Advances in Mathematics Research, Vol. 10

ADVANCES IN MATHEMATICS RESEARCH

ADVANCES IN MATHEMATICS RESEARCH, VOLUME 10

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ADVANCES IN MATHEMATICS RESEARCH

ADVANCES IN MATHEMATICS RESEARCH, VOLUME 10

ALBERT R. BASWELL EDITOR

Nova Science Publishers, Inc. New York

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ISBN 978-1-61209-212-6 (eBook)

Published by Nova Science Publishers, Inc.

New York

CONTENTS Preface Chapter 1

vii Liapunov and Normal-Mode Instability of the Rossby-Haurwitz Wave Yuri N. Skiba

∞

1

Chapter 2

Dimensional Cerebellar Controller for Realistic Human Biodynamics Vladimir G. Ivancevic and Tijana T. Ivancevic

29

Chapter 3

Traveling Waves Arising from Synaptically Coupled Neuronal Networks Linghai Zhang

53

Chapter 4

The Initial-Boundary Value Problems for Parabolic Equations in Domains with Conical Points Nguyen Manh Hung and Nguyen Thanh Anh

205

Chapter 5

The Second Boundary Value Problem for Equations of Viscoelastic Diffusion in Polymers Dmitry A. Vorotnikov

249

Chapter 6

Zigzag and Armchair Nanotubes in External Fields Evgeny Korotyaev and Anton Kutsenko

273

Chapter 7

Global Well-Posedness and Limit Behavior for Some Nonlinear Evolution Equation Lijia Han

303

Chapter 8

Structure of Certain Periodic Near Rings Asma Ali

315

Index

323

PREFACE "Advances in Mathematics Research" presents original research results on the leading edge of mathematics research. Each article has been carefully selected in an attempt to present substantial research results across a broad spectrum. In Chapter 1, the dynamics of perturbations to the Rossby-Haurwitz (RH) wave is analytically analyzed. These waves being of great meteorological importance are exact solutions to the nonlinear vorticity equation describing the motion of an ideal incompressible fluid on a rotating sphere. Each RH wave belongs to a space H1 ⊕ H n where H n is the subspace of homogeneous spherical polynomials of degree n. It is shown that any perturbation of the RH wave evolves in such a way that its energy K(t) and enstrophy η (t ) decrease, remain constant or increase simultaneously. A geometric interpretation of variations in the perturbation energy is given. A conservation law for arbitrary perturbations is obtained n

n

and used to classify all the RH-wave perturbations in four invariant sets M − , M + , H n and

M 0n − H n depending on the value of their mean spectral number χ (t ) = η (t ) / K (t ) . The n

energy cascade of growing (or decaying) perturbations has opposite directions in the sets M − and M + due to a hyperbolic dependence between K (t ) and n

χ (t ) . A quotient space with a

quotient norm of the perturbations is introduced using the invariant subspace H n of neutral perturbations as the zero quotient class. While the energy norm controls the perturbation part belonging to H n , the quotient norm controls the perturbation part orthogonal to H n . It is shown that in the set ( χ (t ) < n(n + 1) ), any non-zonal RH wave of subspace H1 ⊕ H n ( n ≥ 2 ) is Liapunov unstable in the energy norm. This instability has nothing in common with the orbital (Poincaré) instability and is caused by asynchronous oscillations of two almost coinciding RH-wave solutions. It is also shown that the exponential instability is possible only in the invariant set M 0 − H n . A necessary condition for this instability is n

given. The condition states that the spectral number

χ (t ) of the amplitude of each unstable

mode must be equal to n(n+1) where n is the RH wave degree. The growth rate is estimated and the orthogonality of the unstable normal modes to the RH wave are shown in two Hilbert

viii

Albert R. Baswell

spaces.

The

instability

in

the

invariant

set

M +n

of

small-scale

perturbations

( χ (t ) > n(n + 1) ) is still open problem. In Chapter 2 the authors propose an ∞ dimensional cerebellar model of neural controller for realistic human biodynamics. The model is developed using Feynman’s action– amplitude (partition function) formalism. Chapter 3 investigates the traveling wave solutions of the following nonlinear singularly perturbed systems of integral differential equations arising from synaptically coupled neuronal networks:

(I) ut + f (u ) + w = α ∫ K ( x − y ) H (u ( y, t ) − θ )dy,

wt = ε ( g (u ) − γ w). (II) ut + f (u ) + w = α ∫ K ( x − y ) H (u ( y, t ) − θ )dy

+β ∫ K ( x− y ) H (u ( y, t )−Θ)dy,

wt = ε ( g (u ) − γ w). ⎛ ⎛ 1 ⎞ ⎞ (III) ut + f (u ) + w = α ∫ K ( x − y ) H ⎜ u ⎜ y, t − | x − y | ⎟ − θ ⎟ dy, c ⎠ ⎝ ⎝ ⎠ wt = ε ( g (u ) − γ w). These equations involve the convolution product of a synaptic coupling and the Heaviside step function. Some of the model equations involve spatial temporal delays as well, where the speed c of the action potential is finite. The main purpose of this chapter is to establish the existence and stability of traveling wave solutions to the nonlinear systems of integral differential equations. The authors will apply fundamental ideas, methods and techniques in differential equations and functional analysis to establish the existence of the traveling wave solutions and the authors will use linearized stability criterion and Evans functions (i.e complex analytic functions) to accomplish the stability. Let L(ε ) represent a linear differential operator obtained by linearizing the integral differential equations with respect to a traveling wave solution, and let σ (L(ε )) denote the spectrum of the linear differential operator L(ε ) . The traveling wave solution of each nonlinear singularly perturbed system of integral differential equations is stable, if

max{ Re λ : λ ∈ σ (L(ε )), λ ≠ 0} ≤ c0 , and

λ = 0 is algebraically simple, where c0 > 0 is a positive constant. This is a sufficient

and necessary condition for the wave stability of the nonlinear systems of integral differential equations. In Chapter 4 the authors consider an initial-boundary value problem for parabolic equation in a domain with conical points. The authors establish the weak well-posedness of

Preface

ix

the problem and the regularity of the solution. The authors also receive asymptotic formulas for the solution near the conical points The classical approach to diffusion processes is based on Fick’s law that the flux is proportional to the concentration gradient. Various phenomena occurring during propagation of penetrating liquids in polymers show that this type of diffusion exhibits anomalous behavior and contradicts the just mentioned law. However, they can be explained in the framework of non-Fickian diffusion theories based on viscoelasticity of polymers (see e.g. D. A. Edwards and D. S. Cohen, AIChE J. 18 (1995), 2345–2355; T. P. Witelski, J. Polymer Science: Part B: Polymer Physics, 34 (1996), 141–150). Initial-boundary value problems for viscoelastic diffusion equations have been studied by several authors (see D. A. Vorotnikov, J. Differential Equations, 2009, Vol. 246, no. 3, 1038-1056). Most of the studies are devoted to the Dirichlet BVP (the concentration is given on the boundary of the domain). In Chapter 5 the authors study the second BVP, i.e. when the normal component of the concentration flux is prescribed on the boundary, which is more realistic in many physical situations. The authors establish existence of weak solutions to this problem. The authors suggest some conditions on the coefficients and boundary data under which all the solutions tend to the homogeneous state as time goes to infinity. In Chapter 6 the authors consider the Schrödinger operator on the zigzag and armchair nanotubes (tightbinding models) in a uniform magnetic field B and in an external periodic electric potential. The magnetic and electric fields are parallel to the axis of the nanotube. The authors show that this operator is unitarily equivalent to the finite orthogonal sum of Jacobi operators. They describe all spectral bands and all eigenvalues (with infinite multiplicity, i.e., flat bands). Moreover, the authors determine the asymptotics of the spectral bands both for small and large potentials. They describe the spectrum as a function of |B |. For example, if |B | →

16 ⎛ π π k π ⎞ , k = 1, 2,..., N , s ∈ , then some spectral band for + π s ⎟ tan ⎜ − 3 ⎝2 n 2N ⎠

zigzag nanotube shrinks into a flat band and the corresponding asymptotics are determined. In Chapter 7, the authors will talk about global well-posedness and limit behavior for the Cauchy problem of some nonlinear evolution equations with derivative nonlinear term. The authors will introduce some results and show some method for solving this kind of problems. In the first section, the authors introduce generalized finite-depth-fluid equation

∂ t u − G (∂ 2x u ) + ∂ x ( where G = −i F

[coth(2πδξ ) −

−1

1 2πδξ

]

u k +1 ) = 0, u (0) = u0 , k +1 F, k ≥ 4 is an integer. The authors will show

that its solution will converge to those of the Benjamin-Ono and KdV equations as the depth parameter δ → ∞ and δ → 0, respectively. I will introduce some basic idea for solving global well-posedness and limit behavior. Meanwhile, I will emphasize Kato smoothing effect method with Besov space and Littlewood-Paley decomposition here. In the second section, the authors introduce the Cauchy problem for the generalized Benjamin-Ono-Burgers equation

x

Albert R. Baswell

∂ t u − H (u xx ) −ν u xx + ∂ x (

u k +1 ) = 0, u (0) = u0 , k +1

where H denotes Hilbert transform, k ≥ 6 is an integer. The authors will show its solution will converge to that of generalized Benjamin-Ono equation. The authors will emphasize some new estimates here. In the third section, the authors introduce the Cauchy problem for the Korteweg-de VriesBurgers equation

ut + u xxx + ε | ∂ x |2 u + (u 2 ) x = 0, u (0) = u0 , where ε > 0 . The authors will introduce the global well-posedness and inviscid limit for Korteweg-de Vries equation obtained in [10]. The authors will emphasize a l1 variant Bougaintype space method here. Let R be a left near ring and let m = m( x, y ) ≥ 1 , n = n( x, y ) ≥ 1 be positive integers depending on x and y in R. Further, let p( xy ) be an element of R which is finite sum of powers ( xy ) ; k ≥ 2 and additive inverses of such powers. In Chapter 8 the authors establish k

some decomposition theorems for a near ring R satisfying any one of the conditions, (i)

xy = ( xy ) n p ( xy ) , (ii) xy = ( yx) n p( xy ) , (iii) xy = ( yx) n p( yx) , (iv) xy = x m y n p( xy ) , (v) xy = y x p ( xy ) , (vi) xy = x p ( xy ) x and (vii) xy = y p ( xy ) y . m

n

m

n

m

n

In: Advances in Mathematics Research, Volume 10 ISBN 978-1-60876-265-1 c 2010 Nova Science Publishers, Inc. Editor: Albert R. Baswell, pp. 1-28

Chapter 1

L IAPUNOV AND N ORMAL -M ODE I NSTABILITY OF THE ROSSBY-H AURWITZ WAVE Yuri N. Skiba∗ Centro de Ciencias de la Atm´osfera Universidad Nacional Aut´onoma de M´exico Circuito Exterior, CU, C.P. 04510, M´exico, D.F., M´exico

Abstract The dynamics of perturbations to the Rossby-Haurwitz (RH) wave is analytically analyzed. These waves being of great meteorological importance are exact solutions to the nonlinear vorticity equation describing the motion of an ideal incompressible fluid on a rotating sphere. Each RH wave belongs to a space H1 ⊕ Hn where Hn is the subspace of homogeneous spherical polynomials of degree n. It is shown that any perturbation of the RH wave evolves in such a way that its energy K(t) and enstrophy η(t) decrease, remain constant or increase simultaneously. A geometric interpretation of variations in the perturbation energy is given. A conservation law for arbitrary perturbations is obtained and used to classify all the RH-wave perturbations in four inn n variant sets M− , M+ , Hn and M0n −Hn depending on the value of their mean spectral number χ(t) = η(t)/K(t). The energy cascade of growing (or decaying) perturban n tions has opposite directions in the sets M− and M+ due to a hyperbolic dependence between K(t) and χ(t). A quotient space with a quotient norm of the perturbations is introduced using the invariant subspace Hn of neutral perturbations as the zero quotient class. While the energy norm controls the perturbation part belonging to Hn , the quotient norm controls the perturbation part orthogonal to Hn . It is shown that in the n set M− (χ(t) < n(n + 1) ), any non-zonal RH wave of subspace H1 ⊕ Hn (n ≥ 2 ) is Liapunov unstable in the energy norm. This instability has nothing in common with the orbital (Poincar´e) instability and is caused by asynchronous oscillations of two almost coinciding RH-wave solutions. It is also shown that the exponential instability is possible only in the invariant set M0n − Hn . A necessary condition for this instability is given. The condition states that the spectral number χ(t) of the amplitude of each unstable mode must be equal to n(n + 1) where n is the RH wave degree. The growth rate is estimated and the orthogonality of the unstable normal modes to the RH wave ∗

E-mail address: [email protected]

2

Yuri N. Skiba n are shown in two Hilbert spaces. The instability in the invariant set M+ of small-scale perturbations (χ(t) > n(n + 1)) is still open problem.

Keywords: Ideal incompressible flow, Liapunov and exponential instability, RossbyHaurwits wave.

1.

Introduction

As is well known, the dynamics of two-dimensional ideal and incompressible fluid on a rotating sphere is described by the nonlinear vorticity equation [1–5]. The same equation captures many features of the large-scale dynamics of the barotropic atmosphere, and is used in the meteorological literature under the name of barotropic vorticity equation (BVE) [6–10]. Moreover, the Rossby-Haurwits (RH) waves, being exact solutions to this equation [6,7,10] form the main features of daily meteorological fields. Therefore the stability properties of the RH wave are of considerable interest for deeper understanding of the low-frequency variability of the atmosphere [11,12] as well as for developing initial data assimilation methods [13–15]. The RH wave of degree n is a spherical polynomial of subspace H1 ⊕Hn , where Hk is the subspace of homogeneous spherical polynomials of degree k. It represents the simplest two-dimensional infinitely differentiable BVE solution on the sphere. Many works has been devoted to the instability of Rossby waves on the β-plane [16–19] and RH waves on a sphere [20–25]. However, mathematically, the problem of linear and nonlinear stability of the RH wave is still far from its complete solution. Actually, some results on the stability of wave were obtained numerically or approximately [4,18,19] and, therefore, can contain the errors of calculation. For example, severe truncation of perturbations used in [16,17], though leads to very interesting and useful conclusions, does not allow obtaining comprehensive results. A quite detailed analytical study of the RH-wave stability carried out in [20] cannot be considered as in-depth analysis as well, because the measure (13) used in that work is not even a seminorm. It should also be noted the interesting results on the barotropic flow stability obtained in [26, 27]. However, the application of these results to the stability of RH wave on a sphere is not always fruitful. For example, Theorem 2 from [27] can be applied only to the trivial situation with a super-rotation flow. The Liapunov instability of non-zonal RH waves was proved in [22] (see also [23,24]). As to the linear stability of the RH wave, the first necessary condition for the normal-mode instability of this wave was obtained only recently [25]. However, no constructive sufficient condition for the exponential instability of the RH wave has been obtained up to now, and the spectrum of the linearized operator has not been studied. No strict proof of the existence of a critical RH-wave amplitude for the exponential instability has been given. The estimates of the growth rate of unstable mode are rather approximate, and the sensitivity of numerical stability results to variations in the basic parameters has not been analyzed. Although the Liapunov instability of any non-zonal RH wave has been proved [13], its orbital instability is still open mathematical problem. Thus, up to now, the nonlinear instability of the RH wave has not been completely analyzed. This work represents a review of results on the nonlinear and linear instability of the RH waves in an ideal incompressible fluid on a rotating sphere. A conservation law obtained for arbitrary RH-wave perturbations asserts that any perturbation evolves in such a

Liapunov and Normal-Mode Instability of the Rossby-Haurwitz Wave

3

way that its kinetic energy K(t) and enstrophy η(t) decrease, remain constant or increase simultaneously. By the law, all the perturbations to the RH wave of degree n are divided into four invariant sets M−n , M+n , Hn and M0n − Hn depending on the value of their mean spectral number χ(t) = η(t)/K(t), namely, χ(t) < n(n + 1) for perturbations of M−n , χ(t) > n(n + 1) for perturbations of M+n , and χ(t) = n(n + 1) for perturbations of M0n . The set M0n \ Hn is the complement of Hn in M0n . This allows us to study the instability of the RH wave in each invariant set separately [24, 28, 29]. We show that the energy (and enstrophy) of a perturbation to the RH wave varies only if the projection of the perturbation on the RH wave varies. It means that the energy norm (as well as the enstrophy norm) controls the evolution of perturbations only in the subspace Hn , and hence, this norm is useless in the subspace orthogonal to Hn . It is demonstrated that the cascade of the energy of growing perturbations has opposite directions in invariant sets M−n and M+n due to a hyperbolic dependence between the perturbation characteristics K(t) and χ(t). Since invariant subspace Hn contains just neutral (that is, non-increasing and non-decreasing) perturbations, we use Hn as the zero quotient class and introduce a quotient (factor) space and quotient norm of the perturbations. The sum of the energy norm and the quotient norm is suggested to be an appropriate norm for the instability study of the RH wave in the sense that while the energy norm controls the part of a perturbation that belongs to Hn , the quotient norm controls the other part of the perturbation that is orthogonal to Hn . If n = 1 then the RH wave belongs to the subspace H1 , and hence, represents a super-rotation flow in some geographical system of coordinates. It is demonstrated that any such super-rotation flow on a sphere is stable both in the energy norm and the enstrophy norm. The linear stability of this flow was proved in [30]. Then we analyze the stability of the RH wave of subspace H1 ⊕ Hn for n ≥ 2. We prove that any non-zonal wave of this type is Liapunov unstable in the energy norm with respect to relatively large-scale perturbations from invariant set M−n . Besides, it is shown that the growing perturbation is generated by the asynchronous oscillations of the two RH waves, which slightly differ from each other only by their super-rotational components. As a result, these two waves have different phase velocities, and their representative points, which initially are very close to each other in the phase space, will diverge from each other with time. Thus, the Liapunov instability proved here has nothing in common with the orbital (Poincar´e) instability [22], since the orbits of the two RH waves are very close to each other in the phase space. In fact, the instability mechanism described is identical with that of a periodic solution to a nonlinear pendulum equation [31, 32]. It is also shown that the exponential (normal-mode) instability is possible only in the invariant set M0n − Hn [24, 33]. We give a criterion (see below (64)) and a necessary condition (see below (68)) for the normal mode instability of the RH wave of H1 ⊕ Hn , besides, condition (68) states that the spectral number χ of the amplitude of each unstable mode must be equal to n(n + 1). The growth rate of unstable normal modes is also estimated and the orthogonality of such modes to the RH wave are shown both in L2 -inner product and in the energy inner product. It is well-known that both the Legendre-polynomial flow and the RH wave of degree n are linearly stable with respect to relatively small-scale perturbations of invariant set M+n [24,33]. However, the Liapunov stability of the RH wave in invariant set M+n as well as the orbital stability of this wave are still unsolved problems.

