because particles
begin to be trapped by the (growing) waves.
Furthermore one can check
that the assumption of slowly-varying waves remains valid during the motion if valid initially.
For more general initial conditions a solution
of the equations (3.8) - (3.11) can be found by the method of strained coordinates.
Reasons why the method is successful in this case along with
a general description are given in Parkes and Gribben (1978). (b) WARM PLASMA STREAM The evolutionary equations can be obtained for this case, as in (a) above, from the governing equations for warm electrons which are (3.5) and
3t
3 _
/
x
(nv) +
2
9x
cn3
9 _
(3.12)
2m
£
° / 9( f>\ 2 l
where the pressure p = cn 3 (c constant).
Ne
H
n
(3.13)
The appropriate uniform solutions
of these equations have been given by (2.20) and (2.21) in the form of series in amplitude.
Substitution of the solutions into the 'averaged1
equations yields the set (3.8) and
AV
AW
8T
AN 5 V 2 (V+V -3V ) o o o
JL
(1-6)
9T
0,
(3.14)
(3.15)
and the wave conservation equation (3.1) in which k is given by the dispersion relation for warm plasmas, (kV ) o
2
= co 2 p
) + 0(A).
Note these equations are valid only for A << 1 and again the perturbation
NONLINEAR WAVES IN A NON-UNIFORM PLASMA
237
motion of the ions is neglected. As a matter of fact these equations (and therefore also the equations given above for the cold electrons corresponding to 3 = 0) can be derived from the method described in general by Whitham (1974) using an averaged Lagrangian (Parkes 1980).
Thus equations (3.5), (3.12) and the Poisson
equation (equivalent to (3.13)) are the Euler equations obtained by variations in (f) , <j> and <J>, respectively, from the Lagrangian,
T "1
TO
where - — = n, - — = v and only derivatives of <j>- and cf)« appear.
In
similar problems such potentials 1 are associated with pseudo-frequencies Y-., Y 2
and
pseudo-wave numbers 3., 3 2 by Whitham (1974) and they satisfy
compatibility conditions like a) and k.
The Whitham procedure can be used
on L to define an averaged Lagrangian L(k,03,A,31 ,3 2> Y 2 ) and the Euler equation corresponding to the variational principle L dX dT = 0, together with the compatibility conditions can be shown to yield (3.8), (3.14) and (3.15).
An advantage is that the Euler equation
3T \du)
9X [dkl
'
yields directly the conservation of wave action equation, (3.15), which is only obtained by the former method after some tedious manipulation. The characteristic velocities of equations (3.8), (3.14), (3.15).and (3.1) can be written as C = V + V - 3V ± aV Y_ + 0(a 2 ) ± o o o 1 > £v = V + V ± 35V + AV Y +...., ± o o o ± ' where Y 2 = (3+63+68 +3 3 )/12(l-3), Y ± = [8B*±(1+3B)]/[8B*<1"B)]now a splitting of a linear characteristic into C
There is
and C_ but all
characteristics are real implying stability of the type referred to in part (a) above (see also Infeld and Rowlands 1979).
Since the double
238
R.J. GRIBBEN
limit is taken in the order lim lim it is not possible to recover the cold $-K) a-HD case by setting 3 = 0 in these expansions. An initial value problem of a similar type to that described in section 3.3(a) was solved in Gribben and Parkes (1981) but an approximate method using coordinate straining along two of the characteristics was required.
The method is of interest but the results less so chiefly
because of the severe restrictions on the amplitudes that could be considered in the model (cf (2.22)).
In this case, to leading order in A,
the initially symmetric quartic profile distorts asymmetrically at the earlier times, with greater distortion at the trailing edge, but the final distortion is symmetric.
In the cold plasma case the amplitude profile
changes symmetrically for all times until breakdown. 3.4 FORMULATION FOR SCHAMEL-TYPE DISTRIBUTIONS The Schamel solutions of BGK type described in Section 2.2 would seem to provide a good case for application of the present theory to a hot plasma.
Unfortunately the complexity increases markedly for such cases
and it is inevitable that numerical computation will be required at some stage of the work.
However, we are in a position to see how the analysis
goes through in principle and to check on the consistency of the theory with the number of parameters that would need to be determined from the Schamel solutions. Thus to fix ideas we consider an initial value problem in T for some interval of X and we regard the solution (2.15) for as being locally valid in x«
Initially, the electron distributions will be the Maxwellians
(2.10) for the free particles and (2.11) for the trapped ones, with similar expressions for the ions, and the parameters of these distributions N , ., T ei
., V
6j I
. and y e,i
. would be regarded as given functions of e,i
X. The remaining parameters in the solution, whose initial values would be prescribed, are wave amplitude ij;, the wave number k and the phase speed V (together with the frequency a) = kV).
Not all the parameters, however,
are independent since the dispersion relation, the condition v (^) = 0 and the current condition (2.12) must be satisfied.
For T > 0 the
distributions evolve according to the six non-secular conditions (3.2) and simultaneously equations (3.1), (3.3)
(with e = O),(2.15) and (2.16) serve
in principle to obtain self-consistent functions $, k, V and ip. The situation is made somewhat clearer if we make some simplifying
NONLINEAR WAVES IN A NON-UNIFORM PLASMA
assumptions.
239
Thus consider linear waves (excluding trapped particles) and,
in addition, suppose that we seek solutions which change much more slowly with respect to X than with respect to T.
This amounts to obtaining a series
solution in T valid for sufficiently small times and the structure of equations (3.1), (3.2) and (3.3) is such that deviations from the initial values are obtained as quadratures.
The Schamel solution for this linear
case yields $ = iKX,T)(l+cos X )/2,
(3.16)
where iKX,T) - ip(X) + t| ar.d similar expansions are assumed for the other variables.
The solution
(3.16) ensures that the Poisson equation and dispersion relation are automatically satisfied for all X and T.
It follows that the condition
(3.2) yields solutions of the form F . • A i + B V. si si si where A , B
and C
+ C T , s
are all expressible in terms of the initial conditions
and involve terms which can be expressed as complete elliptic integrals. Substitution of these distribution functions into the current condition (2.9) leads to a linear algebraic equation for V- and ip-. A second such equation can be obtained by substituting into (3.3) and the solution of these equations for if/, and V- can then be written down.
The algebra is
heavy and the detailed solutions for particular choices of initial conditions have not yet been worked out;
but such solutions which describe the early
stages of the evolution of the wave ought to provide useful insight into the more general problems. It has been mentioned that the determination of the type of the system of the modulation equations is important especially with regard to the stability of the solutions.
Now a method which lends itself to identifying
the characteristics of the system derived in Section 3.1 uses the property of characteristics in the (X,T)-plane as curves which can support discontinuities in the derivatives (of F , $, k, etc.).
Since the system is
Galilean invariant we may consider steady solutions in which a characteristic speed is zero and without loss of generality assume the discontinuities in X-derivatives occur at X = 0.
Thus, for example, we can define
240
R.J. GRIBBEN
(jump in 3Fs+/8X at X = 0) = iV/ayl L s d A Jyo
=
8s+»
and similarly
Hence, from (3.1), kM + VL = 0. Again to keep the discussion as simple as possible we consider the linear case and exclude trapped particles.
Then
h( X ) = H(l+cos x )/2 where H = and from (3.2) expressions from the g (a ) can be obtained in terms of M and H.
If these are substituted into equations (2.9) and (3.2) two
simultaneous, homogeneous, linear equations for M and K are obtained after some tedious algebra.
The condition for non-trivial solutions should then
provide criteria for the existence of real characteristics.
Again the
calculations have not yet been carried out for particular cases but the results should be interesting when available. 4
DISCUSSION AND OPEN QUESTIONS The understanding of the factors governing the manner in which a
monochromatic wave train propagates in a non-uniform background plasma is an important fundamental problem in plasma physics
In this work a theory
is described which offers one line of approach to this problem.
The
assumptions do not include any restriction on the amplitude of the wave and therefore of particular interest is a knowledge of the detailed processes involved in the trapping and detrapping of particles by the wave. Although the formal classification of mixing and non-mixing cases was described in Section 3.1 at present no application of the theory to distributions where trapping of particles is included has been carried through.
It would be of great interest to apply the theory of Section 3.1
in such cases and in the case of the BGK solutions due to Schamel some progress has been made as indicated in Section 3.3.
But other BGK solutions
might also merit investigation (e.g. Neil and Flynn 1973).
Even the
behaviour of waves of small amplitude in non-uniform plasmas could provide new insights.
Indeed the Schamel solutions take the form of expansions in
amplitude and a corresponding approach to the equations of Section 3.1 is
NONLINEAR WAVES IN A NON-UNIFORM PLASMA
then desirable for consistency.
241
Numerical solution of the resulting partial
differential equations is still likely to be necessary and for that purpose the low resolution of the system should prove to be an advantage, i.e. the detail of the Vlasov description is faveraged outT in the theory and does not directly enter the equations governing the gross properties of the motion. As far as further
f
open questions1 are concerned some have already been
alluded to in Section 3.
The identification of the type of equation
system formulated there and the dependence of type on the distribution F s are of importance because of the association with the stability of BGK waves about which there has been much discussion in the literature (Morse and Nielson 1969, 1971).
Important results have been obtained on stability
of waves in other fields by this method e.g. Stokes waves on water (Whitham 1967).
One possible line of approach was outlined in Section 3.4.
At the present time the foregoing represent what the author considers to be the 'next steps1 in the development of the theory.
The position is
that it has been applied successfully in a number of simple cases but there is an urgent need to apply it to more general problems with hot plasmas and trapped particles.
The difficulty is that the algebra becomes long and
complicated although this could be reduced somewhat by taking simpler models (e.g. immobile ions). Beyond this there remains the possibility of an approach to the problem of a collisionless shock.
Schamel in his work has confined himself to
continuous distributions at the boundaries of the trapped regions on the grounds of 'physical acceptability1 but if discontinuous solutions were allowed (Ozawa et al 1964, Mima and Nishikawa 1972) the idea of a mixing mechanism is introduced (Section 3.1) and hence the formation of a shock. Such a possibility would encourage the search for discontinuous solutions in the leading order theory.
The corresponding structure calculation for
very weak transitions might then be attempted using the first order theory briefly discussed in Section 3.2, although the limitations mentioned there would apply. Finally, the theory in its general form (Butler and Gribben 1968) takes account of relativistic effects and they could be included if necessary. Also the forir.ulation of the equations by using a suitable averaged Lagrangian has not yet been carried out in the general case and, further, adapting the analysis to other coordinate systems might be a possible
242
R.J. GRIBBEN
extension of the approach. From the preceding remarks it will be recognised that the present study is essentially a long-term one.
The subject of nonlinear waves is
mathematically challenging and in the case here, where the governing system is an integro-differential one, even more so.
Nevertheless an
attempt has been made in this article to show that there is hope of making further progress, in which case the resulting information concerning the mathematical description of slowly-varying waves propagating in nonuniform plasmas should help towards a better understanding of the phenomenon. ACKNOWLEDGEMENT.
I am grateful to Dr. E.J. Parkes for reading through the
manuscript. REFERENCES ALLIS, W.P. (1969) Nonlinear, collisionless, plasma waves. In Honour of Philip M. Morse (Ed. H. Feshback and K. Ingard) MIT. ARMSTRONG, T. and MONTGOMERY, D. (1967) Asymptotic State of the TwoStream Instability, J. Plasma Phys. 1, 425-433. BERK, H.L. and ROBERTS, K.V. (1967) Nonlinear Study of Vlasov's Equation for a Special Class of Distribution Functions, Phys. Fluids 10, 15951597. BERNSTEIN, I.B., GREENE, J.M. and KRUSKAL, M.D. (1957) Plasma Oscillations, Phys. Rev. 108, 546-550.
Exact Nonlinear
BUTLER, D.S. and GRIBBEN, R.J. (1968) Relativistic Formulation for Nonlinear Waves in a Non-uniform Plasma, J. Plasma Phys. 2_, 257-281. COFFEY, T.P. (1971) Breaking of Large Amplitude Plasma Oscillations, Phys. Fluids 14, 1402-1406. DAVIDSON, R.C. (1972) New York-London.
Methods in Nonlinear Plasma Theory, Academic Press,
FRANKLIN, R.N., HAMBERGER, S.M., IKEZI, H., LAMPIS, G. and SMITH, G.J. (1972) Nature of the Instability caused by Electrons Trapped by an Electron Plasma Wave, Phys. Rev. Lett. 28, 1114-1117. GRIBBEN, R.J. (1972) Phys. I, 49-65.
Nonlinear Waves in a Non-uniform Plasma, J. Plasma
GRIBBEN, R.J. and PARKES, E.J. (1977) Slowly Varying Nonlinear Waves in a Cold Plasma Stream, J. Plasma Phys. 18, 495-508.
NONLINEAR WAVES IN A NON-UNIFORM PLASMA
243
GRIBBEN, R.J. and PARKES, E.J. (1981) Slowly Varying Nonlinear Waves in a Warm Plasma Stream, J. Phys. A : Math. Gen. 14, 2113-2119. IKEZA, H., KIWAMOTO, Y., NISHIKAWA, K. and MIMA, K. (1972) Trapped-Ion Instabilities in Ion-Acoustic Wave, Phys. Fluids 15, 1605-1612. INFELD, E. and ROWLANDS, G. (1979) On the Stability of Electron Plasma Waves, J. Phys. A : Math. Gen. 1£, 2255-2262. MIMA, K. and NISHIKAWA, K. (1972) On the Theory of Trapped Particle Instability II Stability of a Small Amplitude B-G-K Wave, J. Phys. Soc. Japan 33, 1669-1677. MORSE, R.L. and NIELSON, C.W. (1969) Numerical Simulation of Warm Two-Beam Plasmas, Phys. Fluids 12, 2518-2525. MORSE, R.L. and NIELSON, C.W. (1971) Studies of Turbulent Heating of Hydrogen Plasma by Numerical Simulation, Phys. Rev. Lett. 26, 3-6. NEEL, W.W. and FLYNN, R.W. (1973) Some Properties of Large-Amplitude Electrostatic Waves (Allis Modes), J. Plasma Phys. 9_, 117-130. NEUNZERT, H. (1965) Uber ein Aufangswertproblem fur die stationare Boltzmann-Vlasov-Gleichung. Berichte der Kernforschungsanlage Jtilich Nr. 297. OZAWA, Y., KAJI, I. and KITO, M. (1964) Detailed Solutions of Periodic Waves for Non-Linear Stationary Plasma Waves with Small Amplitude, Plasma Phys. 6_, 227-236. PARKES, E.J. (1980) Propagation of Slowly Varying Nonlinear Waves in Plasmas II: Cold and Warm Plasma Streams. Mathematical Methods of Plasma physics (Ed. R. Kress and J. Wick) Lang, Frankfurt. PARKES, E.J. and GRIBBEN, R.J. (1978) A note on the Application of the Method of Strained Coordinates to a Problem of Wave Propagation in Plasmas, J. Phys. A ; Math. Gen. 11, 2341-2348. ROWLANDS, G. (1969) 2, 567-576.
Stability of Nonlinear Plasma Waves, J. Plasma Phys.
SCHAMEL, H. (1972a)
Nonlinear Electrostatic Plasma Waves, J. Plasma Phys.
SCHAMEL, H. (1972b) Stationary Solitary, Snoidal and Sinusoidal Ion Acoustic Waves, Plasma Phys. 14, 905-924. SCHAMEL, H. (1975) Analytic BGK Modes and their Modulational Instability, J. Plasma Phys. 13, 139-145. SCHAMEL, H. (1979)
Theory of Electron Holes, Physica Scripta, 20, 336-342.
SCHAMEL, H. (1983) Kinetic Theory of Phase Space Vortices and Double Layers, to be published in Physica Scripta.
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R.J. GRIBBEN
THOMPSON, J.R. (1971) Nonlinear Evolution of Collisionless Electron BeamPlasma Systems, Phys. Fluids 14, 1532-1541. WHARTON, C.B., MALMBERG, J.H. and O'NEIL, T.M. (1968) Nonlinear Effects of Large-Amplitude Plasma Waves, Phys. Fluids 11, 1761-1763. WHITHAM, G.B. (1965) A General Approach to Linear and Nonlinear Dispersive Waves Using a Lagrangian, J. Fluid Mech. 22, 273-283. WHITHAM, G.B. (1967) 27_, 399-412.
Nonlinear Dispersion of Water Waves, J. Fluid Mech.
WHITHAM, G.B. (1974) York.
Linear and Nonlinear Waves, John Wiley and Sons, New
CHAPTER 13 ON THE INVERSE SCATTERING TRANSFORM IN TWO SPATIAL AND ONE TEMPORAL DIMENSIONS A. S. FOKAS Institute for Nonlinear Studies Clarkson College of Technology Potsdam, New York 13676, U.S.A.
1.
INTRODUCTION. The goal of this paper is to present a recently developed formalism
for a) solving inverse problems in the plane for potentials decaying at infinity (i.e. given appropriate scattering data reconstruct the potential q(x,y); b) solving the initial value problem (for appropriately decaying initial data) of certain nonlinear evolution equations in two spatial and one temporal dimensions (i.e. given q(x,y,o) find q(x,y,t)).
Several re-
sults obtained by this formalism are also summarized. This formalism has been developed in a series of papers by Fokas and Ablowitz (1982a,b,c; 1983), Ablowitz et al (1982), Fokas (1982), where several inverse problems related to physically significant multidimensional equations have been formally solved.
The inverse problem associated with
a certain differential Riemann-Hilbert problem (in the complex x-plane), which is related to the Benjamin-Ono (BO) equation was considered in Fokas and Ablowitz (1982a).
The BO equation, although an equation in 1 + 1 (i.e.
in one space and one time dimension) has many features similar to problems in 2 + 1 (this results from its nonlocal character). acts as a pivot from 1 + 1 to 2 + 1.
In this sense, BO
The inverse problem associated with
the "time"-dependent Schrodinger equation (see Dryuma (1974)) il/j + \p
+ (q + X)\p = 0
(1.1)
as well as the initial value problem of the related Kadomtsev-Petviashvili (KP)I (1970) (q M
t
+ 6qq
+ q x
)
^xxx x
= 3q H
,
(1.2)
yy'
were considered in Fokas and Ablowitz (1982a,b).
The inverse problem
associated with and the related KPII (q + 6qq + q ) = -3q t x xxx x yy
(1.4)
246
A.S. FOKAS
were considered in Ablowitz et al (1982).
The inverse problem associated
with the matrix equation 4> = XBifr + q^ + Ji|; , x
(1.5)
y
was considered by Fokas (1983) and Fokas and Ablowitz (1983).
In Equation
(1.5), B is a constant n x n diagonal matrix, J is a constant n x n diagonal matrix with elements either all real (hyperbolic case) or all purely imaginary (elliptic case), and q(x,y,t) is a n x n off-diagonal matrix containing the potentials (or field variables).
Equation (1.5) can be used to
solve several physical nonlinear equations in 2 + 1 (Ablowitz and Haberman, 19 7 5).
Among them are the n-wave interaction (Ablowitz and Segur, 1981),
variants of the so-called Davey-Stewartson (DS) equation (1974) (which is the long wave limit of the Benney-Roskes equation (1969)) and the modified KP (MKP) equation. In this paper we first outline the formalism used and then summarize the main results of Fokas and Ablowitz (1982a,b,c; 1983), Ablowitz et al (1982), Fokas (1983).
Only the final results are given and the various
intermediate steps as well as the relevant proofs are omitted.
We feel
that this form of presentation is useful in bringing out the essential ideas and relevant results. The results outlined in this paper were obtained in the process of carrying out a program of study on multidimensional problems, undertaken by Ablowitz and the author.
Later, Bar Yaacov joined us and made important
contributions, in particular with regard to KPII.
Our program of study of
equations in 2 + 1 has by no means been completed.
In particular in Fokas
and Ablowitz (1982b,c), Ablowitz et al (1982), Fokas and Ablowitz (1982, 1983), we make several assumptions about the compactness of certain operators, the existence or not of eigenvalues etc.
Some of these assumptions
can be directly justified by assuming appropriate smoothness of q(x,y) and that a certain norm of q(x,y) is sufficiently small.
Definitely, a rigor-
ous investigation of these and other questions is needed.
Such an inves-
tigation is in progress. 2.
THE GENERAL FRAMEWORK. A necessary condition for a given nonlinear equation to be solvable
by an 1ST formalism, is the existence of a nontrivial pair of equations, the so-called Lax pair (1968), such that the original equation can be viewed as the solvability condition of this Lax pair of equations.
Pro-
vided that this is the case, then most of the analysis of the 1ST is car-
INVERSE SCATTERING TRANSFORM
ried out on the time-independent part of the Lax pair.
247
In this sense, the
investigation of the 1ST for a given equation is essentially tantamount to the investigation of the inverse problem of the underlying time-independent part of its Lax pair. 2.1.
MAIN EIGENVALUE PROBLEMS RELATED TO 1ST IN 1 + 1. In order to put our work in 2 + 1 into perspective, we first review the
main eigenvalue problems (time-independent parts of Lax pairs) related to equations in 1 + 1. The 1ST was discovered by Gardner et al (1967, 1974), who were able to relate the celebrated KdV equation to the classical time-independent Schrodinger equation ty + (u(x,t) + k )ty = 0. XX
(2.1)
The inverse problem associated with (2.1) for potentials decaying as |x|-*°° has been thoroughly investigated by Deift and Trubowitz (1979). The next eigenvalue problem to receive much attention was the socalled AKNS
1 !)
0
(2.2)
This problem was introduced by Ablowitz et al (1973a,b; 1974) as a natural generalization of the eigenvalue problem used by Zakharov and Shabat (1972) in connection with the nonlinear Schrodinger equation.
Equation (2.2) is
the time-independent part of the Lax pair of several important equations; in particular of the nonlinear Schrodinger, of the Sine-Gordon, and of the modified KdV. A third order generalization of (2.1) has been considered by Kaup (1980) and Deift et al (1982). Boussinesq equation.
This eigenvalue problem is related to the
Also a 3 x 3 generalization of (2.2) has been consid-
ered by Kaup (1976) and Zakharov and Manakov (1976) in connection with the 3 wave interaction in 1 + 1. Gelffand and Dikii (1977) proposed an n
order generalization of (2.3)
D n * + qn__2Dn~2* + q n _ 3 D n ~ 3 ^ + ... + qQl^ = A11*; D = i | ^ ,
(2.3)
and investigated many of the "algebraic" properties of the equations solvable by (2.3).
Symes (1979) obtained the recursion operator generating
all equations solvable by (2.3).
Similarly, Ablowitz and Haberman (1975)
proposed an n x n generalization of (2.2)
248
A.S. FOKAS (2.4)
(where J is a constant diagonal matrix and q is an off-diagonal matrix) and established that (2.4) is related to several important equations.
Many
investigators (e.g. Newell (1979), Fokas and Anderson (1982)) obtained the recursion operator generating all equations solvable by (2.4).
However,
in spite of the above progress with regard to (2.3) and (2.4), the question of solving the inverse problems associated with (2.4) and (2.4) remained open for a rather long time, until the significant recent work of Beals (1982) and Beals and Coifman (1980-1981a,b; 1981-1982) respectively (see also Caudrey (1982)). The eigenvalue problems (2.1), (2.2) and their generalization (2.3), (2.4) are, in our opinion, the main differential problems which have been used in connection with 1ST in 1 + 1.
There exists several variants of the
above problems (e.g. Calogero and Degasperis (1976,1977), Wadati et al (1974,1975,1977), Jaulent (1976), Kaup and Newell (1978), Mikhailov (1976), Alonso (1980) which however should be solvable by some simple variation of the procedure used for solving (2.1)-(2.4). So far we have limited our exposition to differential equations.
How-
ever, there exists certain classes of singular integro-differential equations which are also solvable by the 1ST.
The most well known are the so-
called intermediate long wave equation (Kodama et al, 1981,1982) and the BO equation.
Here we only note that the linear eigenvalue problem associ-
ated with the BO is (Bock and Kruskal, 1979; Nakamura, 1979),
U|£ + A(iJ>+ - i|T) = -ui|;+.
(2.5)
Equation (2.5) defines the "Jump condition" for a differential RiemannHilbert (RH) problem; it relates the boundary values of the functions ip , i|; which are meromorphic in the upper, lower half x-complex planes respectively.
Proper generalizations of (2.5) (corresponding to (2.2)-(2.4)) are
still open (some progress is reported in Satsuma et al (1982)). 2.2.
REVIEW OF SOME ASPECTS OF EQUATIONS IN 2 + 1. Prior to our investigation of multidimensional 1ST, the following
facts were established in connection with equations in 2 + 1: i)
Many physically significant multidimensional nonlinear equations
(e.g. N wave interaction in 2 + 1, KP, DS, MKP) are related to Lax pairs. ii)
These multidimensional equations have rich "algebraic" properties.
For example KP possesses infinitely many symmetries and constants of motion (Oevel and Fuchssteiner, 1982; Chen et al, 1982; Chudnovsky, 1979; and
INVERSE SCATTERING TRANSFORM
249
Konopelchenko, 1981, 1982a,b) (for the analogous results for the BO, see Fokas and Fuchssteiner, 1981). iii)
There exist several direct methods for finding particular solu-
tions of the above equations.
Such methods are:
Backlund transformations (Chen et al, 1975).
(a) Appropriate use of
(b) The bilinear approach of
Hirota (Satsuma, 1976; Satsuma and Ablowitz, 1979; Nakamura, 1981a,b,c, 1982).
(c) The more general T-function approach of Kashiwara and Miwa
(1981), Date et al (1981a,b), related to Kac-Moody algebras.
(d) The
"dressing method" of Zakharov-Shabat (1974), which exploits the existence of general linear integral equations of the Gel'fand-Levitan-Marchenko (GLM) type to generate solutions of certain nonlinear equations in 2 + 1. (e) The extension of the "RH direct method" of Zakharov et al (1978,1979) proposed by Manakov (1981).
This method uses certain nonlocal RH boundary
value problems to directly obtain solutions of several nonlinear equations in 2 + 1.
(f) The "direct linearizing method" proposed by the author and
Ablowitz (1981) and (1982), and extended recently by Santini et al (1983). This method is closely related to the perturbation approach of Rosales (1978) and exploits the fact that there exists rather general linear integral equations, involving arbitrary measure-contour, which can be used to linearize certain nonlinear equations in 2 + 1. We note that the above methods, although useful for characterizing particular solutions, are not suitable for solving general initial value problems (since it is not possible given some initial data to find the corresponding measure-contour-scattering data).
Actually the question of
finding a suitable formalism for solving the initial value problems of multidimensional equations was essentially open.
In this regard, we men-
tion that some progress had been made in connection with KPI (Zakharov et al, 1979; Manakov, 1981; Segur, 1982) and the 3 wave interaction in 2 + 1 (Kaup et al, 1980; Cornille, 1979; and Niznik, 1972).
However, it was not
clear from this work how a unified scheme to handle these and other equations could be obtained.
In particular, in Manakov1s treatment of KPI the
usual 1ST had to be supplemented with solving an additional pair of GLMtype equations in scattering space in order to obtain appropriate scattering data; also the lump solutions (algebraically decaying solitons) were excluded (we stress that the manifestation of lumps is one of the novel aspects of 1ST in 2 + 1).
Similarly, Kaup's treatment of the 3 wave inter-
action exploits crucially the existence of characteristic coordinates.
250
A.S. FOKAS
Furthermore, what is perhaps more important, the 1ST had been considered so far, only within the framework of RH problems (local in 1 + 1 , nonlocal in KPI).
However, this framework seems in general inadequate for handling
other multidimensional problems (e.g. KPII).
The role of the RH problem is
now played by a ""3" (DBAR) problem (see Section 2.3). 2.3.
1ST IN 2 + 1. The main steps of the method used in Fokas and Ablowitz (1982abc,1983),
Ablowitz et al (1982), Fokas (1983), can be summarized as follows: (i)
Define an eigenfunction y(x,y,k) which is bounded for all complex
values of the "spectral parameter" k and which is appropriately normalized (]i-*I as k->°°).
This eigenf unction is usually defined in terms of a Fred-
holm linear integral equation of the second type, and it may have different representations in different sectors of the complex k-plane.
To derive
this integral equation, regard the terms involving q(x,y) as a "forcing" and use a suitable Fourier transform or a Green's function formulation. The above integral equation may have homogeneous solutions.
These homogen-
eous modes (corresponding to discrete eigenfunctions) are rather important because they give rise to lumps (i.e. 2 + 1 decaying solitons) . (ii)
Compute 3y/3k.
This is in general expressed in terms of some
other bounded eigenfunction, which we call N(x,y,£,k), and appropriate scattering data.
We note that is some problems (e.g. BO, KPI) y(x,y,k) is
a sectionally meromorphic function of k, i.e. it is holomorphic, modulo poles, in regions of the complex k-plane separated by certain contours and it has a jump across these contours.
In these cases 3y/3k, which measures
the "departure of y(x,y,k) from holomorphicity", will be zero everywhere except on the pole locations and on the above contours. (iii)
Employ a suitable "symmetry" relationship between N and y to ex-
press 3y/3k in terms of y and appropriate scattering data.
If y has homo-
geneous solutions then one needs also to establish a relationship between y and these homogeneous modes. symmetry conditions:
We have so far encountered two types of
"discrete" (KPII, elliptic systems) and "differential"
(BO, KPI). The relationship between 3y/3k, y, scattering data, which we call the scattering equation, is the central equation associated with the inverse problem of a given equation.
This scattering equation defines, in
general, a "3" problem, i.e. given 3y/3k find y.
In the case that y is
sectionally meromorphic this "3" problem degenerates to a RH problem, (iv) 1966)
Use the following extension of Cauchy's formula (Hormander,
INVERSE SCATTERING TRANSFORM
x-
251
k
(where R and C are appropriate region-contour respectively) to solve the ""5" problem.
Its solution is found, in general, in terms of a linear in-
tegral equation for U(x,y,k).
Equation (2,6) is uniquely defined in terms
of the above mentioned scattering data.
If \x has a homogeneous solution
at k. then 3y/9k has a 6 function at k = k.; thus pure soliton solutions are always found in closed form as the solution of a linear system of algebraic equations. (v)
Calculate the potential q(x,y) directly from the solution of the
inverse problem (typically given by integrals over U(x,y,k) and the scattering data).
The above discussion summarizes the steps needed for the
solution of an inverse problem. (vi)
In order to solve the initial value problem of some related non-
linear evolution equation, one needs only to find the evolution of the scattering data.
This can be achieved in a simple manner by employing the
time-dependent part of the Lax pair of the given equation.