4

2.

Yuri N. Skiba

Geometric Structure of RH Waves

It is the well-known fact that, the large-scale barotropic dynamics of the atmosphere can approximately be governed by the nonlinear barotropic vorticity equation (BVE) ∆ψ t + J(ψ, ∆ψ + 2µ) = 0

(1)

which also describes the motion of ideal fluid on a rotating unit sphere S [8, 9, 34–37]. Equation (1) is written in a non-dimensional form in the spherical system of coordinates, besides, ψ(t, x) is the streamfunction, x = (λ, µ) is a point of sphere S where λ is the longitude, µ = sinφ, φ is the latitude, ∆ is the spherical Laplace operator, ∆ψ is the relative vorticity, ∆ψ + 2µ is the absolute vorticity and J(ψ, h) = (b n × ∇ψ) · ∇h = ψ λ hµ − ψ µ hλ b is the unit vector normal to the surface of sphere S, is the Jacobian [4, 5, 9], where n   p 1 2 ∇h = p hλ , 1 − µ hµ 1 − µ2 is the gradient of a function h, and the flow velocity → − b × ∇ψ U =n

(2)

(3)

(4)

with the components

p U = − 1 − µ2 ψ µ , is solenoidal:

1 V =p ψλ 1 − µ2

→ − ∇ · U = 0.

(5)

(6)

Hereafter, ft , fλ and fµ denote the partial derivatives of a function f (t, λ, µ) with respect to t, λ and µ, respectively. In equation (1), the nonlinear term J(ψ, ∆ψ) is responsible for nonlinear interactions of waves [2], while the linear term J(ψ, 2µ) = 2ψ λ describes the super-rotation of sphere S about its polar axis. This is the term that destroys the spherical symmetry of equation (1). The four types of exact solutions to equation (1) are known up to now. The simplest of them are the zonal flows which depend only on the coordinate µ (ψ = ψ(µ)). The solutions of the other three groups generally depend on all the coordinates: ψ = ψ(t, λ, µ). They are the RH waves, Wu-Verkley waves and modons [10, 33]. It should be noted that any solution of these three groups represents a field on the sphere which revolves around the polar axis. This property of the solutions is the direct consequence of the presence of asymmetric term J(ψ, 2µ) in equation (1) [33]. Besides, while the RH waves are the simplest infinitely smooth BVE solutions, the non-stationary Wu-Verkley waves and modons represent weak BVE solutions. However, the steady Wu-Verkley waves and modons are still classic solutions of equation (1). A RH wave of degree n has the form f (t, λ, µ) = −Cµ + fn (λ, µ, t) ≡ −Cµ +

n X

m=−n

fnm Ynm (λ − ω n t, µ),

(7)

Liapunov and Normal-Mode Instability of the Rossby-Haurwitz Wave

5

where Ynm (x) = Pnm (µ)eimλ is the spherical harmonic of degree n and zonal wavenumber m (−n ≤ m ≤ n), Pnm (µ) is the Legendre associated function and fnm are arbitrary amplitudes, though fn−m = (−1)m fnm for a real RH wave. Let k be an integer, and let Hk denote the subspace of all the homogeneous spherical polynomials of degree k satisfying to the spectral problem −∆p = k(k + 1)p. Then the RH wave (7), as the sum of a super-rotation flow −Cµ of subspace H1 (rigid rotation about the polar axis of the sphere) and a homogeneous spherical polynomial fn of degree n of subspace Hn , belongs to subspace H1 ⊕ Hn being direct orthogonal sum of H1 and Hn . This wave is exact solution of (1) under the condition that ωn = C −

2(C + 1) , n(n + 1)

n = 2, 3, 4, . . . .

(8)

Thus the speed ω n of the RH wave depends on the super-rotation velocity C and wave degree n. The RH waves are the typical special feature of the daily weather maps in middle latitudes [6,7,9,11]. Therefore, the stability of the RH wave is the problem of both hydrodynamic and meteorological importance.

3.

Fractional Derivatives and Hilbert Spaces of Smooth Functions on a Sphere

We note that if ψ(t, x) is a solution to equation (1) then ψ(t, x)+const is the solution, too. Therefore, without loss of generality, we will consider in this work only the functions which are orthogonal to any constant function on the sphere S. Let C0∞ (S) denote the set of infinitely differential functions h(x) on the unit sphere S such that Z h(x) dx = 0. (9) S

3.1.

Hilbert Space H 0

The inner product and the norm of these functions are defined as Z hζ, hi = ζ(x) h(x) dx

(10)

S

and khk = hh, hi1/2

(11)

respectively, where h is the complex conjugate of h. By closing the set C0∞ (S) in the norm (11) we obtain the Hilbert space L2 (S) of functions orthogonal to a constant on S which

6

Yuri N. Skiba

for simplicity we will denote as H 0 . It is well known that H 0 = H1 ⊕ H2 ⊕ · · · ⊕ Hk ⊕ · · ·

(12)

is the orthogonal sum of the subspaces Hk = {p(x) : ∆p = −k(k + 1)p}

(13)

of homogeneous spherical polynomials of degree n [38]. As it was noted earlier, any polynomial p(x) of subspace (eigensubspace) Hk is the eigenfunction of spherical Laplace operator −∆ corresponding to the eigenvalue χk = k(k + 1).

(14)

The dimension of Hk is 2k + 1, and 2k + 1 spherical harmonics Ykm (x) = Pkm (µ)eimλ of degree k and zonal number m (−k ≤ m ≤ k) form the orthonormal base in Hk [38]: D E Ykl (x), Ykm (x) = δ lm , (15) where Pkm (µ) is the associated Legendre function, and δ lm is the Kronecker delta. As a result, all the spherical harmonics Ykm (x) (k = 1, 2, 3, . . . ; |m| ≤ k) form the orthonormal base in H 0 . Due to (12), each function h(x) of H 0 is represented by its Fourier series h(x) =

∞ X

Yk (h; x),

(16)

k=1

where Yk (h; x) = (2k + 1)(h ∗ Pk )(x) =

n X

m hm k Yk (x)

(17)

m=−n

is the orthogonal projection of h(x) on Hk , that is, a homogeneous spherical polynomial of degree k. In (17), the operation ∗ is the convolution, Pk (x) is the Legendre polynomial of the degree k and m hm (18) k = hh(x), Yk (x)i is the Fourier coefficient of function h(x) [38, 39].

3.2.

Multiplicative Derivatives and Hilbert Spaces H s

For any real s and each infinitely differentiable function h(x) ∈ C0∞ (S) we define a multiplicative operator Λs as ∞ X s/2 s χk Yk (h; x), (19) Λ h(x) = k=1

where χn and Yk (h; x) are given by (14) and (17) [5,40,41]. In particular, Λ2n h = (−∆)n h

Liapunov and Normal-Mode Instability of the Rossby-Haurwitz Wave

7

for any integer n. Thus, operator Λ can be interpreted as the square root of the symmetric and positive definite spherical Laplace operator: Λ = (−∆)1/2 . Since the Laplace operator is in essence the only differential operator on a sphere invariant under isometries [39], it is natural to use operator Λ as the derivative to characterize differential properties of a smooth function on the sphere S [5]. Besides, unlike the partial derivatives ∂/∂λ and ∂/∂µ with respect to geographical coordinates, the derivative Λ has the advantage that it is independent of the choice of the system of coordinate on the sphere S. It is easy to show that a derivative Λs is a positive definite operator, which commutes with the derivative ∂/∂λ as well as with any other derivative Λr (and in particular, with the Laplace operator). Moreover, for any real s, operator Λs : C0∞ (S) → C0∞ (S) is symmetric, and hence, closable. Definition 1. Let s be a real. The closure of set C0∞ (S) in the s− norm s

khks ≡ kΛ hk =

X ∞ k=1

χsk

k X

m=−k

2 |hm k |

1/2

(20)

is denoted as H s . It is a Hilbert space of functions on S with the inner product hζ, his = hΛs ζ, Λs hi .

(21)

Thus if h ∈ H s then Λs h ∈ H 0 . Definition 2. Let s and r be real. An element z ∈ H s is denoted Λr ψ and called the r th derivative of ψ ∈ H s if hz, his = hψ, Λr his

for any h ∈ C0∞ (S).

The extension operator Λr : H s → H s is closed. Lemma 1. Let s be a real, r > 0 and h ∈ H s+r . Then khks+r = kΛr hks

(22)

khks ≤ 2−r/2 khks+r .

(23)

and Proof. Let first h ∈ C0∞ (S) and ψ = Λs h. Obviously, (22) is fulfilled. Further, using (21), (17), (19), and (14) we get khk2s+r

k ∞ X X

s+r 2 2 2 2 2 r r r r

|ψ m χk = Λ h = kΛ ψk = k | ≥ 2 kψk = 2 khks k=1

m=−k

that is, (23) is also fulfilled for any h ∈ C0∞ (S). Let now h ∈ H s+r . Then there exists a Cauchy sequence {hn } ∈ C0∞ (S) such that khn − hks+r → 0. Note that both {hn } and {Λr hn } are Cauchy sequences in H s , besides, (22) and (23) are valid for their elements. Therefore, the application of lim to khn ks+r = kΛr hn ks and khn ks ≤ 2−r/2 khn ks+r n→∞ proves the lemma.

8

Yuri N. Skiba The derivative Λ is closely related with the gradient due to the following assertion:

Lemma 2. Let h ∈ H 1 . Then k∇hk = kΛhk = khk1 .

Proof. Let first h ∈ C0∞ (S). Applying the partition of unity [5, 39] subordinate to a covering of the sphere S , one can easily get

 k∇hk2 = h∇h, ∇hi = h−∆h, hi = Λ2 h, h = hΛh, Λhi = kΛhk2 = khk21 . (24)

Let now h ∈ H 1 and {hn } ∈ C0∞ (S) such that khn − hk1 → 0. Then the application of lim to both parts of k∇hn k2 = khn k21 proves the lemma. n→∞

Corollary 1. Let s = 0, r = 1 and h ∈ H 1 . Then (23) is the Poincar´e-Steklov inequality.

Lemma 3. Let s be a real. Then Λ−r : H s → H s is the compact operator for r > 0, the Hilbert-Schmidt operator for r > 1, and the trace class operator for r > 2. −r/2

Proof. Let r > 0. By (14), the multipliers χk = [k(k + 1)]−r/2 of the operator Λ−r −r tend to zero as k increases. Therefore Λ is compact due to the compactness criterium [38, Theorem of Section12.2]. Let r > 1. Since kYkm k = 1, we get ∞ k ∞ X k ∞ X X X

−r m 2 X −r

Λ Y = χk = k k=1 m=−k

k=1 m=−k

k=1

2k + 1 . [k(k + 1)]r

(25)

This series is the square of the Schmidt norm of the operator Λ−r [5, 38]. Using inequality k + (k + 1) 1 1 2 = r−1 + r ≤ 2r−1 r r r−1 [k(k + 1)] k (k + 1) k (k + 1) k P∞ −r and comparing (25) with the series k=1 k , we obtain that series (25) converges if r > 1. Therefore, the Schmidt norm of operator Λ−r and its trace norm k ∞ X X

−r/2

χk

k=1 m=−k

are finite if r > 1 and r > 2, respectively. Thus Λ−r is the Hilbert-Schmidt operator for r > 1, and the trace class operator for r > 2 [38, Lemma of Section13.2]. The theorem is proved. Corollary 2. Let s be a real and r > 0. Then a bounded set in H s+r is compact in H s .

3.3.

Jacobian-Containing Formulas

We will need the following properties of the Jacobian proved earlier in [5,28] using the following two relations: Z J(ψ, g)dx = 0 S

[39] and J(hψ, g) = hJ(ψ, g) − ψJ(g, h)

which are valid for any differentiable functions on the sphere S.

Liapunov and Normal-Mode Instability of the Rossby-Haurwitz Wave

9

Lemma 4. Let r be a real, and let ψ(x), g(x) and h(x) be sufficiently smooth complex functions on the sphere S, and let F (ψ) be a continuously differentiable function. Then



 hJ(ψ, g), hi = J(g, h), ψ = − J(ψ, h), g , (26) D E J(ψ, h), F (ψ) = 0, (27) Re hJ(ψ, µ), Λr ψi = 0,

Re hJ(ψ, ∆ψ), µi = 0,

Re hJ(ψ, h), ψi = 0.

(28)

(29)

In particular, if ψ(x) is real then (27) leads to hJ(ψ, h), ψi = 0.

(30)

The motion of ideal fluid is subordinated to the specific limitations (conservation laws): Lemma 5. [5,34] Let ψ(t, x) be a solution of (1). Then each of the following integrals is invariant of motion: Z 1 1 2 2 ∆ψ dx, k∇ψk , k∆ψk , (31) 2 2 ZS Z p F (Ω) dx. (32) u 1 − µ2 dx, h∆ψ, µi = S

S

The values (31) and (32) are the kinetic energy, enstrophy, integral vorticity, angular momentum and Casimir of the flow, respectively. Here Ω = ∆ψ + 2µ is the absolute vorticity.

4.

Conservation of Pseudoenergy and Angular Momentum of RH-Wave Perturbations

Let us considere a real RH wave solution (7), (8) to the BVE (1). Since the streamfunction of the wave f ∈ H1 ⊕ Hn , its absolute vorticity Ω = ∆f + 2µ = −χn f − χn ω n µ

(33)

also belongs to H1 ⊕ Hn , besides, Ω = ∆f + 2µ = −χn f

(34)

for a stationary RH wave (ω n = 0). Arbitrary real perturbation of the RH wave f (t, x) can be presented as ψ ′ (t, x) = ψ(t, x) − f (t, x),

(35)

where ψ(t, x) is some another real solution to the BVE (1). According to Szeptycki’s theorem [42] about the existence and uniqueness of the BVE solution, a suitable norm for estimating the RH wave perturbations must contain the second derivatives of the streamfunction ψ, that is, must be related with the 2-norm (20):

′

ψ = ∆ψ ′ . (36) 2

10

Yuri N. Skiba

The perturbation (35) satisfies equation ∆ψ ′t + J(f, ∆ψ ′ ) + J(ψ ′ , ∆f + 2µ) + J(ψ ′ , ∆ψ ′ ) = 0 with initial condition ψ ′ (0, x) = ψ(0, x) − f (0, x). With (34), the last equation can be rewritten as ∆ψ ′t + J(f + ψ ′ , Dn ψ ′ ) − χn ω n J(ψ ′ , µ) = 0, (37) where Dn ψ ′ ≡ (∆ + χn )ψ ′ .

(38)

In particular, if wave (7) is stationary (ω n = 0) then (37) is reduced to ∆ψ ′t + J(f + ψ ′ , Dn ψ ′ ) = 0.

(39)

Taking the inner product of equation (37) with ∆ψ ′ + χn ψ ′ and using (26), (28) and (29) we obtain η t (t) = χn Kt (t), (40) where χn = n(n + 1) (see (14)), and K(t) =

1

∇ψ ′ 2 = 1 ψ ′ 2 1 2 2

and

η(t) =

1

∆ψ ′ 2 = 1 ψ ′ 2 2 2 2

(41)

are respectively the kinetic energy and enstrophy of perturbation. Thus we obtain the following assertion [5]: Theorem 1. Any perturbation of the RH wave (7), (8) evolves such that its energy and enstrophy decrease, remain constant or increase simultaneously according to (40). For infinitesimal perturbations of a stationary planetary Rossby wave on the beta-plane, the law (40) was established by Gill [36]. For infinitesimal perturbation of a Legendrepolynomial flow it was given by Karunin [43]. Theorem 1 generalizes these results to arbitrary perturbation of any RH wave of H1 ⊕ Hn on a sphere. Unlike the perturbation energy K(t), the perturbation enstrophy η(t) contains information not only on the magnitude, but also on the spectral composition of the perturbation ψ ′ . Due to (23), the 2-norm (the enstrophy norm) is stronger than the 1-norm (the energy norm), and in general, a perturbation of the BVE solution can be stable in the energy norm and unstable in the enstrophy norm. However, according to Theorem 1, the RH wave has the same stability properties in both norms. The value

2 2 (42) χ(t) ≡ χ(ψ ′ (t, x)) = η(t)/K(t) = ψ ′ 2 / ψ ′ 1 is the mean spectral number of the perturbation (the square of the Fj¨ortoft’s spectral number [2]). Due to (40), the functional U [ψ ′ )] ≡ η(t) − χn K(t) = {χ(t) − χn }K(t)

(43)

is conserved with time for any perturbation ψ ′ (t, x) to the RH wave (n ≥ 1), that is, Ut [ψ ′ ] = [{χ(t) − χn }K(t)]t = 0.

(44)

Liapunov and Normal-Mode Instability of the Rossby-Haurwitz Wave

11

Note that conservation law (44) is nothing else that the conservation of pseudoenergy [44]. It was used by Petroni et al. [45] for small perturbations to a harmonic wave on the β-plane (see also [46, 47]). The invariance of the angular momentum h∆ψ, µi for any solution of the BVE (1) gives one more conservation law for arbitrary perturbations to the RH wave:



 ∆ψ ′ , µ = −2 ψ ′ , µ = const.

(45)

Due to (45), the projection of any perturbation ψ ′ on the Legendre polynomial of degree one is invariable. It follows from (45) and convolution formula (17) that the projection of any perturbation ψ ′ on the subspace H1 is invariable too, and hence, this perturbation component is unimportant and may be ignored in the instability study of a BVE solution, and in particular, in the stability study of the RH waves. Although this fact is taken into account, we do not orthogonalize ψ ′ to H1 in this work.

5.

Invariant Sets, Quotient Space, Equivalence Classes and Quotient Norm of RH-Wave Perturbations

The conservation law (44) allows us to divide all possible perturbations to RH-wave (7) of subspace H1 ⊕ Hn into three invariant sets: M+n = {ψ ′ : χ(ψ ′ (t, x)) > χn }, M0n = {ψ ′ : χ(ψ ′ (t, x)) = χn },

M−n = {ψ ′ : χ(ψ ′ (t, x)) < χn },

(46)

where χn is given by (14). Thus if a perturbation to the RH wave initially belongs to one of the sets (46) then it will always belong to this set. By definition, the spectral number (42) is equal to χn for perturbations of M0n , and is bounded for perturbations of M−n . For each n, the set M0n can be considered as a boundary between relatively large-scale perturbations of the set M−n and small-scale perturbations of the set M+n . Note that set M0n contains the subspace Hn of the homogeneous spherical polynomials of degree n, which is one more invariant subset of the RH-wave perturbations. As a result, the stability of the RH wave can be analyzed separately in each of the four invariant sets M−n , M+n , Hn and M0n \ Hn [29], where M0n \ Hn is the complement of Hn in M0n . It is easy to see that any perturbation of invariant set Hn is stable. Indeed, then ∆ψ ′ + χn ψ ′ = 0, and due to (37), this perturbation evolves according to the linear equation ψ ′t + χn ω n ψ ′λ = 0,

(47)

conserving its form, and in particular, its energy and enstrophy. This leads to Theorem 2. All the perturbations to the RH wave (7), (8) may be classified in the four invariant (independent) sets M−n , M+n , Hn and M0n \ Hn , besides, Hn is the linear subspace of stable perturbations.

12

Yuri N. Skiba

Since set Hn is the linear space and all the perturbation of Hn are neutral, we can represent all the perturbations to the RH wave (7) as a quotient space C0∞ (S)/Hn that consists of equivalence classes of perturbations, taking Hn as the zero class. Thus, two elements g and h belong to the same equivalence class only if g − h ∈ Hn , and for any perturbation ψ ′ , the corresponding equivalence class is {ψ ′ + h : h ∈ Hn }. In each equivalence class, the element which has zero projection on the subspace Hn is the representative element of the class. The functional

∆ψ ′ + χn ψ ′ (48) is only a seminorm in the space C0∞ (S), but this is the norm in the quotient space C0∞ (S)/Hn . The quotient space of perturbations to the RH wave (7) consists exactly of the three invariant sets M−n , M+n and M0n \ Hn . It is now convenient to represent each perturbation ψ ′ to the RH wave (7) as the orthogonal sum ψ ′ = ψ ′⊥ + ψ ′n (49)

of two orthogonal components, where ψ ′n is the projection of ψ ′ on the subspace Hn , and ψ ′⊥ is the complementary part of ψ ′ . We now introduce a norm

′



ψ = ∆ψ ′ + χn ψ ′ + ψ ′ (50) ∗

1



which is the sum of the quotient norm (48) and energy norm ψ ′ 1 = ∇ψ ′ . We will show later that norm (50) is appropriate in the instability study of the RH wave (7), besides, the energy norm controls the evolution of the perturbation part ψ ′n , while the quotient norm controls the evolution of its orthogonal part ψ ′⊥

6.

Hyperbolic Law for Perturbations in Invariant Sets M−n and M+n

If we introduce the designation ρ(t) = χ(t) − χn for perturbations of RH-wave of the sets M−n and M+n then law (44) can be rewritten as ρ(t)K(t) = L0 = ρ(0)K(0),

(51)

where L0 is a constant determined by the initial values χ(0) and K(0) of the perturbation ψ ′ (0, x). This constant is negative in M−n and positive in M+n . The law (51) demonstrates hyperbolic dependence between the mean spectral number (42) and kinetic energy (41) of perturbations of sets M−n and M+n , besides, the invariant sets M−n and M+n of perturbations are in turn the family of invariant sets depending on the parameter L0 . Let Kf be the energy of the RH wave (7), and let α > 0. If we consider all the vorticity equation solutions ψ belonging to the energy surface Kψ = α2 Kf then kinetic energy of the perturbations (35) may vary only within the limits Kmin ≡ (α − 1)2 Kf ≤ K(t) ≤ (α + 1)2 Kf ≡ Kmax ,

(52)

where Kmin and Kmax are achieved for ψ = αf and ψ = −αf , respectively [5]. Initial values χ(0) and K(0) of a perturbation determine a certain hyperbola (51), and the point

Liapunov and Normal-Mode Instability of the Rossby-Haurwitz Wave

13

(ρ(t), K(t)), representing the evolution of the kinetic energy K(t) and spectral distribution χ(t) of the perturbation, may move along this hyperbola in both directions just within limits (52). According to (52), the range of variations of the perturbation energy K(t) along a hyperbola decrease with the RH-wave energy Kf and α. Thus, the smaller is Kf the more stable is the RH wave. Also, if the energy Kf of the RH wave is not small then initial perturbation energy K(0) may be small only if α is close to one (Kψ is close to Kf ), and hence, the energetic surfaces of two BVE solutions (ψ and f ) are close to each other in the phase space. This comment can be useful, since the Liapunov stability analyses the behavior of finite but small initial perturbations whose energy norm is small, too. It follows from (42) and (40) that 1 1 [χ(t)]t = [K(t)]t {χn − χ(t)} K(t)

(53)

and hence, the spectral number χ(ψ ′ ) of a growing (or decaying) perturbation ψ ′ grows (decreases) in M−n and decreases (grows) in M+n . Thus, we obtain Theorem 3. The energy cascade of growing (or decaying) perturbations of RH wave (7), (8) has opposite directions in the sets M−n and M+n . Due to (51), the perturbation energy must grow both in M−n and in M+n as mean spectral number χ(t) approximates χn (ρ(t) → 0). Therefore, one can expect the existence of unstable perturbations in M−n and M+n in a vicinity of spectral number χn . Later we show that any non-zonal RH wave of H1 ⊕ Hn is Liapunov unstable in the set M−n if n ≥ 2. We also show that the linear instability is possible only in the invariant set M0n − Hn where χ(ψ ′ (t, x)) = χn . However, till now there is no proof of the instability of the RH-wave in the set M+n .

7.

Geometric Interpretation of Variations in the Energy of Perturbations

The phrase “a perturbation extracts the energy from the basic flow” is not uncommon in the scientific articles devoted to the barotropic instability of flows [11,51,52]. However in some cases this phrase is inaccurate. Indeed, in the case of ideal fluid the kinetic energy of a solution (basic flow) is invariant of motion (Lemma 5), that is, the energy of basic flow is constant. In other words, “a perturbation cannot extract the energy from the basic flow”. Now we will give geometric interpretation to changes in the energy of RH-wave perturbations. By (35), a perturbation ψ ′ (t, x) of the RH wave f (t, x) is the difference between two vorticity equation solutions: ψ ′ (t, x) = ψ(t, x) − f (t, x). We have

2

 2K = ∇ψ ′ = − ∆ψ ′ , ψ ′ = − h∆(ψ − f ), ψ − f i = k∇ψk2 + k∇f k2 + 2Re h∆ψ, f i = 2Kψ + 2Kf + 2Re h∆ψ, f i .

If we differentiate in time the last equation and use the fact that kinetic energy of each of the solutions ψ and f is invariable (see Lemma 5), then we obtain

 Kt (t) = Re h∆ψ, f it = Re ∆ψ ′ , f t .

14

Yuri N. Skiba

Since the inner product h∆ψ, f i depends on the relative position of the solutions in the space, the energy K(t) of a perturbation ψ ′ (t, x) changes in time only due in view of changes in the position of the solution ψ(t, x) with respect to the RH wave f (t, x). Taking into account the geometric form f = −Cµ + fn of the RH wave (7), the invariant of perturbation motion (45) and the symmetry of the Laplace operator, one can rewrite the last equation as  







Kt (t) = Re ∆ψ ′ , f t = Re ∆ψ ′ , fn t = Re ψ ′ , ∆fn t = −χn Re ψ ′ , fn t , (54)

where fn is the polynomial part of the RH wave belonging to subspace Hn . Note that (54) can also be obtained if we steady the RH wave (7) in the rotating coordinate system (λ′ , µ′ ) (where λ′ = λ − ω n t, and µ′ = µ ), then take the inner product of equation (39) with f + ψ ′ and use (29) and (41). In the same way it can be shown that 



η t (t) = Re h∆ψ, ∆f it = Re ∆ψ ′ , ∆f t = χ2n Re hψ, fn it = χ2n Re ψ ′ , fn t . Thus, we have proved the following two assertions:

′ Theorem 4. The energy (or enstrophy)

′ of a perturbation ψ to ′the RH wave (7), (8) changes in time only if the projection ψ , fn of the perturbation ψ (or projection hψ, fn i of the solution ψ = f + ψ ′ ) on the polynomial part fn of the RH wave changes.

Corollary 3. The super-rotation component −Cµ of the RH wave (7), (8) has no influence on its stability in the energy and enstrophy norms, that is, in these norms, the RH wave has the same stability properties as its polynomial part fn of subspace Hn . The convolution (17) shows that any streamfunction of subspace H1 represents a super rotation flow about some axis of symmetry of the sphere. Then Corollary 3 leads to Corollary 4. Let n = 1, that is, RH wave (7), (8) belongs to subspace H1 . Then kinetic energy (and enstrophy) of arbitrary its perturbation is conserved with time. It follows from Theorem 4 that the energy of a perturbation ψ ′ is conserved if the perturbation is orthogonal to fn , and hence, if it is orthogonal to Hn . Thus, the second term

ψ ′ of the norm (50) controls only the part of a perturbation ψ ′ that belongs to subspace 1 Hn and ψ ′ 1 is insensitive to variations of ψ ′ in the subspace orthogonal to Hn . Unlike

′ ′ this, the first term ∆ψ + χn ψ of the norm (50) is insensitive to variations of ψ ′ in Hn , but controls the part of perturbation ψ ′ that is orthogonal to Hn . We obtain Theorem 5. Let ψ ′ = ψ ′n + ψ ′⊥ be the orthogonal decomposition of a perturbation (49)

to the RH wave (7), (8). The quotient norm ∆ψ ′ + χn ψ ′ and the energy norm ψ ′ 1 are the parts of norm (50) controlling the evolution of orthogonal components ψ ′⊥ and ψ ′n , respectively.

8.

Liapunov Instability of Non-Zonal RH Waves in Invariant Set M−n

According to Corollary 4, any RH wave of subspace H1 is stable. We now consider the instability of the RH wave (7) of subspace H1 ⊕ Hn when n ≥ 2. Note that the RH wave

Liapunov and Normal-Mode Instability of the Rossby-Haurwitz Wave

15

is zonal if it has the form f (t, λ, µ) = −Cµ +

n X

fn0 Yn0 (µ).

(55)

m=−n

This wave is steady and depends only on variable µ. We will call RH wave non-zonal if fnm 6= 0 for at least one number m (1 ≤ |m| ≤ n ). A non-zonal RH wave in the general case depends on all three variables: λ, µ and t.The next assertion shows the instability of a non-zonal RH wave to relatively large-scale perturbations in the sense of Liapunov [48]. Theorem 6. Let n ≥ 2. Then any non-zonal RH wave (7) from subspace H1 ⊕ Hn is Liapunov unstable in the set M−n in the energy and enstrophy norms. Proof. By Theorem 1, it is sufficient to prove the instability only in the energy norm. Due to Corollary 3, we assume, without loss of generality, that C = 0, and hence, the RH wave belongs to subspace Hn : f (t, λ, µ) = fn (t, λ, µ) =

n X

m=−n

fnm Ynm (λ − ω n t, µ),

(56)

where ω n = −2/χn , due to (8). Since wave (56) is non-zonal, fnm 6= 0 for some m (1 ≤ m ≤ n). According to Liapunov stability definition [48, 49], we take  1/2 n X m 2 ε = 8χn . (57) |fn | m=1

We now choose a perturbation ψ ′ to wave (56) such that ψ ′ (0) 1 < δ where δ > 0, as small as we like. Then we show that of δ and moment t1 ≥ 0 chosen, there is

regardless

′

always a moment t > t1 such that ψ (t) 1 exceeds ε. For ε given by (57) and arbitrarily small δ, we consider the perturbation ψ ′ = ψ −f generated by one more RH-wave solution: n X δ ψ(t, λ, µ) = − Y10 (µ) + fnm Ynm (λ − ω b n t, µ), 2 m=−n

(58)

also of H1 ⊕ Hn , where according to (8), ω b n = δ/2 − (δ + 2)/χn . Then the perturbation ′ is ψ (t, x) = ψ(t, x) − f (t, x) also belongs to subspace H1 ⊕ Hn , besides, it changes in time subspace and due to Theorem 5, the norm

′ only in the

H n ,

(50)

of√this perturbation,

ψ = ∆ψ ′ + χn ψ ′ + ψ ′ , can be reduced to the norm ψ ′ = 2K, because the ∗ 1

1 quotient norm ∆ψ ′ + χn ψ ′ does not change. That is why we use here the perturbation energy K(t) as the measure of the perturbation ψ ′ (t, x) between the solutions (58) and (56). According to (41) this energy is n X δ2 + 2χn |fnm |2 {1 − cos m(b ω n − ω n )t K(t) = 4

(59)

m=1

and ω b n −ω n = δ(χ 2, χn ≥ 6 and ω b n −ω n 6= 0. Due to (59), K(0) =

n −2)/2χ

n . Since n ≥ 1/2 2 ′

δ /4, and hence, ψ (0) 1 = {2K(0)} < δ at the initial moment t0 = 0. However, if

16

Yuri N. Skiba

t = τ j = π(2j + 1)/(b ω n − ω n ) (where j = 0, 1, 2, . . . ) then K(τ j ) ≥ δ 2 /4 + ε2 /2, and

′

ψ (τ j ) = {2K(τ j )}1/2 > ε. Since the sequence {τ j } tends to infinity with j, it is 1

impossible to find such a moment t1 ≥ 0 so as to satisfy ψ ′ (t) 1 < ε for all t > t1 . Thus both the wave (56) and the wave (7) are Liapunov unstable. The theorem is proved. We take n ≥ 2 in Theorem 6, since due to (8), ω b 1 = ω 1 for n = 1, that is C = 0, and hence, K(t) = const in full agreement with Corollary 4. It should be noted that the mechanism of Liapunov instability examined here is identical to that of a periodic solution of the nonlinear pendulum equation [31] and consists in asynchronous oscillations caused by a non-zero shift ω b n − ω n in the velocities of two RH waves. Indeed, it follows from (59) that for sufficiently small number δ, the orbit of periodic solution (58) will always be inside the tube of the radius ε that envelops the orbit of periodic solution (56). It means that for any time t1 there is a time t2 such that the energy norm of the difference ψ(t1 , x) − f (t2 , x) is not greater than ε. Thus, the mechanism of the Liapunov instability of the non-zonal RH wave described above has nothing in common with the orbital (Poincar´e) instability when trajectories of the BVE solution diverge in the phase space [32]. The problem of Poincar´e instability needs a special study.