Furthermore,
the initial scattering data can always be expressed in terms of the initial data q(x,y,o).
Thus equation (2.6) (and hence the formula for q(x,y,t))is
uniquely defined in terms of the initial data. We conclude this section with the following two remarks: (i)
Our motivation for using the "9" problem comes from the work of
Beals and Coifman (1980-1981ab, 1981-1982) on the 1ST of nonlinear equations in 1 + 1 related to first order systems on the line.
In their treat-
ment, the inverse problem is formulated in terms of a RH problem with respect to certain rays in the complex plane.
However, they indicate (in
particular, see Beals and Coifman (1981-1982)) that the RH problem should be viewed as a special case of a "3" problem. (ii)
RH problems have also been useful in connection with" two other
recent significant discoveries:
(a) the integration of the Ernst equation
(the static, axisymmetric reduction of the vacuum Einstein's equations), in particular with regard to the constructive proof of the Geroch conjecture (Hauser et al,
1980, 1981); (b) the integration of the Self-
dual Yang-Mills (SDYM) equations in 4-dimensional Euclidean space, in particular with respect to the Atiyah-Ward construction (1977).
The SDYM
equations although defining a 4-dimensional model, have many properties
252
A.S. FOKAS
similar to those of two-dimensional problems.
Motivated from the above
discussion we expect that the "9" problem will also be useful for the exact integration of various models in both the fields of relativity as well as that of particle physics.
Furthermore, we also expect these ideas to be
useful for multidimensional difference and differential-difference equations. 3.
THE INVERSE PROBLEM OF ±y
+ y
+ 2ikp
+ uii
=0
(FOKAS AND ABLOWITZ,
1982bc). In this section, we consider the inverse problem associated with ^y
+
^xx
+
2iku
x
=
~uu>
(3
-1}
where we assume that u(x,y) •> 0 for large x,y (Equation (3.1) follows from (1.1) by letting X = 0 and \\) = yexp[i(kx - k 2 y)]). We define an eigenfunction y(x,y,k) which solves (3.1), is bounded + ik , and tends to 1 as k -> °°.
for all complex values of k = k genfunction is definced through y(x,y,k) =
where y
;
y
[
y (x,y,k),
k
y (x,y,k),
kI < o J
This ei-
> o 1 (3.2)
satisfy the following linear Fredholm integral equations y±(x,y,k) = 1 + [g^ u y ± (.,.,k)](x,y), ,oo
.oo
,oo
-y
.0
(3.3)
r°°
[g
f](x,y) = ~ - (- dn dm d£ + dn dm dQ • R U ZTT ' Jyy JQ J-oo -Loo J.oo -Loo exp[im(x - O .00
- im(m + 2k) (y - n) ]u(?,n)f (C,n) , .00
-0
.y
dn dm d^ + u f](x,y) = jz. ('y * y —°° ""°° exp[im(x - Q
.OO
(3.4) +
.0
dn dm ~°° o —
- im(m + 2k)(y - n)]u(5,n)f(?,n).
The function y(x,y,k) is a sectionally meromorphic function of k.
(3.4)" This
follows from the fact that the kernels of (3.3) , (3.3)" are (+), (-) functions respectively (i.e. they are holomorphic functions of k in the upper, lower half k-planes).
Hence y ,y
will also be (+), (-) functions in k,
modulo poles, i.e. y"(x,y,k) = 1 + Z 1
3 J
+
k - kT
+ y-(x,y,k),
(3.5)'
INVERSE SCATTERING TRANSFORM
where y , y
253
are (+) and (-) functions respectively with respect to k (c.
are introduced only for the purpose of normalization). Thus N TTi Z 1
- ; >
N TTi Z 1
9k
-v
*T(x,y)6(k y
+
—y
To compute 3y/9k one needs to calculate y y+(x,y,k) - y~(x,y,k) = |
k
I
k
I
k
I
> 0 < 0 = 0
- y (k real):
T(k,£)N(x,y,£,k)d£,
J —oo
d£ (£,r|,k)exp[0(5,n,k,£)], (3.6) d£| dr]u(^,ri)y dnu(?,n)y+(?,n,
T(k,£) = ^:sgn(k-£)[ — sgn(k-£) ,£) =
J —oo
J —oo
where 6(x,y,£,k) = i(£-k)x - i(£ 2 - k 2 )y, k,£ real.
N(x,y,£,k) solves
(3.1) in k and is defined by N(x,y,£,k) = exp[9(x,y,£,k)] + [g"
N(.,.,£,k)](x,y), k,£ real.
(3.7)
K., U
To formulate a "3" problem (or actually in this case a RH problem) one needs to relate N and (j)7 to y: ^
(N(x,y,£,k)exp[i(kx - k 2 y)]) = -F(£,k)y"(x,y,k)exp[i(kx-k2y) ],
(3.8)
where
i r r F(i,p) = 2 ^
dC J — oo
driu(C,n)N(C,n,«.,p); k,X,,p real,
(3.9)
J —oo
and c
± i
+
+
+
(x - 2kTy + yT)*"
k - kT
(3.10)
± ± ± ± ± where y. are constants and c. = i if 4>. are normalized by (x - 2k.y)(().-> 1 as (x2 + y 2 )1 1//22 -> oo# Equations (3.6), (3.8), (3.9) yield the following scattering equation
y+(x,y,k)=y"(x,y,k)+ [ d£f (k,£)exp[6(x,y,£,k) ]y"(x,y,£) , k real
(3.11)
J —oo
where the scattering data f(k,£) is given in closed form through -oo
f (k,£) = ^- sgn(k - £) J _oo
-co
d£ J —oo
dnu(?,n)N(£,n,k,£).
(3.12)
254
A.S. FOKAS Using (3.11) and (3.10), the solution of the inverse problem of (3.1)
is obtained through the following
+
+
N
+
1
/ *I+ + h,1\ +
(x - 2k"y + YT)*: - i
2771
k real (3.13)
f f J L
f(v,ft)exp[8(x,y,ftav) ]y~q)d&dv kT+ V
_
,
- kT
where y (k) = y (x,y,k) k is real, and S means summation from £ = 1 to N unless any of the denominators vanishes. mine y (k) , {<)>.,<|>.}
Equations (3.13), (3.14)~ deter-
in terms of the scattering data {k
rYj}j-i5
f(k>jl)
'
k>£ real
(3#15)
-
Once y ~ ( k ) , (j)T are found the potential u(x,y) is obtained from u ( x , y ) « y - [2Z(^(x,y)+(()"(x,y))+i[diir dkf (k,£)exp[0(x,y,£,k)y"(x,y,£). 1
J TO
"
-°°
(3.16)
The KPI (for its physical applications see Ablowitz and Segur (1981)) is the compatibility condition of (1.1) and (3
t
+ 4 3 J + 6u3 + 3(u - i u d x f ) + 4ikJ)i|; - 0. x x x J^ y
(3.17)
To solve the initial value problem of KPI one can directly employ (3.13), (3.14)~, (3.16) provided one finds tthe evolution of the scattering data (3.15).
This evolution is given by
f(k,£,t) = f(k,Jl,0)exp[4i(£3 - k 3 )t],
k,£ real.
(3.18)
Pure lump solutions correspond to f(k,£,0) = 0 and are then characterized by u(x,y,t) = 2 ^
E ((()|(x,y,t) + f. (x,y,t)),
(3.19)
where (j)T(x,y,t) satisfy the following system of linear algebraic equations J / A+ A\ Z (x - 2kTy + 12(kT) t + YT(O))<J>: - i £ I + * + * _ I = 1. (3.20) * -1 \ k j
^
k
j
k
H/
INVERSE SCATTERING TRANSFORM 4.
THE INVERSE PROBLEM OF -y
+ y y
+ 2iky xx
255
+ uy - 0 AND KPII (ABLOWTTZ ET x
AL, 1982). In this section we consider the inverse problem associated with -y v + y w + 2iky - -uy, (4.1) y
xx
x
and we assume that u(x,y) •> 0 for large x,y.
As before, we introduce an
eigenfunction y(x,y,k) which solves (4.1), is bounded for all complex values of k
a
k
+ ik T , and which tends to 1 as k •* °°.
This eigenf unction has
two different representations according to whether k R is > 0 or < 0: y(x,y,k) = 1 + [gk j k >u y(.,.,k)](x,y), (4.2) where
k
'KI»
u
f](x y)
'
/
T^F I TJ \
exp[im(x - Q and for k
dn y
d
y
( 4J
J
-°°
dm+fJ dnf dC{fdm+[ J J J
-2k
-°°
0
0
- m(m + 2k) (y - n)]u(£,n)f (£,n) ,
v
dm} I /
kR > 0
(4.3)
< 0 the integrals with respect to m are replaced by K.
r-2k
J0
K
rO dm,
{
dm +
J-~
Equation (4.2) depends explicitly on k where with respect to k. g,
-°°
R
dm}
J
and hence y(x,y,k) is analytic no-
We emphasize this by writing g,
,
instead of
, but for convenience of notation we still write y(x,y,k) instead of
K,U
y(x,y,k R ,k I ). Computing 3y/3k, we find >7>k) = 9k
F (k
R
k )N(x >y> k ,k ) , X
R
X
(4.4)
where the scattering data F is defined by FO^.kj.) i - -^ sgn(kR)j
d?j
dne3cp[i(2kR^-4kIkRn)u(C,n)y(5,n,k),
(4.5)
and the eigenfunction N solves (4.1) and is defined by N(x,y,kR,kI) = exp[-i(2k R x-4k I k R y)] + [gk
>k
R
I
u N(.,.
,kR,k;[) ] (x,y).
(4.6)
Using the discrete symmetry relationship N(x,y,kR,k].) = y(x,y,-£)e
i(-2k x + 4k k )y K *
we obtain the following scattering equation
(4.7)
256
A.S. FOKAS F ( kk
9k
R R
I
i(-2k x + 4k k )y R IR m
)y(x>y>.k)e
Assuming that (4.2) has no homogeneous solutions, we obtain the solution of the inverse problem of (4.1) through the following equations:
^j j
r r F(x
1 1
i(-2x
x )y(x,y,-T)e
p(x,y.k) = 1 + ^ j j — ± - * oo
u(x
' y ) = f f ^ j I F(VTI)uCx'y'"T)e
X+4T K
T
y)
K 1
dx A dx
^
, (4.9)
i(-2x x + 4x x y)
*
T)e
Ri
d x A d x . (4.10)
In equations (4.9), (4.10) R^ denotes the entire complex T-plane and diA
dT = -2idT dT . R 1
We note that equations (4.9), (4.10) are uniquely de-
fined in terms of the scattering data F(k ,k ) (specified by (4.5)). The KPII is the compatibility equation (1.3) and of (3.17) where the -i in front of / X TO is replaced by 1.
This implies the following evolution
for the scattering data: -4i(k3 + Kk3)Z )t F ( k R , k r t ) = F(kR,k].,0)e * 1 U C . 5.
(4.11)
ON THE INVERSE PROBLEM OF HYPERBOLIC SYSTEMS, N-WAVE, DSI (FOKAS, 1982; FOKAS AND ABLOWITZ, 1983). In this section, we consider the inverse problem associated with the
hyperbolic system y
= ikJy + qy + Jy ; Jf = Jf - fJ, x
(5.1)
y
wher3 y(x,y,k) is an n x n matrix, J is a real diagonal matrix with elements J- > J« > . • • > J , and q(x,y) is an n x n off-diagonal matrix containing the potentials q..(x,y).
We assume that q..(x,y) -*• 0 for large x,y.
tion (5.1) is obtained from (1.5) by letting B = 0 and ty =
Equa-
exp[ik(Jx+ y)].
Let IT y, IT y, IT y denote the diagonal, strictly upper diagonal and strictly lower diagonal parts of the matrix y. ed for all complex values of k = k given by
+ ik
A solution of (5.1) bound-
and tending to I as k -*• °° is
^_ y (x,y,k),
k
> 0
/x"(x,y,k),
k T < 0,
(5.2) where y~(x,y,k) satisfy the following linear integral equations,
INVERSE SCATTERING TRANSFORM
dC E
"Wf +
In equations (5,3)
(E
x-C,J e i k < X " 5)J'ir_
the linear operator E
r
X—c,, J
Jf(.))(x-C,y)
257
(5.3)*
is defined by
= f dnf dme^^-^^-'^-^fCn) = f(y+(x-QJ), (5.4)
^'
J —00
J —00
where f(y + (x - Q J ) denotes the matrix obtained from f(x) by evaluating the £
row of the matrix f(x) at y + (x - £)J, 5)Jn-
Also, from the definition
of J it follows that exp[J]f = exp[J]fexp[-J]. Assuming that the linear integral equations (5.3)~ have no homogeneous solutions it follows that y , y and k <0 respectively.
are holomorphic functions of k, for k > 0
Hence the function y(x,y,k) defined by (5.2) is a
sectionally holomorphic function of k having a jump across k
= 0.
Thus
± 0 and 3y/3k = y + - y~ for k = k^.
3y/3k = 0 for all k with k I
is.
Rather than following in detail the method of Section 2, i.e. computing y
- y
in terms of some other bounded eigenfunction N and then es-
tablishing a "symmetry11 condition relating y and N, we find directly a scattering equation. Using (5.3)~ it can be directly shown that y (x,y,k) - y~(x,y,k) n i \ -ikJx - iky J £,k)e ,
. . k = k ,
/c _N (5.5)
£,k real,
(5.6)
where the scattering data f(£,k) can be found via oo
f(£,k) -
dmT,(£,m)f(m,k) = T,(£,k) - T (£,k); J —oo
and ,k) = -rZ
d^
" J-oo
J-c
i -i£J?-i£n ,r N +/r IN ik^J + ikn s dne iT+q(^,n)y (C,n,k)e s '.
/c (5.7)
Equation (5.5) defines a RH problem for the sectionally holomorphic function y(x,y,k) in terms of the scattering data f(£,k).
It is remarkable
that f(£,k) can be expressed in terms of T+(£>k) in closed form, i.e. equation (5.6) is solvable explicitly.
This is because the kernel of (5.6) is
strictly upper triangular. EXAMPLE.
Suppose that n = 2, then (5.6) implies
f 2 2 U , k ) = 0,
f n ( £ , k ) = -T + 1 2 (£,k),
f12(Jl,k) = T_ 1 2 (£,k),
258
A.S. FOKAS
-[
dnff
J —00
2 (Jt,m)T_ 21 (m f k).
Splitting equation (5.5) into its (+) and (-) part with respect to k, it follows that the RH problem defined by (5.5) is equivalent to the following linear integral equation v 1 r \ +
1
,y>k) + 2 i i
roo f00 I AQ\ .111
d
* —00
*
(&-V)y
U
dV
v
" I-
- k + iO
(5#8)
J —OO
Equation (5.8) defines U~(x,y,k) in terms of the scattering data ,v).
Once y (x,y,k) is found the potential q(x,y) is easily obtained:
q(x,y) = - j- j f dlf dv y-(x,y,il)e UJx fa,v)e- iVJx+i ^- V >y. J —00
(5.9)
J —00
The n-wave interaction equations are (Ablowitz and Segur, 1981)
where a.., 3.. are real constants related to the x and y components of the underlying group velocities.
Equations (5.10) are the compatibility con-
ditions of the following Lax pair * x - qip + Ji|;y, * t - A j * + A 2 * y ,
(5.11)
where A
l l±:j
"
a
^ ij
q
ii
( i
+
ij
j )
'
A
1 li±
" °>
A
- diag(c , . . . , c ) 2
(5.12)
In
and J o , c 0 , A = l , . . . , n a r e g i v e n i n terms o f a , , , $ . . v i a
The above Lax pair yields the following time evolution for the scattering data: iJltA -iktA l Z fU,k,t) - e f(£,k,O)e .
(5.14)
Hence the initial value problem of (5.10) can be solved via (5.8), (5.9). In these equations J is defined by (5.13), the initial scattering data by (5.6), (5.7)"", and the evolution of the scattering data is given by (5.14). The DSI equations are
iQt +\
(Qxx+Qyy)--a|Q|2Q+())Q,
o - ±1, * x x " * yy
INVERSE SCATTERING TRANSFORM
259
Equations (5,15) are the compatibility conditions of the Lax pair
*x = q* + Jijy *t = Afl + A2l^y + A 3 * y y
(5.16)
with J± - 1,
J 2 = -1, A 2 = iq, A 3 = diag(i,-i),
(5.17)
) • where the entries of A
A
are defined by:
+ A
= ^
x
t(QQ )
- (QQ ) ].
The function <(> is related to A
, A 2 2 via <|) » i^ii "
Hence, in this case assuming that A
(5.19)
y
y
,A
A
2o^
+ cr
IQ | .
tend to zero for large x,y,
the evolution of the scattering data is uniquely specified by (U) 2 tA. f(Jl,k,t) - e
J
-(ik)2tA
J
f(£,k,0)e
(5.20)
Thus the initial value problem of (5.15) can be solved via (5.8), (5.9) with J = diag(l,-l).
Furthermore, the scattering data is found from (5.6),
1
(5.7) , (5.20). 6.
ON THE INVERSE PROBLEM OF ELLIPTIC SYSTEMS, DSII (FOKAS, 1982; FOKAS, AND ABLOWITZ, 1983). We now consider the inverse problem associated with the elliptic sys-
tem u x where }i(x,y,k) is an n
- ikJy + qu - iJU , y
(6.1)
order matrix, J is a constant real diagonal matrix
with all its entries different from each other, and q(x,y) is an n off diagonal matrix containing the potentials q «(x,y). ^.f-itejy) "* 0 rapidly enough for large x and y.
order
We assume that
Equation (6.1) is obtained
from the well known equation * x - q* - iJ*y,
(6.2)
through the transformation ^ - yexp[ikJx - ky]. For the solution of the inverse problem associated with (6.1), we follow very closely the steps outlined in Section 2.
We first consider step
260
A.S. FOKAS
i), i.e. we introduce an eigenfunction M(x,y,k) which solves (6.1), is bounded for all complex values of k, and tends to I as k->°° . Such an eigenfunction satisfies the following linear integral equation M(x,y,k) = I + (G
vvq
The operator G, following:
,
a /4.,.,k))(x,y).
is a linear integral matrix operator defined by the
th
Let f(x,y) be some nxn matrix, then the ij
operator G,
,
(6.3)
entry of the
applying on f(x,y) i s
VV q
)}i_>
J i > 0
(6>4)
where C.. = (J - J.)/J. and for J. < 0 the integrals with respect to dm
f are replaced by
J
fiJI k
i
dm and J -"°
{f}.. denotes its i j t h entry.
respectively.
If f is a matrix
Sometimes it will be convenient to work with
the column vectors of the matrix y.
Letting y = (y_ ,... ,y_ ,... ,y^ ) it fol-
lows from (6.3) that jp satisfies yj(x,y) = I j + (£j* where JL denotes the j
>q
yj
unit vector, and
J_oo £-Ji)](x-Q
Jx
+ im(y-i
^c .k )} £ ;
dm
dr|
J £ > 0, (6.6)
(for J. < 0 the integrals with respect to dm must be altered just as in (6.4)); if f is a vector {f} 0 denotes its s} entry. Comparing (6.3) to (5.3), it follows that: contrast to (5.3), has no jump across K~J- - 0. plicitly on k T .
a) Equation (6.3), in Equation (6.3) depends ex-
We emphasize this dependence by writing G, ..q
VV
instead
of G. . However, for simplicity of notation, we still write y(x,y,k) inK,q stead of vKx^k^k-j.). From the above follows that the solution U (x,y,k) although bounded everywhere (i.e. for all complex values of k ) , it is analytic nowhere with respect to k, since 3u/3k ^ 0. The "departure from holomorphicity11 of p(x,y,k) is measured by 3y/3k.
INVERSE SCATTERING TRANSFORM
261
Hence we are lead to step ii) of Section 2, namely compute 3y/3k.
Differ-
entiating (6.3) with respect to k and assuming that (6.3) has no homogeneous solution, it follows that 3y(x
>r> k) = ^(x,y,kR,kT) + (G. R 3k * V V
q
M^Al 3k
)(x , y)>
(6 . 7)
where the matrixftis defined by
te}±i = 0; {n} l j -T l j Ck R > k I )e
±j
^ ^
R> T
, i*
j ;
9±j ( x . y . k ^ )
- i C y U ^ x + kj-y), and T.. i s given by
(6.8)
j
,k)}1Je
«
R
X
(6.9)
Equation (6,7) motivates the introduction of another bounded eigenfunction, which we call N(x,y,k ,k ) : K
1
N C x ^ k ^ k p = w C x ^ , ^ , ^ ) + k I ) ) ( x , y ) , R I where w i s the matrix ft with T = 1, i . e . ^ . ( x , y , kK ,kL ) {w}±i = 0; w±j = e 1 J , i^j.
(6.10)
(6.11)
To show that N(x,y,k ,k ) also solves (6.1) one needs only to show that the w above satisfies the "homogeneous" version of (6.1), i.e. W
this is straightforward.
x • (±k R " k I ) 3 w " " V
Equations (6.8), (6.11) imply n
Gwhere w
I
. .
T w
(612>
is a matrix with zeros everywhere, except at its ij { w i j } £ v = 0, ±*l and/or j^v,
entry,
{wij}±. = e ij .
(6.13)
Hence, Equation (6.7) implies
%3k where the matrix N N^
T^N ±j ,
I
(6.14)
j
is defined by
/ v v I, If 1 = w ^ f(x,y,k v v k R,k k I)+G^ }+ClJ V,x,y,K. ,iv y w ^x,y,K.p,K. ^T-IJ
,
iJJ q NN""" ((
R* I *
IM
\c ^ . 9 # ,K.
lr ^ H y v l J J x^x.yyj •
9K.
(f* 1 5^ v. D «- L -'y
262
A.S. FOKAS
Step iii) of the method of Section 2 consists of finding a relationship between y(x,y,k) and N J(x,y,k ,k ) . This crucial relationship is as K
i.
follows Nlj(x,y,k_,k ) = y(x,y,kR + i -=1 k )w ij (x,y,k_,k T ). K 1 K J. 1 K 1
(6.16)
To derive (6.16) w e only comment that one uses the following property of i (
j
e6ij(.,.,kR,kI)
K_
- eij(x,y,kR,kI)^
, &_, q
^
^
~~ =
(E^k
iLk
f(.,-)(x,y)
(6.17)
Equations (6.14), (6.16) yield y ( x , y , k 4-i^ik )^ij(x,y,k ,k ) , R J± I R I i,j=l
3k where the matrix Q
J
has a nonzero element only at its ij 0 9 -(x T..(kR,k_)e J
(6.18)
entry,
; k ^ i and/or £ ^ j . k ) (6.19) ; k — i, X< = J
k
Substituting (6.18) in (2.6), we find the following linear integral equation for
:
)fi I ly(x,y,x R +i-=ix J I
ij
±
U(x,y,k)-IiI f f-^J
(x,y,T ,x )dx A dx R I
r
27T1 J R J
= 1 (6.20)
x- k
(R00 is the entire x-complex plane, dxA dx = 2idX_dx_). L
is.
The function U is
uniquely determined in terms of the scattering data T..(k R ,kp which is defined by (6.9).
Hence given T.., Equation (6.20) yields y and then the
potential q can be reconstructed from
q(x,y) = ^ jj J ^ I y(x,y,x R +ijix l )fi i:3 (x,y,x R ,x I )dx A dx.
(6.21)
The DSII equations are O
"1
i OH
t
+ — V( 0
^xx
2
+
>y
-
0
V
) ° O\Q\ IWI
*yy " ^ ^ ^ ^ x x '
0 + d>0. 9
^
° ~
^
±1-
(6#22)
The above equations are the compatibility conditions of the Lax pair
INVERSE SCATTERING TRANSFORM
263
with J
±
- 1>
J 2 - "-1'
A
2
=
"q>
^ M / 1 "12)» >1
A
3
=
d^S^1*"1)*
« - UJ -
99 /
(6 23)
-
where the entries of A are defined by:
A
ll
+
"ll - f [±™ >x + <« V
x
A
22 " 1A22 V
(6-24)
y
=
I[~i(QQ" >x+ (QQ" KU
V
Hence (assuming that A--,A 22 tend to zero for large x,y) the initial value problem of (6.22) can be solved via (6.20), (6.21).
In these equa-
tions J = diag(l,-l); furthermore, the evolution of the scattering data (defined by (6.9)) is given by
k ±. O (' v v t If f" ^ — c» 06 \ A , jr , lv_ , lv_ , L y c
2
-k 2 A t
At O at
('v v 1r Tr O^ O ^ X , y , K._ , i\._ , U^
fci
^A 9*^»^ ^D.^-Jy
where k = k + i(J./J )k . In the above analysis, we have assumed that (6.3) has no homogeneous solutions.
If homogeneous modes exist, then lumps (i.e. solitons in 2 + 1 )
arise and the above analysis must be modified. to that found in BO and KPI.
The situation is similar
Details are given in Fokas and Ablowitz
(1983). REFERENCES ABLOWITZ, M.J., BAR YAACOV, D., and FOKAS, A.S. (1982) On the 1ST for KPII, INS//21 to appear Stud. Appl. Math. ABLOWITZ, M.J. and HABERMAN, R. (1975) Nonlinear Evolution Equations - Two and Three Dimensions, Phys. Rev. Lett. 35, 1185-1188. ABLOWITZ, M.J., KAUP, D.J., NEWELL, A.C. and SEGUR, H. (1973a) Method for Solving the Sine-Gordon Equation, Phys. Rev. Lett. 30, 1262-1264. ABLOWITZ, M.J., KAUP, D.J., NEWELL, A.C. and SEGUR, H. (1973b) Nonlinear Evolution Equations of Physical Significance, Phys. Rev. Lett. 31,125-127. ABLOWITZ, M.J., KAUP, D.J., NEWELL, A.C. and SEGUR, H. (1974) The Inverse Scattering Transform-Fourier Analysis for Nonlinear Problems, Stud. Appl. Math. 53, 249-315.
264
A.S. FOKAS
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CHAPTER 14 LINEAR EVOLUTION EQUATIONS ASSOCIATED WITH ISOSPECTRAL EVOLUTIONS OF DIFFERENTIAL OPERATORS ANTONIO DEGASPERIS Dipartimento di Fisica & I.N.F.N., Sez. di Roma, Universita di Roma "La Sapienza" 00185 Roma, Italy
1.
INTRODUCTION. The spectral transform of a (matrix-valued) function of a (space)
variable x is a powerful tool to solve, and investigate, several (classes of) wave equations that model both dispersion and nonlinear interaction. An introduction to this method can be found in Calogero and Degasperis (1982).
Here, in order to keep the following discussion within a simple
context, consider the prototype nonlinear evolution equation (u = — *i
t
u t = a o u x + a 1 ( u x x x - 6uu x ), u = u*, u = u(x,t),
,
dt
(1.1)
that can be solved by the spectral transform technique; this is the celebrated Korteweg-de Vries equation with two arbitrary parameters, a o and a.. Two necessary ingredients of the spectral transform method are the following:
the linear ordinary differential equation H(t)i|> = k2i[i, $ - iKx f k,t),
(1.2)
and the linear partial differential equation <|>t = MT|I,
$ - i|;(x,k,t),
(1.3)
where H(t) and M are differential operators, and k is the spectral variable. In the case of the KdV equation (1.1), these operators read H(t) = -D 2 + u(x,t), D = •—• M = aoD + a ^ D 3
(l.Aab)
- 6u(x,t)D - 3u x (x,t)].
(1.5)
Note that the variable t enters in the ODE (1.2) only parametrically, and that the validity of both the equations (1.2) and (1.3) for all values of k implies the operator evolution equation (Lax, 1968) H t (t) = EM,H(t)]; indeed this equation, with (1.4) KdV equation (1.1).
and
(*1.5),
(1.6) is equivalent to the
LINEAR EVOLUTION EQUATIONS In
269
general the nonlinear PDE (1.1) is the equation of applicative
and/or mathematical interest, while the linear equations (1.2) and (1.3) merely play an auxiliary role in the solution of (1.1).
Regarding these
two equations most of the attention has been devoted to the direct and inverse spectral problems associated with the ODE (1.2) with (1.4a), their solution being essential to the very application of the spectral transform method.
Here our attention is instead focussed on the other linear equa-
tion, i.e., the PDE (1.3) with (1.5).
More precisely, the aim of this
paper is to investigate a class of linear PDE's, with x and t dependent coefficients, that includes the evolution equation (1.3); however, we first confine our consideration to the linear PDE v t = (M - ib o - i b ^ v , v = v(x,t),
(1.7)
or, more explicitly, (see (1.4a) and (1.5)) v
= a v X-
+ a.C4v X
J-
- 6u(x,t)v
- 3u (x,t)v] - ib v + ib-Cv X
XXX
X
X
-u(x,t)v], XX
(1.8) v=v(x,t), and
then a larger class of linear PDE's is reported in the last section.
Here, and in the following, we consider evolution equations of dispersive type; this implies that the four constants a o , a-, b Q and b. be real. Despite the linearity of the equation (1.8), the dependence of its coefficients on the variables x and t through the solution u(x,t) of the KdV equation (1.1) seems to be sufficiently interesting to motivate the following investigation.
In section 3 we first show how to solve the
Cauchy problem, v(x,to)-> v(x,t), for an initial value v(x,t Q ) that vanishes sufficiently fast as x •*• ± °° ; this is done by using the well-known results, that are very tersely reported in the next section, on the direct and inverse spectral problems associated with the Schroedinger equation (1.2) with the condition (see (1.4a)) I (l+|x|)|u(x,t)|dx < oo.
(1.9)
—CO
Then we consider the evolution of a wave-packet solution v(x,t) of the equation (1.8); because of its interaction with the solution u(x,t) of the KdV equation, this wave-packet undergoes a scattering process.
In section
3, we define the corresponding scattering operator for any solution u(x,t) of the KdV equation (1.1) that satisfies (1.9).
In the special case with
a o = a 1 = b o = 0 and b. = 1 u(x,t) = u(x) is time-independent, and the evolution equation (1.8) coincides with the time-dependent Schroedinger
270
A. DEGASPERIS
equation of wave mechanics; of course, in this case the scattering operator associated with (1.8) is the usual quantum scattering matrix corresponding to a (static) potential u(x), and its expression is tersely derived.
In
the more interesting case with a. 4 0 we consider the scattering process only for those functions u(x,t) that are pure N-soliton solutions of the KdV equation; in this case the expression of the scattering operator is explicitly given. Finally, section 4 is devoted to various generalizations, and extensions of the results of section 3, and a list of other linear PDE's, that can be investigated by similar means, is included. 2.
THE SCHROEDINGER OPERATOR AND ITS ISOSPECTRAL EVOLUTION. Here we merely collect the formulae that are used in the next section.
For the continuum spectrum of H(t): H(t)f = k 2 f, f (x,k,t)->exp(ikx),
f = f(x,k,t),
Imk - 0,
(2.1)
x + + ~
Imk - 0,
(2.2)
Imk = 0,
(2.3)
Imk - 0,
(2.4)
f(x,k,t)-v[l/T(k)]exp(ikx)-[R(-k,t)/T(-k)]exp(-ikx), xR(k,t) R(-k,t) + T(k) T(-k) = 1,
ft = Mf - ik(ao- 4a x k )f,
f = f(x,k,t),
Imk - 0,
(2.5)
Rt - 2ik(aQ-4a1k2)R,
R
Imk = 0,
(2.6a)
Imk = 0,
(2.6b)
Imk - 0
(2.7)
= R(k,t),
R(k,t) = R(k,0) exp[2ik(a o - 4 a i k 2 ) t ] , T t (k)
- o,
For the discrete spectrum (if any) of H(t)i
HI:tnn
= -pn\. •v* Ll
LI
LL
(x.t),
LL
LI
[ n (x,t)] 2 +p n (t)exp(-2p n x),
•nt
P n t (t) = -2p n (a o+ 4a lP 2 n ) p n ( t ) , P n (t) - P n (0)exp[-2p n (a o +4a 1 p^)t],
0,
1,2,...,N,
(2.8)
1,2,...,N,
(2.9)
n = 1,2,...,N
(2.10)
n = 1,2,...,N
(2.11)
n = 1,2,...,N
(2.12)
n =•
1,2,...,N
(2.13a)
n
1,2,...,N
(2.13b)
•
LINEAR EVOLUTION EQUATIONS
271
The function f(x,k,t), that satisfies the asymptotic condition (2.2), is the usual Jost solution of the Schroedinger equation (2.1) (with (1.4a)); its asymptotic behaviour (2.3) defines the reflection and transmission coefficients R(k,t) and T(k) that satisfy the so called unitarity relation (2.4).
On the other hand, the function <J> (x,t) is the normalized (see
(2.9)) eigenfunction of the operator H(t) corresponding to the discrete 2 eigenvalue -p (see (2.8)); its asymptotic behaviour (2.10) defines the spectral parameter p (t). In these definitions the variable t enters only n parametrically; the actual t-dependence, consistently with (1.3), is characterized by the evolution equations (2.5) (that implies (2.6) and (2.7) via (2.3)) and (2.11) (that implies (2.12) and (2.13) via (2.10)), together with (1.5). The spectral transform of u(x,t) is defined as the set of quantities S[u(x,t)] - {R(k,t), -«, < k < + oo; p^, p n (t), n = 1,2,..,N},
(2.14)
and the solvability of the Cauchy problem associated with the KdV equation is based, via the one-to-one correspondence u(x,t)«->S[u(x,t)], on the following steps:
(2.15)
u(x,0)->S[u(x,0)]->SCu(x,t) ]-m(x,t) . A detailed
exposition of this techniques can be found f.i. in Calogero and Degasperis (1982). For future reference, we finally introduce the scalar product (g,h) = | dxg(x)h(x),
(2.16)
—00
and we notice that, with respect to it, the differential operator M, see (1.5), is skew-symmetric, while the Schroedinger operator (1.4a) is symmetric, M T - -M,
H T (t) - H(t).
(2.17)
Moreover, we display below the orthogonality relations satisfied by the Jost, and discrete spectrum, solution (6(x) is the Dirac distribution):
3.
(f(k,t), f(q,t)) - 2Tr[T(k)T(-k)]"1[6(k+q)-R(-k,t)6(k-q)],
(2.18a)
(f(k,t), <j>n(t)) = 0,
(2.18b)
(n(t), ^ ( t ) ) = 6 nm .
(2.18c)
THE ASSOCIATED LINEAR PDE AND THE SCATTERING OPERATOR. To a given solution u(x,t) of the KdV equation (1.1), one can then
associate the linear PDE (1.8).
This is a third order PDE, just as the
272
A. DEGASPERIS
Kdv equation; of course we could consider also an higher order equation by adding to the right side of (1.7) higher powers of the operator H (see below).
Indeed this arbitrarity merely reflects the fact that the operator
M is defined by the Lax equation (1.6) modulus an additional arbitrary (entire) function of H.
In this respect we also note that the connection
between the operators M and H expressed by the Lax equation (1.6) implies that, if v(x,t) is a solution of the linear PDE (1.8), then also v(x,t) = f(H)v(x,t), f(z) being an arbitrary (entire) function, is a solution of (1.8). The Cauchy (initial value) problem associated with the evolution equation (1.8) can be solved by generalizing the familiar Fourier transform method of solving a linear evolution equation with constant (i.e., xindependent) coefficients.
To show this we first introduce the spectral
components of v(x,t), namely (see (2.16)) v(k,t) = (f(-k,t), v(t)) = I dx f(x,-k,t)v(x,t),
Imk = 0,
(3.1)
for the continuum spectrum, and 4co
v n (t) = (n(t),v(t)) = [ dx (J)n(x,t)v(x,t), n = 1,2,..,N,
(3.2)
for the discrete spectrum (if any); here of course we assume that v(x,t) be such that these integrals exist, a sufficient condition being that +00
f |v(x,t)|dx < «> .
(3.3)
These integrals define the mapping v(x,t)-Kv(k,t), -«>; v (t), n=l,2,...,N} that can be easily inverted be using the orthogonality relations (2.18) (and the unitarity condition (2.4)); the inversion formula reads +00
f
v(x,t) - j-
N dk[v(k,t)+R(k,t)v(-k,t)]f(x,k,t)+£ v (t) <j> (x,t), * n=l
(3.4)
— 00
where R(k,t) is the reflection coefficient defined by (2.3). The relevant property of this transformation is that, if v(x,t) satisfies the PDE (1.8), its spectral components v(k,t) and v (t) satisfy the following ODE f s: v = v(k,t), v n t (t) = -i(bo - b ^ ) v n (t),
(3.5) (3.6)
LINEAR EVOLUTION EQUATIONS
273
where we have set o)(k) = ~k(ao - 4a.k ) +b Q +b-k .
(3.7)
These evolution equations for the spectral components of v(x,t), that can be trivially solved, are easily derived by first differentiating with respect to the variable t both sides of (3.1) and (3.2), and then by using the evolution equations (2.5), (2.11) and (1.7), together with the symmetry properties (2.17). These results provide therefore a method to solve the linear PDE (1.8), that is similar to the Fourier analysis.
In fact, the associated Cauchy
problem, for a given solution u(x,t) of the KdV equation (1.1), can be solved via the usual steps: v(x,0)-Kv(k,0) ;v (0) }->{v(k,t) ;v (t) }->v(x,t) . n n Here it is however required, as a preliminary task, to compute the functions f(x,k,t) and
This is done by first computing a particular
solution u(x,t) of the KdV equation (1.1)» via the spectral transform technique, and then by solving the Schroedinger equation (see(2.1) and (2.8) with (1.4a), (2.2) and (2.9)).
For instance, if u(x,t) is a pure multi-
soliton solution of the KdV equation (1.1), these computations can be explicitly performed to yield an analytic expression of all the relevant spectral quantities. Let us consider now a scattering process for the linear evolution equation (1.8); this process is described by a solution v(x,t) of (1.8) that behaves, as t-> ± °°, as a wave-packet v
(x,t) satisfying the "free"
evolution equation
v ( ± ) = a v(±> + 4a,v(±) - ib v ( ± W v ( ± ) , v (±) = v (±) (x,t), t
OX
1 XXX
o
1 XX *
that obtains by setting u(x,t) - 0 in (1.8). of the variable x, both v(x,t) and v
(3.8)
' *
2 If v(x,t) is a L -function
(x,t) vanish as t •*• ± » (for fixed
x); therefore the scattering asymptotic condition should more precisely read +00
lim t ^
±oo
j |v(x,t) - v (±) (x,t)| 2 dx = 0.
(3.9)
—00
This condition implies that, if the solution u(x,t) of the KdV equation (1.1) has a solitonic component (i.e. its spectral transform has a discrete part), the solution v(x,t) satisfying (3.9) cannot have a discrete spectrum component, v n (t) = 0 (see (3.2)).
Indeed a solution of (1.8) given
by the sum in the r.h.s. of the spectral decomposition (3.4) would travel
274
A. DEGASPERIS
together with the solitons contained in u(x,t), and therefore it could not satisfy asymptotically (as t -> ± <» ) the free equation (3.8).
A scattering
process is then described by a solution v(x,t) of (1.8) whose spectral decomposition has only the continuum spectrum component, namely (see(3.4)) v(x,t) = (2ir) L\ dk[v(k,O) + R(k,O)v(-k,O)]exp[-io)(k)t]f (x,k,t) ;
(3.10)
—00
here the expression on the right side follows from (2.6b), (3.5) and (3.7). We introduce now the following Fourier integral representations: +00
v
(±)
(x,t) = (2ir)""1| v(±)(k)exp[ikx-ico(k)t]dk,
(3.11)
—00
+00
v(x,t) = (27T)"1] dkZ(k,t)exp[ikx-iu>(k)t],
(3.12)
—00
the first one being of course implied by the evolution equation (3.8) (see (3.7)), while the second one does allow us to rewrite the scattering condition (3.9) in the form +00
lim [ |z(k,t) - v ( ± ) (k)Pdk - 0 t->±« l or,
(3.13)
for a sufficiently regular Z(k,t), lim Z(k,t) = v ( ± ) ( k ) .
(3.14)
t-V±oo
As usual, the scattering operator associated with the evolution equation (1.8) is defined by the formula +co +co
) J S(k,q) ,q) v v (""(q)dq, ( )
that specifies the transformation of the initial wave-packet v t = -oo into the final wave-packet v
(3.15) (x,t) at
(x,t) at t = -H» (see (3.11)).
This
operator is of course constrained to satisfy the unitarity condition +00
J S(k,s) S*(q,s)ds = <5(k-q),
(3.16)
—00
if the solution u(x,t) of the KdV equation (1.1) is real, u = u*; indeed the equation (3.16) easily follows from the conservation law
LINEAR EVOLUTION EQUATIONS
275
P t (x,t) + J x (x,t) = 0,
p = v*v, J - -a o p-a-(v*v
(3.17a)
+v* v-v*v -6up) + ib.(v*v
- v*v),
(3.17b)
that is associated with the evolution equation (1.8). In order to compute the scattering operator S(k,q) we first invert the Fourier integral (3.12) and use the spectral representation (3.10) to obtain +00
Z(k,t) = | dqCv(q,O) + R(q,o)v(-q,o)]exp{i[a)(k)-a)(q)]t}g(k,q,t),
(3.18)
where we have set g(k,q,t) = (27T)"1! exp(-ikx)f(x,q,t)dx.
(3.19)
—00
By standard techniques, namely by using (2.1),(2.2), (2.3) and the representation lim
Pk^expUkx) = ±±TT 6(k)
(3.20)
of the Dirac distribution, it is convenient to derive for the integral (3.19) the following expression g(k,q,t) = %[l+l/T(k)]5(k-q)-^[R(k,0)/T(k)]» 1 2 2 1 •exp[2ik(a^-4 ai k )t]6(q+k) +(2Tr)" p(q -k )" h(k,q,t) ,
(3.21)
with +00
h(k,q,t) = j
u(x,t)exp (-ikx)f(x,q,t)dx.
(3.22)
—oo
In fact the principal value distribution that appears in the r.h.s. of (3.21), together with the boundedness of the function h(k,q,t)(see(3.22)), readily implies that in the integral (3.18) only the values q=±k give^a contribution to the limit (3.14)(see (3.20)).
Therefore, the general
formula v ( ± ) (k) «= A±(k)v(k,0)+B±(k)v(-k,0)
(3.23)
holds for the limit (3.14), and this, by using the definition (3.15), implies the following expression of the scattering operator: S(k,q) = Sf(k)6(k-q) + Sb(k)6(k4q),
(3.24)
276
A. DEGASPERIS
where S f(k) is the forward scattering amplitude, S-(k) = [A,(k)A_(-k)-B,(k)B_(-k)]/[A_(k)A (-k)-B (k)B (-k)l,
(3.25)
and Sb(k) is the backward scattering amplitude, S b (k) = CA_(k)B+(k)-A+(k)B_(k)]/CA_(k)A_(-k)-B_(k)B_(-k)].
(3.26)
Moreover, because of (3.16), the two amplitudes S_(k) and S, (k) are of r b course related to each other by the unitarity equations |S f (k)| 2 + |S b (k)| 2 = 1,
(3.27a)
S f (k) S*(-k) + S*(-k)Sb(k) = 0.
(3.27b)
From (3.18) and (3.21) it is evident that the actual computation of the scattering amplitudes S_(k) and S, (k) requires the evaluation of the r b function h(k,q,t), see (3.22), at least in the limit as t -> ± «> . This is easily done in the special case with a, = 0, since in this case (see (1.1) and (2.2)) u(x,t) = u(x+a g t,0), f(x,q,t) • exp(-iqaQt)f(x+aQt,q,0)
(3.28)
and therefore (see (3.22)) h(k,q,t) - expCiaot(k-q)] h(k,q,0);
(3.29)
by using then the wronskian integral relations h(k,k,t) - 2ik[l-l/T(k)],
(3.30)
h(k,-k,t) = 2ik R(k,t)/T(k),
(3.31)
the previous formulae yield the expressions Sf(k) = T(n|k|), S b (k) = R(k,0)[e(nk)-6(-Tik)T(-k)/T(k)],
(3.32)
n - b 1 /|b 1 |,
(3.33)
where
and 0(x) is the step function (9(x) = 1 for x > 0, 6(x) = 0 for x < 0 ) . It should be noted that, consistently with their obvious physical meaning, the constants a
and b
the scattering operator.
(see (1.8))
do not appear in the expression of
Moreover the signature constant (3.33) in the
expressions (3.32) merely indicates that changing the sign of b^ (see (1.8)) is equivalent to changing the sign of the time variable t, and therefore to inverting the scattering operator. a
o
=b
o
In the subcase
= 0 , b- = 1, the evolution equation (1.8) is the Schroedinger
equation of wave mechanics, and the expressions (3.32), with n ~ 1, give the corresponding scattering operator (see (3.24)).
LINEAR EVOLUTION EQUATIONS
277
As for the general case, with a. 4 0 , we compute below the scattering operator only for those solutions of the KdV equation (1.1) that are pure N-soliton solutions. In this case therefore R(k,t) = 0 and N T(k) - II T (k); n-1 n here we have introduced the convenient notation T n (k) = (k+ipn)/(k-ipn)s
(3.34)
n = 1,2,...,N.
(3.35)
In order to compute the function h(k,q,t), see (3.22), as t ->• ± °°, we use the property of the N-soliton solution u(x,t) to break up into N single-soliton solutions, N ( . 2 U; osh {p [x-C (t)]>, t -> ± co, u(x,t) + I (-2p2)/ C n n=l n n
(3.36)
with
^n ±)(t) = ^nf + V 1 C n - -(ao + 4p£ a x ) , £
being constants; for the velocities C
n= 1 2
> >"-»N>
(3 37)
-
n = 1,2,...,N,
(3.38)
of the asymptotic solitons we
conventionally assume that C x < C 2 <...< C N ,
(3.39)
and, just to be definite,, we consider the case a o > 0 and a_ > 0, so that C
< 0 for n = 1,2,...,N.
In this instance, the single-soliton bumps are
ordered ( from left to right) from N to 1 as t ->-«>, and from 1 to N as t •*• + °°. The other cases, with different choices for the sign of a and a1 can, of course, be similarly treated. To the breaking up process for u(x,t) as t -*• ± «, there corresponds a simple behaviour of the Jost solution f(x,k,t) defined by (2.1), (2.2) and (2.3); in fact, if f^(x,k,t) is the Jost solution n soliton solution
for the one-
u(x,t) = ^ p V c o s h ^ p ^ x - C ^ O O ] } ,
(3.40)
f^fr.k.t) = {Ck+ipntgh{pn[x-^±)(t)]}]/(k+ipn)}exp(ikx),
(3.41)
namely
it is easily seen that, as t •»••», the Jost solution f (x,k,t) behaves, in the vicinity of x = £
(t), as
f(x,k,t) -> Tn(k)[ n T (-^Df^Cx.k.t), t + - oo, x - ^ - ^ t ) ; m=l
(3.42)
278
A. DEGASPERIS
here the definition (3.35) is used, and the factor in front of the Jost solution in the r.h.s. originates from the matching condition for f(x,k,t) between two adjacent one-soliton bumps.
Inserting now the asymptotic be-
haviours (3.36) and (3.42) into the definition (3.22) finally yields the asymptotic formula N N ( . h(k,q,t) - I T n (k) n T (-k)hn(k,q)exp[-i(k-q)^~;(t)], n=l m^l
(3.43)
for t -*• - °°, where hn(k,q)=-2p2J dx{{[q+ipntgh(pnx)]/(q+ipn)}/cosh2(pnx)}exp[i(q-k)x]. (3.44) —00
By the same arguments the asymptotic behaviour of h(k,q,t) as t -> + » can be similarly derived, and it reads N N ( h(k,q,t) + I T n (k) n T m (-k)h n (k,q)exp[-i(k-q)^Y(t)].
(3.45)
Combining now these asymptotic expressions with (3.21), (3.18), (3.7), (3.37), (3.38) and the formulae hn(k,k) = 2ik[l-Tn(-k)], hn(k,-k) = 0 ,
n = 1,2,...,N,
(3.46)
n - 1,2,...,N,
(3.47)
we obtain the following scattering amplitudes (see (3.34) and (3.35)) Sb(k) - 0, N
,. v
(3.48) n
Sf(k) = {1+T(k) + I r,nW[l-Tn(-k)] n Tm(k)}.
"N
V
,
.{l+T(k) - I nn(k)Cl-Tn(-k)] n T (k)F. n^l
m=n
(3 49)
-
m
In this formula n (k) is the signature function defined as follows (see (3.38)): n (k) =
1
if
da)(k)/dk > C ,
n = 1,2,...,N,
(3.50)
n n (k) = -1
if
da)(k)/dk < C n ,
n = 1,2,...,N,
(3.51)
n (k) =
if
dio(k)/dk - C ,
n = 1,2,...,N.
(3.52)
0
The physical meaning of the function n (k) is quite apparent if one considers that do)(k)/dk is the group velocity of the asymptotic wave-packets with average wave number k, and C (see (3.37)) is the velocity of the asymptotic solitons. (see(3.7) and (3.38))
Indeed the equality (3.52), that can occur only if
LINEAR EVOLUTION EQUATIONS
\
279
^"1,
implies that a soliton with velocity C
(3.53)
does not contribute to the scat-
tering amplitude at those wave-numbers such that the group velocity is equal to the soliton velocity C . Thus, in the particular case with b o = b = 0 (and a- > 0) these special values of k, such that (3.52) holds, never occur, and r\ (k) * 1 for n = 1,2,...,N.
In this case the forward scattering amplitude reduces
to the transmission coefficient, S f (k) - T(k);
(3.54)
this formula, together with (3.34), shows that each soliton contributes to the total scattering amplitude S.(k) with the partial amplitude
T
n0O»
For completeness the scattering operator S(k,q) should be computed also for those solutions u(x,t) of the KdV equation (1.1) that have a non vanishing continuum spectrum component (i.e., with R(k,t) ^ 0 ) . This problem is however left to future investigation. 4.
EXTENSIONS AND REMARKS. Several extensions of the results presented in the previous sections
are possible.
They are pointed out in the following remarks.
REMARK 1.
An immediate generalization obtains by replacing the KdV
equation (1.1) with the generic nonlinear evolution equation u t = a(L)ux, of the KdV family.
u = u(x,t),
(4.1)
Here L is the integro-differential operator defined
by the equation Lf(x) - fxx(x)-4u(x,t)f(x)+2ux(x,t) | dyf(y),
(4.2)
x that specifies its action on an arbitrary (except for obvious restrictions) function f(x) (see Calogero and Degasperis 1982). is an arbitrary polynomial of z. for a(z) = a o + a.z.
Moreover in (4.1) a(z)
In fact the KdV equation (1.1) obtains
The associated linear evolution equation (that re-
places (1.7)) now reads v
- Mv - ib(H)v,
v = v(x,t),
(4.3)
where M is the differential operator corresponding to (4.1) via (1.4a) and (1.6) (its explicit expression can be found in Calogero and Degasperis, 1982), and b(z) is an arbitrary real polynomial of z (the equation (1.8) obtains with b(z) = b Q + b-z).
In addition to other obvious changes with
respect to the previous formulae, we note the following.
The evolution
280
A. DEGASPERIS
equation for the spectral components of v(x,t) is still (3.5) for the continuum, but with the dispersion law a)(k) = -ka(-4k2) + b(k 2 )
(4.4)
in place of (3.7); while for the discrete spectrum the equation (3.6) should now read v n t (t) - -ib(-p2)vn(t). The asymptotic wave-packets v
(A.5)
(x,t), see (3.9), satisfy, in place of
(3.8), the evolution equation (see (1.4b)) (± a ( 4 D )vv^^> - i b (D2)v ib(D ) v >,
consistently with (3.11) and (4.4).
± v >(x,t), v<±> = vv((± >
(4.6)
Moreover the scattering operator
S(k,q) has still the form (3.24), and the expressions (3.48) and (3.49) of the scattering amplitudes corresponding to a pure N-soliton solution of (4.1) are still valid, with the signature function n (k) given by (3.50), (3.51) and (3.52) where however a)(k) is now given by (4.4) and the soliton velocity C R has the expression ),
n
n = 1,2,...,N,
(4.5)
instead of (3.38). REMARK 2.
The class of linear PDE !s (4.3) can be derived also by
applying the AKNS approach (Ablowitz, et al., 1974) to the system of equations ^xx " £u(x,t)-k2]i|H-v(x,t), i[»t - A(x,k,tH+B(x,k,tH x +C(x,k,t)
* -
(4.6a)
lp = i|;(x,k,t);
(4.6b)
in fact the evolution equations (4.1) and (4.3) obtain via the integrability condition
ty
ty
, by asking the functions A(x,k,t), B(x,k,t)
and C(x,k,t) to be polynomial in the spectral variable k.
It should be
emphasized that, since this technique involves only local operations, it applies to other ODE's such as the Zakharov-Shabat system (Ablowitz, et al., 1974) or to any other higher rank matrix ODE.
By starting with non
homogeneous equations, such as (4.6), one can therefore produce at the same time both the nonlinear evolution equations (that one usually obtains via the homogeneous equations), and the associated linear PDE's; however one should know how to solve the direct, and inverse, spectral problem in order to solve the initial value problem for the associated (linear and nonlinear) evolution equations.
LINEAR EVOLUTION EQUATIONS
REMARK 3.
281
By the spectral method discussed here one can solve also
the non homogeneous equation v t = Mv-ib(H)v+s(x,t),
v = v(x,t),
(4.7)
that differs from (4.3) only for the additional source term s(x,t). REMARK 4.
The method of solution for the scalar equation (4.3) can
be trivially applied to solve the system of linear PDE fs v«> Z
=
Mv
(j)
- i I bj (H)v (h) , h h
for a multi-component field v
v ( j ) = v ( j ) (x,t),
j = 1,2,..,
(4.8)
(x,t); here, of course, M is the same
differential operator as in (4.3), and bj!(z) is a real polynomial (H is of course defined by (1.4)). REMARK 5.
The method of solving the initial value problem for the
linear PDE (1.8), or, more generally, (4.3), can be generalized to the case with more than one space coordinate, in the following way.
For the
sake of simplicity, let us consider the simplest extension to two space coordinates, say x.. and x«; in a self-evident notation (i.e., we attach the index j, j - 1,2, to symbols defined in the previous sections), let H ( j ) (t) = -D|+u ( j ) ( X j ,t),
Dj - 3/3 Xj , j
Schroedinger operators, operators, the the f be two Schroedinger function u of the nonlinear evolution equation X
-t
j = 1,2,
(4.9)
(x.,t) being a solution
J
and let •',Hw)v,
v = v(xrx2,t)
(4.11)
be the associated linear PDE for a (complex) function v of the three variables x,,x2 and t, where b(z.,z2) is an arbitrary real polynomial in zand z 2 .
Consider now the four spectral components of v(x-,x 2 ,t), +00
+00
t) - I d x J dx 2 v(x 1 ,x 2 ,t)f ( 1 ) (x 1 ,-k 1 ,t)fB 2 ,-k 2 ,t)^
j dxJ —00> >
(4.12a)
dx2v(x1,x2,t)^1)(x1,t)f(2)(x2,-k,t),n=l,2,...,N1,(4.12b)
—00 —00 _|_oo
dx1jdx2v(x1,.x2,:t)f(1)(xL,-rk,t)^2)(x2,O, n=l,2,...,N2,(4.12c) — 00
_OC
282
A. DEGASPERIS +00
+CO
—00
—00
n. = 1 , 2 , . . . , ^ ,
j -= 1,2; 1,2;
(4.12d)
they satisfy the following ODEfs:
(4.13)
(4.14)
(4.15) vn
2)
2)
= v^ (k,t),
n = 1,2,...,N2,
As these equations are trivially solved, the solution of (4.11) obtains via the inversion formula +OO
+00
v ( X l , x 2 , t ) - (27r)J d k j dk 2 Cv(k 1 ,k 2 ,t)+R ( 1 ) (k 1 ,t)v(-k 1 ,k 2 ,t)
+ R ( 2 ) (k 2 > t)v(k i r k 2 ,t)
R ( 1 ) (k 1 ,t)R ( 2 ) (k 2 ,t)-v(-k 1 ,-k 2 ,t)^ 1 ) (x 1 ,k 1 ,t)f ( 2 ) (x 2 ,k 2 ,t)
n-1
283
LINEAR EVOLUTION EQUATIONS
(2)
(-k.t)
(x2>k,t)
n=1
+ I1 r
(4
ln2
For instance, if a '(z) = a
-17)
(z) = z and b(z ,z ) = 0, the linear evo-
lution equation (4.11) reads v t = 4(vs
HKu^fe.t
+v
'(x,,t)vr ] 2
(4.18)
v = v(Xl,x2,t).
It should be finally noted that, if v depends on M space coordinates, the number of its spectral components is (at most) 2 . REMARK 6.
By replacing the Schroedinger spectral problem with an
other spectral problem, one can similarly investigate other (classes of) nonlinear evolution equations, and their associated linear evolution equations.
Below we merely report a few simple examples of linear evolu-
tion equations that are associated with well-known nonlinear PDE f s that are solvable via the generalized Zakharov-Shabat spectral problem (Ablowitz, et al., 1974).
Of course the initial value problem associated with them
can be solved by a technique that parallels that described in the previous sections.
Thus, for the Nonlinear Schroedinger equation:
^t^xx"2^'2^ " °
* - *(x,t),
(4.19a)
16 -26 +i|> e*+2if;e*+|iH20 = 0,
9 = e(x,t);
(4.19b)
u = u(x,t),
(4.20a)
v = v(x,t);
(4.20b)
t
XX
X
X
for the Modified KdV equation: u
t
= u
v XXX
2 -6u u , XXX X -2(3u +u 2 )v , X X
284
A. DEGASPERIS
for the Sine-Gordon equation: •xt " A
Vf XL
'
= —*s4> B +Js(cos<j>A-sin<|>B),
Xt B
Sini)>
Xt =
1 1 7 * At-+-(cos(t)B+sin(J)A), 2
A
• = *(x,t),
(4.21a)
A = A(x,t),
(4.21b)
B = B(x,t);
(4.21c)
z = z(x,t),
(4.22a)
w = w(x,t).
(4.22b)
L. Lj.
for the Sine-Gordon equation: z
= sinh z
xt
w___ = ^z w^4^w e x p ( z ) , ACKNOWLEDGEMENTS.
The hospitality of the International School for Advanced
Studies, Trieste (Italy), where part of this work has been done, is gratefully acknowledged. REFERENCES ABLOWITZ, M.J., KAUP, D.J., NEWELL, A.C., SEGUR, H.,(1974) The inverse scattering transform - Fourier analysis for nonlinear problems, Stud. Appl. Math. 53, 249-315. CALOGERO, F # and DEGASPERIS, A. (1982), Spectral Transform and Solitons I, North-Holland Publ. Co., Amsterdam. LAX, P.D.(1968), Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21, 467-490.
CHAPTER 15 INVERSE SCATTERING FOR THE MATRIX SCHRODINGER EQUATION WITH NON-HERMITIAN POTENTIAL PETER SCHUUR Mathematical Institute Rijksuniversiteit Utrecht 3508 TA Utrecht, The Netherlands
1.
INTRODUCTION. Kamijo and Wadati (1974) developed an inverse scattering formalism for
the matrix Schrodinger equation with Hermitian potential matrix U(x).
As-
suming a similar formalism in the non-Hermitian case the authors and later on Calogero and Degasperis (1976) found several classes of solvable nonlinear evolution equations.
To our knowledge this assumption has nowhere
been justified in the literature.
Indeed, Wadati (1980) writes:
"At the moment, it is an open question as to what are the most general conditions for which the inverse scattering problem can be solved. I have considered three cases: I
U(x) is diagonal (and can be complex), ~ 2
II
U(x) = JU(x)J; J a constant matrix, J
III
U*(x) = JU(x)J; J a constant matrix, J
=1, 2
= I.
The first crucial point in the discussions is the definition of the Wronskian.
Although we need extra assumptions, similar arguments seem to be
valid". The present paper is an attempt to settle this question.
We consider
the Schrodinger scattering problem for any continuous potential matrix U(x) decaying sufficiently rapidly for |x| -*• °°.
Superimposing some rather na-
tural regularity conditions on the right transmission coefficient that are partly familiar from the Zakharov-Shabat scattering problem (Eckhaus and van Harten, 1981), we show that - apart from the unique solvability of the Gel1fand-Levitan-Harchenko equation - the inverse scattering problem can be solved completely.
Though initially the approach parallels that of
Kamijo and Wadati (1974), our ways split up in Section 5.
The reason is
that the Wronskians employed by Kamijo and Wadati to extend the scattering coefficients to the upper half plane are of no use in the non-Hermitian case.
To circumvent this difficulty, we perform the extension by means of
integral representations in terms of the potential and the Jost functions.
286
2.
P. SCHUUR
NOTATION. Throughout, we shall use the following notation:
If A = (a. .) is an n x n matrix with a
e t, then we write:
n
|A| = max I |a |. j
(2.1)
±J
1-1
The determinant of A will be denoted by det A. matrix.