9.

Normal-Mode Instability of RH Wave in Invariant Set M0n − Hn

In this section, we consider the normal mode (exponential) instability of a steady RH wave (7), (8) with n ≥ 2. Since ω n = 0, and hence, C = 2/(χn − 2), the stationary RH wave belongs to subspace H1 ⊕ Hn . Infinitesimal perturbations of such a wave are governed by the linearized equation (39): ∆ψ ′t + J(f, ∆ψ ′ + χn ψ ′ ) = 0.

(60)

Exponentially growing and decaying solutions to (60) are searched as a normal mode ψ ′ (t, x) = Ψ(x) exp {νt} ,

(61)

where ν = ν r + iν i is a complex number, and amplitude Ψ(x) is the solution of the spectral problem (62) J(f, ∆Ψ + χn Ψ) = −ν∆Ψ. The real part ν r of eigenvalue ν determines the growth rate of the mode, while its imaginary part ν i characterizes the mode period. A mode is called unstable if ν r > 0, decaying if ν r < 0, neutral if ν r = 0, and stationary if ν i = 0. We note that conservation law (40) is valid for solutions to equation (60) as well, and hence, all the normal-mode disturbances (61) can be divided again into sets M−n , M+n , Hn andM0n − Hn .

9.1.

Conditions for the Normal-Mode Instability

Taking the real part of the inner product of equation (62) with function ∆Ψ and using (29), we obtain ν r kΨk22 = −χn Re hJ(f, Ψ), ∆Ψi . (63)

Liapunov and Normal-Mode Instability of the Rossby-Haurwitz Wave

17

Thus Re hJ(f, Ψ), ∆Ψi < 0

(64)

is the necessary and sufficient condition (i.e. a criterion) for the instability of mode (61). Taking the real part of the inner product of equation (62) with function ∆Ψ + χn Ψ and using (29) we obtain ν r [η Ψ − χn KΨ ] = ν r [χΨ − χn ] KΨ = 0,

(65)

where

1 1 kΨ(x)k21 , η Ψ = kΨ(x)k22 2 2 are the energy and enstrophyof the mode amplitude Ψ(x), and KΨ =

χΨ = η Ψ /KΨ

(66)

(67)

is its mean spectral number ((see (42)). Since ν r 6= 0 for any growing or decaying mode, equation (65) leads to a necessary condition for the exponential instability of normal mode (61) [24,25]: χΨ = χn . We recapitulate these two results as Theorem 7. Let n ≥ 2. A normal mode (61) of stationary RH wave (7), (8) of subspace H1 ⊕ Hn is unstable if and only if the inequality (64) is satisfied. Besides, χΨ = χn = n(n + 1)

(68)

for each unstable mode where χΨ is the spectral number of mode amplitude Ψ(x). Equation (65) is the requirement that the pseudoenergy [χΨ − χn ] KΨ (see [50]) associated with a growing normal mode vanishes, as it is required to guarantee the fulfilment of the conservation law (44) simultaneously with the exponential growth of the mode. It should be noted that if the spectral method is used for the numerical instability study of a steady RH wave (7), (8) of degree n, besides the truncation number N for the disturbance series N X Yk (ψ ′ ; x) (69) ψ′ = k=1

is taken less than or equal to n, then condition (68) cannot be fulfilled. This leads to Corollary 5. In the spectral normal-mode instability study of a steady RH wave (7), (8) of degree n, the triangular truncation number N for the disturbance series (69) must be taken greater than n, because no unstable mode can appear if N ≤ n. For a zonal RH wave (55), one can use the instability condition by Rayleigh-Kuo [8] related with the structure of the basic flow. However no analogeous condition has existed for a non-zonal RH wave. Although the new instability condition (68) refers, instead of basic flow, to the structure of the growing perturbation, it is related to the basic flow by means of its degree n. Moreover, this condition (as a precise equality) is useful for testing

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Yuri N. Skiba

a computational program developed and used for the normal-mode instability study of arbitrary flow on a sphere. It also characterizes the spatial (geometric) structure of growing perturbations to the RH wave. Theorem 7 asserts that unstable modes may exist only in subset M0n − Hn . Thus, exponential growth (as well as exponential decay) of perturbations is impossible in the sets M−n , M+n and Hn , and hence, all the modes of these three sets are neutral. In practice, the probability of the fulfilment of condition (68) is small enough, because the set M0n is only the boundary hypersurface separating the basic sets M−n and M+n . Therefore it is difficult to expect that real atmospheric disturbances will frequently meet the precise instability condition (68). Moreover, the Liapunov instability proved in section 8 is not orbital. Probably, these two arguments can be used to explain the fact that the RH waves are regularly observed on the daily meteorological maps.

9.2.

Estimation of Maximum Growth Rate of Unstable Modes

The knowledge of the highest possible growth rate of unstable modes sometimes is important for evaluating the degree of instability of a flow. We now estimate the maximum growth rate of unstable modes to the RH wave. Taking into account the inequality kJ(f, Ψ)k ≤ max |▽f | · kΨk1 S

and applying the Schwarz inequality to the inner product in (63), we obtain → kΨk1 − 1/2 |ν r | ≤ χn max |▽f | = χn max U (x) , S kΨk2 S

(70)

− → → − where U = k × ∇f is the velocity of the RH wave. Finally, the use of condition (68) leads to Theorem 8. The growth rate of unstable modes (61) of the stationary RH wave (7), (8) is bounded, besides, − p → (71) |ν r | ≤ n(n + 1)max U (x) . S

Baines [4] points out that the growth rate is proportional to the amplitude of a Legendrepolynomial flow. In the case of a RH wave (7), (8), Theorem 8 specifies this result stating → − that the growth rate of a mode decreases directly with the maximum velocity value U (x) and degree n of the RH wave. In addition to conditions (64) and (68), the estimate (71) serves as one more test of the computational program used in the normal mode stability study.

9.3.

Orthogonality of Unstable Modes to RH Wave

Taking the inner product of equation (62) with the steady RH wave f and using the formula (30) we obtain ν h∆Ψ, f i = 0, (72) where Ψ is the mode amplitude, and f is the basic wave. Thus, hΨ, f i1 = hΛΨ, Λf i = − h∆Ψ, f i = 0

(73)

Liapunov and Normal-Mode Instability of the Rossby-Haurwitz Wave

19

that is, the RH wave is orthogonal in the energy inner product (see (21) for s = 1) to the amplitude of any non-neutral (ν r 6= 0) or non-stationary (ν i 6= 0) mode. Since h∇Ψ, ∇f i = − h∆Ψ, f i = 0, equation (73) means that integrally (over the whole sphere → − → S), the mode amplitude velocity − u = k × ∇Ψ is orthogonal to the basic flow velocity → → − − U = k × ∇f : Z → − → − u · U dS = 0. (74) S

Indeed, using Lagrange’s identity → → − − → → − − → → − − → → → → → → (− a × b ) · (− c × d ) ≡ (− a ·− c )( b · d ) − (− a · d )( b · − c) [46; p. 169], we obtain − → → − → − → − u · U = ( k × ∇Ψ) · ( k × ∇f ) = ∇Ψ · ∇f

→ − → − because k · ∇f = 0 and ∇Ψ · k = 0. Therefore, it follows from (73) that Z Z → − → − u · U dS = ∇Ψ · ∇f dS = h∇Ψ, ∇f i = 0. S

S

We now show that amplitude Ψ of any non-neutral or non-stationary mode is orthogonal to the RH wave f in the L2 -inner product (10) as well: hΨ, f i = 0.

(75)

Indeed, f = −Cµ + fn where fn belongs to Hn , i.e., is a homogeneous spherical polynomial of degree n. Therefore, using operator (38), Dn Ψ ≡ (∆ + χn )Ψ, and formula (30) we obtain hJ(f, Dn Ψ), fn i = −C hJ(µ, Dn Ψ), fn i = C h(Dn Ψ)λ , fn i = 0, (76) since (Dn Ψ)λ is orthogonal to Hn , that is, orthogonal to any homogeneous spherical polynomial of the degree n. Taking the inner product of equation (62) with fn and using (76) we get ν h∆Ψ, fn i = 0. (77) Since ∆ is the symmetric operator, (77) leads to

ν hΨ, fn i = 0.

(78)

ν h∆Ψ, −Cµi = ω h∆Ψ, f i − ω h∆Ψ, fn i = 0

(79)

Also, due to (72) and (77), and hence, ν hΨ, −Cµi = 0.

(80)

Thus, (75) follows from (78) and (80). We recapitulate the results in Theorem 9. Let f (x) be a steady RH wave (7), (8). Then amplitude Ψ(x) of each unstable, decaying, or non-stationary mode is orthogonal to the basic wave f both in the L2 -inner product (10) and in the energy inner product h·, ·i1 defined by (21) for s = 1.

The orthogonality properties (73), (75), (78) and (80) also serve well for testing the computational programs in the normal mode stability study.

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Figure 1. The profile of zonal flow velocity U (µ) of flow (81) (fig.1a), and isolines of the real part of amplitude of its most unstable mode (Fig. 1b).

9.4.

Examples

Example 1. Zonal RH wave. Let us consider a zonal flow in the form of Legendre polynomial of degree 5: e ψ(µ) = −0.06P50 (µ).

(81)

e In the case of a zonal flow ψ(µ), we can use equation Z Z p dU µ 2 p 1 − µ (uv) dS − Kt = − (uv)U dS dµ 1 − µ2 S S

(82)

that describes the evolution of the kinetic energy (41) of a disturbance ψ ′ (λ, µ, t) to the zonal flow on sphere S where p e , U = − 1 − µ2 ψ µ

p u = − 1 − µ2 ψ ′µ ,

1 v=p ψ ′λ 1 − µ2

(83)

are the velocity components of the basic flow and disturbance, respectively. Usually, the maximum values of the amplitude of unstable mode are localized in the regions related with characteristic features of the basic velocity U (µ). Unlike the equation for perturbation energy on the β-plane [53], its spherical analogue (82) contains one more integral of the product of basic velocity U with uv. Whereas the first integral dominates p principally at the sides of jets (where Uµ is large) and far off the polar regions (where 1 − µ2 is not small), the second integral can be significant in the central parts of strong jets p (where U is large), especially when such jets are located in the polar regions (where µ/ 1 − µ2 is also large). By (82), the sign of Kt depends on the signs of the products (uv)Uµ and (uv)U in various regions of the sphere S. In the case when the first integral is dominant, one can say that the

Liapunov and Normal-Mode Instability of the Rossby-Haurwitz Wave

21

growth of the perturbation energy takes place in the regions where the inclination of closed isolines of the streamfunction of disturbance is opposite to the inclination of the profile of basic flow velocity U (µ), that is in the regions where product (uv)Uµ is negative [53].

e µ) = −0.2µ + aP 2 (µ) cos 2λ for a = 0.013 (a), Figure 2. Isolines of the RH wave ψ(λ, 3 a = 0.014 (b); a = 0.015 (c) and a = 0.023 (d). The profile of velocity U (µ) of the basic flow (81) is shown in Fig. 1a, while isolines of the real part of the amplitude of its most unstable mode are presented in Fig. 1b. As it is seen from these figures, the basic flow contains two subtropical jets, and the zonal wavenumber of the mode is m = 2. It is known [5] that if the streamfunction of the basic flow is antisymmetric with respect to equator then all the normal modes of such flow are divided in two groups: symmetric or antysymmetric around equator. One can see from Fig. 1b that the mode amplitude is symmetric around equator. The maximum values of the mode amplitude are located in a neighborhood of two subtropical jets, so the first integral

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is dominant in the formula (82). The inclination of closed isolines of the streamfunction of disturbance is opposite to the inclination of the profile of basic flow velocity U (µ) so that the negative product (uv)Uµ generates the instability. The mean spectral number χΨ is equal to 30for the unstable modes in full accordance with condition (68) which is also valid for polynomial flows (81) [25]. Example 2. Non-zonal RH wave. Let us consider a steady non-zonal RH wave e µ) = −ωµ + aP 2 (µ) cos 2λ, ψ(λ, 3

(84)

where ω = 2/(χ3 − 2) = 0.2 [54]. In order to show numerically the existence of a critical amplitude for the RH-wave instability, the modes have been calculated for different amplitude values a of the RH wave: a = 0.013, a = 0.014; a = 0.015, a = 0.023. The basic flows for different a are presented in Fig. 2a-d, while the real parts of amplitudes Ψr (x) of the unstable modes are given in Fig.3a-d. The numerical results show that all the modes to the RH wave with a ≤ 0.013 are neutral (ν r = 0), whereas for a ≥ 0.014, there appears one unstable mode. Thus, the critical value acr of amplitude a for the instability of the RH-wave (81) is in the interval (0.013, 0.014). In particular, the mode presented in Fig.3a for a = 0.013 is neutral and stationary. The amplitude Ψ(x) of the modes is mainly formed by the spherical harmonics Y2±1 (x), Y4±1 (x) and Y4±3 (x). It is in full agreement with the work [21] where it is noted that for the wave under consideration, the main nonlinear interaction triad is formed by the basic wave and two spherical harmonics (1,2) y (1,4). The mean spectral number χΨ is equal to 12 for all unstable modes, and hence, the instability condition (68) is fulfilled. It should be noted that the growth rate ν r (or the e-folding time τ e ) of the unstable mode increases (decreases) with amplitude a in accordance with the estimate (71). The number of unstable modes also increases with a. At the same time, period T of the unstable mode is changed slowly with a.

10.

Conclusions

In the present work, some results have been given on the linear (normal-mode o exponential) and nonlinear (Liapunov) instability of the RH wave (7), (8) being exact solution to the nonlinear barotropic vorticity equation describing the motion of ideal incompressible fluid on a rotating sphere. This wave belongs to subspace H1 ⊕ Hn , where Hk is the subspace of homogeneous spherical polynomials of degree k. The RH waves form the main features of daily meteorological fields, and their stability study is of considerable interest for deeper understanding of the low-frequency variability of the atmosphere. Conservation laws for perturbations to the RH-wave are obtained and used to divide all the perturbations into four invariant non-intersecting sets: M−n , M+n , Hn and M0n − Hn (Theorem 2). The main conservation law asserts that any perturbation evolves in such a way that its energy and enstrophy decrease, remain constant or increase simultaneously (Theorem 1). Besides, the energy cascade of growing (or decaying) perturbations has opposite directions in the sets M−n and M+n (Theorem 3). A geometric interpretation of variations in the perturbation kinetic energy is given, according to which the energy of a perturbation ψ ′ (t, x) to the RH

Liapunov and Normal-Mode Instability of the Rossby-Haurwitz Wave

23

wave f (t, x) changes only if the projection of ψ ′ (t, x) onto the main RH-wave component fn (t, x) changes (Theorem 4). We have tried to analyze the instability in each invariant set separately. Since any perturbation of Hn is neutral, a quotient space is introduced with Hn as the zero-class. Then we show that the norm (50), which is the sum of the two norms, the quotient norm and the energy norm, is appropriate in the instability study of the RH wave, because the energy norm controls the evolution of the perturbation part belonging to Hn , while the quotient norm controls the evolution of its orthogonal part (Theorem 5).

Figure 3. Isolines of ReΨ(x) of unstable modes to the RH waves given in Fig.2. It has been shown that the RH wave is stable to any perturbation of invariant set Hn , and any non-zonal RH wave of subspace H1 ⊕ Hn where n ≥ 2 is Liapunov unstable with respect to perturbations of invariant set M−n (Theorem 6). Besides, the growing perturbations used in the proof, also belong to the subspace H1 ⊕ Hn , and the mechanism of the

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Liapunov instability is identical with the instability mechanism of a periodic solution to the nonlinear pendulum equation, namely it consists in asynchronous oscillations of two RH waves caused by a non-zero shift of their velocities. The set M+n is the only invariant set, where we have been unable to get instability results. A criterion for the normal-mode instability of a stationary RH wave of H1 ⊕ Hn has been given (see (64)), besides, according to the necessary instability condition (68), a mode can be unstable only if the mean spectral number of its amplitude is equal to n(n + 1) (Theorem 7). Moreover, unstable normal modes can exist only in the set M0n − Hn . The growth rate of the unstable modes is estimated (Theorem 8), and the orthogonality of the normal mode amplitude to the RH wave is proved in the two inner products defined by (21) for s = 0 (the L2 -inner product) and s = 1 (the energy inner product). In the normal-mode instability study of a zonal RH wave (55), one can use the condition by Rayleigh-Kuo [8] related only with the structure of the basic flow. Unfortunately, the utility of Rayleigh-Kuo condition is rather limited, since it does not give any information about the growing perturbations. Moreover, this necessary condition is satisfied in many cases, for example, for any wave (55) if n > 2 and wave amplitude fn0 is large enough. For a non-zonal RH wave, no instability condition analogeous to that by Rayleigh-Kuo has been developed. The instability conditions (64) and (68), instead of the structure of basic flow, refer to the structure of the growing perturbation and relate with the basic flow only by means of its degree n. Nevertheless, as a precise equality, condition (68) is useful for testing any computational program used for the normal-mode instability study of arbitrary flows on a sphere. This condition also characterizes the spatial (geometric) structure of growing perturbations to the RH wave. Both the inequalities (64) and (71) and the orthogonality conditions (75) and (80) can also serve for testing the computational program. Theorem 7 asserts that unstable modes may exist only in subset M0n − Hn . Thus, exponential growth (as well as exponential decay) of perturbations is impossible in the sets M−n , M+n and Hn , and hence, all the modes of these three sets are neutral. In practice, the probability of the fulfilment of instability condition (68) is quite small, because the set M0n is only the boundary hypersurface separating the basic sets M−n and M+n . Therefore it is difficult to expect that real atmospheric disturbances will frequently meet the precise instability condition (68). Moreover, the Liapunov instability proved in section 8 is not orbital one. Probably, these two arguments can be used to explain the fact that the RH waves are regularly observed on the daily meteorological maps. In this connection, the next problem of significant importance is to know if the RH wave is orbitally stable.

Acknowledgments This work was supported by the PAPIIT grant IN105608 (UNAM, Mexico), projects 46265-A1 and FOSEMARNATd-2004-01-160 (CONACyT, Mexico), and SNI grant 14539 (CONACyT, Mexico).

References [1] L. Rayleigh, On the stability or instability of certain fluid motions, Proc. London Math. Soc. 11 (1880), 57–70.

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[2] R. Fj¨ortoft, On the changes in the spectral distribution of the kinetic energy for twodimensional nondivergent flow. Tellus 5 (1953), 225–230. [3] V. I. Arnold, Conditions for nonlinear stability of stationary plane curvilinear flows of an ideal fluid. Sov. Math. Doklady 6 (1965), 331–334. [4] P. G. Baines, The stability of planetary waves on a sphere. J. Fluid Mech. 73 (1976), 193–213. [5] Yu. N. Skiba, Mathematical Problems of the Dynamics of Viscous Barotropic Fluid on a Rotating Sphere. Dept. Numerical Mathematics, USSR Academy of Sciences, Moscow, 1989, 178 pp. (English transl.: Indian Inst. Tropical Meteorology, Pune, 1990, 211 pp.). [6] C. -C. Rossby, Relation between variations in the intensity of the zonal circulation of the atmosphere and the displacements of the semi-permanent centers of action. J. Marine Res. 2 (1939), 38–55. [7] B. Haurwitz, The motion of atmospheric disturbances on the spherical earth. J. Marine Research 3 (1940), 254–267. [8] H. L. Kuo, Dynamic instability of two-dimensional nondivergent flow in a barotropic atmosphere. J. Meteor. 6 (1949), 105–122. [9] I. Silberman, Planetary Waves in the Atmosphere. J. Meteor. 11 (1954), 27–34. [10] Ph. D. Thompson, A generalized class of exact time-dependent solutions of the vorticity equation for nondivergent barotropic flow. Mon. Wea. Rev. 110 (1982), 1321–1324. [11] A. J. Simmons, J. M. Wallace, G. W. Branstator, Barotropic wave propagation and instability, and atmospheric teleconnection patterns. J. Atmos. Sci. 40 (1983), 1363– 1392. [12] D. T. Crommelin, Regime transitions and heteroclinic connections in a barotropic atmosphere. J. Atmos. Sci. 60 (2003), 229–246. [13] V. P. Shutyaev, Solvability of the data assimilation problem in the scale of Hilbert spaces for quasilinear singularly perturbed evolutionary problems. Russian J. Numer. Analysis & Math. Modelling 12 (1997), 53–66. [14] V. P. Shutyaev, Data mastery in a scale of Hilbert spaces for quasilinear evolution problems. Differential Equations 34 (1998), 382–388. [15] V. P. Shutyaev, Initial condition reconstruction problem for a viscous barotropic fluid dynamic equation on a sphere. Russian J. Numer. Analysis & Math. Modelling 17 (2002), 457–468. [16] E. N. Lorenz, Barotropic instability of Rossby wave motion. J. Atmos. Sci. 29 (1972), 258–264.

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[17] B. J. Hoskins, A. Hollingsworth, On the simplest example of the barotropic instability of Rossby wave motion. J. Atmos. Sci. 30 (1973), 150–153. [18] S. A. Coaker, The stability of a Rossby wave, Geophysical & Astrophysical Fluid Dynamics. 9 (1977), 1–17. [19] J. L. Anderson, The instability of finite amplitude Rossby waves on the infinite betaplane. Geophys. Astrophys. Fluid Dynamics 63 (1992), 1–27. [20] B. J. Hoskins, Stability of the Rossby-Haurwitz wave. Quart. J. R. Met. Soc. 99 (1973), 723–745. [21] R. S. Haarsma, J. D. Opsteegh, Barotropic instability of planetary-scale flows. J. Atmos. Sci. 45 (1988), 2789–2810. [22] Yu. N. Skiba, Liapunov Instability of the Rossby-Haurwitz Waves and Dipole Modons. Sov. J. Numer. Analysis & Math. Modelling 6 (1991), 515–534. [23] Yu. N. Skiba, Rossby-Haurwitz Wave Stability. Izvestiya, Atmos. Ocean. Physics 28 (1992), 388–394. [24] Yu. N. Skiba, Nonlinear and linear instability of the Rossby-Haurwitz wave. Journal of Mathematical Sciences 149 (2008), 1708–1725. [25] Yu. N. Skiba, On the normal mode instability of harmonic waves on a sphere. Geophys. Astrophys. Fluid Dynamics 92 (2000), 115–127. [26] G. Wolansky, M. Ghil, Nonlinear stability for saddle solutions of ideal flows and symmetry breaking. Comm. Math. Phys. 193 (1998), 713–736. [27] M. Subbiah, M. M. Padmini, Note on the nonlinear stability of equivalent barotropic flows. Indian J. Pure Appl. Math. 30 (1999), 1261–1272. [28] Yu. N. Skiba, Instability of the Rossby-Haurwitz wave in invariant sets of perturbations. J. Math. Analysis & Applications 290 (2004), 686–701. [29] V. I. Zubov, Methods of A. M. Liapunov and Their Application. Leningradsky University Press, Leningrad (1957) (English translation - 1964, Noordhoff, Groningen). [30] Yu. N. Skiba, J. Adem, On the linear stability study of zonal incompressible flows on a sphere. Numer. Methods Partial Differential Equations 14 (1998), 649–665. [31] A. A. Andronov, A. A. Vitt, S. E. Khaikin, Theory of Oscillators. Fiz. Mat. Lit., Moscow, 1959 (Russian) (translated by F. Immirzi, Dover Publications, Inc., New York, 1987). [32] F. Verhulst, Nonlinear Differential Equations and Dynamical Systems. Springer, Berlin, 1996. [33] Yu. N. Skiba, Linear instability of ideal flows on a sphere. Mathematical Methods in the Applied Sciences 32 (2009), 284–306.

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[34] L. A. Dikiy, Hydrodynamic Stability and Atmosphere Dynamics. Gidrometeoizdat, Leningrad, 1976 (in Russian). [35] B. Machenhauer, Spectral methods. In: Numerical Methods Used in Atmospheric Models, Vol. 2, GARP Publication Series, Vol. 17, WMO/ICSU, 1977. [36] A. E. Gill, The stability of planetary waves on an infinite beta plane. Geophys. Fluid Dyn. 6 (1974), 29–47. [37] G. W. Platzman, The analytical dynamics of the spectral vorticity equation. J. Atm. Sci. 19 (1962), 313–328. [38] R. D. Richtmyer, Principles of Advanced Mathematical Physics. Vol.2. SpringerVerlag, New York, 1981. [39] S. Helgason, Groups and Geometric Analysis, Integral Geometry. Invariant Differential Operators and Spherical Functions, Academic Press, Orlando, 1984. [40] Yu. N. Skiba, On the Spectral Problem in the Linear Stability Study of Flows on a Sphere. Journal of Mathematical Analysis and Applications 270 (2002), 165–180. [41] A. D. Gadzhiev, On differential properties of the symbol of multi-dimensional singular integral operator. Matem. Sbornik 114 (1981), 483–510. [42] P. Szeptycki, Equations of hydrodynamics on manifold diffeomorfic to the sphere. Bull. L’acad. Pol. Sci., Ser. Sci. Math., Astr., Phys. XXI (1973), 341–344. [43] A. B. Karunin, On Rossby waves in barotropic atmosphere in the presence of zonal flow. Izvestiya, Atmos.Ocean. Physics 6 (1970), 1091–1100. [44] T. G. Shepherd, Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics. Advances in Geophysics 32 (1990), 287–338. [45] R. Petroni, S. Pierini, A. Vulpiani, The double cascade as a necessary mechanism for the instability of steady equivalent-barotropic flows. Nuovo Cimento C 10 (1987), 27–36. [46] T. G. Shepherd, Remarks concerning the double cascade as a necessary mechanism for the instability of steady equivalent-barotropic flows. Nuovo Cimento C 11 (1988), 439–442. [47] R. Petroni, S. Pierini, A. Vulpiani, Reply to a note by T.G. Shepherd. Nuovo Cimento C 12 (1989), 271–275. [48] A. M. Liapunov, Stability of Motion. Academic Press, New York, 1966. [49] J. La Salle, S. Lefschetz, Stability by Liapunov’s Direct Method. Academic Press, New York (1961). [50] P. Ripa, Positive, negative and zero wave energy and the flow stability problem in the Eulerian and Lagrangian-Eulerian descriptions. Pure and Applied Geophysics 133 (1990), 713–732.

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[51] Mankin Mak, Ming Cai, Local barotropic instability. J. Atm. Sci. 46 (1989), 3289– 3311. [52] M. D. Borges, P. D. Sardeshmukh, Barotropic Rossby wave dynamics of zonally varying upper-level flows during northern winter. J. Atm. Sci. 52 (1995), 3779–3796. [53] J. Pedlosky, Geophysical Fluid Dynamics. Springer-Verlag, New York, 1979. [54] Yu. N. Skiba, I. Peres-Garcia, On the structure and growth rate of unstable modes to the Rossby-Haurwitz wave. Numerical Methods for Partial Differential Equations 21 (2005), 368–386.

ISBN 978-1-60876-265-1 In: Advances in Mathematics Research, Volume 10 c 2010 Nova Science Publishers, Inc. Editor: Albert R. Baswell, pp. 29-52 °

Chapter 2

∞−D IMENSIONAL C EREBELLAR C ONTROLLER FOR R EALISTIC H UMAN B IODYNAMICS Vladimir G. Ivancevic1,∗ and Tijana T. Ivancevic2 1 Human Systems Integration, Land Operations Division, Defence Science & Technology Organisation, P.O. Box 1500, Edinburgh SA 5111, Australia 2 School of Electrical and Information Engineering, University of South Australia, Mawson Lakes, S.A. 5095, Australia

Abstract In this paper we propose an ∞−dimensional cerebellar model of neural controller for realistic human biodynamics. The model is developed using Feynman’s action– amplitude (partition function) formalism.

Keywords: realistic human biodynamics, cerebellum motion control, ∞−dimensional neural network

1.

Introduction

Realistic human biodynamics (RHB) is a science of human (and humanoid robot) motion in its full complexity. It is governed by both Newtonian dynamics and biological control laws (see [42, 27, 28, 29, 30]). There are over 200 bones in the human skeleton driven by about 640 muscular actuators (see, e.g., [49]). While the muscles generate driving torques in the moving joints,1 ∗

E-mail address: [email protected] Here we need to emphasize that human joints are significantly more flexible than humanoid robot joints. Namely, each humanoid joint consists of a pair of coupled segments with only Eulerian rotational degrees of freedom. On the other hand, in each human synovial joint, besides gross Eulerian rotational movements (roll, pitch and yaw), we also have some hidden and restricted translations along (X, Y, Z)−axes. For example, in the knee joint, patella (knee cap) moves for about 7–10 cm from maximal extension to maximal flexion). It is well–known that even greater are translational amplitudes in the shoulder joint. In other words, within the realm of rigid body mechanics, a segment of a human arm or leg is not properly represented as a rigid body 1

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Vladimir G. Ivancevic and Tijana T. Ivancevic

subcortical neural system performs both local and global (loco)motion control: first reflexly controlling contractions of individual muscles, and then orchestrating all the muscles into synergetic actions in order to produce efficient movements. While the local reflex control of individual muscles is performed on the spinal control level, the global integration of all the muscles into coordinated movements is performed within the cerebellum [28, 29]. All hierarchical subcortical neuro–muscular physiology, from the bottom level of a single muscle fiber, to the top level of cerebellar muscular synergy, acts as a temporal hout|ini reaction, in such a way that the higher level acts as a command/control space for the lower level, itself representing an abstract image of the lower one: 1. At the muscular level, we have excitation–contraction dynamics [16, 18, 17], in which hout|ini is given by the following sequence of nonlinear diffusion processes [28, 29]: neural−action−potential à synaptic−potential à muscular−action−potential à excitation−contraction−coupling à muscle−tension−generating.

Its purpose is the generation of muscular forces, to be transferred into driving torques within the joint anatomical geometry. 2. At the spinal level, hout|ini is given by autogenetic–reflex stimulus–response control [24]. Here we have a neural image of all individual muscles. The main purpose of the spinal control level is to give both positive and negative feedbacks to stabilize generated muscular forces within the ‘homeostatic’ (or, more appropriately, ‘homeokinetic’) limits. The individual muscular actions are combined into flexor–extensor (or agonist–antagonist) pairs, mutually controlling each other. This is the mechanism of reciprocal innervation of agonists and inhibition of antagonists. It has a purely mechanical purpose to form the so–called equivalent muscular actuators (EMAs), which would generate driving torques Ti (t) for all movable joints. 3. At the cerebellar level, hout|ini is given by sensory–motor integration [10]. Here we have an abstracted image of all autogenetic reflexes. The main purpose of the cerebellar control level is integration and fine tuning of the action of all active EMAs into a synchronized movement, by supervising the individual autogenetic reflex circuits. At the same time, to be able to perform in new and unknown conditions, the cerebellum is continuously adapting its own neural circuitry by unsupervised (self-organizing) learning. Its action is subconscious and automatic, both in humans and in animals. Naturally, we can ask the question: Can we assign a single hout|ini measure to all these neuro–muscular stimulus–response reactions? We think that we can do it; so in this Letter, we propose the concept of adaptive sensory–motor transition amplitude as a unique measure for this temporal hout|ini relation. Conceptually, this hout|ini−amplitude can be formulated as the ‘neural path integral’: Z hout|ini ≡ hmotor | sensoryi = D[w, x] ei S[x] . (1) amplitude

fixed at a certain point, but rather as a rigid body hanging on rope–like ligaments. More generally, the whole skeleton mechanically represents a system of flexibly coupled rigid bodies. This implies the more complex kinematics, dynamics and control then in the case of humanoid robots.