I is the n x n identity
We set t+ = {k e 0|lm k > 0}
f
- {k e t\Im k > 0}.
(2.2)
Our notation for the Jost functions, scattering coefficients, etc. closely resembles that used in Eckhaus and van Harten (1981). different from the one in Kamijo and Wadati (1974).
This notation is For convenience, we
specify here the relation between these different notations: F
l "V
F =
2 V
C
ll= V
C
12= R-> C22 ' L-> C21= V
R
- \>
Kx(x,y) - N(x,y-x). 3.
(2.3)
THE MATRIX SCHRODINGER EQUATION. We consider the differential equation i|/f + (k2- U(x))i|i = 0,
f
" ^
»
-oo< x <-|
(3.1)
where U(x) is a complex n x n matrix and ty is a vector function with n components. tion.
Equation (3.1) is called the n
Accordingly, the complex parameter k
x
n matrix Schrodinger equaand the matrix U(x) are re-
ferred to as energy and potential, respectively. We shall impose the following conditions on the potential: U(x) is continuous on -°° < x < 4^°,
r
lim
U(x) - llm
U(x) - 0,
(1 + |x|)£|u(x)|dx < -H»,
I - 0 or 1.
(3.2a) (3.2b)
(3.2c)
We call (3.2) a growth condition of order I and write [[£]] to indicate which growth condition is meant. [[0]].
For many results it suffices to assume
However, we need [[1]] to prove existence and regularity of the
Jost functions for k = 0 (see Section 4 ) , a result which is essential in the derivation of the Gelffand-Levitan-Marchenko equation in Section 7.
INVERSE SCATTERING FOR MATRIX SCHRODINGER EQUATION
287
Clearly, any system of n solutions of (3.1) can be represented as an n x n matrix ¥ satisfying the equation ¥" + (k2 - U(x))Y = 0 ,
-oo <
x
< -H».
(3.3)
Conversely, the columns of any solution of (3.3) are solutions of (3.1). Following Kamijo and Wadati (1974) we shall not study (3.1) directly, but instead focus our attention on (3.3).
This enables us to exploit analogies
with the scalar Schrodinger equation with real potential, of which the inverse scattering mechanism is now perfectly understood (Deift and Trubowitz, 1979; Eckhaus and van Harten, 1981). 4.
JOST FUNCTIONS. In this section we generalize some results that are familiar from the
n = 1 case with real potential as described in Eckhaus and van Harten (1981).
Since the proofs of these generalizations are analogous, we omit
them. For Im k ^ 0, we introduce the Jost functions ^(x,k) and ¥ 0 (x,k),
288
P. SCHUUR
H(x,y,k) =
fv
tw
"-'
— (y-x)U(y)
( THEOREM 1. (i)
v.".
k ,
k = 0.
(4.5b)
If [[0]] and Im k > 0, k + 0, then one has:
The problem (4.4) for R has a unique solution that is continuous in x and bounded for x -* -°°.
This solution satisfies (4.2) in
classical sense and is given by the Neumann series R =
E
G
m=0
m' G 0
= I>
G
m+l(x'k) "
G(x,y,k)G (y,k)dy,
•'-»
m > 0.
(4.6)
The G f s satisfy the estimate: J m
|G (x,k)| < (mD^UQCx)/^!}111, u (x) = [ (ii)
|U(y)|dy.
(4.7)
The problem (4.5) for L has a unique solution that is continuous in x and bounded for x •> °°.
This solution satisfies (4.3) in
classical sense and is given by I H ,H m=0
|H
- I, H
(x,k) =
0
THEOREM 2.
H(x,y, H(x,y,k)Hm(y,k)dy,
m > 0.
(4.8)
• 'x 'x
f
m
, v (x) - Jf|u(y)|dy. Jx
x
(4.9)
Let the potential U satisfy a growth condition of order 0.
Then the matrix functions R,R* ,R n ,L,L f , and Lfl are x
(i)
continuous in x and k on I
(ii)
analytic in k on t+ for each x € E..
THEOREM 3.
Assume [[0]].
(j5,\{0}) , and
Then the limits prescribed in (4.2) and
(4.3) for R,Rf,L and L 1 are uniform in k on compacta c jfi \{0}. For the limits in the nonprescribed directions, we find if k e t,: llm R(x,k) - I - (2ik)
x
U(y)R(y,k)dy,
lim R1(x,k) = 0, (4.10a) X-x»
lim L(x,k) - I - (21k)" 1 [ U(y)L(y,k)dy,
lim l/(x,k) - 0. (4.10b)
The limits in (4.10) are uniform in k on compacta c 0 . THEOREM 4.
Let U satisfy a growth condition of order 1.
Then the
problems (4.4), (4.5) for R,L with k = 0 have a unique solution that is continuous in x and bounded for x -> -°°, x -»• °° respectively. (4.2), (4.3) in classical sense for all (x,k) e H x 0,.
R,L satisfy
INVERSE SCATTERING FOR MATRIX SCHRODINGER EQUATION
289
The functions R,Rf ,R",L,Lf and L" are continuous in (x,k) on i x " # .
The
limits prescribed in (4.2) and (4.3) for R,Rf,L and L 1 are uniform in k on THEOREM 5.
Suppose [[1]].
Then the matrix L has the Fourier repre-
sentation e i k s N(x,s)ds,
L(x,k) = I + |
J
x e 3R ,
k e 1,, +
0
2 2 with N(x,s) € L* n L n (0 < s < -+«>) for each x. 2 ment of C £ ( ] R 2 ) = {w e Cn2(]R x [0,°°))|Vx e ]R: U + is differentiable with respect to x as well as to
(4.11)
The kernel N is an elelim W(x,s) = 0 } , which S -K» ? —-— s and N ,N e Cj^ ( H 2 , ) . x s U •
Moreover, N has the important property
N(x,0) = \ J U(y)dy,
(4.12)
so that U(x) = -2 N x (x,0), 5.
x € ]R.
(4.13)
SCATTERING COEFFICIENTS. Let us assume [[0]].
For k € ]R\{0} the pairs ¥„ (x,k), ¥ (x,-k) and
y (x,k), V (x,-k) constitute fundamental systems of solutions of equation (3.3).
In particular, we have (5.1a) (5.1b) A/
j.
—
i.
n
where the scattering coefficients R+,R_,L_ and L + are n x n matrices depending on k e H \ { 0 } . Substituting (5.1a) into (5.1b) and vice versa, we find R+(k)L_(k) + R_(-k)L+(k) = I,L_(k)R+(k) + L+(-k)R_(k) = I, R._(k)L_(k) + R + (-k)L + (k) = 0,L+(k)R+(k) + L_(-k)R_(k) = 0.
(5.2)
From (4.2), (4.3) and (5.1), we obtain the asymptotic behavior for |x| + » off., X/
f*
XJ
f
r
and 4" with k € H\{0} fixed:
^ r (x,k)
r
X
e" i k x I
for x + -«
«
ikx e i k x R, R (k) R + ((k) + e"
for x ^ -H»
e" l k x I
for x + -~
e i k x R + (k) - e" ikx R_(k)
for x -> -H»
(5.3a)
(5.3b)
290
P. SCHUUR
^(x,k)
-1
«
e1KXI
for x + +»
«
e i k x L + (k) + e~ ikx L_(k)
for x + -»
.
(5.3c)
-fkv
(ik) ^^(x,k)
»
e1KXI
for x •* 4~
a
e i k x L + (k) - e" ikx L_(k)
for x * -».
(5.3d)
Here the « sign denotes that the difference between the left and right hand sides tends to 0. Using (5.3), (4.2), (4.3) and (4.4), (4.5) we find for k € H \ { 0 } : X
R (k) = lim (21k) "e x-x»
{iky (x,k) - Yf(x,k)} r r
= lim R(x,k) - (2ik)"1Rf(x,k) -lim I- ^ i k ) " ^ - I - (2ik)~ 1 I
U(y)R(y,k)dy
U(y)R(y,k)dy
(5.4a)
J —OO
R ( k ) - (21k)" 1 [
e " 2 i k y U(y)R(y,k)dy
(5.4b)
J —00 -00
L (k) - I - (21k)" 1
U(y)L(y,k)dy
(5.4c)
e 2 i k y U(y)L(y,k)dy.
(5.4d)
J-oo L (k) - (21k)" 1 [
We now observe that the right hand sides of (5.4a) and (5.4c) are well defined for all k e #\{0}.
Thus we extend the domain of R
and L, from
H\{0} to "^{0} by defining: R_(k) = I - ^Ik I
U(y)R(y,k)dy
* —00
L + (k)
E
1 f00 I - 2lk I U(y)L(y,k)dy —00
It follows from Theorem 2, that R on ? + \{0}.
and L
k € £ + \{0}
(5.5a)
k e t+\{0}.
(5.5b)
are analytic on #
and continuous
By (4.10), one has for k e (C+: lim R(x,k) - R (k),
lim L(x,k) - L,(k),
X-*»
X-^-OO
~
(5.6)
+
where the limits are uniform in k on compacta c £ m Furthermore, the determinants of the matrices defined in (5.5) satisfy det R (k) - det L,(k) - (2ik) x e *,
A (x,k)
detl
k £ ^ + \{0}.
To prove this, we first note that any pair of solutions ^-.^o satisfies
(5.7) of
(3.3)
INVERSE SCATTERING FOR MATRIX SCHRODINGER EQUATION
291
/i
i.e. the determinant of the 2n x 2n matrix I x € R.
* | is constant for
By continuity it suffices to prove (5.7) for k e 0., in which case:
detl
\V
VQ I
= det( \ Rf-ikR
**'
\
L'+ikL/
/R
L
= lim detl V Rf
A
\ = detl Lf+2ikL / \0
R \ /I lim detl I - det I 00 f t x^ \-L -R +2ikR/ \0 6.
+ ]= (2ik)ndet L 2ikL_J
R " 2ikR
)= (2iK)ndet R_. (5.9)
BOUND STATES. We now study the bound states of equation (3.1) in 0 + , i.e. values of
k with Im k > 0 for which (3.1) possesses a non-trivial quadratically integrable solution I|J. They are characterized by: LEMMA 2.
If k
€ (5 and [[0]], then the following statements are
equivalent (i)
det R_(kQ) = 0
(6.1)
(ii)
There exists a non-trivial \p e L (E. ) n such that l 2 V + (kQ" - U(x))* - 0.
(6.2)
PROOF.
Suppose Suppo det R (k ) = 0. Then by (5.7) we can find
n
a,b e £ \{0} with 0
Vx £ H .
(6.3)
Putting *(x) = ¥r(x,kQ)a = -^(x,k Q )b we obtain from (4.1), (4.2), (4.3) that \p decays exponentially for x -•• +00 and therefore belongs to L 2 (E.) . Furthermore, it is evident that \p satisfies (6.2) andtyt 0. Conversely, suppose (ii) holds.
If det R (kQ) ^ 0, then by (5.7) the
columns of ¥ (x,k Q ), ¥.(x,kQ) form 2n linearly independent solutions of (6.2).
Thus there are a,b e 0 n , (a,b) + (0,0), such that \(x,k Q )a + y £ (x,k 0 )b.
(6.4)
Using (4.1), (4.2), (4.3) and (5.6), (5.7) we conclude that for x + ±*> the function ip grows exponentially in at least one direction. the fact that * is in L 2 ( E ) n .
This contradicts •
292
P. SCHUUR
We next present a property of bound states that will play an important role in Section 7. THEOREM 6.
Let the potential U satisfy a growth condition of order 0
and let k e jC be a bound state of (3.1). Assume furthermore that the ma-1 trix R (k) has a simple pole at k = k. with residue D(k ) = lim (k - k J R - V ) . u
k-*kQ
(6.5)
~
Then there is a unique matrix C(kft) such that for x e I
^r(x,k0) D(kQ) = V * , k 0 ) C ( V PROOF.
(6 6)
-
Since for potentials with compact support the proof is imme-
diate, we shall apply a truncation procedure.
For N »
1 we define
N
U (x) E aN(x)U(x) with a N (x) = 1 for |x| < N - 1, a ^ x ) = N - |x| for N - 1 < |x| < N, a N (x) = 0 for |x| > N. £ = 1. /,R
N
Plainly, U N satisfies (3.2) with
From (4.1), (4.6), (4.8) and (5.4a,b) it is clear that the functions
JJ,LV
and R^ associated with U N can be extended to analytic func-
tions with respect to k on £\{0}.
Moreover, the relation (5.1a), which we
obtained for k e IR\{0}, remains valid for k e j£\{0} by analytic continuation.
Thus for x £ 1R and k e A i O } one has: ^
^
^
^ ^ ( k ) .
(6.7)
Concerning the behavior as N -> °° we first note that |UN(y) - U(y)|dy < _oo
|u(y)|dy ^ 0
for N + ~.
(6.8a)
, ,J
|y|>N-l Furthermore, for x e ~R and Im k > 0, the following estimate holds max{|RN(x,k) - R(x,k)|, |LN(x,k) - L(x,k)|, |R^(k) - R_(k)|} < -T-T- exp( -r—r )f ^ ( y ) - U(y)|dy with A - [ |u(y)|dy. I'M | K | J.OO J-OO
(6.8b)
Indeed, consider the integral equation RN(x,k)-R(x,k) = f
(2ik)" 1 (e 2 ± k ( x " y ) - l)(UN(y) - U(y))R(y,k)dy
—oo X
(2ik)""1(e2ik(x""y)-l)UN(y)(RN(y,k)-R(y,k))dy.
(6.9)
By (4.6), (4.7) we have the estimate |R(y,k)| < exp( -rj-r ) , which yields
INVERSE SCATTERING FOR MATRIX SCHRODINGER EQUATION
293
|RN(x,k) - R(x,k)| <-|iyexp( -|£|- ) j ^ ( y ) - U(y)|dy
Iterating (6.10), we find
|RN(x,k) - R(x,k)| < J^j-exp( j ^ ) | 11^(7) - U(y)|dy.
(6.11)
Similarly, we derive (6.11) with left hand side |LN(x,k) - L(x,k)| .
The
proof of (6.8b) is completed by taking x "*• °° in (6.11) and using (5.6). We shall apply the fundamental relation (6.7) for suitably restricted values of N,x and k.
First, we choose ft = {k| |k - k | < e}, e > 0, with
ft c £ + and det R_(k) + 0 for k e ft\{kQ}. By (6.8), there is an N Q , such that for N > N Q (i)
det R^(k) + 0 for k e 3ft
(ii)
|LN(x,k) - L(x,k)| < j- for x e I
(6.12a) and k e
ft.
(6.12b)
Finally, by virtue of Theorem 3, we can pick xfi such that for x > x_. and k € ft one has |L(x,k) - l| < — .
Combination with (ii) yields that for
N > N Q , x > x Q and k e ft, it holds that |LN(x,k) - l| <j9
so that ^(x,k)
is invertible. Now let us take N > N
and x > x Q .
For k e 3ft we can rewrite (6.7)
as follows: ^ k ) ; f V ( x , k ) [R^(k) J"1 = R^(k) [R^(k) f1 + [^(x,k) ]""1^(x,-k) . Since [^(xjk)]
(6.13)
^ p (x,-k) is continuous on ft and analytic on ft, we obtain
|
| R^kMRV)]"^,
3ft
(6.14)
3ft
where 3ft is traversed counterclockwise. Next we let N •> °°.
By (6.8) the integrand on the left converges uniformly
in k on 3ft, yielding ^ ( x . l O Y (x,k)R"1(k)dk = lim
f R^(k)[RN(k)]~1dk.
(6.15)
N
3ft 3ft A p p l y i n g Cauchy's residue theorem, w e find R"1(k)dk = 2-rri ^ ( X . I C Q ) \ ( x , k Q ) D(k Q ).
(6.16)
Hence E (2-rri)"1 lim
[ R^(k) [RN(k) T ^ k . (6.17) 3ft
294
Thus 7.
P. SCHUUR
(6.6) holds for x > x_. and therefore necessarily for all x e R. , THE GEL'FAND-LEVITAN-MARCHENKO
EQUATION.
In this section we shall derive a linear integral equation for N(x,s), the Fourier kernel introduced in
(4.11).
Let the potential U satisfy a growth condition of order 1.
We shall
make the following extra assumptions: det R_(k) + 0 for k e H \ { 0 } and for |k| < e,
Im k > 0
with e > 0 small enough
(7.1a)
lim R""1(k) exists k->0 " Im k>0
(7.1b)
All poles of R ~ 1 ( k ) in j£+ are simple.
(7.1c)
Since det R (k) is continuous on jfi,\{0} and analytic on £ , whereas lim det R (k) = 1, it follows from (7.1a) that det R (k) has at most |k|-*» Let us denote these by k., j = 1,2,...,N.
finitely many zeros in t,.
cause of (5.7) and Lemma 2, we can characterize k ,k ,...,k ments k of ~£+\{0] such that
Be-
as the ele-
(3.1) admits a non-trivial quadratically
inte-
grable solution ip, i.e. k-jk^, •••!&« are the bound states of (3.1) in j£,\{0}. We now introduce A (k) = R~ ( k ) , the right transmission coefficient
(7.2)
1
B r ( k ) = R + ( k ) R ~ ( k ) , the right reflection coefficient By (7.1) the matrix A k
k
#
^+\"f l> 2' •• A priori B
>k
N^
w i t h
is continuous on 0 + \{k sim
P
le
,k 2 ,...,k } and analytic on
poles at k ^ ^ , . . . ,1^.
is defined on H \ { 0 } .
(7.3)
Let us examine B^.
However, Theorem 4 tells us that
lim L(x,k) = I uniformly in k on (C, and thus for x_ sufficiently large we x-*» + 0 obtain from
(5.1a)
Br(k) = ^ 1 ( x o , k ) H ' r ( x o , k ) A r ( k ) - ^ 1 ( x o , k ) ^ £ ( x o , - k )
k 6 H\{0},
(7.4)
where the right hand side is continuous in k on all of ]R .
Therefore, in
a natural way we extend B
From
to a continuous function on H .
(4.6),
(4.7) and (5.4a,b) it is clear that
Br(k) - 2^7- [ J-oo
e"2ikyU(y)dy + 0( -y )
k + ±~.
(7.5)
k
2 2 Since U e L^ n L« ( R ) , Fourier theory implies that k B (k) is 2 x z r of L2 ( R ) with the property ,lim k BT.(k) = 0. Note that both |k|-K» r
an element -r and K.
INVERSE SCATTERING FOR MATRIX SCHRODINGER EQUATION k B (k) are quadratically integrable on |k| ^ 1. B
r
Hence by Cauchy-Schwarz,
is absolutely integrable on |k| £ 1. Thus we have 2 2 2 B r £ L^ n L* n C n (R ) and B r (k) = o( ~ ) for k -^ ±».
As a consequence, the function B
t 2 belongs to L n
H
295
(7.6)
defined by
f
zeB
r
(7.7)
2 n C^ (!R).
Next, we introduce the normalization coefficients C(k.), j • 1,2,...,N, determined as in Theorem 6, i.e. ^ ( x , ^ ) D(k.) - V D(k.) J
=
X
'
k
j
}
C ( k
j)
lim (k - k ) A (k). r k-^kj J
(7#8)
(7.9)
In terms of these we form the discrete counterpart of (7.7), namely N B d i s c r ( z ) - -i J
e l k j Z C(k j )
z € H.
(7.10)
Finally, we define B = B,. + B . discr cont
(7.11)
We shall call the aggregate of quantities {B (k),k.,C(k.)} the scattering data of the potential U. Our main result is now given by: THEOREM 7.
If the conditions (3.2) with I = 1 and (7.1) are fulfilled,
then the Fourier kernel N(x,s) introduced in (4.11) satisfies the following integral equation for x e 3R , s > 0:
f N(x,
N(x,s) + B(2x + s) + f N(x,t) B(2x + s + t)dt - 0, Jo
(7.12)
Jo
where the variable x appears as a parameter. The matrix B that governs this integral equation is completely determined by the following scattering data:
the right reflection coefficient B (k),
the bound states k. of (3.1) in ^ + \{0} and the normalization coefficients C(k j ). PROOF.
For a matrix $(x,.), depending parametrically on x, we shall
use the following notation of the Fourier transform
(R>)(x,k) - [ e"iks$(x,s)ds J— oo
k eH
(7.13)
296
P . SCHUTJR
f
oo
e iks O(x,k)dk
s 6 * .
(7.14)
_oo
The proof i s based on the fundamental identity (5,1a) which we rewrite as Y (x,k)A (k) = Y (x,k)B (k) + V x > " k > r r X/ r x> Substitution of the formulas ¥
—1kv
R, ^ 0 - e r X/ 2ikx RAr - e L B r + L,
where L(x,k) = L(x,-k).
- e
x e R, ikv
^
k e » .
(7.15)
(see (4.1)) yields (7.16)
According to Theorem 5, we can represent L as
L -. I + 27T F ^ J with J(x,s) = 0 for s < 0 and J(x,s) - N(x,s) for s > 0. Inserting this we obtain: RA
- I = e 2 i k x B r + 27Te2ikx(F"1J)B
+ Fj.
2 Note that each term in (7.17) belongs to L^ (-00 < k < +°°).
(7.17) Applying F~
to both sides, one gets F"1(RAr - I) - F" 1 (e 2 i k x B r ) + 27rF" 1 (e 2ikx (F" 1 J)B r ) + J.
(7.18)
We shall show that for s > 0 {r 1 (e 2 i k x B r )}(x,s) = B c o n t (2x + s)
(7.19a)
{2TrF"1(e2ikx(F"1J)Br)}(x,s) -
(7.19b)
N(x,t)B cont (2x + s + t)dt
N(x,t)Bdlgcr(2x+s+t)dt.(7.19c) Clearly the integral equation (7.12) is an immediate consequence of (7.18), (7.19). ad (7.19a)
Trivial, since the integral in (7.7) converges absolutely.
ad (7.19b)
Because the integrals in (4.11), (7.7) are absolutely
ad (7.19c)
Observe, that RA - I is continuous on (C \{k1 ,k«,.. .k^}
convergent, we may interchange the order of integration. and analytic on # \{k-,k2,... ,1^} with simple poles at ^ , ^ , . . . , 1 ^ , whereas RAr - I - 0( £• ) for |k| + «, Im k ^ 0.
Thus, using standard limit procedures, we find
by Cauchy's residue theorem: {F"1(RA
r
- I)}(x,s) = f- 27Ti 27T
N I lim (k - k )e1KS(R(x,k)A (k) - I) J r j-l-k^k.
INVERSE SCATTERING FOR MATRIX SCHRODINGER EQUATION = i
N I
e 1 1 C j S R(x,k.)D(k.)
= i
N I
ik. (2x+s) e J L(x,k.)C(k ) J
- j
297
J
N(x,t)B dlscr (2x+s+t)dt.(7.20)
D 1
We shall refer to (7.12) as the Gel fand-Levitan-Marchenko equation. Though it remains to be proved, we expect that under mild conditions, equation (7.12) is uniquely solvable for all x e K. .
In that case, one can
reconstruct the potential from the scattering data by the following procedure: Given the scattering data {B (k), k., C(k.)} one calculates the function B(z) = -i
N ik.z l f00 *lc I e J C(k.) + je 1 K Z B (k)dk Z7T J -Leo j=l
z € *
and then solves the Gelffand-Levitan-Marchenko equation (7.12).
(7.21) The po-
tential U is found from N by the formula U(x) = -2N (x,0) as we saw in (4.13).
In the literature, the above procedure is known as inverse scat-
tering. ACKNOWLEDGEMENT.
I would like to thank Professor W. Eckhaus and Dr. A. van
Harten for their enthusiasm, encouragement and advice.
It was the proof-
reading of their book Eckhaus and van Harten (1981), that formed the impetus for this research. REFERENCES CALOGERO, F. and DEGASPERIS, A. (1976) Nonlinear Evolution Equations Solvable by the Inverse Spectral Transform Associated with the Multichannel Schrodinger Problem, and Properties of their Solutions, Lett. Nuovo Cimento 15, 65-69. DEIFT, P. and TRUBOWITZ, E. (1979) Inverse Scattering on the Line, Comm. Pure Appl. Math. 32^, 121-251. ECKHAUS, W. and VAN HARTEN, A. (1981) The Inverse Scattering Transformation and the Theory of Solitons, North-Holland, Amsterdam. KAMIJO, T. and WADATI, M. (1974) On the Extension of Inverse Scattering Method, Prog. Theor. Phys. 52^, 397-414. WADATI, M. (1980) Generalized Matrix Form of the Inverse Scattering Method, In Solitons (Ed. R.K. Bullough and P.J. Caudrey) Topics in Current Physics 17, Springer Verlag, New York.
CHAPTER 16 A GENERAL n t h ORDER SPECTRAL TRANSFORM P. J. CAUDREY Department of Mathematics The University of Manchester Institute of Science and Technology Manchester, M60 1QD, England
1.
INTRODUCTION. The spectral transform was first used by Gardner et al. (1967) to
solve the Korteweg-de Vries equation qt " 6q q x - q x x x
(1.1)
Their method was to solve the Schrodinger equation ^ - J - (q(x) - C ) u and extract the "spectral data", S from this solution.
(1.2) Then they discov-
ered that if the "potential" q(x) evolved with respect to t according to (1.1) each item of spectral data evolved as the solution of a first order linear ordinary differential equation with t as independent variable. Thus they were able to find S at a later time and by using the Gel'fandLevitan-Marchenko equation to reconstruct q(x) at this later time. For some years it was thought that this was a freak result until Zakharov and Shabat (1971) solved the nonlinear Schrodinger equation
i
S + ^f+ M2* =°
d.3)
oX in a similar way using the spectral problem
9T+1Cui 3u 2 3x
= q(x)
V
i£uo = r(x)u- , 2 1
This was followed by the famous "AKNS" paper (1974) which firmly established the spectral transform (or inverse scattering method) as a very powerful weapon in the armoury of mathematicians and physicists who were studying the "soliton" (Bullough and Caudrey, 1980).
GENERAL SPECTRAL TRANSFORM
299
Matrix generalizations of (1.2) and (1.4) greatly extended their scope (Wadati, 1980; Calogero and Degasperis, 1980), but it was 1980 before a third order spectral transform was discovered (Kaup, 1980; Caudrey, 1980). Higher order ones have been found since then (Mikhailov, 1981). This paper describes a fairly general n (Caudrey, 1982) and is organized as follows.
order spectral transform Section 2 describes the
spectral problem and shows how it can be solved. the spectral transform and its inverse.
Sections 3 and 4 give
Section 5 shows that the original
spectral transform of Gardner et al. is just a special case of this general transform (as is that of Zakharov and Shabat).
Finally, Section 6
shows how a system of nonlinear Klein-Gordon equations can be solved. 2.
THE SPECTRAL PROBLEM. A general n t n order spectral problem can be written in the form of a
set of first order equations. |
+-B(x,C)}u
(2.1)
300
P.J. CAUDREY
the boundary conditions (i) and (ii) is equivalent to the Fredholm integral equation
^(x.s) - v ^ O + | i((x,y,C) • £,(y,C) dy
(2.5)
where the kernel is jL(x,y,£) = exp{(x - y) (A(£) - A. (£,)!)} • {6(x - y ) ^
- E 6(Re{A (C) - A.(£)})£. (O> • B(y,O (2.6) 3
j^i
X
J
A simple extension to the standard Fredholm theory shows that, provided there exists a real function k.(x,£) satisfying ...,n and
f
—oo J —<
k ± (y,O dy = M ± (C)
(2.7)
(2.8)
where M ^ C ) is bounded throughout the complex £-plane, then the Fredholm determinant f.(£) and minor F.(x,y,C) are defined.
Thus, except when
f.(£) = 0 the resolvent is uniquely defined by J^xyC) =f (C) F^x^C) F^x^C) J^x.y.C) =
(2.9)
and the solution of (2.5) is
[ R±(x,y,C) • ^ ( O dy
(2.10)
The Fredholm theory also shows that, provided K.(x,y,£) is analytic with respect to £, then so is ]L(x,y,O and hence $.(x,£) and ^ ( x , ^ ) except where f. (C) = 0. Thus the singularities in (j). (x,C) are of two types. There are poles at the zeros of f (£) and jump singularities where Re{A.(£) - A.(£)} = 0 for some j ^ i because JL(x,y,C) has jump singularities at these points.
The complex £~plane can be divided into regions so
that within each region the order of the numbers Re A (£), i = l,2,...,n is fixed.
As we cross from one region to another this order changes and
on the boundary Re A.(£) = Re A.(£) for at least one pair i ^ j. Apart from poles, the singularities of (J).(x,C) are confined to finite jumps on these boundaries.
Conditions (2.7) and (2.8) are sufficient but not neces-
sary for these results to be true.
They can be satisfied by imposing
appropriate conditions on J|(x,£;).
If we impose the further condition
(again sufficient, but not necessary) that M±(O + 0
as
C + °°,
(2.11)
GENERAL SPECTRAL TRANSFORM
301
we can show that £.(x,£) - v.(£) •* 0
as £ •> °°
(2.12)
This property will be needed later. 3. THE SPECTRAL TRANSFORM.
(k) Let them be at C. >
Let us now examine the poles of 6.(x,C). k = l,2,...,m..
For simplicity we will assume that they are simple, that
they do not occur on the boundaries between the regions and that only one JJ.(X,C) has a pole at a given point. Res ^. (x,£.
Then
) = E y. .
^.(x,£.
).
(3.1)
This follows from an examination of the Wronskian
exp[{Tr{A(O}x +
rx
Tr{B(y,C) }dy]det [v^^,... .vj
2) (3.
J —OO
Multiplying by (£ - ? . ^ )
and letting ? •* ? ^
det^ 1 (x,? 1 ( k ) ),i 2 (x,q ( k ) )
Res ^ ( x , ^ ^ )
gives ^ ( i , ^ 1 ) ] =0. (3.3)
(k) The result follows. The constants y.. , k = l,2,...,m.; i,j = l,2,...,n 1 (k) with the locations £.
of the poles form the discrete part of the spec-
tral data. To understand what happens at a boundary let us consider the following (non-rigorous) argument. As x -> -°°, 6. (x,£) behaves like exp(A x)v. + Z a exp(X.x)v.
(3.4)
where j runs over all values such that Re X. > Re A. and as x -* +00 it behaves like 3 ± ± exp(A i x)v i + E 3 ± j exp(A jX )v j
(3.5)
where 3.. ^ 0 and this time j runs over all values such that Re A. < Re A . On the boundary there are some values of j ^ i such that Re A.= Re A u £ + ^ and so we divide the numbers j ^ i into four sets J , J , J and J~ where if j e JU, Re A. > Re X 9 if j e J £ , Re A. Re A on one side of the boundary (labelled +ve) and Re A. < Re A. on the other side (labelled -ve) and finally, if j e j " , Re A. < Re A on the positive side and Re A. > Re A. on the negative side.
Thus, if we use the notation
302
P.J. CAUDREY
that a superfix ± denotes the limit of a quantity as the boundary is approached from the ±ve side, we have that as x -> -00. ^
) ^+
Z
a^ exp^x)^
(3.6)
jeJ u uJ + and exp(X x)v, + 1
x
Z a" exp(A x)v . J J jeJuuJ~ 3
(3.7)
Thus we can see that
it
^
S + ±j ij
and by examining the behavior as x -*• +00 we find that Q. . = 0 for jeJ thus ^
?) =
E j + Q ±j (?) i|(x,c) +
Z
Qi;j(?) ^ ( x , C ) .
(3.9)
This argument can either be made rigorous or the same result can be proved by some (extremely tedious) manipulation of the Fredholm equations. The quantities Q..(C) along all the boundaries form the continuum part of the spectral data S - (C i ( k ) , Y ± j ( k ) , Q ± J ( O ; i.J " 1,2,...,n, i + j, k « l,2,...,mi; (3.10) and £ runs over all the boundaries between the regions} The mapping from the space of matrix functions j^XjC) to S is the spectral transform. 4.
THE INVERSE SPECTRAL TRANSFORM. To invert the spectral transform we make use of the following proper-
ties of the ^ ( X J C ) .
(i)
i^x,^) - v ± (C) + 0
as C + °°
(ii) j^Cx.C) has simple poles at £ - C ± ( k ) , k = lf2,...,m± with residues: Res ^ ( x . q 0 0 ) - ^ Y ± j ( k ) exp{(Xj(Ci(k))-Xi(Ci(k)))x}$j(x,Ci(k))
(4.2)
(iii) On a boundary Z Q ±j (C) exp{(Xj(C) - X±(C))x} £^(x,C). These give
(4.3)
GENERAL SPECTRAL TRANSFORM Yl1(k)exp{(X1(Ci(k))-Xl(Sl<1
*
* Z
303
* ^
\
„ (fc) i
Q^(C') exp{(A (C) - X (
"^
V±T'
+
^(x.C')dC', (4.4)
where the integration is along the boundaries with the positive side on the left.
Giving 5 the appropriate values gives a set of linear matrix/ (k) i Fredholm equations with unknowns _*. (x,C. ) and _* (x,C f ). In practice there appears to be no difficulty but the existence and uniqueness of the solution to these has yet to be proved. hence j^.(x,£) throughout the C-plane.
Equation (4.4) gives ^ ( x , £ ) and Finally, the spectral equation (2.1)
gives B(x,£). 5.
THE SCHRODINGER SPECTRAL EQUATION. The Schrb'dinger spectral equation in dimensionless form is Jl 9 -2-ji- q(x)u = - T u 3x
(5.1)
This takes the form (2.1) if we put
2 giving
X 1 (C) - A 2 ( - O - iC
)md S=(x'C) = I and
I'
^ ( c ) = X 2 (-C) - |
(5 2)
j .
'
(5.V3)
In this case there are only two regions separated by the real axis. The kernels of the Fredholm equations are given by
K^x.y,?) = K2(x,y,-S) = | g 2 [ e(x - y) / - e"2±C(x "
y)
j
{6(x - y) - d d m OH \-ic
)] 0/
(5.4)
and we can put ±,(x,C) = i,(x,-C) = 1
2
/•(x,C) \ I \*x(x,C)/
(5.5)
Thus we need only consider the scalar function • (x,C) = e
i U
«(x,c)
(5.6)
304
P.J. CAUDREY
where $(x,£) satisfies the scalar Fredholm equation
lf f [ 6(x-y)-e" 2ii;(x " y) {e( X -y)-e(Im £)}]q(y) $(y,C)dy (5.7) For Im £ < 0 this is a Volterra equation and hence $(x,£) and (x,C) have no poles in this --plane. Also, if q(x) is real, the operator — = - - q(x) 2 9x 2 is Hermitian and hence has real eigenvalues -£ . Thus the poles occur on the upper half of the imaginary axis £ = in,,
k = l,2,...,m; n k real and > 0.
Res (|)(x,ink) = Y k 4>(x,-ink),
Y k real.
(5.8) (5.9)
The jump singularity of the real axis C = C(real) is given by <|>(xf5 + 10) - 4>(x,C - iO) - Q(£) *(x,-5 - ±0).
(5.10)
The spectral data reduces to S = (n k ,Y k ,Q(O; k = l,2,...,m; -°° < K < °°}
(5.11)
which is the standard spectral data for this problem. For the inverse problem (4.4) reduces to m
Y- exp(2n, x)
*(x,« - 1 + S ~ k-1
C
T £ — *(x,-inlc) k " \ (21?X)
(x,-5 - ±O)dE
(5.12)
In the lower --plane, $(x,Q can be Fourier transformed L(x,y) = ^
[[
e
i5(x
~ y ) {$(x,^ - 10) - l } d &
(5.13)
This vanishes for x < y and so the inverse is given by nd so + f
e " i C ( x " y ) L(x,y)dy.
(Im ^ < 0)
(5.14)
J —00
The Fourier transform of (5.12) is the Gel1fand-Levitan equation. fx L(x,y) - F(x+y) + | L(x,z)F(z+y)dz
(x > y)
(5.15)
J-00
where
m n k x 1 (°° -i£f*r F(x) = i Z Y e K - ±- \ Q ( ? ) e ^ X d? f k=l J -°°
The easiest way to find q(x) is to examine the asymptotic expansion
(5.16)
305
GENERAL SPECTRAL TRANSFORM
(5.17) Substituting in (5.6) and (5.1) gives q(x) = 2i ^ r
(5.18)
and from (5.12) a(x) -
E y , exp(2n,x)*(x,-in,) - ^ r k_^ K
K-
K.
Q( 0exp(-2i?x)(x,- E, -10)d£
Z/IIJ^
fX =-iF(2x) - i
L(x,z) F(z -f x)dz J —oo
= -iL(x,x- 0)
(5.19)
Thus q(x) = 2 ^ L ( x , x - 0) 6.
(5.20)
SYSTEMS OF NONLINEAR KLEIN-GORDON EQUATIONS. In this section, we show how the spectral transform can be used to
solve a system of nonlinear Klein-Gordon equations (6.1)
Xt where j = l,2,...,n and 9 ( 0 ) = 6 ( n ) and 6 ( n
+ 1)
= 6(1).
The spectral
problem is of type (2.1) with (Fordy and Gibbons, 1980)
o o
e o o c (6.2)
7
\;» and
diag(e
0/
v
x
x
x
(6.3)
Since ^(xjt,^) now depends additionally on the variable t (which is nothing more than a parameter as far as the spectral problem is concerned) the solution ii also depends on t.
We assume that this dependence takes the form u
where
= M(x,t,O
• u
(6.4)
306
P.J. CAUDREY 0
0 •\
0 3
2
'
(6.5)
V' 7 \. >
and aKJ'
.
(j = exp(eVJ
.
T M
.
a 0/ n-1 /
' - 6VJ/)
(6.6)
The compatibility condition for (6.4) and the spectral equation (2.1) I t - Mx = [M,A + £ ]
is (6.7)
which is equivalent to (6.1). The eigenvalues and eigenvectors of A(£) are A.(£) = co.^ and
(6.8)
where
0). = exp(2frij/n) are the n
roots of unity.
Thus the complex C~
plane is divided into 2n sectors with boundaries arg £ = m7r/n, m^Ojl^.^n-l. The t-evolution of the spectral data £ is easily found once it is realized that (6.4) is satisfied by (6.9) The results are
dt S
(t) = 0,
it^r «
(6.10)
0). -
(k)
Y,0W(t),
(6.11)
and (6.12)
GENERAL SPECTRAL TRANSFORM
307
Thus to solve the initial value problem for (6.1), we first take the spectral transform at the initial t with A(£) and ^(x,t,C) given by (6.2) and (6.3), then allow the spectral data to evolve according to (6.10) (6.12) and finally take the inverse spectral transform. REFERENCES ABLOWITZ, M.J., KAUP, D.J., NEWELL, A.C., and SEGUR, H. (1974) The Inverse Scattering Transform - Fourier Analysis for Nonlinear Problems, Stud. Appl. Math. 53_, 249-315. BULLOUGH, R.K. and CAUDREY, P.J. (1980) Solitons, Topics in Current Physics 17, Springer Verlag, Berlin, Heidelberg, New York. CALOGERO, F. and DEGASPERIS, A. (1980) Nonlinear Evolution Equations Solvable by the Inverse Spectral Transform Associated with Matrix Schrodinger Equation, Solitons, (Ed. R.K. Bullough and P.J. Caudrey), Springer Verlag, 301-323. CAUDREY, P.J. (1980) The Inverse Problem for the Third Order Equation u + q(x)u + r(x)u - -i£3 u, Phys. Lett. 79A, 264-268. CAUDREY, P.J. (1982) The Inverse Problem for a General N x N Spectral Equation, Physica 6D, 51-66. FORDY, A.P. and GIBBONS, J. (1980) Integrable Nonlinear Klein-Gordon Equations, Commun. Math. Phys. 77, 21-30. GARDNER, C.S., GREENE, J.M., KRUSKAL, M.D., and MIURA, R.M. (1967) Method for Solving the Korteweg-de Vries Equations, Phys. Rev. Lett. 19, 1095-1097. KAUP, D.J. (1980) On the Inverse Scattering Problem for Cubic Eigenvalue Problems of the Class ty + 6ty\) + 6Rip = \ty9 Stud. Appl. Math. 62, XXX X 189-216. MIKHAILOV, A.V. (1981) The Reduction Problem and the Inverse Scattering Method, Physica 3D, 73-117. WADATI, M. (1980) Generalized Matrix Form of the Inverse Scattering Method, In Solitons (Ed. R.K. Bullough and P.J. Caudrey) Springer-Verlag, 287-299. ZAKHAROV, V.E. and SHABAT, A.B. (1971) Exact Theory of Two-Dimensional Self-Modulation of Waves in Nonlinear Media, Zh. Eksp. Theo. Fiz. 61, 118-134 and Soviet Phys, JETP 34, 62-69.
CHAPTER 17 RECURRENCE PHENOMENA AND THE NUMBER OF EFFECTIVE DEGREES OF FREEDOM IN NONLINEAR WAVE MOTIONS A.THYAGARAJA Theoretical Physics Division Culham Laboratory, Abingdon, Oxon, 0X14 3DB, England (Euratom/UKAEA Fusion Association)
10
INTRODUCTION The physics of nonlinear wave motions is an extremely rich subject
with applications to many fields0
Several excellent accounts of it (eogo
Lighthill (1978), Whitham (1974) Yuen and Lake (1980) or Lonngren and Scott (1978)) are available in the literature.
These describe both the classical
foundations originating in continuum mechanics and electrodynamics and the more recent work on exact solutions by the inverse scattering transformo An extensive literature of numerical work including comparison with experiment also exists (see, for example, Yuen and Ferguson (1978), Martin and Yuen (1980) and Fornberg and Whitham (1978)). The aim of the present article is relatively modest: it is to describe recent results on the qualitative dynamics of nonlinear wave systems with particular emphasis on asymptotic or long time behaviour of the system motion.
This area is the
subject of active current research by many workers and hence no attempt will be made at a comprehensive presentation.
Instead, the first principles will
be emphasized and the standpoint restricted to elementary, analytical methods. The main theme of the article is simply stated in the form of a question: "under what conditions does the state of a nonlinear wave system recur?"
This question remains vague unless the class of systems of
interest is precisely specified0
Furthermore, one must be quite clear about
what is meant by fstate1 and 'recurrence of states f0 It is useful at this point to review briefly the historical origins of qualitative (or, equivalently, topological) dynamics0
Apart from common-
sense pronouncements like "History repeats itself", the concept of recurrence goes back to Poincare^s famous theorem on recurrent motions in bounded Hamiltonian systems (Poincare* (1899)).
This theorem states that in
a bounded Hamiltonian system with a finite number of degrees of
freedom,
almost all (in the sense of the invariant phase volume) initial phase points
RECURRENCE PHENOMENA IN NONLINEAR WAVE MOTIONS
309
will evolve in time such that they will return to an arbitrarily small neighbourhood of the initial point after a finite time interval„
The
1
'recurrence time will depend, in general, on the initial state and the pre-assigned smallness of the neighbourhood0
A readable account of this
theorem can be found in Kac (1959) and also in Halmos (1956).
The concept
of recurrence itself seems to be due to Poisson who was interested in improving upon Lagrange's work on the stability of the Solar system0 While Poincare*fs theorem applies to dynamical systems which preserve the phase-volume, Birkhoff (1927) proved a much deeper recurrence theorem for more general dynamical systems (still with a finite number of degrees of freedom)o
A complete exposition of BirkhoffTs results and other
relevant modern results can be found in the treatise of Nemytskii and Stepanov (1960)o
A brief statement will suffice for our purposeo
A motion is recurrent in Birkhofffs sense, if for any e > 0 , there can be found a T (e) > 0 such that any arc of the trajectory of this motion of time length T approximates the entire trajectory with a precision to within e with respect to a suitable norm defined on the state space of the system. Birkhoff showed that in general dynamic systems with a finite number of degrees of freedom, bounded motions are 'genericallyf recurrent; i o e., almost all such motions are recurrent in the sense definedo A crucial requirement in the recurrence theorems of Poincare" and Birkhoff is the fact that the number of degrees of freedom must be finite and that the motion must take place in a bounded region of phase space.
It
would seem at first sight that these theorems could not possibly be relevant to continuum systems which (at least formally) possess an infinite number of degrees of freedomo
Consequently, even if we consider closed systems (i.e.
those that do not interact with the external world) and energy is conserved, the distribution of energy among the infinity of available modes could change irreversibly in time.
This process is called thermalizationo
it occurs, the motions of the system are unlikely to be recurrent0
If Indeed,
it is not hard to see that continuum dynamical systems (linear or not) which relate to physically unbounded regions of space and therefore possess a continuous infinity of independent degrees of freedom do exhibit irreversible behaviour ('phase mixing1 or Landau damping) even in the absence of explicit energy dissipation.
However, the case is less clear cut if the
system is enclosed in a finite 'box1 or periodicity domaino
In many cir-
cumstances, although the number of modes of oscillation is infinite, one
310
A. THYAGARAJA
has now a countable infinity.
Yet even in this case, examples of non-
recurrent, dispersive approach to thermalization are known (Thyagaraja (1976); see po162 for a very elementary and explicit model)„
The purpose
of the present article is to exhibit several examples of interesting nonlinear wave systems which appear to have only a finite number of 'effective1 degrees of freedom (the actual number is a countable infinity) and which d£ exhibit recurrence of states.
All these systems are of great interest in
several fields of physics and engineeringo
In every case, the number of
effective degrees of freedom is a definite functional of the constants of the motiono Before we proceed to a study of specific examples, it should be noted that the mechanisms which limit the number of effective modes of a continuum dynamical system depend on whether the system is conservative or dissipative0
While it is noteworthy that the concept of the number of
effective degrees of freedom of a continuum system was first put forward by Landau and Lifshitz (1959) in connection with turbulent fluid flow (a driven, dissipative system) 9 in this review we mainly concentrate on nonlinear dispersive wave systems in which dissipation is ignorable to a good approximation0 The contents of the article are briefly summarized.
In the next
section the qualitative dynamics of the one-dimensional cubic nonlinear Schrbdinger equation is considered. the method are discussed. cated.
In Section 3 certain extensions of
Applications to problems in physics are indi-
In the fourth and final section the general implications of the
results are examinedo
Some of the major unsolved problems are listedo
The main conclusions are briefly re-stated0 2,
QUALITATIVE DYNAMICS OF THE NONLINEAR SCHRODINGER EQUATION As a first (and typical) example of what can be accomplished by
elementary analytical techniques, we consider the nonlinear Schrodinger equation with a cubic nonlinearity in a single space dimension,.
Since
recurrent behaviour is typical only for bounded or periodic domains, we restrict attention to the consideration of the initial value problem with periodic boundary conditions0
With a suitable choice of length and time
units, the equation takes the following form:
i | * . £ l + v |*|*,, 9t ax2 in the periodic domain o £ x ^ 1.
(2.1)
RECURRENCE PHENOMENA IN NONLINEAR WAVE MOTIONS
p is a real constant which can be positive or negative. of physical interest, it tends to be positive.
311
In applications
In problems where the
periodicity conditions are appropriate, we set Y(o,t) = n i , t ) ; y
t ) = V x (l f t).
O o
(2O2)
Most of the results apply with only very minor changes to the 'homogeneous1 case when one replaces (2.2) by either Y(o.t) - Y(l,t) - o;
(2.3)
Y (o,t) = ¥ (l,t) = oo
(2.4)
or,
In the present article, we assume without proof that equation (2O1) possesses suitably smooth solutions to the initial-boundary value problem formulated for appropriate initial data for |t| >o.
We further assume
that the solution is uniquely determined by the initial wave function ¥(x,o).
Our task is to examine and characterize the qualitative properties
of the solution.
This will be done by deriving certain elementary jl priori
bounds involving the integral invariants associated with equation (2 O 1). It is easy to verify by direct substitution that the functionals I(t) and J(t) defined below are constants of the motion.
I(t) * /V(x,t)|2dx,
(2.5)
o
J(t) s / V (x,t)| 2 -^ /1|Y(x,t)|ifdx. o
x
z
(2.6)
o
In fact, it is known from the work of Zakharov and Shabat (1972) that equation (2O1) possesses an infinite set of integral invariants, provided ¥(x,t) is sufficiently smooth. first two, I and Jo
However, we shall only make use of the
Let us now consider any function ¥(x,t) (not necess-
arily a solution of (2.1)) which is defined for |t| >o and sufficiently smooth,,
We require that ¥(x,t) satisfy the boundary conditions and evolve
in time in such a manner that the functionals I(t) and J(t) formed with it using the definitions Eq0 (2O5) and Eq0 (2O6) are constant in time. the initial values of these invariants are denoted I
and J
If
respectively,
we can derive bounds on the Rayleigh quotient Q(t) defined by, Q(t) I(t) s J k (x,t)|2dx. x o When y < o , it is plain from Eq. (2.6) that
(2.7)
312
A. THYAGARAJA
Q(t) < J O / I Q O
(2.8)
The situation is more complicated when y > o o
At any instant t, let x be
the point at which | ¥ (x,t)| 2 takes its minimum value0
We then have the
identity, ¥ 2 (x,t) = ¥ 2 (x Q ,t) + 2 f w d x
(2O9)
Clearly, Eq0 (2.9) leads to the inequality, 2j2j| W Y (x,t) | 2 < | V(x,t)| ,t)|2 + | W |dx. o x 0 o x 2 2 Obviously, |Y(X ,t)| <J |Y(x ,t)| dx<J Y(x,t)|2dx,
(2o10)
J | Wx |dx< I(t)Q' 2 (t) 0 (Schwarz inequality) 0 o
and,
Multiplying (2.10) by |Y(x,t)| 2 and integrating, we obtain the result, J1 |Y(x,t) l^dx < I 2 + 2I 2 Q 1/2 O o Substituting in (2O6) we get the quadratic inequality, Q(t) * ( J O / I O ) + (y/2) I o+ y I o Q / 2 ( t ) -
(2.11)
(2
-12)
The inequality (2 o 10) can be re-arranged to give, | Y ( x , t ) | 2 < I o + 2IQQ/2(t)0
(2 O 13)
We note that Q(t) is non-negative by definitiono 'solve
1
Hence, it is easy to
the inequality (2.12) and obtain the following results: Q(t) <M 2 (I Q ,J o ,y), IMI
2
«
^
Max | n x , t ) | 2 < I o<x<1
(2.14) (1+2M),
(2.15)
where M(I ,J ,y) is the positive root of the quadratic,
It is important to note that the results derived apply to any Y(x,t) which satisfies the boundary conditions and evolves in t such that I(t) and J(t) are constant. Although solutions of (2.1) belong to this class of
RECURRENCE PHENOMENA IN NONLINEAR WAVE MOTIONS
313
functions, clearly the equation has only been used in the results in a very weak sense0 Let us now investigate the principal implications of the J. priori bounds (2.14) and (2 O 15) O
First, we give an important definition embody-
ing the notion of Lagrange stability.
Definition 1:
We call a solution
¥(x,t) of (2oO Lagrange stable if there exists a constant K independent of t but possibly dependent on initial data such that |M|2 for all
=
Max |y(x,t)| 2
(2J-7)
|t| > 0 .
It is clear from (2O 15) that with K = I
(1 + 2 M ) , we have established
Lagrange stability of the solutions of the one-dimensional nonlinear SchrSdinger equation (2ol)o The interpretation of the inequality (2O14) is more subtle and involves the notion of the number of effective degrees of freedom of a solution V(x,t) of (2.1)o
Given any function ¥(x,t) which satisfies the
conditions necessary to obtain the foregoing bounds, we can always expand it in a Fourier series in x.
Thus, we have,
00
V(x,t) = I C n (t) exp (2mrix),
(2.18)
—00
where,
1
O
s
00
I \cn^ I2>
(2 0 19)
— 00
00
and
Q(t) = 4TT2 £ |c (t)|2n2
~°°
n
(2,20)
I Ic n (t)| 2
— 00
In analogy with quantum mechanics, we may interpret Q(t)/4ir2 as the instantaneous average of n 2 , M#2
is then interpretable as an upper bound
to the rms value of n, the number of modes carrying the "wave energy11 I If we call Q ^ / ^
0
the number of 'effective modes' and denote it by N
the inequality (2.14) says that N ~~ is bounded by a number M/« depends only on the initial values I
and J
which
of the integral invariants
and on the 'interaction strength', u. The boundedness of Q(t) has an interesting alternative interpretation,. Let € be any preassigned, arbitrarily small, positive number. Since, for positive integer N,
314
A. THYAGARAJA
"N -co
M2 n
4^2
I IC I2+ I iC I2 ) < ~ I o -»
n
N
n
'
4,2
°
If we choose N2(€) so that N2(€) >( - — ) f — V \ 47r2 / V e / O < I
o " I lCnl2<€-
(2 21)
-
-N
The inequality (2.21) holds uniformly for all t. 3e
DISCUSSION OF THE RESULTS AND GENERALIZATIONS We have established that every motion described by the equation (2O1)
with given I and J can be assigned N r , as a measure of the number of o o ° eff effective modes0
It is straightforward to apply Birkhoff's theorem to
conclude the generally recurrent character of the motiono We recall that the wave function ¥(x,t) has the Fourier expansion, oo
Hf = I C (t) exp (2niTix)0
(3 O 1)
—oo
Consider the partial sum ¥
defined by,
N % » ! C n (t) exp (2mrx), —N
where
(3.2)
N2(e) > ( £- ) ( .£ ).
We have proved that
Thus if we picture the system motion as taking place in the complex Hilbert space spanned by the infinite set of amplitudes Cn(t),Y(x,t) represents a point on a hypersphere of radius I
2
. ¥
represents a
point in a 2N+ 1 dimensional complex unitary sub space. point representing \
Furthermore, the
is restricted always to lie in the interior of a 1/ 1/ spherical annulus bounded by hyperspheres of radii I z and (I - €) . It
RECURRENCE PHENOMENA IN NONLINEAR WAVE MOTIONS
315
is evident that V always lies within an € - neighbourhood of Y respect to the usual norm of the Hilbert space.
with
Clearly Birkhoff's
theorem is applicable to the motion of ¥ . This fact implies the generally recurrent character of ¥, since ¥X7 is a uniform (in t) approximant of Y.
w
1
Consider the 'error
function E N (x,t) defined by E N (x,t) = Y - H^.
By definition, J !E N 2 (x,t)|dx<€ 0
and
(3.3)
J1|EN (x,t)|*dx = J J n 2 ||c c ||2++ || nn* |cj* |cj <<-£• IIQ
N ox The inequality (3.3) says that ¥
-°°
(3.4)
4TTZZ
is an approximant of V in the L« norm.
However, it is possible to deduce a much stronger resulto
Applying
inequality (2.10) to E (x,t), we get |E N (*,t)|2<e + 2 6 1 k | F I o 1 *
(3.5)
Hence as £ may be chosen arbitrarily small, and
|| Y - Y N I L =
Max
o<x<1
|Y-Y I is 0(€ / 2 ) o
We therefore have the result that ¥ in the sense of the maximum normo
approximates ¥ uniformly in x and t In other words, the Fourier series (3.1)
converges to Y uniformly, (not merely in the mean, as implied by (2 O 21)) O The system motion is effectively confined to a neighbourhood of a 2N + 1 dimensional hypersphere in Hilbert space. It is interesting to compare the results derived above using elementary inequalities with numerical simulations and experiment.
Yuen and
Ferguson (1978) considered spatially periodic solutions of (2 O 1).
They
numerically evolved initial data for which the linearized form of (2.1) is unstable to the modulational instability of Benjamin and Feir (1967).
The
long time behaviour of these solutions revealed that the energy sharing effectively occurred between a finite number of modes. depended on the initial datum.
The number
Furthermore, the solution 'reconstructed1
itself after appearing to show a tendency to 'break up' due to linear instability (Yuen and Lake (1980))0
Both the failure of the energy to
thermalize and the tendency to recur have been known since the early (and now classic) work of Fermi, Pasta and Ulam (1955).
More recently, Zabusky
and Kruskal (1965) and Fornberg and Whitham (1978) reported similar observations. Yuen and Lake (1980) in a series of careful experiments,
316
A. THYAGARAJA
showed that in circumstances where (2O1) is a good description of physical phenomena, recurrence is a generic phenomenon. For a long time it was thought mistakenly chat the recurrence properties of (2.1) are attributable to the fact that it possesses multisoliton solutions and can be solved exactly using the inverse scattering transform0
The thrust of the arguments developed here is to show that
recurrence of solutions has little to do with soliton solutions or exact integrability.
Thus these additional and strong requirements are not
necessary (although of course they are sufficient) to demonstrate the failure of thermalization or recurrence.
To see this, we only need
observe that any ¥(x,t) for which I(t), J(t) are invariant has been shown to have recurrent behaviour, not merely solutions of (2.1).
This point is
further demonstrated by certain other examples we shall consider in this section. The physical picture of the phenomena observed by Yuen et al 0 is relatively easy to understand.
Although nonlinearity renders the modes
with low n values unstable, and dispersion transfers the energy to shorter length scales, the "cascade" process does not go on forever.
Thus, if the
energy content of the sufficiently short wavelengths was small to begin with, it remains small for all time.
Since the total energy is constant,
it is perpetually redistributed among a finite set of modes and the motion overall appears recurrent0
In fact, this is precisely what is observed
in numerical simulations and in experiments (after allowing for the inevitable losses due to the small dissipation). Evidently the elementary bounds derived in this paper for quantities such as N
ff
(or Q(t)) are sufficient to explain the qualitative proper-
ties of the system motion.
However, it is interesting to examine whether
f
the inequalities are the best possible10
An attempt to answer this by
numerical studies was made by Martin and Yuen (1980)0
They considered a
class of initial data which represent modulations of a nearly monochromatic wave train and are as such infinitely differentiable and strongly localized in wave number spaceo
Their numerical work leads to
the estimate N
eff " ( ^
) (2^o)
(3
'6)
This is considerably more stringent (for large N -,.) than that implied by (2.14), i.e., N -- 2! ( Vi^ )pl • This discrepancy was resolved by the
RECURRENCE PHENOMENA IN NONLINEAR WAVE MOTIONS
author (in Thyagaraja (1981))o F(x,t) of (2.1).
317
Firstly, let us consider any solution
Let m be an arbitrary integer.
It is readily verified
2
that mF(mx,m t) satisfies the equation (2.1) and the boundary conditions (periodic!)o
If we keep F fixed and vary m through integer values, I
varies like m 2 as does Q.
It follows that Q is proportional to I for
large m and the Martin-Yuen estimate (3.6) applies to this special class of solutions, at least.
On the other hand it is not difficult to show
(cf. Thyagaraja (1981)) that initial wave functions strongly localized in position space, even if infinitely differentiable, must satisfy N .. £ I -s— J yl . Thus, in the sense that initial data are not particularly restricted, the estimates of the present paper are the 'best possible1o
It must however be remembered that in applications, intial
data localized in wave-number space are commonly encountered and for these the Martin-Yuen estimates give more useful results.
The Martin-Yuen
estimates can often be obtained by a heuristic physical argument. N
s the number
eff *"
of
Thus
linearly unstable modes clustered about the most
unstable linear mode.
One must be careful in using this argument in nu-
merical work as the linearly stable modes are coupled to the unstable ones and provide a stabilizing influence.
Furthermore, if the nonlinearity is
not weak, it is arbitrary to take only modes which are initially linearly unstable.
As a general rule, it seems advisable to monitor N
ff
(or Q(t))
throughout the calculation and make sure that the number N of modes used in the calculation satisfies N > N ,-,.. We now consider whether the results obtained for equation (2O1) (i.e., the one-dimensional, cubic nonlinear Schrb'dinger equation with periodic or homogeneous boundary conditions) can be generalized.
Plainly, if we
replace y in (2.1) by a bounded, periodic function of x, the results still apply.
Indeed, y can also be an essentially arbitrary, real, bounded
function of Y and ¥* without affecting the conclusions0 2
replace \i\v\
Thus we could
in (2O1) by V(x,Y,Y*) such that V is real and |v| < k |^|2
for all x,Y,Y* where k is the positive constant.
The following example
shows that there are other cases when the method yields useful and interesting results. . |f
= i
Consider, instead of (2.1), the equation 2 | 3x
+
(
|y|2 +
l ¥ |^ > Y >
2
with periodic boundary conditions; V p V ^ a r e r e a l constants,, It is easy to verify the following facts:
(3.7)
318
A. THYAGARAJA
a.
I(t) s J k| 2 dx is a constant of the motion: so is o
J(t) S /o V b.
^
x
| n x , t ) | 2 < 2I(t)Q1/2(t) o
/ |v|6dx < 2IQ /2 J k
o
o
where
Thus,
{
I(t)Q(t) • J
J
1/1
/'/
^Po
2>
^
Q < { ^ + u,IoQ1/2 }/(• " - £ I*)
Evidently when — s — I
(3.8)
< 1, we can infer the boundedness of Q as before,
irrespective of the size or sign of p.. ation can be obtained by the method0
If —^- I 2 > 1, no useful inform-
Indeed, in this case it seems likely
that Q(t) is unbounded and the motion is not Lagrange stable; it 'collapses1 (Goldman et al. (1978), (1980)).