∞−Dimensional Cerebellar Controller for Realistic Human Biodynamics

31

Here, the integral is taken over all activated (or, ‘fired’) neural pathways xi = xi (t) of the cerebellum, connecting its input sensory−state with its output motor−state, symbolically described by adaptive neural measure D[w, x], defined by the weighted product (of discrete time steps) n Y D[w, x] = lim wi (t) dxi (t), (2) n→∞

t=1

in which the synaptic weights wi = wi (t), included in all active neural pathways xi = xi (t), are updated by the standard learning rule new value(t + 1) = old value(t) + innovation(t). More precisely, the weights wi in (2) are updated according to one of the two standard neural learning schemes, in which the micro–time level is traversed in discrete steps, i.e., if t = t0 , t1 , . . . , tn then t + 1 = t1 , t2 , . . . , tn+1 :2 1. A self–organized, unsupervised (e.g., Hebbian–like [19]) learning rule: wi (t + 1) = wi (t) +

σ d (w (t) − wia (t)), η i

(3)

where σ = σ(t), η = η(t) denote signal and noise, respectively, while superscripts d and a denote desired and achieved micro–states, respectively; or 2. A certain form of a supervised gradient descent learning: wi (t + 1) = wi (t) − η∇J(t),

(4)

where η is a small constant, called the step size, or the learning rate, and ∇J(n) denotes the gradient of the ‘performance hyper–surface’ at the t−th iteration. Theoretically, equations (1–4) define an ∞−dimensional neural network (see [25, 26, 36]). Practically, in a computer simulation we can use 107 ≤ n ≤ 108 , roughly corresponding to the number of neurons in the cerebellum [31, 32]. The exponent term S[x] in equation (1) represents the autogenetic–reflex action, describing reflexly–induced motion of all active EMAs, from their initial stimulus−state to their final response−state, along the family of extremal (i.e., Euler–Lagrangian) paths ximin (t). (S[x] is properly derived in (8–9) below.)

2.

Sub-Cerebellar Biodynamics and Its Spinal Reflex Servo–Control

Subcerebellar biodynamics includes the following three components: (i) local muscle–joint mechanics, (ii) whole–body musculo–skeletal dynamics, and (iii) autogenetic reflex servo– control. 2 Note that we could also use a reward–based, reinforcement learning rule [60], in which system learns its optimal policy: innovation(t) = |reward(t) − penalty(t)|.

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2.1. Local Muscle–Joint Mechanics Local muscle–joint mechanics comprises of [43, 28, 29]): 1. Synovial joint dynamics, giving the first stabilizing effect to the conservative skeleton dynamics, is described by the (x, x)–form ˙ of the Rayleigh – Van der Pol’s dissipation function n 1X i 2 R= (x˙ ) [αi + βi (xi )2 ], 2 i=1

where αi and βi denote dissipation parameters. Its partial derivatives give rise to the viscous–damping torques and forces in the joints Fijoint = ∂R/∂ x˙ i , which are linear in x˙ i and quadratic in xi . 2. Muscular dynamics, giving the driving torques and forces Fimuscle = muscle Fi (t, x, x) ˙ with (i = 1, . . . , n) for RHB, describes the internal excitation and contraction dynamics of equivalent muscular actuators [18]. (a) Excitation dynamics can be described by an impulse force–time relation Fiimp = Fi0 (1 − e−t/τi ) Fiimp

=

Fi0 e−t/τi

if stimulation > 0 if stimulation = 0,

where Fi0 denote the maximal isometric muscular torques and forces, while τi denote the associated time characteristics of particular muscular actuators. This relation represents a solution of the Wilkie’s muscular active–state element equation [66] µ˙ + γ µ = γ S A,

µ(0) = 0,

0 < S < 1,

where µ = µ(t) represents the active state of the muscle, γ denotes the element gain, A corresponds to the maximum tension the element can develop, and S = S(r) is the ‘desired’ active state as a function of the motor unit stimulus rate r. This is the basis for the RHB force controller. (b) Contraction dynamics has classically been described by the Hill’s hyperbolic force– velocity relation [20] ¡ 0 ¢ Fi bi − δij ai x˙ j Hill Fi = , (δij x˙ j + bi ) where ai and bi denote the Hill’s parameters, corresponding to the energy dissipated during the contraction and the phosphagenic energy conversion rate, respectively, while δij is the Kronecker’s δ−tensor. In this way, RHB describes the excitation/contraction dynamics for the ith equivalent muscle–joint actuator, using the simple impulse–hyperbolic product relation Fimuscle (t, x, x) ˙ = Fiimp × FiHill . Now, for the purpose of biomedical engineering and rehabilitation, RHB has developed the so–called hybrid rotational actuator. It includes, along with muscular and viscous forces, the D.C. motor drives, as used in robotics [65, 43, 28] Fkrobo = ik (t) − Jk x ¨k (t) − Bk x˙ k (t),

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with lk ik (t) + Rk ik (t) + Ck x˙ k (t) = uk (t), where k = 1, . . . , n, ik (t) and uk (t) denote currents and voltages in the rotors of the drives, Rk , lk and Ck are resistances, inductances and capacitances in the rotors, respectively, while Jk and Bk correspond to inertia moments and viscous dampings of the drives, respectively. Finally, to make the model more realistic, we need to add some stochastic torques and forces [38, 31] Fistoch = Bij [xi (t), t] dW j (t) where Bij [x(t), t] represents continuous stochastic diffusion fluctuations, and W j (t) is an N −variable Wiener process (i.e. generalized Brownian motion), with dW j (t) = W j (t + dt) − W j (t) for j = 1, . . . , N .

2.2.

Hamiltonian Biodynamics and Its Reflex Servo–Control

General form of Hamiltonian biodynamics on the configuration manifold of human motion is formulated in [38, 41, 42, 27, 28, 30]) using the concept of Euclidean group of motions SE(3)3 (see Figure 1), Briefly, based on affine Hamiltonian function of human motion, formally Ha : T ∗ Q → R, in local canonical coordinates on the symplectic phase space (which is the cotangent bundle of the human configuration manifold Q) T ∗ Q given as Ha (x, p, u) = H0 (x, p) − H j (x, p) uj ,

(5)

where H0 (x, p) = Ek (p) + Ep (x) is the physical Hamiltonian (kinetic + potential energy) dependent on joint coordinates xi and their canonical momenta pi , H j = H j (x, p), (j = 1, . . . , m ≤ n are the coupling Hamiltonians corresponding to the system’s active joints and ui = ui (t, x, p) are (reflex) feedback–controls. Using (5) we come to the affine 3

Briefly, the Euclidean SE(3)–group is defined as a semidirect (noncommutative) product of 3D rotations and 3D translations, SE(3) := SO(3) B R3 . Its most important subgroups are the following (for technical details see [30, 55, 33]): Definition Subgroup Set of all proper orthogonal SO(3), group of rotations in 3D (a spherical joint) 3 × 3 − rotational matrices Set  of all 3 × 3 − matrices:  cos θ sin θ rx SE(2), special Euclidean group  − sin θ cos θ ry  in 2D (all planar motions) 0 0 1 Set of all proper orthogonal SO(2), group of rotations in 2D subgroup of SE(2)–group 2 × 2 − rotational matrices (a revolute joint) included in SE(2) − group R3 , group of translations in 3D Euclidean 3D vector space (all spatial displacements)

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Figure 1. The configuration manifold Q of the human musculoskeletal dynamics is defined as an anthropomorphic product of constrained Euclidean SE(3)–groups acting in all major (synovial) human joints. Hamiltonian control RHB–system, in deterministic form x˙ i = ∂pi H0 − ∂pi H j uj + ∂pi R,

(6)

j

p˙i = Fi − ∂xi H0 + ∂xi H uj + ∂xi R, oi = −∂ui Ha = H j , xi (0) = xi0 ,

pi (0) = p0i ,

(i = 1, . . . , n;

j = 1, . . . , Q ≤ n),

(where ∂u ≡ ∂/∂u, Fi = Fi (t, x, p), H0 = H0 (x, p), H j = H j (x, p), Ha = Ha (x, p, u), R = R(x, p)), as well as in the fuzzy–stochastic form [38, 31] ¢ ¡ dq i = ∂pi H0 (σµ ) − ∂pi H j (σµ ) uj + ∂pi R dt, dpi = Bij [xi (t), t] dW j (t)+ ¡ ¢ + F¯i − ∂xi H0 (σµ ) + ∂xi H j (σµ ) uj + ∂xi R dt,

(7)

d¯ oi = −∂ui Ha (σµ ) dt = H j (σµ ) dt, xi (0) = x ¯i0 ,

pi (0) = p¯0i .

In (6)–(7), R = R(x, p) denotes the joint (nonlinear) dissipation function, oi are affine system outputs (which can be different from joint coordinates); {σ}µ (with µ ≥ 1) denote fuzzy sets of conservative parameters (segment lengths, masses and moments of inertia), dissipative joint dampings and actuator parameters (amplitudes and frequencies), while the ¯ over a variable denotes the corresponding fuzzified variable; Bij [q i (t), t] denote bar (.) diffusion fluctuations and W j (t) are discontinuous jumps as the n–dimensional Wiener process.

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In this way, the force RHB servo–controller is formulated as affine control Hamiltonian–systems (6–7), which resemble the autogenetic motor servo (see Appendix), acting on the spinal–reflex level of the human locomotion control. A voluntary contraction force F of human skeletal muscle is reflexly excited (positive feedback +F −1 ) by the responses of its spindle receptors to stretch and is reflexly inhibited (negative feedback −F −1 ) by the responses of its Golgi tendon organs to contraction. Stretch and unloading reflexes are mediated by combined actions of several autogenetic neural pathways, forming the so–called ‘motor servo.’ The term ‘autogenetic’ means that the stimulus excites receptors located in the same muscle that is the target of the reflex response. The most important of these muscle receptors are the primary and secondary endings in the muscle–spindles, which are sensitive to length change – positive length feedback +F −1 , and the Golgi tendon organs, which are sensitive to contractile force – negative force feedback −F −1 . The gain G of the length feedback +F −1 can be expressed as the positional stiffness (the ratio G ≈ S = dF/dx of the force–F change to the length–x change) of the muscle system. The greater the stiffness S, the less the muscle will be disturbed by a change in load. The autogenetic circuits +F −1 and −F −1 appear to function as servoregulatory loops that convey continuously graded amounts of excitation and inhibition to the large (alpha) skeletomotor neurons. Small (gamma) fusimotor neurons innervate the contractile poles of muscle spindles and function to modulate spindle–receptor discharge.

3. 3.1.

Cerebellum: The Adaptive Path–Integral Comparator Cerebellum as a Neural Controller

Having, thus, defined the spinal reflex control level, we proceed to model the top subcortical commander/controller, the cerebellum (see Appendix). The cerebellum is responsible for coordinating precisely timed hout|ini activity by integrating motor output with ongoing sensory feedback (see Figure 2). It receives extensive projections from sensory–motor areas of the cortex and the periphery and directs it back to premotor and motor cortex [8, 9]. This suggests a role in sensory–motor integration and the timing and execution of human movements. The cerebellum stores patterns of motor control for frequently performed movements, and therefore, its circuits are changed by experience and training. It was termed the adjustable pattern generator in the work of J. Houk and collaborators [10]. Also, it has become the inspiring ‘brain–model’ in robotic research [59, 61, 62]. The cerebellum is known to be involved in the production and learning of smooth coordinated movements [64, 5]. Two classes of inputs carry information into the cerebellum: the mossy fibers (MFs) and the climbing fibers (CFs). The MFs provide both plant state and contextual information [3]. The CFs, on the other hand, are thought to provide information that reflect errors in recently generated movements [39, 40]. This information is used to adjust the programs encoded by the cerebellum. The MFs carry plant state, motor efference, and other contextual signals into the cerebellum. These fibers impinge on granule cells, whose axons give rise to parallel fibers (PFs). Through the combination of inputs from multiple classes of MFs and local inhibitory interneurons, the granule cells are thought to provide a sparse expansive encoding of the incoming state information [2]. The large number of PFs converge on a much smaller set of Purkinje cells (PCs), while the PCs,

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Vladimir G. Ivancevic and Tijana T. Ivancevic

Figure 2. Schematic hout|ini organization of the primary cerebellar circuit. In essence, excitatory inputs, conveyed by collateral axons of Mossy and Climbing fibers activate directly neurones in the Deep cerebellar nuclei. The activity of these latter is also modulated by the inhibitory action of the cerebellar cortex, mediated by the Purkinje cells.

in turn, provide inhibitory signals to a single cerebellar nuclear cell [5]. Using this principle, the Cerebellar Model Arithmetic Computer, or CMAC–neural network has been built [2, 48] and implemented in robotics [63], using trial-and-error learning to produce bursts of muscular activity for controlling robot arms. So, this ‘cerebellar control’ works for simple robotic problems, like non-redundant manipulation. However, comparing the number of its neurons (107 − 108 ), to the size of conventional neural networks (including CMAC), suggests that artificial neural nets cannot satisfactorily model the function of this sophisticated ‘super–bio–computer’, as its dimensionality is virtually infinite. Despite a lot of research dedicated to its structure and function (see [10] and references there cited), the real nature of the cerebellum still remains a ‘mystery’. The main function of the cerebellum as a motor controller is depicted in Figure 3. A coordinated movement is easy to recognize, but we know little about how it is achieved. In search of the neural basis of coordination, a model of spinocerebellar interactions was recently presented in [1], in which the structural and functional organizing principle is a division of the cerebellum into discrete micro–complexes. Each micro–complex is the recipient of a specific motor error signal, that is, a signal that conveys information about an inappropriate movement. These signals are encoded by spinal reflex circuits and conveyed to the cerebellar cortex through climbing fibre afferents. This organization reveals salient features of cerebellar information processing, but also highlights the importance of systems level analysis for a fuller understanding of the neural mechanisms that underlie behavior.

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Figure 3. The cerebellum as a motor controller.

3.2.

Hamiltonian Action and Neural Path Integral

Here, we propose a quantum–like adaptive control approach to modeling the ‘cerebellar mystery’. Corresponding to the affine Hamiltonian control function (5) we define the affine Hamiltonian control action, Z tout £ ¤ dτ pi q˙i − Haf f (q, p) . (8) Saf f [q, p] = tin

From the affine Hamiltonian action (8) we further derive the associated expression for the neural phase–space path integral (in normal units), representing the cerebellar sensory– motor amplitude hout|ini, Z ­ i ® i in qout , pout |q , p = D[w, q, p] ei Saf f [q,p] (9) i in i ¾ ½ Z tout Z £ i ¤ dτ pi q˙ − Haf f (q, p) , = D[w, q, p] exp i Z with

tin

Z Y n wi (τ )dpi (τ )dq i (τ ) D[w, q, p] = , 2π τ =1

where wi = wi (t) denote the cerebellar synaptic weights positioned along its neural pathways, being continuously updated using the Hebbian–like self–organizing learning rule (3). Given the transition amplitude out|in (9), the cerebellar sensory–motor transition probability is defined as its absolute square, |hout|ini|2 . in = i i i (t), q i In the phase–space path integral (9), qin = qin out = qout (t); pi in out out pi (t), pi = pi (t); tin ≤ t ≤ tout , for all discrete time steps, t = 1, . . . , n → ∞, and we are allowing for the affine Hamiltonian Haf f (q, p) to depend upon all the (M ≤ N ) EMA–angles and angular momenta collectively. Here, we actually systematically took a discretized differential time limit of the form tσ − tσ−1 ≡ dτ (both σ and τ denote discrete (q i −q i

)

σ−1 i For technical details regarding the path integral caltime steps) and wrote (tσσ −tσ−1 ) ≡ q˙ . culations on Riemannian and symplectic manifolds (including the standard regularization procedures), see [46, 45].