We shall have more to say about collapsing
motions when we consider equations in higher space dimensions.
Returning
to one dimension, it is a trivial observation that the results derived apply to the linear Schrb'dinger equation as well.
However, recently Hogg
and Huberman (1982) have proved recurrence in the linear Schrb'dinger equation (not necessarily one-dimensional) for the case when the potential is actually time-periodic.
At the time of writing, this result does not
seem to have been generalized to the nonlinear Schrb'dinger equation, even in the one dimensional case.
The generalization needs more subtle methods
than those described here, since, in the time-dependent case, J(t) is not a constant of the motion. As a generalization in a different sense, the writer (Thyagaraja (1979)) showed that the Korteweg-de Vries equation can also be handled by this method.
Historically (Zabusky and Kruskal (1965)) recurrence
phenomena were first observed in numerical solutions of the Kortew#g~de Vries equation.
As this equation possessed multi-soliton solutions, it is
understandable that recurrence was thought to be due to this special feature of the equation.
Our examples amply demonstrate that recurrence
RECURRENCE PHENOMENA IN NONLINEAR WAVE MOTIONS
319
is a very general phenomenon in wave motions and is not necessarily associated with exact integrability (although the latter feature enables us to describe the recurrent motions as almost periodic; Lax (1976)). We now investigate properties of the nonlinear SchrSdinger equation in higher space dimensions.
It is easy to see that the boundedness of Q
or I (suitably interpreted as multiple integrals) does not necessarily imply the boundedness of ¥ itself.
For instance, in three dimensions, a
function behaving like log |r| near the origin is unbounded whereas both J M 2 d v and J|w| 2 dv are bounded.
In fact, Zakharov (1972) showed
that in more than one spatial dimension, there are Lagrange unstable or collapsing solutions, at least in the infinite space domain.
It is not
yet known whether collapse can occur in bounded or periodic domains. However, it is plain that the methods of the present paper cannot apply as they stand to higher dimensions.
On the other-hand, the writer
(Thyagaraja (1981)) proved a result valid in any number of dimensions: if y > o , and I(t) = J|¥ |2dv, J(t) = J|v^|2dv - u / l ^ d v are constants of the motion, all noncollapsing motions have a bounded Q(t) and are hence generically recurrento
The proof is very simple.
collapsing motion, Jl^l^dv is bounded0
If ¥(x,t) is a non-
It follows that Q(t) is bounded
and the number of effective modes is finite.
Application of Birkhoff's
theorem establishes recurrenceo It is of interest to seek conditions under which collapse can occur. We note that there are two types of collapse: it seems natural to call a motion weakly collapsing if ¥(x,t) is defined for all t >o but ||^|1^ is unbounded; in contrast to this, a strongly collapsing motion is such that there exists a t z
Lim
c
crit
.. >o such that crit
I M I =«
One can draw some elementary but illuminating deductions about collapse in bounded or periodic domains from the invariance of I(t) and J(t). Consider two space dimensions. of ¥.
2
Collapse requires the unbounded behaviour 2
However, J|^| dv s /|^| rdrd0 must be bounded, where it is con-
venient to shift the origin to the singularity (there may be more than one and the position of these might well vary in time; these complications do not alter the argument)o r>o.
|^|2 is a periodic (bounded) function of 0 for
Since the integral I converges, |y|2 could only be unbounded (at
most) like
— r
where v < 1.
This means that | w | 2 is unbounded like
320
A. THYAGARAJA
v and l^l4 like — ^ — — „ •
Since J is a constant of the motion, if
Jl^l^dv is unbounded, its 'singular part1 must be exactly cancelled by that of /|w| 2 dv.
For this to be the case, it is necessary that
v + 3 = 2v + 2, i.e., v = l0
We have seen that for I to be bounded, v < 1.
Thus, we conclude that collapse cannot occur in two dimensions for the cubic nonlinear SchrSdinger equation. The same argument leads in the case of three space dimensions to |y|2 varying like 2
Iv^l varies like 1 1
-»« with v< l0
4
. and l^l like —* /0 v+4 ' ' 2v + 4 r . r
This means that |v|
Then
We then obtain v = o o
at most near singularity0
Thus, in three space
dimensions, the cubic nonlinear SchrSdinger equation could have collapsing solutions although it is not yet known which (if any) initial conditions lead to such motions.
Returning to the two dimensional equation,
it seems that this is a critical case in the following sense: suppose Q(t) is boundedo |V¥|2 ~ —
Then J|w| 2 dv must converge«
where X < 2 O
|¥| must be boundedo
At a singularity (if any)
This means that \v\2 ~ r
+ Const; clearly
Although this does not amount to a rigorous proof,
it seems that in two dimensions, the boundedness of Q does imply that of |\v\l^o
The same argument fails in higher dimensions.
The ideas discussed in this section have many applications.
Some of
these relate to the behaviour of modulational instability and filamentation in plasma physics.
Bingham and Lashmore-Davies (1982) review
this work in their contribution to this volume.
The following application
to inviscid, two dimensional hydrodynamics is typical. Consider a bounded, two-dimensional region R. inviscid, incompressible fluid fills R. 3R) is assumed to be a streamline.
We assume that an
The boundary of R (denoted by
This continuum system is described by
the Eulerian equations of motion:
i£. + a"> = o 8t
where
3(x,y)
(3.9)
*
¥ is the stream function and w, the vorticity.
The vorticity is
related to Y by the Poisson equation, 10 = V2 **
(3.10)
The boundary condition is ¥ = 0 on 9R. It is well-known that J|V¥|2dA, /wndA (n= 1,2..) are a family of constants of the motion for this system. 2
Let us consider the energy /|w| 2 dA s I
the so-called enstrophy Ja) dA » J.
In view of (3•10) it is
natural to
RECURRENCE PHENOMENA IN NONLINEAR WAVE MOTIONS
321
expand ¥ in terms of the solutions of the eigenvalue problem, V 2 $ - -X 2 $, with $ s 0 on "3R.
(3O11)
These can clearly form a complete orthonormal set.
Thus, we have the
expansions, no
n=l
n
n
00
a) • The X
I -X 2 C (t)* . u . n n n n= 1
rise monotonically with n. 00
(3813) It is clear from the invariance of
00
I 2 Y X 2 C 2 and J 3 > X^C2, that the number of effective modes is n n n n ^ n-\ n-1 limited and is indeed a constant of the motion? A finite set of modes can describe the motion which must consequently be recurrent. The example illustrates that making the system inviscid is a decisive approximation.
If one wishes to describe two-dimensional turbulent flow
properly, both viscosity (however small) and the imposed external forces (necessary to obtain a statistically stationary state) must be taken into account.
Incidentally, it should be noted that the present analysis would
not work in three dimensional Euler flow.
The reason is that in three
dimensions, although the kinetic energy is conserved, there is no conservation of enstrophy. 4.
In fact, vortex stretching amplifies vorticity,
CONCLUDING REMARKS The possibility that the dynamics of nonlinear wave motions can be
understood by the study of a finite number of modes is of great interest from the point of view of numerical simulation.
In effect, the problems
of continuum mechanics are rendered more manageable by a reduction to ordinary dynamics.
However, the practical application of this principle
requires some subtlety. A physical system can be adequately described by conservative equations only in an approximationo
Such approximations are, by their
nature, not valid for very long time-scales.
Thus predictions about the
long time behaviour of conservative systems are only valid provided the recurrence times are not so long that the neglect of dissipation becomes invalid.
It has been shown that in conservative systems, the existence of
invariants plays a vital role in establishing recurrence properties. Dissipative systems also exhibit recurrence behaviour for quite different reasons.
In such systems, the dissipation rate is essentially fixed by
322
A. THYAGARAJA
the energy input to the system.
The system is in a statistically
stationary state which is once again characterized by a finite number of effective modes.
These ideas were due originally to Landau and Lifshitz
(1959) but have been developed and applied recently by Tre"ve and Manley (1982).
In physical applications, the neglect or otherwise of dissipation
must be carefully considered in every case.
The ultimate arbiter is of
course experiment. It is perhaps useful in a review such as the present one to list some of the unsolved problems of the subject. character of recurrent motions.
The first of these concerns the
We have seen that Lagrange stability
implies the boundedness of N ~r and the recurrence of initial stateso This property corresponds to Poisson1s notion of stability (Nemytskii and Stepanov (I960)).
Thus if P
represents an initial state and N is a neigh-
bourhood of P , we call P Q Poisson stable if for any T > o , there exists a t such that |t| >T and P
belongs to N.
In other words, however long we
wait, the system does not wander off but eventually returns to an arbitrarily small neighbourhood of the initial state.
Birkhoff recurrence
implies Poisson stability in general, and is possibly best visualized in terms of it.
It is an interesting fact that Poisson stable motions need
not in general be quasi-periodic or even almost-periodic.
Thus, Poisson
stable motions cannot be represented in general as a finite or infinite sum of periodic functions.
According to a theorem of general dynamics
(c.f. Nemytskii and Stepanov (1965)), a motion must be Lyapunov stable in order to be almost periodic.
This means that any two sufficiently close
initial phase points must remain close to each other for all time.
This
notion of stability is clearly much stronger than Lagrange or Poisson stabilityo
It is not known whether the solutions of the nonlinear
SchrSdinger equations are generally Lyapunov stable.
Lax (1976) suggests
that in the one dimensional case where for suitable initial data, the equation (2.1) can be shown to be exactly integrable, the solutions are almost-periodico
If this were indeed the case, in numerical solutions
this fact would manifest itself in power spectra with a series of sharp peaks;
the continuous component would be absento
At present it is not
known in higher dimensions which initial data (if any) lead to Lagrange stability and under what conditions almost-periodic recurrence is expected. Furthermore, there is at present no algorithm for estimating the recurrence time T(€) in terms of N
ff
and €.
RECURRENCE PHENOMENA IN NONLINEAR WAVE MOTIONS
323
The second problem of interest concerns the statistical characteristics of the solutions of nonlinear wave motions0
The simplest question
one can formulate in this area is the following one: given that the Fourier expansion (3»1) represents the wave function ¥, do averages such as
exist?
If they do, can they be calculated in terms of I ,J
and y with-
out actually solving the problem for C (t) and evaluating (4.1) numerically?
The averages need not of course be 'equal time1 ones; in
fact, the time correlation function defined by the equation,
$
mn (t) " T ^ i J C m (u + t)C;(u)du
is of great interesto
(4.2)
Its Fourier transform supplies complete information
about the spectral characteristics in the asymptotic limit„
As far as is
known to the writer, even the existence of the relevant time averages has not been theoretically establishedo
Furthermore, at present the only
method of calculating such quantities is by solving for C (t) numerically and evaluating the mean values directly0
Such evaluations suffer from
the usual difficulties of numerical analysis, particularly those associated with the asymptotic limit of large time0 As a final problem, we might mention that sufficiently reliable numerical techniques (particularly error estimates and stability limits) ara yet to be established for the calculation of solutions in the asymptotic limito
Careful control of numerical errors is necessary if one wants to
obtain reliable results0 In conclusion, it has been shown that many nonlinear wave systems behave as if their effective number of degrees of freedom is limited.
This
finite set of modes interact nonlinearly, perpetually exchanging energy with each other.
To a good approximation the total energy of the system
is a constant on timescales of experimental interesto
This results in a
recurrent motion which may in special cases be periodic or almost-periodic, but generally involves continuous power spectra0
The problem of the
analytic characterization and effective calculation of the power spectra and the average energy distribution among the modes remains open. ACKNOWLEDGEMENT:
The author thanks his colleague, Dr. C.N.Lashmore-Davies
324
A. THYAGARAJA
for his active encouragement during many phases of the present worko REFERENCES BENJAMIN, ToBo and FEIR, J 0 E 0 (1967) The Disintegration of Wave Trains in Deep Water, Part 1, Theory, Jo Fluid Mech0 27, 4l7-430o BINGHAM, Ro, and LASHMORE-DAVIES, C 0 N 0 (1982) To be published in ADVANCES IN NONLINEAR WAVES, edited by L. Debnath, Pittman Publishing Company, 1983. BIRKHOFF, GoDo (1927) Dynamical Systems, American Mathematical Society, New Yorko FERMI, Eo, PASTA, Jo, and ULAM, So (1955) Studies of Nonlinear Problems, reprinted in Lectures in Applied Mathematics (Volo 15, 1974) American Mathematical Society, New Yorko FORNBERG, B., and WHITHAM, G O B O (1978) A Numerical and Theoretical Study of Certain Nonlinear Wave Phenomena. Philoso Trans0 Roy. Soc. A289, 373-404o GOLDMAN, M.Vo, and NICHOLSON, D O R. (1978) Virial Theory of Direct Langmuir Collapse0 Phys0 Rev0 Letts0 41, 406-410. GOLDMAN, M.Vo, RYPDAL, K o , and HAFIZI, Bo (1980) Dimensionality and Dissipation in Langmuir Collapse0 Phys0 Fluids. 23, 945-955. HALMOS. P.R. (1956) New York.
Lectures on Ergodic Theory, Chelsea Publishing Co.
HOGG. To, and HUBERMAN, BOAO (1982) Recurrence Phenomena in Quantum Dynamicso Physo Revo Letts, 48, 711-714. KAC, Mo (1959) Probability and Related Topics in Physical Sciences, Interscience Publishers Inco New Yorko LANDAU, L.D., and Lifschitz, E.M0 (1959) Fluid Mechanics, Vol o 6 of Course of Theoretical Physics, (Trans. J O B O Sykes and W.H o Reid) o Pergamon Press, Londono LAX, P.Do (1976) Almost Periodic Behaviour of Nonlinear Waves, in Surveys in Applied Mathematics (Eds. No Metropolis, So Orszag and G.-C. Rota) 259-270o Academic Press, New York LIGHTHILL, Jo (1978) Cambridge0
Waves in Fluids, Cambridge University Press,
LONNGREN, K., and SCOTT, A. (1978) New Yorko
Solitons in Action, Academic Press,
MARTIN, D o Wo, and YUEN, H 0 C o (1980) Spreading of Energy in Solutions of the Nonlinear SchrSdinger Equation. Phys. Fluids, 23 0 1269-1271„ NEMYTSKII, V0V., and STEPANOV, V O V O (1960) Qualitative Theory of Differential Equations, Princton University Press, New Jerseyo POINCARE, Ho (1899) Les Me*thodes Nouvelles de la Me*canique Celeste, Volo 3, Cho26o (Repreinted 1957) Dover Publications Inco New York. THYAGARAJA, Ao (1976) Prototurbulent Motions in Dissipative Model Systems, Math0 Proco Camb0 Philo Soc. 80, 153-163O THYAGARAJA, Ao (1979) Recurrent Motions in Certain Continuum Dynamical Systems. Physo Fluids, 22, 2093-2096
RECURRENCE PHENOMENA IN NONLINEAR WAVE MOTIONS
325
THYAGARAJA, Ao (1981) Recurrence, Dimensionality, and Lagrange Stability of Solutions of the Nonlinear Schrodinger Equation0 Physo Fluids, 24. 1973-1975O TR^VE, YoMo, and MANLEY, 00Po (1982) Galerkin Approximations for Benard Convection, Physica Do 42, 319-342o WHITHAM, G.B. (1974)
Linear and Nonlinear Waves, John Wiley, New York,
YUEN, H o C , and FERGUSON, W.E. (1978) Relationship between BenjaminFeir Instability and Recurrence in the Nonlinear SchrSdinger Equation. Phys0 Fluids, 21, 1275-12780 YUEN, H.C. and LAKE, B0M. (1980) Instabilities of Waves on Deep Water, in Ann. Rev0 Fluid Mech., _T2, 303-334. ZABUSKY, N.J., and KRUSKAL, M 0 D 0 (1965) Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States0 Phys0 Rev. LettSo 15, 240-2430 ZAKHAROV, VoEo, and SHABAT, A.BO (1972) Exact theory of Two-Dimensional Self-Focusing and One-Dimensional Self-Modulation of Waves in Nonlinear Media. Soviet Physics JETP, 34, 62-69. ZAKHAROV, V.E., (1972) 35, 908-914.
Collapse of Langmuir Waves.
Soviet Physics JETP,
CHAPTER 18 ACTION-ANGLE VARIABLES IN THE STATISTICAL MECHANICS OF THE SINE-GORDON FIELD R. K. BULLOUGH and D. J. PILLING Department of Mathematics, UMIST, PO Box 88, Sackville Street, Manchester M60 1QD, U.K.
and J. TIMONEN
Department of Physics, University of Jyvaskyla, SF-4O72O Jyvaskyla, Finland and NORDITA, Blegdamsvej 17, Copenhagen 0, Denmark. 1.
INTRODUCTION The sine-Gordon field in one space dimension and covariant form has
Hamiltonian HL"4>] = y ^ 1 / {Jn 2 y o 2 + Jc|>x2 + m 2 (1-cos 4>) dx} In this Hamiltonian chosen that 3<j>/3x.
y
.
(1.1)
is a dimensionless coupling constant, units are so
11= c = 1,
and
m
is the mass of the field:
It is easily checked that if
{II(x), <J>(xf)} - 6(x-x f ),
so that the Poisson bracket
by <j>
we mean
(II(x), <J>(x)) are conjugate variables, Hamilton's equations
yield the familiar sine-Gordon equation ^xx " *tt
= m2
Sin
(1
•
*2)
It is well known that the spectral transform which solves the sineGordon equation <|>§n - m2 sin
(1.3)
in light cone co-ordinates is one example of the now famous Zakharov-ShabatAblowitz-Kaup-Newell-Segur (ZS-AKNS) or Dirac spectral problem (see, for example, Bullough and Caudrey (1980) and especially the references to ZS and AKNS there).
Indeed the solution of (1.3) by the Dirac spectral pro-
blem was published first by AKNS (1973a) before the more general case was described (AKNS, 1973b). It is also well known that the spectral transform is a canonical transform (Flaschka and Newell, 1974; 1980).
In the case of
(1.3) new canonical co-ordinates can be found so 1
that the 'total momentum ates
Dodd and Bullough, 1979, Faddeev,
P = / { £2d£; and
H,
in light cone co-ordin-
(£,n),can be expressed in the forms
"k - •
\
£K p ' j=l J
+
IK p j=l J
+
n
b
£D n ' j=*l J
r +
;
-o
(1.4a)
STATISTICAL MECHANICS OF SINE-GORDON FIELD
16m2Y-2 sin2G' n/"1 + f ^ k k
°
P (O
327
.
(1.4b)
i <5)
By taking the Lorentz covariant combinations |{m 2 (p(£)) -1 ± p(£))> etc., defining a new mass
mf = 2m, and a new coupling constant
then dropping this primed notation one finds
y ' = 4y , and
(Bullough, 1980) that one
can obtain the following Hamiltonian in covariant form
+ Evidently The number
E b (4M2 sin 2 0. + n 2 ) 1 / 2 J J j=l
+( -oo
j
(1.5) is an equivalent of M
is a rest mass:
(m 2 +(p(O) 2 ) 1 / 2 P(OdC
.
(1.1) under canonical transformation.
M = Smy""1.
The kink solution
4 = 4 tan*"1 exp {m(x-Vt) / (1-V 2 ) 1 / 2 } of (1.2) also has rest mass turn 2
IL . . = MV(1-V 2 )" 1 ' 2
(M + JL .
2
1
2
) ' .
<|> = 4tan~
1
U-5)
M = 8my ~ 1 o
(1.6a)
and there is an associated momen-
so a kink moving with velocity
V
has energy
The same argument applies to the anti-kink exp {-m(x-Vt) / (1-V2)1/2}
(1.6b)
while a similar one can be made for the breather solution (j) = 4 tan"1 [tan 0 sin {mcos 0 (t-Vx) (1-V2)""1/2}
•
sech {msin 0 (x - Vt) (l-V2)"1/2} which has energy
(1.6c)
{4M2 sin2 0 + IL 2 } x / 2 . orea
One would conclude from the results (1.6) that (1.5) is the sum of independent kink, antikink, breather and 'radiation1 parts —
the radia-
tion part being the integral in (1.5) which is associated with the continuous part of the eigenspectrum of the spectral problem.
However, the
co-ordinates in (1.5) are actually collective co-ordinates with no indication of any interaction between them via phase shifts.
In contrast, the
multisoliton solutions of (1.2), which can be derived from (1.5) by inversion, contain explicit phase shifts induced between any pair of solitons, kinks, antikinks or breathers Gibbon, 1975):
(see e.g. Caudrey, Eilbeck and
these solitons also phase shift the small amplitude
328
R.K. BULLOUGH, D.J. PILLING and J. TD1ONEN
harmonic modes which, for small enough amplitude, become solutions of (1.2) (Rubinstein, 1970; Currie, 1977). identification
Thus it is not correct to make a complete
of (1.5) with contributions from
n,
kinks,
it- antikinks,
and so on, if these kinks are the kinks (1.6a) the antikinks (1.6b), and so on. The set {p., p , II , 4y "* 1 0 O ; P(O> J
K
O
Kr
actually forms a set of action
J6
variables for the sine-Gordon system (1.1) and, for example, the quantities 4y ~l 0, O
are collective co-ordinates representing the internal momenta of
K
the breathers.
The conjugate variables are
that {p., q.f } = 6 . . t , etc. and
{q., q* , 0 , $ ; Q(£)}
f
t
{P(O, Q(S )> = 6(£-£ ).
The phase spaces
-°° ; o <
prove to be 1
<2TTY O " ; 0 < $ £ l <8TT; Bullough, 1980),
f
0 < P(£) <°°, 0 < Q(C )<2iT
such
^ " ^
(Dodd and Bullough, 1979;
Faddeev and Takhtadzhyan (1974) first found (1.5); Flaschka
and Newell (1975) stressed the Hamiltonian structure of flows described by the AKNS spectral problem.
Zakharov and Faddeev (1971) first demonstrated
that the Korteweg-de Vries equation
u
+ 6uu
+ u
= 0 was completely
integrable with an infinite dimensional Hamiltonian like (1.5) expressible in terms of a complete set of action variables (momenta) only.
Our own
demonstration (Dodd and Bullough, 1979) that all AKNS systems had the same property was slow in publication; but the results (1.4a, b) which lead to (1.5) were reported in 1974 and 1975, and appeared subsequently in meetings1 proceedings (Bullough and Dodd, 1977,1979). The Hamiltonian (1.5) has several remarkable properties; noted it appears to describe
n, kinks,
a continuous infinity of harmonic modes;
n^
antikinks,
(i)
as
nfe breathers, and
these objects are independent in
the sense that no interactions or phase shifts are present, (ii)
The expression
Q
{m2 + (p(£))2}l/2P(£)d£
is just the Hamiltonian
for the free Klein-Gordon equation in action-angle variables; so the canonical transformation from (1.1) appears to linearise the problem and set the 'nonlinearity1 in the remaining terms for kinks, antikinks, and breathers. (iii) The coupling constant 1
M = 8my
(iv)
appears (correctly) to play no role in
part of the problem; but it remains in the 'nonlinear1 part
the 'linear in
y
-1
o
.
The kinks, antikinks, etc. in (1.5) are independent in the sense that
each depends on a single action variable; the set of action variables is
STATISTICAL MECHANICS OF SINE-GORDON FIELD
329
complete since complete integrability can be demonstrated by integrating Hamilton's equations from (1.5) and then using the Marchenko equation to invert the problem.
Thus the set of action variables forms a complete set
of first integrals, and the number of degrees of freedom for the field <j>(x) (-«> <
x
<«>)
i s correctly counted:
no degrees of freedom are lost in the
canonical transformation from {II(x), <J>(x); -°° < x <»} to
j, j 1 , k, k 1 , A, £' (> 0 ) 6 , 2 ; - ~ < S < ~ } .
(1.8)
This correct counting of degrees of freedom is important to what follows. We turn now to the statistical mechanics of the sine-Gordon (s-G) field.
An important object to compute is the partition function
Because of the continuous infinity of degrees of freedom nal integral.
I
where 1
jtj
is a functio-
We shall take it to be
:J9]
3 = l/k_T, l
Z
Z.
(1.9)
k,. is Boltzmann's constant and
means functional integration.
ment on the form adopted for
Z
For the s-G
T
the temperature;
H[<|>] is (1.1).
We com-
in (1.9) later (see §6 and the APPENDIX).
Here we want to explain the essential character of the problem studied (necessarily less than comprehensively) in this paper. the following one:
since
This problem is
(1.9) is defined through (1.1) which can be
written in the form (1.5), and since the transformation involved is canonical (with determinant unity), can (1.9) be evaluated in the form
W V
5
f £ *j d«j k J d*k *t jj exp-3H
where
H
(1.10b)
is given by (1.5) and a suitable measure is chosen for the final
functional integral?
The form of (1.10) is intuitive and the combinatorial
factor avoids overcounting of kinks, antikinks, etc. an unimportant one:
The question is not
it raises further questions about changes of variables
in functional integration; but it also offers in the results (1.10) means of evaluating partition functions for integrable fields which can be extended to the quantum case. presented in
§6
This is already apparent from the analysis
below (and the APPENDIX) and more details are given in
our forthcoming paper (Builough, Pilling and Timonen, 1983).
330
R.K. BULLOUGH, D.J. PILLING and J. TIMONEN
At first sight the answer to the question posed is a free energy
1
- 3"" In
Z
from
Z
f
No f .
One derives
and, from this, number densities (num-
bers of particles per unit length) of kinks, anti-kinks, etc. in thermal equilibrium at temperature 00
(L + )
T.
But the support -|L <x < iL
is infinite
so that finite number densities require infinite numbers of parti-
cles.
On the other hand the canonical transformation from (1^1) to (1.5)
(as analysed by Dodd and Bullough (1979) anyway) is worked in light-cone co-ordinates and uses the infinite support on the variable £ with / (1+|C | )r (£)d < °° for some
k.
Evidently this condition does not per-
s
—oo
mit any but a finite set of kinks (for which asymptotically 1
2msech m(£A + nX"" )).
<J>r(5) ~
The use of light cone coordinates is a complication.
But the conclusion must be that the 'exponential1 type boundary conditions on infinite support associated with the spectral transform lead to zero soliton (kink, antikink, breather) densities.
We therefore need periodic
or box boundary conditions, for which action-angle variables are scarcely available, followed by a thermodynamic limit to finite particle densities. Even so we believe that the Hamiltonian (1.5) can actually be used to compute the partition function Z.
The argument starts from the Hamiltonian
(1.5) and uses the action-angle variables. we need to discretize it.
But in order to define (1.9)
This then involves a finite number of degrees
of freedom and a finite number of action-angle variables.
Consequently we
find an effect on the measure associated with canonical transformation to this finite number of variables. finally goes over to (1.8). (1.9) survive.
Of course this finite set of co-ordinates
But consequences to the functional integral
These consequences are that contributions of the free (if
collective) kinks, antikinks and breathers to (1.10) are 'dressed1 in a procedure which in some sense effects the change from 'exponential' to periodic boundary conditions.
The further consequence is that, while
(1.10a) proves to be a correct form from which to compute
Z, the separable
form of (1.5) does not mean that Z itself is in general expressible in simple separable form: and
3
-1
certainly it is not the case (except for T •* 0
<< m as we shall show for the kinks at the end of §6) that
exp-3 J(m2+ (p(0«l/2P(OdC with, in particular, Z. = 7Tk
27r
dq
LIT
J
Z,
U.ID
given by the simple expression dp
exp [-3 (M 2 +p 2 ) x / 2 ]
(1.12)
STATISTICAL MECHANICS OF SINE-GORDON FIELD
as might have been expected (the 2TT arises because
"h = 1
331
so
h = 2TT).
The formidable piece of analysis which leads to this conclusion res* olves a lot of problems but (at the time of writing) still contains two difficulties.
These concern discrepancies between certain sets of coe-
fficients mentioned again in
§3.
The purpose of the present paper is
therefore to use the case of the kinks to show how, in principle, all such problems may be resolved (for the kinks one discrepancy between coefficients still remains).
The case of the breathers is more difficult than the kinks
in that a unique expression for the whole contribution to a function of
3
is still to be found.
theory itself, are treated elsewhere
Z(n, ,nr-,n, ) as
This aspect, and the general
(Bullough, Pilling and Timonen, 1983).
One particular result of the theory is a justification of the form (1.10a) and particular expressions for factored from
Z(n, ,n=-,n, )
general given by (1.12).
Z(n, ,nr-,n, ) .
We also show how a Z^ can be
as in (1.11) although this
Z,
is not in
The present paper already shows how such a factor-
isation becomes possible and is concerned to calculate expressions for Z, itself valid in different ranges of temperature. 2.
NAIVE CALCULATION OF THE KINK CONTRIBUTION TO THE PARTITION FUNCTION The single kink contribution (1.12) follows directly from (1.5) (with
h - 2TT) providing the q-coordinates have the range which is the range of the original x-coordinate.
-JL < q < +JL as L •*- °° That this is the case can
be demonstrated and will be proved elsewhere (Bullough, Pilling and Timonen, 1983).
Alternatively one uses the energy
(M2 + IL.2
kink solution of (1.1) which has the momentum intuition about the variable
x
) * / 2 of a single
IL . , , together with some
'conjugate1 to
IT, . , :
we prefer to
base (1.12) on (1.5). The result for
Z,
obtained from (1.12) is
Z k = if^K (M3)LM - 2mL(7ry )~ 1 / 2 (m3)"^ e" 3M (l+ •&- (M3)"1 + 2JL (M3)~ 2 O
•>•«
128 +
for where
K (y)
)
T + 0
(2.1)
is a modified Bessel function of the second kind.
is true, the contribution of
Z
exponentiates in
Z.
If (1.10a)
Then since the
contribution to the free energy per unit length can be put in the form -3""1P1 , where
p, is the number of kinks per unit length, this contribution
332
R.K. BULLOUGH, D.J. PILLING and J. TIMONEN
for the free energy per unit length is (M3)~ 2 ) The result for
p,
.
(2.2)
(amongst others) was reported by Timonen and Bullough
(1980, 1981) before the critical fact about discretization of (1.9) was properly realised. correct? 3.
This density
The argument
p,
is certainly not zero.
But is it
which follows suggests otherwise.
TRANSFER INTEGRAL CALCULATION FOR THE KINK DENSITY. We work with the Hamiltonian (1.1) which can be defined for periodic
boundary conditions for plain that, with ta
0 < x < L:
ultimately the period
n(x) in (1.9) can be carried out immediately:
is divided into discrete steps of length Na - L.
L •> °°.