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Now, motor learning occurring in the cerebellum can be observed using functional MR imaging, showing changes in the cerebellar action potential, related to the motor tasks (see, e.g., [47]). To account for these electro–physiological currents, we need to add the source term Ji (t)q i (t) to the affine Hamiltonian action (8), (the current Ji = Ji (t) acts as a source Ji Ai of the cerebellar electrical potential Ai = Ai (t)), Z Saf f [q, p, J] =

tout

tin

£ ¤ dτ pi q˙i − Haf f (q, p) + Ji q i ,

which, subsequently gives the cerebellar path integral with the action potential source, coming either from the motor cortex or from other subcortical areas. Note that the standard Wick rotation: t 7→ t (see [46, 45]), makes our path integral real, i.e., Z Z D[w, q, p] ei Saf f [q,p]

W ick −−− →

D[w, q, p] e− Saf f [q,p] ,

while their subsequent discretization gives the standard thermodynamic partition function (see Appendix), X −wj E j /T Z= , (10) j

where E j is the energy eigenvalue corresponding to the affine Hamiltonian Haf f (q, p), T is the temperature–like environmental control parameter, and the sum runs over all energy eigenstates (labelled by the index j). From (10), we can further calculate all statistical and thermodynamic system properties (see [6]), as for example, transition entropy S = kB ln Z, etc.

3.3.

Entropy and Motor Control

Our cerebellar path integral controller is closely related to entropic motor control [12, 13], which deals with neuro-physiological feedback information and environmental uncertainty. The probabilistic nature of human motor action can be characterized by entropies at the level of the organism, task, and environment. Systematic changes in motor adaptation are characterized as task–organism and environment–organism tradeoffs in entropy. Such compensatory adaptations lead to a view of goal–directed motor control as the product of an underlying conservation of entropy across the task–organism–environment system. In particular, an experiment conducted in [13] examined the changes in entropy of the coordination of isometric force output under different levels of task demands and feedback from the environment. The goal of the study was to examine the hypothesis that human motor adaptation can be characterized as a process of entropy conservation that is reflected in the compensation of entropy between the task, organism motor output, and environment. Information entropy of the coordination dynamics relative phase of the motor output was made conditional on the idealized situation of human movement, for which the goal was always achieved. Conditional entropy of the motor output decreased as the error tolerance and feedback frequency were decreased. Thus, as the likelihood of meeting the task demands was decreased increased task entropy and/or the amount of information from the environment is reduced increased environmental entropy, the subjects of this experiment employed

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fewer coordination patterns in the force output to achieve the goal. The conservation of entropy supports the view that context dependent adaptations in human goal–directed action are guided fundamentally by natural law and provides a novel means of examining human motor behavior. This is fundamentally related to the Heisenberg uncertainty principle [35] and further supports the argument for the primacy of a probabilistic approach toward the study of biodynamic cognition systems. The action–amplitude formalism represents a kind of a generalization of the HakenKelso-Bunz (HKB) model of self-organization in the individual’s motor system [11, 44], including: multi-stability, phase transitions and hysteresis effects, presenting a contrary view to the purely feedback driven systems. HKB uses the concepts of synergetics (order parameters, control parameters, instability, etc) and the mathematical tools of nonlinearly coupled (nonlinear) dynamical systems to account for self-organized behavior both at the cooperative, coordinative level and at the level of the individual coordinating elements. The HKB model stands as a building block upon which numerous extensions and elaborations have been constructed. In particular, it has been possible to derive it from a realistic model of the cortical sheet in which neural areas undergo a reorganization that is mediated by intra- and inter-cortical connections. Also, the HKB model describes phase transitions (‘switches’) in coordinated human movement as follows: (i) when the agent begins in the anti-phase mode and speed of movement is increased, a spontaneous switch to symmetrical, in-phase movement occurs; (ii) this transition happens swiftly at a certain critical frequency; (iii) after the switch has occurred and the movement rate is now decreased the subject remains in the symmetrical mode, i.e. she does not switch back; and (iv) no such transitions occur if the subject begins with symmetrical, in-phase movements. The HKB dynamics of the order parameter relative phase as is given by a nonlinear first-order ODE: φ˙ = (α + 2βr2 ) sin φ − βr2 sin 2φ, where φ is the phase relation (that characterizes the observed patterns of behavior, changes abruptly at the transition and is only weakly dependent on parameters outside the phase transition), r is the oscillator amplitude, while α, β are coupling parameters (from which the critical frequency where the phase transition occurs can be calculated).

4. 4.1.

Appendix Houk’s Autogenetic Motor Servo

About three decades ago, James Houk pointed out in [21, 14, 22, 23] that stretch and unloading reflexes were mediated by combined actions of several autogenetic neural pathways. In this context, “autogenetic” (or, autogenic) means that the stimulus excites receptors located in the same muscle that is the target of the reflex response. The most important of these muscle receptors are the primary and secondary endings in muscle spindles, sensitive to length change, and the Golgi tendon organs, sensitive to contractile force. The autogenetic circuits appear to function as servo-regulatory loops that convey continuously graded amounts of excitation and inhibition to the large (alpha) skeletomotor neurons. Small (gamma) fusimotor neurons innervate the contractile poles of muscle spindles and function to modulate

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Vladimir G. Ivancevic and Tijana T. Ivancevic

spindle–receptor discharge. Houk’s term “motor servo” [22] has been used to refer to this entire control system, summarized by the block diagram in Figure 4.

Figure 4. Houk’s autogenetic motor servo. Prior to a study by Matthews [50], it was widely assumed that secondary endings belong to the mixed population of “flexor reflex afferents,” so called because their activation provokes the flexor reflex pattern – excitation of flexor motoneurons and inhibition of extensor motoneurons. Matthews’ results indicated that some category of muscle stretch receptor other than the primary ending provides important excitation to extensor muscles, and he argued forcefully that it must be the secondary ending. The primary and secondary muscle spindle afferent fibers both arise from a specialized structure within the muscle, the muscle spindle, a fusiform structure 4–7 mm long and 80– 200 µ in diameter. The spindles are located deep within the muscle mass, scattered widely through the muscle body, and attached to the tendon, the endomysium or the perimysium, so as to be in parallel with the extrafusal or regular muscle fibers. Although spindles are scattered widely in muscles, they are not found throughout. Muscle spindle contains two types of intrafusal muscle fibers (intrafusal means inside the fusiform spindle): the nuclear bag fibers and the nuclear chain fibers. The nuclear bag fibers are thicker and longer than the nuclear chain fibers, and they receive their name from the accumulation of their nuclei in the expanded bag-like equatorial region-the nuclear bag. The nuclear chain fibers have no equatorial bulge; rather their nuclei are lined up in the equatorial region-the nuclear chain. A typical spindle contains two nuclear bag fibers and 4-5 nuclear chain fibers. The pathways from primary and secondary endings are treated commonly by Houk in Figure 4, since both receptors are sensitive to muscle length and both provoke reflex excitation. However, primary endings show an additional sensitivity to the dynamic phase of length change, called dynamic responsiveness, and they also show a much–enhanced sensitivity to small changes in muscle length [51].

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The motor servo comprises three closed circuits (Figure 4), two neural feedback pathways, and one circuit representing the mechanical interaction between a muscle and its load. One of the feedback pathways, that from spindle receptors, conveys information concerning muscle length, and it follows that this loop will act to keep muscle length constant. The other feedback pathway, that from tendon organs, conveys information concerning muscle force, and it acts to keep force constant. In general, it is physically impossible to maintain both muscle length and force constant when external loads vary; in this situation the action of the two feedback loops will oppose each other. For example, an increased load force will lengthen the muscle and cause muscular force to increase as the muscle is stretched out on its length-tension curve. The increased length will lead to excitation of motoneurons, whereas the increased force will lead to inhibition. It follows that the net regulatory action conveyed by skeletomotor output will depend on some relationship between force change and length change and on the strength of the feedback from muscle spindles and tendon organs. A simple mathematical derivation [53] demonstrates that the change in skeletomotor output, the error signal of the motor servo, Should be proportional to the difference between a regulated stiffness and the actual stiffness provided by the mechanical properties of the muscle, where stiffness has the units of force change divided by length change. The regulated stiffness is determined by the ratio of the gain of length to force feedback. It follows that the combination of spindle receptor and tendon organ feedback will tend to maintain the stiffness of the neuromuscular apparatus at some regulated level. If this level is high, due to a high gain of length feedback and a low gain of force feedback, one could simply forget about force feedback and treat muscle length as the regulated variable of the system. However, if the regulated level of stiffness is intermediate in value, i.e. not appreciably different from the average stiffness arising from muscle mechanical properties in the absence of reflex actions, one would conclude that stiffness, or its inverse, compliance, is the regulated property of the motor servo. In this way, the autogenetic reflex motor servo provides the local, reflex feedback loops for individual muscular contractions. A voluntary contraction force F of human skeletal muscle is reflexly excited (positive feedback +F −1 ) by the responses of its spindle receptors to stretch and is reflexly inhibited (negative feedback −F −1 ) by the responses of its Golgi tendon organs to contraction. Stretch and unloading reflexes are mediated by combined actions of several autogenetic neural pathways, forming the motor servo (see [28, 29, 30]). In other words, branches of the afferent fibers also synapse with with interneurons that inhibit motor neurons controlling the antagonistic muscles – reciprocal inhibition. Consequently, the stretch stimulus causes the antagonists to relax so that they cannot resists the shortening of the stretched muscle caused by the main reflex arc. Similarly, firing of the Golgi tendon receptors causes inhibition of the muscle contracting too strong and simultaneous reciprocal activation of its antagonist.

4.2.

Cerebellum and Muscular Synergy

The cerebellum is a brain region anatomically located at the bottom rear of the head (the hindbrain), directly above the brainstem, which is important for a number of subconscious

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Vladimir G. Ivancevic and Tijana T. Ivancevic

and automatic motor functions, including motor learning. It processes information received from the motor cortex, as well as from proprioceptors and visual and equilibrium pathways, and gives ‘instructions’ to the motor cortex and other subcortical motor centers (like the basal nuclei), which result in proper balance and posture, as well as smooth, coordinated skeletal movements, like walking, running, jumping, driving, typing, playing the piano, etc. Patients with cerebellar dysfunction have problems with precise movements, such as walking and balance, and hand and arm movements. The cerebellum looks similar in all animals, from fish to mice to humans. This has been taken as evidence that it performs a common function, such as regulating motor learning and the timing of movements, in all animals. Studies of simple forms of motor learning in the vestibulo–ocular reflex and eye– blink conditioning are demonstrating that timing and amplitude of learned movements are encoded by the cerebellum. When someone compares learning a new skill to learning how to ride a bike they imply that once mastered, the task seems imbedded in our brain forever. Well, imbedded in the cerebellum to be exact. This brain structure is the commander of coordinated movement and possibly even some forms of cognitive learning. Damage to this area leads to motor or movement difficulties. A part of a human brain that is devoted to the sensory-motor control of human movement, that is motor coordination and learning, as well as equilibrium and posture, is the cerebellum (which in Latin means “little brain”). It performs integration of sensory perception and motor output. Many neural pathways link the cerebellum with the motor cortex, which sends information to the muscles causing them to move, and the spino–cerebellar tract, which provides proprioception, or feedback on the position of the body in space. The cerebellum integrates these pathways, using the constant feedback on body position to fine–tune motor movements [39]. The human cerebellum has 7–14 million Purkinje cells. Each receives about 200,000 synapses, most onto dendritic splines. Granule cell axons form the parallel fibers. They make excitatory synapses onto Purkinje cell dendrites. Each parallel fibre synapses on about 200 Purkinje cells. They create a strip of excitation along the cerebellar folia. Mossy fibers are one of two main sources of input to the cerebellar cortex (see Figure 5). A mossy fibre is an axon terminal that ends in a large, bulbous swelling. These mossy fibers enter the granule cell layer and synapse on the dendrites of granule cells; in fact the granule cells reach out with little ‘claws’ to grasp the terminals. The granule cells then send their axons up to the molecular layer, where they end in a T and run parallel to the surface. For this reason these axons are called parallel fibers. The parallel fibers synapse on the huge dendritic arrays of the Purkinje cells. However, the individual parallel fibers are not a strong drive to the Purkinje cells. The Purkinje cell dendrites fan out within a plane, like the splayed fingers of one hand. If we were to turn a Purkinje cell to the side, it would have almost no width at all. The parallel fibers run perpendicular to the Purkinje cells, so that they only make contact once as they pass through the dendrites. Unless firing in bursts, parallel fibre EPSPs do not fire Purkinje cells. Parallel fibers provide excitation to all of the Purkinje cells they encounter. Thus, granule cell activity results in a strip of activated Purkinje cells. Mossy fibers arise from the spinal cord and brainstem. They synapse onto granule cells and deep cerebellar nuclei. The Purkinje cell makes an inhibitory synapse (GABA) to the

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deep nuclei. Mossy fibre input goes to both cerebellar cortex and deep nuclei. When the Purkinje cell fires, it inhibits output from the deep nuclei.

Figure 5. Stereotypical ways throughout the cerebellum. The climbing fibre arises from the inferior olive. It makes about 300 excitatory synapses onto one Purkinje cell. This powerful input can fire the Purkinje cell. The parallel fibre synapses are plastic—that is, they can be modified by experience. When parallel fibre activity and climbing fibre activity converge on the same Purkinje cell, the parallel fibre synapses become weaker (EPSPs are smaller). This is called long-term depression. Weakened parallel fibre synapses result in less Purkinje cell activity and less inhibition to the deep nuclei, resulting in facilitated deep nuclei output. Consequently, the mossy fibre collaterals control the deep nuclei. The basket cell is activated by parallel fibers afferents. It makes inhibitory synapses onto Purkinje cells. It provides lateral inhibition to Purkinje cells. Basket cells inhibit Purkinje cells lateral to the active beam. Golgi cells receive input from parallel fibers, mossy fibers, and climbing fibers. They inhibit granule cells. Golgi cells provide feedback inhibition to granule cells as well as feedforward inhibition to granule cells. Golgi cells create a brief burst of granule cell activity. Although each parallel fibre touches each Purkinje cell only once, the thousands of parallel fibers working together can drive the Purkinje cells to fire like mad. The second main type of input to the folium is the climbing fibre. The climbing fibers go straight to the Purkinje cell layer and snake up the Purkinje dendrites, like ivy climbing a trellis. Each climbing fibre associates with only one Purkinje cell, but when the climbing fibre fires, it provokes a large response in the Purkinje cell. The Purkinje cell compares and processes the varying inputs it gets, and finally sends

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Vladimir G. Ivancevic and Tijana T. Ivancevic

its own axons out through the white matter and down to the deep nuclei. Although the inhibitory Purkinje cells are the main output of the cerebellar cortex, the output from the cerebellum as a whole comes from the deep nuclei. The three deep nuclei are responsible for sending excitatory output back to the thalamus, as well as to postural and vestibular centers. There are a few other cell types in cerebellar cortex, which can all be lumped into the category of inhibitory interneuron. The Golgi cell is found among the granule cells. The stellate and basket cells live in the molecular layer. The basket cell (right) drops axon branches down into the Purkinje cell layer where the branches wrap around the cell bodies like baskets. The cerebellum operates in 3’s: there are 3 highways leading in and out of the cerebellum, there are 3 main inputs, and there are 3 main outputs from 3 deep nuclei. They are: The 3 highways are the peduncles. There are 3 pairs (see [52, 15, 49]): 1. The inferior cerebellar peduncle (restiform body) contains the dorsal spinocerebellar tract (DSCT) fibers. These fibers arise from cells in the ipsilateral Clarke’s column in the spinal cord (C8–L3). This peduncle contains the cuneo–cerebellar tract (CCT) fibers. These fibers arise from the ipsilateral accessory cuneate nucleus. The largest component of the inferior cerebellar peduncle consists of the olivo–cerebellar tract (OCT) fibers. These fibers arise from the contralateral inferior olive. Finally, vestibulo–cerebellar tract (VCT) fibers arise from cells in both the vestibular ganglion and the vestibular nuclei and pass in the inferior cerebellar peduncle to reach the cerebellum. 2. The middle cerebellar peduncle (brachium pontis) contains the pontocerebellar tract (PCT) fibers. These fibers arise from the contralateral pontine grey. 3. The superior cerebellar peduncle (brachium conjunctivum) is the primary efferent (out of the cerebellum) peduncle of the cerebellum. It contains fibers that arise from several deep cerebellar nuclei. These fibers pass ipsilaterally for a while and then cross at the level of the inferior colliculus to form the decussation of the superior cerebellar peduncle. These fibers then continue ipsilaterally to terminate in the red nucleus (‘ruber–duber’) and the motor nuclei of the thalamus (VA, VL). The 3 inputs are: mossy fibers from the spinocerebellar pathways, climbing fibers from the inferior olive, and more mossy fibers from the pons, which are carrying information from cerebral cortex (see Figure 6). The mossy fibers from the spinal cord have come up ipsilaterally, so they do not need to cross. The fibers coming down from cerebral cortex, however, do need to cross (as the cerebrum is concerned with the opposite side of the body, unlike the cerebellum). These fibers synapse in the pons (hence the huge block of fibers in the cerebral peduncles labelled ‘cortico–pontine’), cross, and enter the cerebellum as mossy fibers. The 3 deep nuclei are the fastigial, interposed, and dentate nuclei. The fastigial nucleus is primarily concerned with balance, and sends information mainly to vestibular and reticular nuclei. The dentate and interposed nuclei are concerned more with voluntary movement, and send axons mainly to thalamus and the red nucleus.