It is
H[(j>] given by (1.1), the functional integral over momenthe support 0 < x < L
a; x -> ja with
0 < j £ N and
We find )n
->
exp f
{-SYQ" 1
JII 2 (X)Y
2
dx
N
N
i
( n C.dn.) e x p - B a y ^ 1 I J j=l J J j=l
o
o
o
Jn. 2 Y o 2 a" 2 J
- (27ra C 2 / 3 Y Q ) 2
with normalisations with spacing
a,
C. * C
(3.1)
for all j. Notice that, after discretization
it is necessary to define new momenta
order to preserve the equations of motion:
n. = all(ja) in
these equations are (3.3)
below. The remaining part of the functional integral is discretized to
r
N ( n
r
_
d<j>.) exp ^~3Y o " 1 a
N E
o
1
ia2 (<}>.- . + 1 ) 2 + m 2 (1-cos (j>.) >
. (3.2)
Using the discretized form of (3.1) and the new canonical momenta one readily finds the equation of motion d)
- a~2 (<j>
j>tt which becomes the
+ d>
j+1
j-1
s-G
when
- 2d> ) • - m 2 sin 6 j a •> 0.
(3 3)
j However the lattice sine-Gordon sys-
tem (3.3) is not integrable and no action-angle variables are available. It is an undetermined and interesting question what damage (if any) is done to a functional integral for an bility is broken by discretization.
integrable field when the integra-
STATISTICAL MECHANICS OF SINE-GORDON FIELD
333
On the other hand the realisation that the functional integral (1.9) is not actually defined until it is discretized and expressed as the limit of an N dimensional ordinary integral is profoundly important. consequences are:-
Immediate
(i) the effective number of degrees of freedom becomes
finite for each member of the sequence to the limit; (ii) the action-angle variables we have used must be modified for each member of this sequence both in number (and presumably) in form;
(iii) if integrability is bro-
ken for each member of the sequence the Hamiltonian is not expressible in terms of action-variables alone.
A further difficulty for the sine-Gordon
system is that the contribution of (3.1) to the free energy is -JlT 1 a"1 In (27TaC2/3To)
(3.4)
which is not defined as the discretization parameter normalisation
C has to be chosen:
we choose
to an elementary volume of phase space
h
a •> 0. Finally the
C = (2TT)~1 corresponding
for each of the finite number
of degrees of freedom. The transfer integral method like (3.2) in the following way: on the
<J> so that
evaluates
N-dimensional integrals
periodic boundary conditions are placed
N+1 = (J^. Put / <*N+1 S(N+1 " ^
- 1: the range of
<j>. is -co < (f). < oo. Introduce a complete set of functions
¥
so
that
6(
(3 5)
w*i>=! v <w v v •
-
The N dimensional integral (3.3) becomes
\ j d W V ( W I d*N exP{"B f < W V } ' \
j d ^
exp C-gf
Ot'j,,'^..!)}
< j d<|>2exp {-gf
d<j) exp {-^ f (<j> ,<j) )} m ((j) )
(3.6a)
in which "(p •
)
T* m
ij.—cos(i)»jj
•
I J «OD )
The
¥ are then chosen as eigenfunctions of the 'transfer integral m operator1 eigen condition *j exp {-0 f<+ j+1 ,*jj> Y ^ ) = e"6aYo~lm2EJl \ ( * j + 1 ) •
O.7)
From this the contribution to the partition function is t The transfer integral method has a long history but we refer the reader to Scalapino, Sears and Ferrell (1972) and Krumhansl and Schrieffer (1975) as useful references.
334
R.K. BULLOUGH, D.J. PILLING and J. TIMONEN
as
L
+
.
ro
(3.8)
y ~1m2e
Hence the contribution to the free energy per unit length is and it remains to calculate the smallest eigenvalue
e
of (3,7),
This eigenvalue problem can be replaced, to terms
m* E 3 2 m 2 y ~ 2
in which V
o ~ Yoa"lm"2
ln
As well as the range Schrodinger Evidently
an
0(a), by
^
(3Yo"1/2
.
-°° < <J> < °° for
(3.10)
<j> it is assumed the usual
eigenvalue problem conditions are imposed on the V
shifts the zero of
be removed and simply added to
e_,
(3.4):
V (<J>).
which is bounded below, and it can V
and (3.4) together provide
-£ e ^ a " 1 in (2TraC2/3Y ) + i 3a"1 ln (3y ~"l/2'na) = 3"1 a"1 ln(3a"!) to the free energy once tion still diverges as
C
has been set to
a •*• 0;
C
(3.11)
= (27T)""1.
but it is independent of
This contribuy
which suggests
it is a contribution from the linear part of the problem namely the KleinGordon (K-G) equation
(1.7) about which it should be scaled.
We show
below that this is the case. It remains to find the smallest eigenvalue tion (3.7) (with V Q removed).
e
of the Mathieu equa-
It is well known (Stone and Reeve, 1978 and
references) that under the requirement that (3.9) has periodic solutions of period
2TT an asymptotic
series solution for
e
exists, and the
contribution to the free energy per unit length then proves to be m3" x CJ - i(MB)" 1 - 4(M3)" 2 - -i- (M3)~ 3 ... ] .
(3.12)
However, both an instanton calculation (Coleman, 1977) and sophisticated use of WKB methods (Goldstein, 1929; DeLeonardis and Trullinger, 1979, 1980;
Stirland, Dinsmore and Bullough, 1983) show that within harmonic
approximation the contribution to the eigenvalue is |m* - 8m*n~t
(3.13a)
and the contribution to the free energy per unit length is
me-1:* - sdry^" 1 / 2 (me) 1 / 2 e"6M: In the
WKB analysis a factor
1 2
(e/iT) '
o.isb) must be removed and the method
for this is Goldstein's method of matched asymptotic expansions.
Further
STATISTICAL MECHANICS OF SINE-GORDON FIELD
335
analysis outside harmonic approximation currently yields
mS"1 C U - i (MB)"1
) 1 2
(mB) / e"eM ( l - ^ W
1
)]
(3.13c)
but some doubt still surrounds the coefficient- 7/16. Comparison with (2.2) suggests that the exponential term is due to kinks and antikinks:
the Jm* in (3.13a) (which becomes imft""1 in (3.13c))
is the smallest eigenvalue of the oscillator and is surely K-G:
so the
remaining power series in C3.12), which has begun to emerge in the anharmonic term form (3.13c) of (3.13b), is the contribution of the breathers. We check on the K-G below and defer consideration of the breathers for the further publication.
Here we note that the exponential term in (3.13c)
differs from the asymptotic form (2.2) of (2.1) in the remaining points: there is an additional factor tional factor kinks; is
(m3) in the prefactor;
there is an addi-
even when (2.2) is doubled to count both kinks and anti-
the coefficient A in the power series development (1 + A(M3)" 1 + ...)
+3/16
factor
2
in (2.2) but is
-7/16
(m$) and the factor
§§5 and 6 below.
2
in (3.13c).
are introduced to correct
The same methods change the
and do not agree with (3.13c)
We shall show how the (2.2) in the
+3/16 in (2.2) to +5/8
in this respect.
introduces a non-analytical factor neglected in
However, the WKB analysis (3.13c).
There is doubt
about the 7/16 coefficient because of present difficulty in extending the matching of the asymptotic expansions to the anharmonic case. believe that
WKB
is now the argument in doubt for the kinks.
actually able to find the same coefficient
+5/8
In all we For we are
from the classical limit
of the Bethe ansatz (Bullough, Pilling and Timonen, 1983) whilst Maki (1981) finds the same number.
The final results of the WKB investigation
will be reported elsewhere (Stirland, Dinsmore and Bullough, 1983).
The
present paper is solely concerned with details of the dressing procedure which produces the factor X
(1 + A(M3)" + ...)
(m$), the factor
2, and a power series
without non-analytical terms.
The series for the breathers (3.12) has not yet been found identically by this dressing procedure although an asymptotic series with the same structure has been found. even powers 1980).
of
(MB)"1
It solves the problem of finding both odd and in the series for the breathers
(compare de Vega,
The problem of establishing (3.12) identically is the 'non-unique-
ness1 of the breather contributions and the discrepant coefficients referred to in §1.
336
R.K. BULLOUGH, D.J. PILLING and J. TIMONEN
Before we show, in different ways, how these dressings for the kinks arise we first analyse the 4.
K-G
field separately.
FREE ENERGY OF THE KLEIN-GORDON FIELD We do this calculation in three different ways finishing finally with
a fourth calculation. tion for the quantum (i)
In the APPENDIX we carry out an equivalent calculaK-G field,
Calculation in k-space We evaluate the functional integral (1.8) with H[(J>] E 4TT2
The
J
{|TT 2 + £
r
2
x
+ Jm2<j>2} dx
(27T)-1 {£.47T2II(k)2 + £ (k2+m2)<j>(k)2} dk H[cJ>]
discretizes
.
{II(k) ,<j>(kf)} - 6 ( k - k f ) .
i s introduced so t h a t
( 4 . 1 b ) of
(4.1a) (4.1b)
The second form
to (4.2)
with
k. = 2J7TL"1.
{n.,, } - 6., . all
j
The new canonical momenta
n.
are introduced so that
We take the measure on each integral as
as before.
C. = h - 2TT for
The result for the (2n+l) dimensional integral from
(4.2) (which is simply a product of integrals) can be put in the form (Tr3)- 2n a 2n e 2n (Lm)- 1 .n^
(1 + m 2 L 2 /47T 2 j 2 )" 1
.
(4.3)
For a check on invariance of functional integrals under canonical transformation ((4.1a) -> (4.1b)) and subsequent discretizations note that the 2TTL~1 -> 0
discretization parameter here is troduce a second parameter 2 2
4TT J )"
1
a
by
na = £L.
as
L -* °°.
The product
However we in#n
(l+m2L2/
~ JmL/sinh £mL, for small enough a, so that the contribution to
the free energy per unit length is a"^" 1 The
£ m3"
1
(In £n_ + £ ma - 1) . a is identifiable with that in (3.13).
with (3.11) with, however, the addition of a" this discrepancy in (iv) below.
1
(4.4) The remainder agrees 1
3" (l.n IT - 1 ) . We look at
Here we conclude that (3.11) arises in
the transfer integral calculations as the contribution of free K-G modes, (ii)
Calculation through action-angle variables. We evaluate exp -
STATISTICAL MECHANICS OF SINE-GORDON FIELD
where
337
= [ (n^ + p 2 ) 1 / 2 P(p)dp.
H[<j>(p)]
(4.5b)
'—00
The phase spaces are - <5(p-pf).
We use
ting product j=-n With C.f
J
0 < P(p) < °° , 0 ^ Q(p) < 2TT, and p. = 2J7TL"1
(P(p), Q(pf)}
and JL = na to discretize
H.
The resul-
of integrals for (4.5a) is
{LC. C.f 3" 3 1((p. p .2 +m 2 )" 1 / 2 } JJ
JJ
.
(4.6a)
2 - I T L " 1) ( 2 7 T ) "1 = C = (2-ITL" ) (27T)" 1
1
[[ ** **In 3(p2+
J = exp i~
m 2 ) l/2\ dp
'-ir/a
(4.6b)
'
and the contribution to the free energy per unit length is - S ^ L " 1 In J = a ^ a " 1
(In &. + jma-1) + 0(a)
which agrees to 0(a) with (4.4).
(4.7)
The conclusion here is that canonical
transformation on the Hamiltonian (4.1a) to k-space ((4.1b)) or to actionangle variables ((4.5b))-yields the same result.
We also see that the
integral in the s-G Hamiltonian (1.5) would yield the same contribution. Thus if (1.10) were true the free energy per unit length contains contri-3" 1 L"1 In Z^, free antikinks -3""1 L"1 In Z^, free
butions from free kinks 1
breathers (iii)
1
-3" L"
In Z, , together with (4.7) from free K-G modes.
Method of steepest descents
This method uses the Hamiltonian (4.1a) without canonical transformation.
The
n
integration can be done immediately and we are left to
evaluate Jm2cj)2} dx
exp -3
.
(4.8)
We first consider briefly the general case in which the K-G potential im22 minima
is replaced by <\> . . y min
V((j>).
This potential will have, perhaps degenerate,
The boundary condition is taken to beY J
(j)T -> (j) . , x ->-<». min'
For steepest descents dominant contributions to (4.8) arise from neighbourhoods of those
(J) which minimise
H[<J>],
These § satisfy 0
=
6 H O ] / 6(f> =
-<> XX
+ ^-
'
(4.9)
CKp
Notice that this equation determines static, that is time independent, solutions of the nonlinear K-G
<J> - . We functionally expand xx 11 H[(J>] about such solutions <j> .. (say). Since 6H/6cJ> = 0 H[cf>] = H U c l a s s 3 + J(<5,<52H/6(j>26(|>) + .... (4.10)
338
R.K. BULLOUGH, D.J. PILLING and J. TIMONEN
for real variations 6<> j = <>j - ^ d a s s ' formally the operator
|!H = -^L + 0
Tlie s e c o n d
functional derivative is
(4<11)
cb T = yd) _
class w2
And this operator has eigenvalues
and real eigenf unctions
(for appropriate boundary conditions).
n (x)
CO
If we write
S<|> = E e n (x) n=-« n n
00
class n——°° ** ** ^ (we have taken a normalisation a"1 / n n dx = 6 ) . We suppose the •'-oo m n mn measure i s such t h a t 3u (J> •* m IT dc with -°° < c < °° . Then (4.8) becomes n=—°° n n
""'
n
/ ^
and k
( )
t cias
T I Tz L a = m 2 , and (4.9) h a s eigenvalues dc() '(f)=(j)ciass = 2mTL~1,
on a length k
(4.13)
2
n=-~ __ n (providing n o u = 0 ) . In the case of the K - G
= nirL"
B-oo
1
n L
°>
H
GO2 = k 2 + m 2 , w i t h n n
integer in -°° < n < °° , for periodic boundary conditions (for b o x boundary conditions
with 0 < n < °° ) .
IT" 2 —* a W
=
exp
<|> = 0
>
x-0,
L , and
W e replace
1
n
^
27r
^/a
to see that the free energy per unit length is again given by (4.4) once the contribution from the II- integration, namely (3.4) with
y - 1, is
included. (iv) The effect of discretization We expect to need to find the eigenvalues and eigenfunctions of which goes over to (4.11) in continuum limit.
Under periodic boundary
conditions the eigenvalues are n
sinjkal2
n*
and (4.14) becomes ,-n/a exp "^ dk in N
#
;
^
,
2irn
k = t^L n
L
(4.16)
STATISTICAL MECHANICS OF SINE-GORDON FIELD
With the result of the
339
II - integration also included the contribution to
the free energy per unit length is $-la~l (In | +
Jma)
(4.18)
which coincides with the transfer integral result.
Since the dispersion
relation (4.16) obtains for periodic boundary conditions and becomes o>n = ( m 2 + k n 2 ) 1 ' 2
for
a •* 0
under the same conditions, the transfer
integral result exhibits specific effects from discretization but not obviously any through the choice of boundary conditions. 5.
STEEPEST DESCENTS CALCULATION FOR THE KINK CONTRIBUTION TO THE FREE ENERGY The argument which follows is substantially due to Rajaraman and
Raj Lakshmi (1982).
It is an instanton version of the
WKB
methods intro-
duced by Dashen, Hasslacher and Neveu (DHN, 1975). For a direct comparison with (4.8) it is convenient to take
H
for
the s-G in the form
HC(J>] = Uk*t2 with
+
^x2
+
{<{>t,} = 6(x-x').
(exp-mx).
mi+X-1(l-cos(X1/2m-1(J)))} dx
A solution of (4.9) is
(5.1)
= 4> cla8S
=
4mX" 1 / 2 tan" 1
This is the static form of the kink solution (1.6a) for which
2 H[<J>class ] = M = 8m 3 X"»1 so the coupling r & constant is YiQ = Xm~ . The cosine in the form (5.1) can be expanded to see that (5.1) is then linear K-G
plus non-linearity, though this is not significant to what follows. The
II- integration
ning part.
is done as before and we are left with the remai-
The eigenvalue problem corresponding to (4.11) is
{ " | ^ 2 + ni 2 (l~2sech 2 mx)} n n (x) There is a non-degenerate eigenvalue 2
1
2m X~ '
2
sech mx = 9<j> -
inace of x (-00 < x
H[<J> -
/8x.
] = M::
< oo) when
= V\<*> w
2
= 0
with eigenfunction
a static kink can be placed at any point
L •> °°. Accordingly one uses variables
-*o> + J l
{l
c used in §4 n
(5 3)
'
is associated with a
The proper choice, which is important even to a factor
2 in what follows, can be inferred from Coleman (1977):
tion
{x ; b ,
and expands
Vn< x - x o) + b -n^-n (x - X o )} '
Dimensional arguments suggest that the element dx
.Ildb , the factor
n (x) =
It corresponds to the translational invar-
n = ± 1, ± 2 , ... ,±°°) instead of the
Jacobian <* m 1 / 2 .
<5-2>
•
ndc -> (a~1M)1/2dx . n o
a"1 coming from our particular scaling. The contribu-
to the functional integral is then
340
R.K. BULLOUGH, D.J. PILLING and J. TIMONEN db
n
2 2 expv 2Ja$5 b n n
(( J( db-ne Fx p 2- J a -n 3 w 2-nj b2] <«> a*1 / n n dx = 6 as before. —oo n m ntn
We have used the normalisation
It is convenient to scale the result (5.4) against the free KleinGordon result (4.14). 3
M
2
The ratio is (a2* £ r 1 /
n
n
n-1 RM
L e ~ * m (SMI ll2 | J
"n
2
(a2u) u>
r+iT/a
exp -_L_ ^
2TT
/ a
dk I n u ( k ) u,(k)
e ~ 3 ( M + a ) E Zk
E 2L with
f/a
a E 3"1 L
We show below that
a
(5.5a)
dk In does not depend on
The translation mode with eigenvalue cial role: that both
it introduces the factor oo = (m2 + k 2 ) ! / n n
2
and
w
n
L. UJ = 0 1
evidently plays a cru2
2(3m/fTy ) / .
Otherwise it is assumed
become dense in k-space when
These too play an important role since they dress the mass 3-dependent correction
a.
The dressed quantity
Z,
M
L -> °°.
with the
now compares with
(1.12) and this comparison will be drawn in detail in a moment.
Certainly
we can expect that a comparable but substantially more general analysis would show that
l(jL9vtr,rL) in (1.10a) factors into
contribution in which
Zk,Z-^-,Zb
are now all dressed.
Z, Z=-Z, x free K-G We demonstrate else-
where just such a result within the much more general framework we have developed (Bullough, Pilling and Timonen, 1983). show next that
Z,
partition function
Here we merely want to
in (5.5a) makes an exponential contribution to the final Z
exactly as in (1.10a).
Consider two well separated kinks for which + (f)
1
(x-x ) with |x -x I >>m~ .
Then H[<J>
cj> (x) =
problem for the operator (4.11) has two well separated but identical single kink potential wells. 2-fold degenerate
i.e.
The bound state eigenvalues are unchanged but become CO Q 2 = 0
is
2-fold degenerate.
ponds to independent translation of the two kinks;
This corres-
and this is
not
STATISTICAL MECHANICS OF SINE-GORDON FIELD
341
actually possible for the S-G since at most one kink can be brought to rest while the second phase shifts as it passes through the first at rest. |x - x I >> m" 1 .
the argument will be good for
1
co-ordinates
x
1
and x ,
2
r
and it will be assumed
2
,
executed independently —
Then the contribution to
Z
allows for invariance under interchange of is finally energy
is
I
(n,.
)"
1
n
(Z,) k =
2
x
can be
There is an for the single
the factor (2.1)-1
and x .
Since any number of
H ) , the contribution to
Z
and the contribution to the free
'^l\-
We have now to show the them.
exp Z
Idx
i
a = 2 x(a
is (2!)" 1 Z£ :
well separated kinks solves (4,9) (minimises 1
and
1
|x - x | >> m" 1 .
thus ignoring
error here perhaps O( (mL)" ). We show below that kink).
t
Jdx .
1
But
There are now two location
a's
We require the eigenvalues
add for individual kinks and calculate co2 of (5.2) which are eigenvalues of
K-G type modes in the presence of the kinks (and indeed the anti-kinks and breathers as well in principle). values are accurately of the form tial boundary conditions.
The result (1.5) shows that these eigen(n^ + k 2 ) 1 / 2
L -*• °°
under exponen-
The problem is to infer them for
with periodic (or box) boundary conditions. with discretization:
when
L < °° and (say)
There may also be a problem
the point is the number of degrees of freedom is now
finite and we are in effect, trying to find a form of (1.5) which goes over to (1.5) itself when
L -> °° and the discretization parameter
a ->• 0
(for some order of these limits). Izergin and Korepin (1981, 1982) report a lattice model on infinite support which goes to the s-G with Hamiltonians (1.1) or (1.5) as and action-angle variables have been found (Tarasov, 1982). integrable system which reduces to the previous case when ently available (Kulish, 1982) therefore follow
DHN(1975)
ary conditions the
a)
L •> °° is appar-
but action-angle variables are not.
and argue as follows: n
We
under periodic bound-
for the K-G satisfy u> - (m2 + k
n
a -• 0,
A discrete
2
) 1 / 2 ; k L = 2™.
n
n
In the presence of kink-like objects under periodic boundary conditions the K*-G-like contributions, which will provide the integral in (1.5) when L •*- °°, will be assumed to have eigenvalues 2im + A(k n ).
In the present case
ca = (m2 + k 2 ) * ' 2 ; n n
but K L = n
A (k ) will be the phase shift induced by
the single kink in the K-G mode labelled by k . We show that a is linear in A(k ) ; and, since the A(k ) add, the a add as was assumed n n above. Thus we need consider only the 1-kink phase shifts. After substituting for con and co and one integration by parts
342
R.K. BULLOUGH, D.J. PILLING and J. TIMONEN
we find 3a
= ±-
(
ZTT
In ^ 4
-auk)
-7r/a
dk
=
j -
ZTT
-1
[A(k)
In
f+7T/a dk dA I n
27 I
which is indeed independent of L. (Rubinstein, 1970)
(5
dk
*6)
-it/a
For one kink at rest the phase shift is
A(k) = irsgnk - 2 tan""1 (km"1) = 2 tan"1 (mk"1)
;
(5.7)
there is no contribution from the square bracket in (5.6) and the integral is precisely
In 2 (for a ->• 0 ) .
This introduces a factor 2 to Z, and a
further factor 2 follows by including antirkinks.. The contribution to the free energy per unit length is -3 w l L f c l Z.
-3- 1 8m /^-
=
K
e~3M
(5.8)
Y TTY o
and this agrees with (3.13). 1
kinks, M -> M -3" In 2m3
Compared with (2.1), now doubled for anti-
a result found by Maki (1981) from a quantum theory
and by Fowler (1982) through the quantum Bethe ansatz.
This analysis
therefore seems to resolve the discrepancies between (3.13) and (2.1) although it provides no estimate of the coefficient A in the power series (1 + A(M3)""1 + ••••) 6.
and no comment on the non-analytical factor.
QUANTUM EFFECTS. Consider the quantum propagator X
=
s[] S[cf)]
' & 4 £ ) n expiS[]
(6.1a)
I 1 J n < x H t | ( x ) d x " H[^] j
dtf
can be integrated over
(6 lb)
•
o is the classical action in Hamiltonian form.
-
The n(x)cf> , (x) term
n(x) (compare the APPENDIX).
When
H[<}>] is given
by (1.1) the result is ft f(j)(x,t) G(, ) <57f<J> e x p i L(cf>,cj) , ) d t f
°
J
J
c
(6.2a)
where L(<M t ) = Y^ 1 k i ^ t 2 - i (f> x 2 -m 2 (l-cos (())] dx.
(6.2b)
Or
The measure associated with pC/f ® $
i s not the same as that associated with
and we suggest that t h i s may be important to what follows. In order to compare with
§5
we use
STATISTICAL MECHANICS OF SINE-GORDON FIELD
dx [ U 2 - J J)x2
L(<J>,O So f
t
y
o
=
^m~2
and 1
•> t" = -it
the trace
-mH"1
343
(l-cosCA^m- 1 ^))]
.
(6.3)
mA * <j> •* (f). To make connection with (1,9) we must set
in (6.1a) and take a classical limit (n" -»• 0 ) :
/G(<j> ,0)d<|> .
we also need
We look at these points in the APPENDIX and for
more details refer the reader to our forthcoming paper (Bullough, Pilling and Timonen, 1983).
Similarly we are obliged to move (6.2)
to imaginary
time and we do this below. We are attempting to apply the WKB analysis of DHN (1975) to the calculation of the partition function (1.9). of §5
by working with moving kinks.
nection with (1.5) in the work kink was at rest.
of
The idea is to extend the work
There was a conspicuous lack of con§5,
since the result (5.8) assumed the
It is not sufficient to Lorentz transform the argument
as it stands since we must (surely) finally sum over kinks of arbitrary momenta
p.
We first expand (6.3) about the classical trajectory for a moving kink
= 4mA"~* tan"1 {exp m(x-Vt) (1-V?) * } . To apply the
(j)
lysis we perturb about periodic orbits of length V = LT""1 .
L
and period
The classical trajectories minimise the action
372 - &
*
l'/2 Sin
S
(xim 1
T:
then
and satisfy
<6'4a>
" *> - °
This is the analogue of (4.9).
WKB ana-
The analogue of (4.11) and its eigen-
condition is
x t} =
(6 5)
' °
with
(j) E Y(j> . Y class
is at rest, are plainly (5.2). we shall
n (fc)e ^
where
In the original frame for period have
eia)nT(l-V ) ^
r\ (x) T
(T - VL) (1 - V 2 ) ~ * = T (1 - V 2 )* The
-
The solutions of this, in the frame in which the kink .~ '
quantities
v
is an eigenfunction of
of length and
E w T(1-V2)^
L
and
V = LT"1
nn(x,t + x) = n n (x,t) * are the 'stability angles1
(DHN, 1975; Rajaraman, 1975). Evidently there will be the eigenvalue a) = 0 in the frame in which the kink is at rest and this is associated o with the translation of the kink in that frame. original frame
v
This means that in the
=0.
From (6.3), following periodic orbits of period
DHN (1975), Rajaraman (1975) shows that, for x£, (6.2a) can be reduced, with Gaussian quan-
tum fluctuations described by the n (x,t), to (cf. (4.63))
Rajaraman (1975), eqn.
344
R.K. BULLOUGH, D.J. PILLING and J. TIMONEN
I I & Sclass I
,I ^ {p n }
^
in which S
and
class
S is a counter term. The p are excitation numbers for the K-G c.t. n type modes. We have some problems in applying this sort of analysis further in the present context for reasons not yet understood.
We shall therefore
sketch a heuristic development which is based in a rather flimsy way on (6.6) as a starting point. A better argument is needed. First of all we shall assert that the correct way to interpret (6.6) is to sum over sets
{p }
subsequently to the power
for the orbit of period £.
T
and raise the result
This way we certainly get for the exponen-
tial factor exp
Since
3
l U
2
S
class
+ l S
2
c.t." £ C i i v n " l n (1 " n = exp i I S (say) 2
3
2
2
3
2
S / 3 T « ML T~ (1 - L T ~ ) ~ / , c .Lass
ex
P
T, L fixed
with the temperature of freedom
I
T
n)]} .
(6.8)
we then find for (6.6)
We are concerned with periodic orbits period V = L T " 1 with
iv
T = T £ and speeds
(T used briefly here is not to be confused
used elsewhere).
There is therefore one degree
(say) and we have to sum (6.9) over
I.
In a correspon-
ding calculation for the eigenspectrum DHN (1975) find, in the two degree of freedom case in particular, that a remarkable cancellation of factors like those lying in (6,9) eventually occurs. Although for % =• 1
V = 0
and
one can already see the connection of (6.9) with (5.4) and (5.5)
((-i) T " *rL ->• 3 *M*L
as the remarks below will make plain) we have been
obliged to assume in the present context that the measure must be adjusted so as to make contact with (6.1a).
Cancellation is thus achieved here by
fiat. To achieve that cancellation we deliberately replace (6.9) by
J
T"* r*M*L (1 - V2)"3/1*} exp U S
.
(6.10)
STATISTICAL MECHANICS OF SINE-GORDON FIELD
345
We note that the extra factor introduced is i /
^
/HT,
=1
T-3/2*-1/^1/^
1
d S
(1 - V 2 ) - 3 / 1 *
class / d T i1/2, (1 - V 2 ) 1 i
' 2 7 ' T "(1 - v 2 ) 3 / 2 '
'
T
'
K
*
}
In the second form the result is remarkably close to that given by DHN (1975) (eqn. (3.6)) for the two degree of freedom case (the change is the IT" 1 (1 - V 2 ) 1 / 2 ! ) .
factor
The point of this otherwise arbitrary step is that we believe we know the answer we must reach for the contribution of kinks to the free energy. -3" 1 L times
This answer is
i r°
•5^
where
JL-00
S(p)
dp In U + exp-6 (B~xS(p))} i s derived from
canonical momentum 3" 1 S(p) and
=
p
S
(6.12a)
of (6.8) by working in terms of a
and moving to imaginary time so that
[CM2 + p 2 ) 1 / 2 + o ( p ) ]
a(p) generalises (5.5b).
(6.12b)
The reasons for (6.12) we give below.
First
we start from (6.10) and sketch a derivation of (6.12). Since (6.10) is to be summed over orbits labelled by j^
I L T - 2 (1 - V 2 )~ 3 / 2 M I"1 exp US
.
£, it becomes (6.13)
We introduce the WKB condition | with
MV2 (1 - V 2 ) " 1 / 2 dt
-
(2m + 1)TT
(6.14)
m an i n t e g e r , so that 2 1 2 MV(1 - V ) " /
=
(2m + DTTL" 1
(6.15)
Then US . class
= - U ( E T - (2m + l)ir) m
(6.16)
t An alternative procedure avoiding some difficulties follows DHN all the w a y t o G(E), the Fourier transform t = T -> E of (6.2). Then Z = Tre" 3 = (2-iri)"1 / dE e""3EG(E). But a(p) must still emerge from in (6.8) and the In found in (6.12a).
S
346
R.K. BULLOUGH, D.J. PILLING and J. TIMONEN
and E
= M U - V 2 ) " " 1 / 2 = (M2 + p 2 ) x / 2 m m
(6.17)
where p m - (2m + DTTL" 1
(6.18)
The reason for odd integers in (6.14) and (6.18) is apparently because for
C
solutions under periodic boundary conditions, a kink on support
L must be paired with an anti-kink on support is
L
and the effective period
2L. Note again that (6.11) means that (6.10) is close to the form
given by DHN (1975) for the two degrees of freedom case* We now see that if
V
is fixed by the condition (6.15), but %
remains free, (6.13) is - L
Lx""2(l - V )"3/2M I {exp U(S-S . ClaSS
"|j
L"1 V 2 (1 - V 2 ) " 3 / 2 M
2i
i
-E x)} {exp -i(£+l)w}r 1 m
£n (1 + exp i (S-S c l a g g (6.19)
Evidently we have relaxed the connection between T
in
1
Lx" = V, Ultimately
T, L •*• ~
x
in
&x • T and
separately so that all V are
possible. This relaxation allows us to sum over periods T by the condition (6.15) with the factor
x
in S .
Vm = L x ^
1
determined
keeping the time period
x,
equation (6.7), for example, fixed. This way
the contribution from (6.10) becomes
M
(6.20) Noting from (6.15) and dp
=
V m
= Lx ~l m
that
M(l - V m 2 ) " 3 / 2 (-iT1 V m 2 )dx m
(6.21)
we find finally that the contribution from (6.10) has become L_
[
dp £n(l + exp i(S-S c l a g g - Em(p)x)
(6.22)
STATISTICAL MECHANICS OF SINE-GORDON FIELD
347
where E(p)
=
(M2 + p 2 ) 1 / 2
(6.23)
We now deform the time contour so that the fundamental time period T -* -i3 . Then (6.22) is
L
•7TZ7T
C )
J
-1 S(p)))
dp £h(l + exp -3(3
(6.24a)
mmCO
with 3"1 S(p) E [(M2 + p 2 ) l / 2 + and, from ( 6 . 8 ) , -(ix)
a(p)
(6.24b)
a ( p ) ]
i s determined from
L^ijv - in (1-exp iv ) ] . n n n -T"1 £ \ v
It is evident that the term action (6.7) by correcting
(6.25) in this corrects the classical
M . We have to scale against the vacuum, that
is against the free Klein-Gordon modes at 3"1 • 0.
This can be done exactly
as in §5, and we find (as DHN (1975) did for the stationary kink) that M -*- (1 - Y Q /8TT)M Y
which is equivalent to renormalising
(1 - Y /8TT) : the S
y
to
y
f
=
looks after the divergent contribution and is
a simple renormalisation of the mass Rajaraman (1975)). The renormalised
m (as in DHN (1975), or see M now appears in the WKB condition
(6.14). The new feature in the analysis is therefore the contribution I in (1 - exp iv ) n is 1
in S . Plainly the contribution to 3"1S(p)
3" I £n(l-exp-3w n ) n
in which we interpret
GO
in (6.25)
(6.26)
as the Klein-Gordon ei gen frequency evaluated
for (that is phase shifted by) the kink moving with momentum
p . This
interpretation is correct in that (scaled against the free K-G)
it is
exactly what we find from the Bethe ansatz. If the contribution of the i I ivn
to
a
(p) *-sa l s o included we find of course that (6.27) becomes exp-.Bw )> n
(6.27)
n
which is just the sum of the free energies of the K-G modes shifted by the kink moving with momentum
p . Surprisingly in this form we find a
problem with the renormalisations of mass
m
Y o "*• Y o ' > but we report on this elsewhere.
and coupling constant
348
R.K. BULLOUGH, D.J. PILLING and J. TIMONEN The result (6.24) still concerns one kink.
the partition function
Z
-ft""^""1 times
as in §5, and this means that
(6.24) is the free energy per unit length. hazarded in (6.12) with
So it is exponentiated for
This is just the expression
a(p) given by (6.26).
It is an open question
whether the WKB method can be developed in detail so that (6.24) with (6.26) follows from it.
Nor do we know whether there is any significance to our
present starting point (6.10) and the two degrees of freedom (almost) implied by it.
In the absence of such details our concern has been to present
(6.12) as correct and indicate incidentally some of the problems which may surround the proving of it. that when multiplied by
Thus (6.12) is so far the assertion, simply,
-3"
1
it provides the correct contribution of
quantum kinks to the free energy per unit length:
is given by (6.26).
a(p)
We believe (6.12) with (6.26) is correct for two reasons.
The first
of these is that the equivalence of the quantised sine-Gordon and the massive Thirring model, first realised by Coleman (1975), means that the quantised sine-Gordon can be described either as a bose system or as an equivalent fermi system (see e.g. Luther, 1980).
When
y
= 4TT,
the
fermi problem becomes a free fermion problem and the result (6.12) follows directly - although the
a(p) vanishes in this case
(cf. e.g. Puga and
Beck (1982)). The second reason for wanting (6.12) is that we have derived it using Fowler's and Zotos's (1981, 1982) solution of the Bethe ansatz and Imada's work (1982): for
a(p)
1983).
the expressions (6.12a) with (6.12b), and the specific forms
we give below, emerge identically (Bullough, Pilling and Timonen,
Evidently if the semi-classical WKB method of DHN (1975) is to be
generalised to finite temperatures we need a much deeper analysis than that sketched for (6.24) above.
Still the form for
in (6.24) with
a(p)
S
in (6.8) (which leads
to the form for
S
given in (6.26)) at least seems
to be correct.
We therefore develop the expressions (6.24) and (6.26) in
more detail now. In the APPENDIX we derive the result equivalent to (6.27) for the free K-G system:
the calculation offers no difficulty in formal terms and
it confirms the work of §4 in the classical limit* that
w
in (6.27) is simply replaced by
u^.
Since
We find as expected a(p)
must be scaled
against the free K-G we find finally that
a(p) = IIfi fi<SL-« <SL-« ))•• S" S"11 InIn rara-e*P-BVf] exp-BWn)
.
(6.28)
The classical limit of this is obtained for k_T = 3 " » m and the In 15 reduces to Z B""1 (In $u) - In8w ) a form assumed empirically by Bishop and n n n
STATISTICAL MECHANICS OF SINE-GORDON FIELD
349
collaborators (1978,1979,1980 and references) for the classical problem. We derive this classical result from first principles in our forthcoming paper (Bullough, Pilling and Timonen, 1983). Evidently in the quantum case (6.12), that is (6.24), applies with given by (6.28): term.
M = 8my '~* so
3"1 «
For
M (6.12a) is approximated by
i- f dp exp -SC(M2 + p 2 W 2 - a(p)] )
a(p)
a(p) in (6.28) now omits the zero point
(6.29)
—00
which (multiplied by
L)
is the natural generalisation of (1.12).
It is important to realise that a contribution from the zero stability angle limit
v
to
3"1 >> m
a(p)
is missing from (6.28).
that expression reduces to T
a(p) = -6" 1 In (6m) + fT1 jThe argument here is that the laced by the kink: which explains the
-3"
dense in the limit.
~fu\
dk In ^ £ f
k = 0
accordingly 1
r+T\/a
Hence in the classical
.
mode is omitted from
(6.30) w(k) and rep-
u>(0) must be removed from the
In(3m).
The remaining
a)(k)
This way we see that the factor
and (3m)
E3" 1 In 3"wn
u)(k) become distinguishing
the exponential term in (3.13) from (2.2) emerges as expected from the zero stability angle
v ,
that is from the translational mode.
The phase shift analogous to (5.7) is A(k,p) = TT sgn(k + mVy) - 2 tan"1 = 2 tan"1
|-
LY in which
y = (1-V 2 )"" 1 / 2 ,
lated by Currie (1977):
\—5-M/O
I
(6.31)
k+V(m 2 +k 2 ) 1/2 J and
p = MVy.
This is the phase shift calcu-
it reduces to (5.7) for p = V = 0.
We then find
r
a = j P ln(l+k2m"2) 6 (k+mVy) dk -my f° ln(l+k2m""2) „ n-i fo N o " 7 7 — T ~ l — dk - 3 In (3m) +m ioo ^ Y = -ln(l+y) " 3"1 In (3m)
.
(6.32)
It is important to realise the role of the P-integral which is keeping track of K-G modes: of
k
the 6-function (which appears to describe scattering
into the K-G field mode of momentum -mVy) is excluded by that
P-integral:
this means one K-G mode (2 degrees of freedom) is excluded
and replaced by the 2 degrees of freedom of the kink of momentum
MVy.
This way the finite number of action variables (= constants of the motion) associated with the discretization of the original functional integral (1.9) is preserved.
350
R.K. BULLOUGH, D . J . PILLING and J . TIMONEN From ( 6 . 3 2 ) ,
( 6 . 2 9 ) p r o v e s t o be
tft
f I" dp (l +
= MiT 1 (MS) [K (MS) + K (M0) + (MS)"XK (MS)] 1 o 1 - 4m(7r ))" J (3m)* e" M 3 [ 1 + | Yo Yo
(MS)^ 1 + • • • ]
(6.33)
the result also found by Maki (1981); it is also found by Bishop (1981) but both he and Maki (1981) (see also Maki and Taykayama, 1979) really use the same methods - namely dressing by (6.32) - as ourselves.
After the
equal contribution of anti-kinks has been included the expression (6.33) agrees with (3.13) except that the coefficient A rather than the
- 7/16 found for (3.13c).
in §3 suggesting
of
(MS)""1
is 5/8
We have commented on the result
that the problem rests with the WKB expansion of the
Mathieu equation rather than the dressing procedure.
Of course we find
exactly the result (6.33) from the Bethe ansatz; but in this analysis the phase shift
A(k,p) of (6.31) arises naturally from within the argument,
and it is not inserted as the correct interpretation of the present analysis.
a(p)
as it is in
This seems to justify the argument, traceable to
DHN (1974, 1975), that dressing is due to K-G modes phased shifted by the kinks (or antikinks).
Some details of understanding remain nevertheless:
one of these may be that
co
in (6.26) or (6.27) is not used consistently
in terms of the stability angle
v
from which its existence was inferred:
this raises a further problem associated with generalising the theory from (6.8). Despite the missing details our conclusion must be that there will be an equivalent procedure for correcting (that is dressing) the whole of the independent particle analysis based on the Hamiltonian (1.5).
And that
ultimately complete agreement with the transfer integral results can be found this way.
The problem of * exponentialf, as against periodic
boundary conditions raised in §1 is apparently eliminated by such dressing in that complete agreement with the Bethe ansatz, which involves periodic boundary conditions, has been obtained for the kink and antt-kink contributions at least.
In our forthcoming paper (Bullough, Pilling and Timonen,
1983) we start from (1.5) and show how the dressings of kinks, anti-kinks and breather contributions all emerge as corrections to the measure associated with the canonical transformation to action-angle variables.
The
argument is wholly deductive, unlike those of the present paper which is concerned to relate a number of different calculations relevant to the dressing of kinks and anti-kinks alone.
Broadly speaking all these methods
STATISTICAL MECHANICS OF SINE-GORDON FIELD
351
have been shown to be in agreement the key role being played by the 'translational mode 1 . One interesting point remains:
for small enough
T, (3" 1 £ m) the
quantum analysis shows that only the zero point energies of the K-G modes contribute to a(p). Thus, except for the quantum renormalisation
y •+ y f
the free particle Hamiltonian (1.5) correctly describes the statistical mechanics of the kinks and anti-kinks and the contribution of each is Z given by (1.12) and (2.1)!
This conclusion agrees with that of Maki and
Takayama (1979) and Maki (1981) — but really exactly for the same reasons. APPENDIX THE QUANTUM KLEIN-GORDON FIELD Consider the partition function (1.8) namely
5H
exp - BHC*D
(A.I)
For the K-G H[] = | U n 2 + £ c f > x 2 + Jm2<J>2] dx
.
(A.2)
The form (A.I) is in classical limit. exp -3H[] is to be replaced by
The analysis in §6 shows that 3 > exp - J [H[<|>] - i j II ,dx]dtf. The
integrand is the classical action in Hamiltonian form evaluated along the purely imaginary contour goes to
-i3.
If
"h
t = -it1 (t1 = real).
The period
x
of §6
is included explicitly we note that the correspon-
exp-t""1 J [....] dtf and this reduces to exp-3H[<j>] o This is so because the trace which must be taken for Z closes
ding expression is as "ft •> 0.
initial and final data the contribution of the
(J>(x,O) = <j>(x,i): 1
/ II(x)cJ) , (x) dt
these closed paths mean that -> n(x)d<J)(x), and this is negli-
gible as .the paths shrink to nothing We now observe that much as in §6 Jn(x,t f ) 2 - in(x,t f H t l (x,t f ) - ^ t t ( x , t f ) 2 = Hn(x,t')-i<J> tl (x,t f )) 2 The
n
.
(A.3)
integration can then be performed to eliminate these quantities
from the exponent in (A.I).
This way that partition function becomes
e x p - | dt' | dx [Jc^, 2 + !x2 + Jm 2 * 2 :
.
(A. 4)
o There must be doubt about the measure to be assigned tofcj<\> —
and it
was this that led us to suggest in §6 that an adjustment of the effective
352
R.K. BULLOUGH, D.J. PILLING and J. TIMONEN
measure in (6.9) might be acceptable. If we work from (A.4) directly we see that for periodic behaviour of period
3
d t ' J dx [Jcf> t l 2 + £x2 + Jin 2 * 2 ]
j o
<*> I
- ^
r+Tl/a
iC47r 2 v 2 3" 2 + k 2 + m2]cj>2(k,v) dk
.
(A.5)
The situation here is comparable with that used in (i) of §4 (where however only an integral for x -> k
is involved).
Thus, by using
both cut-off in (A.5) and the additional discretization parameter L —
a for (na =
see (i), §4), and without any possible correction of the measure for
yQ ^(Xjt 1 ) -> ptj <J) (k,v)
under the canonical transform
<J)(x,tf) -* <|>(k,v) we
expect the result for the partition function (A.4) to be r+Tr/a °° exp =± dk I In {[47T 2 v 2 3" 2 + k 2 + m 2 ] 1 / 2 } • (A.6) v= -°° -ir/a The sum Z In [4TT 2 V 2 3" 2 + k 2 + m 2 ] 1 / 2 i s a c t u a l l y d i v e r g e n t : but i t ~\) ^
—CO
can be formally evaluated by a trick already used by others (e.g. Dolan and Jackiw, 1974).
Set u>2 = k 2 + m 2 and (A.7)
(A. 8)
(A. 9)
(A. 10)
S(w) - 20CJW + 3" 1 In (1 - e" 3a) )] up to neglect of constants. (3
As a check one might notice that at
(A. 11) T = 0
•*»)
2^1
In (-z2 + a ) 2 - i e ) 1 / 2 dz = i—o
t Evidently the classical limit of (A.4) is / S ) > exp-3C£<|> 2 + Jm2(|)2] which is (4.8) and the n-integration plays the role described finally by the contribution (3.4).
STATISTICAL MECHANICS OF SINE-GORDON FIELD
353
It follows from (A. 6) that the partition function for the free quantum Klein-Gordon field is f+Tr/a
I dk {£ (n^ + k 2 ) 1 / * + 3" 1 In (1 - exp -S(m 2 + k 2 ) 1 / 2 } —7r/a Thus the free energy per unit length i s exp ~ | £
±
V
( V
2
Ci(m + k ) /
'-ir/a This shows that the scaling of
+ 3-l a
ln (1
.exp -aCmZ
in §6 is chosen in proper form.
In the classical limit 3"1 >> m T r+ir/a 3"1 -i dk l n e ^ + k 2 ) 1 / 2 ;
.
we have to evaluate
—rr/a
= 3"1 a"1 ( l n y
+ j ma - 1) + 0(a)
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The
FOWLER, M. and ZOTOS, X. (1982) Bethe ansatz quantum sine-Gordon thermodynamics: The specific heat, Phys. Rev. B25, 5806-5817. GOLDSTEIN, S. (1929) On the Asymptotic Expansion of the Characteristic Numbers of the Mathieu Equation, Proc. Roy. Soc. (Edinburgh) 49, 210-223. IMADA, M., HIDA, K., and ISHIKAWA, M. (1982) The Finite Temperature Properties of the Massive Thirring Model and the Quantum SineGordon Model, Journ. Phys. C16, 35-48.
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IZERGIN, A.G. and KOREPIN, V.E. (1981) The lattice quantum sine-Gordon model Lett. Math. Phys. 5^f 199-205. IZERGIN, A.G. and KOREPIN, V.E. (1982) Lattice regularizations of quantum field theory models in two dimensions, In Problems in Quantum Field Theory and Statistical Physics. 3 (Eds. P.P. Kulish and V.N. Popov) Chap. IX, 75-91, Notes on Scientific Seminars, LOMI Vol. 120 "NAUKA", Leningrad. KRUMHANSL, J.A. and SCHRIEFFER, J.R. (1975) Dynamics and Statistical Mechanics of a One-dimensional Model Hamiltonian for Structural Phase Transitions, Phys. Rev. Bll, 3535-3545. KULISH, P.P. (1982)
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TARASOV, V.O., (1982) Classical version of the lattice sine-Gordon model, In Problems in Quantum Field Theory and Statistical Physics. 3 (Eds. P.P. Kulish and V.N. Popov) Chap. XVI, 173-187, Notes on Scientific Seminars, LOMI Vol. 120 "NAUKA", Leningrad. TIMONEN, J. and BULLOUGH, R.K. (1980) Solitons in physics: an application to spin waves in the one dimensional ferromagnet CsNiF3, In Problemes Inverses et Evolution Nonline^aire (Ed. P.C. Sabatier) Chap. V, 133-187, Comptes rendus de la rencontre R.C.P. 264, Editions du CNRS, Paris. TIMONEN, J. and BULLOUGH, R.K. (1981) Breather Contributions to the Dynamical Form Factors of the sine-Gordon Systems CsNiF3 and (CH ) NMnCl 3 (TMMC), Phys. Lett. 82A, 183-187. DE VEGA, H., (1980) Functional integration through inverse scattering variables II, Phys. Rev. D22, 2400-2406. ZAKHAROV, V.E. and FADDEEV, L.D. (1981) Korteweg-de Vries equation: completely integrable Hamiltonian system, Funkt. Anal, i Ego Prilozh. 5, 18-27.
a
INDEX Ablowitz 35, 44, 45, 50, 58, 245, 246, 247, 250, 254 Acheson 119 Acrivos 44, 46, 123 Adam 119, 123, 125 AKNS approach 280 AKNS spectral problem 328 Akylas 119 Alexander 95 Alfven solitons 217 Alonso 248 Anderson 248 Andreev 159, 160 Anker 32, 34 anti-kink 327 Armstrong 124, 229 Ashour-Abdalla 177, 178 asymptotic solitons 277 Atiyah-Ward construction 251 averaged Lagrangian 120 Ball 123 Beaudry 126 Beals 248, 251 Beck 348 (3- effect 88 Benjamin 27, 30, 44, 46, 315 Benjamin-Feir instability 30, 31 Benney 30, 34, 119 Berezin 35 Berger 69, 70 Berggren 84 Berk 229 Bernstein 222, 225 Bingham 320 Birkhoff recurrence 322 Birkhoff's theorem 309, 314 Bishop 348, 350 bisoliton 140 Bock 248 Bohm diffusion coefficient 186 Boltzmann distribution 202 Bondeson 218 Bozatli 124 breather solution 327 Bretherton 124 Brevig 1 Brillouin effects 217 Brindley 86, 89, 93, 94, 97 Bryant 110, 111 Bullough 298, 326, 327, 328, 329, 330, 331, 334, 340 Burns 27
Butler 221, 231, 232 Byatt-Smith 39 Cairns 119, 122 Calogero 33, 248, 271, 279, 285 299 Candler 28 Case 123 Cauchy's formula 250 Caudrey 248, 298, 299, 326, 327 Chen 203, 218, 248 Chian 170 Chiu 123 Cho 217 Chu 128 Chudnovsky 248 Chukbar 218 Clemmow 165, 169, 170, 171, 172, 173 Coffey 228 Coifman 248, 251 Cokelet 1 Coleman 334, 339, 348 conservation law 120 Cornille 249 Costabile 94 Courant 75 Craik 119, 122, 123, 124, 125, 128 Cumberbatch 40 Currie 328, 349 Dashen 339 Davey 32, 34 Davidson 201, 228 Davis 44, 46, 123 Decoster 163 Degasperis 33, 248, 271, 279, 285, 299 Deift 247, 287 Deleondaris 334 Dikii 247 Dinsmore 334 Dirac spectral problem 326 Dirichlet hyperbola 3 dispersion relation 48, 122, 197 Dodd 326, 328, 330 Drazin 86, 87, 93 Drummond 177, 179, 183, 186 Dryuma 32, 245 Dysthe 217 Eady 58 Eckhaus 285, 286, 287
INDEX eigenvalue problem 321 Eilbeck 327 electron plasma wave 197 Elsasser 212 equation Benjamin-Davis-Acrivos (BDA) 62 Benjamin-Ono (BO) 245 Benney-Roskes 246 Boltzmann and Maxwell 221 Boussinesq 29 concentric KdV 36 conservation of waves 229 cylindrical KdV 33 Davey-Stewartson 246 Euler 26, 237, 320 evolution 46, 275 Fredholm 304 Fredholm integral 252 Gel'fand-Levitan 304 Gel'fand-Levitan-Marchenko 286, 297, 298 Ginzburg-Landau 95 Hamilton-Jacobi 78 Hamilton 326, 329 hyperbolic 224 KdV 27, 28, 44, 46, 49, 78, 80, 202, 268, 328 Klein-Gordon 305, 328, 334 Korteweg-de Vries -6£e KdV Korteweg-de Vries Burgers 38, 46, 62 Laplace 26 Lax 272 Lorenz Strange Attractor 95 Marchenko 329 matrix Schrodinger 286 Maxwell 138, 222 modified KdV 283 modified KP 246 multidimensional KdV 202 NLS,
4 c e VIOVILLYHLCUI SchA'dcting&i
n-wave interaction 258 nonlinear evolution 268, 281 nonlinear Klein-Gordon 299, 305 nonlinear Schrodinger (NLS) 30, 93, 107, 112, 202, 283, 310, 313 operator evolution 268 Painleve 151, 152 Poisson 214, 230 Schrodinger 269, 298, 303 Schrodinger-KdV 218 shallow water 70 sine-Gordon 92, 284, 326 time-dependent Schrodinger 245, 247
357 Vlasov 232, 235 Volterra 304 Weber 120 Yang-Mills (SDYM) 251 Estabrook 159 Eubank 178 Faddeev 326, 328 Farnell 86 Feir 30, 315 Ferguson 308, 315 Fermi 315 Ferrell 333 Flaschka 326, 328 Fleierl 84 Flynn 227, 240 Fokas 245, 246, 250 Fordy 305 Fornberg 100, 308 Fourier transform 52, 224, 295 304 Fowler 342 Fowlis 85 Franklin 229 Freeman 27, 32, 33, 34 Friedrichs 71 Fuchs 126 Fuchssteiner 248, 249 Function correlation 323 elliptic 145 Jacobi elliptic 124 Jost 329, 351, 352 Riemann 75, 225 smooth 57 functional integral 330 Galilean invariance 40 Gardner 202, 247, 298 Gelffand 247 Gibbon 86, 92, 327 Gibbons 305 Gildenburg 159 Goldman 217, 318 Goldstein 334 Gollub 129 Gorshkov 45, 50 Greenhow 5, 7, 19, 22 Gribben 221, 231, 232, 233, 238 Grimshaw 44, 45, 46, 47, 49, 50, 56, 63, 66 group speed 30 Gurtin 69, 72 gyro frequencies 165 Hs-norm 81
358 Habennan 246, 247 Haken 95 Halmos 309 Hamiltonian 326, 327, 328, 330, 332, 337, 341 Hart 85 Hasimoto 30 Hasslacher 339 Hauser 251 Hendel 177, 178 Herbert 124 Hide 85, 86, 94 Hilbert 75 Hilbert space 314 Hirota 30, 249 Ho 69, 74, 75 Hogg 318 Holmes 95 Hormander 250 Huberman 318 Ikezi 211, 229 Imada 348 Infeld 237 inverse problem 256 inverse spectral transform 302 Izergin 341 James 86 Jaulent 248 Jeffrey 69 Jeffreys 38 John 1, 3 Johnson 27, 28,31, 32, 34, 35, 37 39, 44, 45, 52, 53, 55, 70 Jost solution 277 Kac 309 Kac-Moody algebras 249 Kadomstev 32 Kadomstev-Petviashvili (KP) 245 Kakutani 28, 70 Kamijo 285, 286, 287 Kano 69 Karney 128 Karpman 28, 35, 45, 50, 58, 207, 209, 210 Kashiwara 249 Kato 70, 80 Kaufman 217 Kaup 28, 45, 50, 58, 124, 125, 247, 248, 299 Kaw 44, 213 Keller 69, 78 Kells 124 Kelly 123
INDEX Kennel 177 Kindel 177 kink solution 327 Kintner 177 Klebanoff 123 Knickerbocker 28, 70 Kodama 45, 50, 58, 248 Konopelchenko 249 Korepin 341 Kotetishvili 213, 215 Kozlov 123, 213, 217 Krumhansl 333 Kruskal 248, 315, 318 Kulish 341 Kuznetsov 202 L -norm 81 Laedke 217 Lagrange stability 322 Lagrangian 3, 7, 29, 237 Lake 34, 103, 106, 129, 308, 315 Lamb 93 Landau 310, 322 Landau damping 213, 309 Langmuir oscillations 214 Lashmore-Davies 119, 320 Lax 268, 319, 322 Lax pair 246, 258 Leblond 123 Lee 187 Levchenko 123 Lifshitz 310, 322 Lighthill 113, 308 Lions 81 Liu 218 Longuet-Higgins 1, 3, 5, 12, 13 23 Lonngren 308 Lorenz 84 Luther 348 Luxembourg effect 162 Lyapunov stable 322 Madsen 28 Makhankov 207, 209 Maki 335, 342, 350, 351 Manakov 125, 247, 249 Mandelstam-Brilluen scattering 158 Manley 322 Martin 308, 316 Maslov 28, 45, 50, 58 Mason 85 Matveev 94 Maxon 33 Maxworthy 84 McEwan 126
INDEX McGoldrich 123 McGuinness 91 Mclver 1 McLaughlin 93 Mei 28 Meyer 69, 74, 75, 77, 129 Mikhailov 248, 299 Miles 32, 35, 37, 70 Miller force 133 Mima 241 Miwa 249 Montgomery 229 Moroz 86, 89, 93, 94, 97 Morse 241 Mvungi 69 Mysak 123 n-soliton 30, 142 Nakamura 248, 249 Nayfeh 124 Neel 227 Neil 240 Neilson 241 Nemytskii 309, 322 Neumann problem 79 Neumann series 288 Neunzert 224 Neveu 339 New 3, 19 Newell 28, 30, 45, 50, 58, 70, 91, 248, 326, 328 Nishida 69, 70 Nishikawa 44, 207, 209, 241 Niznik 249 Number Berger 88 Internal Rotational Froude 88 Mach 205, 206 Oevel 248 Okuda 177, 178, 186, 187 Ono 30 Orszag 124 Ostrovsky 44, 45, 50 Ozawa 229, 241 Painleve 35 parametric instabilities 128 Parkes 233, 236, 237, 238 ParsevalTs theorem 52 Pasta 315 Pedlosky 87, 91 Pelinovksy 44 Peregrine 1, 103, 106 Perkins 177 Petviashvili 32
359 Pfirsch 39 phase speed 30 Phillips 85, 122, 123 Pilling 329, 331, 340 Poincare 308 Poisson stability 322 Porkolab 217 Puga 348 quantum Bethe ansatz 342 quantum K-G field 336, 353 Quon 211 Raetz 123 Rajaraman 339, 343, 347 Raj Lakshmi 339 Rao 207 Rayleigh quotient 311 recurrence phenomena 318 Redekopp 84 Reeve 334 Riemann-Hilbert problem 245, 248 Rienecker 103 Roberts 103, 106, 229 Rosales 35, 249 Rosenbluth 177, 179, 183, 186 Roskes 34 Rowlands 227, 237 Rubinstein 328, 342 Rudakov 207, 209 Sagdeev 197, 199 Sagdeev potential 199 Santini 249 Saric 123 Sato 123 Satsuma 32, 248, 249 Schamel 202, 203, 210, 212, 222 225, 227 Schamel solution 239 Schrieffer 333 Schwartz 100, 103 Schwarz inequality 312 Scott 308 Sears 333 Segur 35, 44, 246, 249, 254 semi-Langrangian 3 Shabat 247, 298, 299, 311 Shen 69, 77, 78, 79 Shukla 202, 210, 213, 217, 218 Shuto 44, 70 Silin 159 similarity solution 58 Simmons 122, 123 Sobolev space 80 soliton 27, 30, 140, 142, 209,217
360 sonic soliton 209 Spatschek 207, 217, 218 spectral problem 298, 305 spectral transform 271, 298 Stenflo 217 Stepanov 309, 322 Stewartson 34 Stirland 334 Stokes flow 3 Stokes wave 30 Stoker 69 stratification parameter 88 Stone 334 Sudan 39 Swinney 129 Symes 247 Tagare 202 Takhtadzhyan 328 Tanaka 217 Taniuti 201 Tappert 28 Tarasov 341 Taykayama 305, 351 Thirring model 348 Thomas 123 Thompson 35, 229 Thorpe 123 three-parametric soliton 210 Thyagaraja 310, 317, 318, 319 Timomen 329, 331, 340 Treve 322 trisoliton 140 Trubowitz 247, 287 Trullinger 334 Tsintsadze 213 Tskhakaya 213 Ulam 315 Ursell 30 Usher 122, 123 Van Harten 285, 286, 287 variational principle 237 Varley 40 Varma 207 Velthuizen 28 Viecelli 33 Vinje 1 Vlasov-Poisson problem 225 Wadati 248, 285, 286, 299 Washimi 201 Wave BGK 241 EIC 177
INDEX electroacoustic nonlinear 146 electromagnetic 214 ion-acoustic 197, 227 ion-cyclotron 193 Langmuir 202, 203, 209, 210, 227 light 197 overturning 1 radiating 64 solitary 50, 52, 57, 202 subluminous 163 superluminous 163 Tollmein-Schlicting 123 wave action 120 wave energy 120 Weissman 91, 92 Wersinger 127 West 100, 129 Wharton 229 Whitham 44, 74, 100, 221, 222 Wielami 125, 126, 127 Wijngaarden 28 Wilhelmsson 125, 126, 127 Wong 198, 211, 212 Yamada 177, 178 Yankov 218 Yu 202, 213, 217 Yuen 103, 106, 129, 308, 315, 316 Zabusky 28, 315, 318 Zakharov 125, 126, 202, 209, 247 249, 298, 299, 311, 219, 328 Zakharov-Shabat spectral problem 283, 285 Zhong 78, 79 Zotos 348