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Figure 6. Inputs and outputs of the cerebellum. The main function of the cerebellum as a motor controller is depicted in Figure 3. A coordinated movement is easy to recognize, but we know little about how it is achieved. In search of the neural basis of coordination, a model of spinocerebellar interactions was recently presented in [1], in which the structure-functional organizing principle is a division of the cerebellum into discrete micro–complexes. Each micro–complex is the recipient of a specific motor error signal, that is, a signal that conveys information about an inappropriate movement. These signals are encoded by spinal reflex circuits and conveyed to the cerebellar cortex through climbing fibre afferents. This organization reveals salient features of cerebellar information processing, but also highlights the importance of systems level analysis for a fuller understanding of the neural mechanisms that underlie behavior. The authors of [1] reviewed anatomical and physiological foundations of cerebellar information processing. The cerebellum is crucial for the coordination of movement. The authors presented a model of the cerebellar paravermis, a region concerned with the control of voluntary limb movements through its interconnections with the spinal cord. They particularly focused on the olivo-cerebellar climbing fibre system. Climbing fibres are proposed to convey motor error signals (signals that convey information about inappropriate movements) related to elementary limb movements that result from the contraction of single muscles. The actual encoding of motor error signals is suggested to depend on sensorimotor transformations carried out by spinal modules that mediate nociceptive withdrawal reflexes. The termination of the climbing fibre system in the cerebellar cortex subdivides the paravermis into distinct microzones. Functionally similar but spatially separate microzones converge onto a common group of cerebellar nuclear neurons. The processing units formed as a consequence are termed ‘multizonal micro-complexes’ (MZMCs), and are each related

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Vladimir G. Ivancevic and Tijana T. Ivancevic

to a specific spinal reflex module. The distributed nature of microzones that belong to a given MZMC is proposed to enable similar climbing fibre inputs to integrate with mossy fibre inputs that arise from different sources. Anatomical results consistent with this notion have been obtained. Within an individual MZMC, the skin receptive fields of climbing fibres, mossy fibres and cerebellar cortical inhibitory interneurons appear to be similar. This indicates that the inhibitory receptive fields of Purkinje cells within a particular MZMC result from the activation of inhibitory interneurons by local granule cells. On the other hand, the parallel fibre–mediated excitatory receptive fields of the Purkinje cells in the same MZMC differ from all of the other receptive fields, but are similar to those of mossy fibres in another MZMC. This indicates that the excitatory input to Purkinje cells in a given MZMC originates in non–local granule cells and is mediated over some distance by parallel fibres. The output from individual MZMCs often involves two or three segments of the ipsilateral limb, indicative of control of multi–joint muscle synergies. The distal–most muscle in this synergy seems to have a roughly antagonistic action to the muscle associated with the climbing fibre input to the MZMC. The model proposed in [1] indicates that the cerebellar paravermis system could provide the control of both single– and multi–joint movements. Agonist-antagonist activity associated with single–joint movements might be controlled within a particular MZMC, whereas coordination across multiple joints might be governed by interactions between MZMCs, mediated by parallel fibres. Two main theories address the function of the cerebellum, both dealing with motor coordination. One claims that the cerebellum functions as a regulator of the “timing of movements.” This has emerged from studies of patients whose timed movements are disrupted [37]. The second, “Tensor Network Theory” provides a mathematical model of transformation of sensory (covariant) space-time coordinates into motor (contravariant) coordinates by cerebellar neuronal networks [56, 57, 58]. Studies of motor learning in the vestibulo–ocular reflex and eye-blink conditioning demonstrate that the timing and amplitude of learned movements are encoded by the cerebellum [4]. Many synaptic plasticity mechanisms have been found throughout the cerebellum. The Marr–Albus model mostly attributes motor learning to a single plasticity mechanism: the long-term depression of parallel fiber synapses. The Tensor Network Theory of sensory–motor transformations by the cerebellum has also been experimentally supported [7].

4.3.

Feynman’s Partition Function

Recall that in statistical mechanics, the so–called partition function Z is a quantity that encodes the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas. Other thermodynamic variables of the system, such as the total energy, free energy, entropy, and pressure,

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can be expressed in terms of the partition function or its derivatives.4 P The partition function of a canonical ensemble5 is defined as a sum Z(β) = j e−βEj , where β = 1/(kB T ) is the ‘inverse temperature’, where T is an ordinary temperature and kB is the Boltzmann’s constant. However, as the position xi and momentum pi variables of an ith particle in a system can vary continuously, the set of microstates is actually uncountable. In this case, some form of coarse–graining procedure must be carried out, which essentially amounts to treating two mechanical states as the same microstate if the differences in their position and momentum variables are ‘small enough’. The partition function then takes the form of an integral. For instance, the partition function of a gas consisting of N molecules is proportional to the 6N −dimensional phase–space integral, Z Z(β) ∼ d3 pi d3 xi exp[−βH(pi , xi )], R6N

where H = H(pi , xi ), (i = 1, . . . , N ) is the classical Hamiltonian (total energy) function. More generally, the so–called configuration integral, as used in probability theory, information science and dynamical systems, is an abstraction of the above definition of a partition function in statistical mechanics. It is a special case of a normalizing constant in probability theory, for the Boltzmann distribution. The partition function occurs in many problems of probability theory because, in situations where there is a natural symmetry, its associated probability measure, the Gibbs measure (see below), which generalizes the notion of the canonical ensemble, has the Markov property. Given a set of random variables Xi taking on values xi , and purely potential Hamiltonian function H(xi ), (i = 1, . . . , N ), the partition function is defined as X Z(β) = exp [−βH(xi )] . xi

The function H is understood to be a real-valued function on the space of states {X1 , X2 , · · · } while β is a real-valued free parameter (conventionally, the inverse temperature). The sum over the xi is understood to be a sum over all possible values that the random variable Xi may take. Thus, the sum is to be replaced by an integral when the Xi are continuous, rather than discrete. Thus, one writes Z Z(β) = dxi exp [−βH(xi )] , 4 There are actually several different types of partition functions, each corresponding to different types of statistical ensemble (or, equivalently, different types of free energy.) The canonical partition function applies to a canonical ensemble, in which the system is allowed to exchange heat with the environment at fixed temperature, volume, and number of particles. The grand canonical partition function applies to a grand canonical ensemble, in which the system can exchange both heat and particles with the environment, at fixed temperature, volume, and chemical potential. Other types of partition functions can be defined for different circumstances. 5 A canonical ensemble is a statistical ensemble representing a probability distribution of microscopic states of the system. Its probability distribution is characterized by the proportion pi of members of the ensemble which exhibit a measurable macroscopic state i, where the proportion of microscopic states for each macroscopic state i is given by the Boltzmann distribution,

pi =

1 −Ei /(kT ) e Z

= e−(Ei −A)/(kT ) ,

where Ei is the energy of state i. It can be shown that this is the distribution which is most likely, if each system in the ensemble can exchange energy with a heat bath, or alternatively with a large number of similar systems. In other words, it is the distribution which has maximum entropy for a given average energy < Ei >.

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for the case of continuously-varying random variables Xi . The Gibbs measure of a random variable Xi having the value xi is defined as the probability density function P (Xi = xi ) =

1 exp [−βH(xi )] . exp [−βE(xi )] = P Z(β) xi exp [−βH(xi )]

where E(xi ) = H(xi ) is the energy of the configuration xi . This probability, which is now properly normalized so that 0 ≤ P (xi ) ≤ 1, can be interpreted as a likelihood that a specific configuration of values xi , (i = 1, 2, . . . , N ) occurs in the system. As such, the partition function Z(β) can be understood to provide the Gibbs measure on the space of states, which is the unique statistical distribution that maximizes the entropy for a fixed expectation value of the energy, hHi = −

∂ log(Z(β)) . ∂β

The associated entropy is given by X S=− P (xi ) ln P (xi ) = βhHi + log Z(β). xi

The principle of maximum entropy related to the expectation value of the energy hHi, is a postulate about a universal feature of any probability assignment on a given set of propositions (events, hypotheses, indices, etc.). Let some testable information about a probability distribution function be given. Consider the set of all trial probability distributions which encode this information. Then the probability distribution which maximizes the information entropy is the true probability distribution, with respect to the testable information prescribed. Now, the number of variables Xi need not be countable, in which case the set of coordinates {xi } becomes a field φ = φ(x), so the sum is to be replaced by the Euclidean path integral (that is a Wick–rotated Feynman transition amplitude in imaginary time), as Z Z(φ) = D[φ] exp [−H(φ)] . More generally, in quantum field theory, instead of the field Hamiltonian H(φ) we have the action S(φ) of the theory. Both Euclidean path integral, Z Z(φ) = D[φ] exp [−S(φ)] , real path integral in imaginary time (11) and Lorentzian one, Z Z(φ) = D[φ] exp [iS(φ)] ,

complex path integral in real time,

(12)

are usually called ‘partition functions’. While the Lorentzian path integral (12) represents a quantum-field theory-generalization of the Schr¨odinger equation, the Euclidean path integral (11) represents a statistical-field-theory generalization of the Fokker–Planck equation.

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References [1] R. Apps, M. Garwicz, Anatomical and physiological foundations of cerebellar information processing. Nature Rev. Neurosci 6 (2005), 297–311. [2] J. S. Albus, A theory of cerebellar function. Mathematical Biosciences 10 (1971), 25–61. [3] J. R. Bloedel, J. Courville, Cerebellar afferent systems. In: J. Brookhart, V. Mountcastle, V. Brooks, and S. Geiger (eds.), Handbook of Physiology, Sect. 1. The Nervous System. Motor Control. American Physiological Society, Bethesda, MD, 1981. [4] E. S. Boyden, A. Katoh, J. L. Raymond, Cerebellum-dependent learning: The role of multiple plasticity mechanisms. Annu. Rev. Neurosci. 27 (2004), 581–609. [5] A. H. Faag, N. Sitkoff, A. G. Barto, J. C. Houk, Cerebellar learning for control of a two-link arm in Muscle space. Proc. IEEE Int. Conf. Rob. Aut. (1997) [6] R. P. Feynman, Statistical Mechanics. A Set of Lectures, WA Benjamin, Inc., Reading, Massachusetts, (1972). [7] C. C. Gielen, van E. J. Zuylen, Coordination of arm muscles during flexion and supination: application of the tensor analysis approach. Neurosci. 17 (1986), 527–39. [8] C. Ghez, Introduction to motor system. In: Kandel, E.K. and Schwarz. J.H. (eds.), Principles of neural science (2nd ed.) Elsevier, Amsterdam, 1990, 429–442. [9] C. Ghez, Muscles: Effectors of the Motor Systems. In: E. R. Kandel. J. H. Schwartz, and T. M. Jessell (eds.), Principles of Neural Science. (3rd ed.) Appleton and Lange, Elsevier, 1991, 548–563. [10] J. C. Houk, J. T. Buckingham, A. G. Barto, Models of the cerebellum and motor learning. Behavioral and Brain Sciences 19(3) (1996), 368–383. [11] H. Haken, J. A. S. Kelso, H. Bunz, A theoretical model of phase transitions in human hand movements. Biol. Cybern. 51 (1985), 347–356. [12] S. L. Hong, K. M. Newell, Entropy conservation in the control of human action. Nonl. Dyn. Psych. Life. Sci. 12(2), 163–190. [13] S. L. Hong, K. M. Newell, Entropy compensation in human motor adaptation. Chaos 18(1) (2008), 013108. [14] J. C. Houk, J. J. Singer, M. R. Goldman, An evaluation of length and force feedback to soleus muscles of decerebrate cats. J. Neurophysiol. 33 (1970), 784–811. [15] J. K. Harting, The Global Anatomy. Medical School, Univ. Wisconsin, (1997). [16] H. Hatze, A myocybernetic control model of skeletal muscle. Biol. Cyber. 25 (1977), 103–119.

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[17] H. Hatze, A complete set of control equations for the human musculoskeletal system. J. Biomech. 10 (1977b), 799–805. [18] H. Hatze, A general myocybernetic control model of skeletal muscle. Biol. Cyber. 28 (1978), 143–157. [19] D. O. Hebb, The Organization of Behavior. Wiley, New York, (1949). [20] A. V. Hill, The heat of shortening and the dynamic constants of muscle. Proc. Roy. Soc. B76 (1938), 136–195. [21] J. C. Houk, Feedback control of skeletal muscles. Brain Res. 5 (1967), 433–451. [22] J. C. Houk, Participation of reflex mechanisms and reaction-time processes in the compensatory adjustments to mechanical disturbances. Progr. Clin. Neurophysiol. 4 (1978), 193–215. [23] J. C. Houk, Regulation of stiffness by skeletomotor reflexes. Ann. Rev. Physiol. 41 (1979), 99–114. [24] J. C. Houk, Regulation of stiffness by skeletomotor reflexes. Ann. Rev. Physiol. 41 (1979), 99–114. [25] V. Ivancevic, E. Aidman, Life-space foam: A medium for motivational and cognitive dynamics. Physica A 382 (2007), 616–630. [26] V. Ivancevic, E. Aidman, L. Yen, Extending Feynman’s Formalisms for Modelling Human Joint Action Coordination. Int. J. Biomath. (to appear). [27] V. Ivancevic, N. Beagley, Brain–like functor control machine for general humanoid biodynamics. Int. J. Math. Math. Sci. 11 (2005), 1759–1779. [28] V. Ivancevic, T. Ivancevic, Human–Like Biomechanics. Springer, 2006. [29] V. Ivancevic, T. Ivancevic, Natural Biodynamics. World Scientific, 2006. [30] V. Ivancevic, T. Ivancevic, Geometrical Dynamics of Complex Systems. Springer, 2006. [31] V. Ivancevic, T. Ivancevic, Neuro–Fuzzy Associative Machinery for Comprehensive Brain and Cognition Modelling. Springer, Berlin, 2007. [32] V. Ivancevic, T. Ivancevic, Computational Mind: A Complex Dynamics Perspective. Springer, Berlin, (2007). [33] V. Ivancevic, T. Ivancevic, Applied Differential Geometry: A Modern Introduction. World Scientific, Singapore, (2007). [34] V. Ivancevic, T. Ivancevic, Complex Nonlinearity: Chaos, Phase Transitions. Topology Change and Path Integrals, Springer, (2008).

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[35] V. Ivancevic, T. Ivancevic, Quantum Leap: From Dirac and Feynman Across the Universe to Human Body and Mind. World Scientific, Singapore, 2008. [36] V. Ivancevic, T. Ivancevic, Nonlinear Quantum Psychodynamics with Topological Phase Transitions. NQJ (to appear). [37] R. B. Ivry, S. W. Keele, H. C. Diener, Dissociation of the lateral and medial cerebellum in movement timing and movement execution. Exp. Brain. Res. 73(1) (1988), 167–80. [38] V. Ivancevic, M. Snoswell, Fuzzy–stochastic functor machine for general humanoid– robot dynamics. IEEE Trans. SMCB 31(3) (2001), 319–330. [39] M. Ito, Cerebellum and Neural Control. Raven Press, New York, 1984. [40] M. Ito, A New Physiological Concept on Cerebellum. Rev. Neurol, 1990. [41] V. Ivancevic, Generalized Hamiltonian biodynamics and topology invariants of humanoid robots. IJMMS 31(9) (2202), 555–565. [42] V. Ivancevic, Symplectic Rotational Geometry in Human Biomechanics. SIAM Rev. 46(3) (2004), 455–474. [43] V. Ivancevic, Lie–Lagrangian model for realistic human bio-dynamics. IJHR 3(2) (2006), 205–218. [44] J. A. S. Kelso, Dynamic Patterns: The Self Organization of Brain and Behavior. MIT Press, Cambridge, MA, 1995. [45] J. R. Klauder, Beyond Conventional Quantization. Cambridge Univ. Press, Cambridge, 2000. [46] J. R. Klauder, Understanding Quantization. Found. Phys. 27 (1997), 1467–1483. [47] M. Mascalchi, et al.: Proton MR Spectroscopy of the Cerebellum and Pons in Patients with Degenerative Ataxia. Radiology 223 (2002), 371. [48] W. T. Miller, F. H. Glanz, L. G. Kraft, CMAC: An associative neural network alternative to backpropagation. In C. Lau (ed.), Neural Networka. Theoretical Foundations and Analysis, IEEE Press, New York, 1992, 233–240. [49] E. N. Marieb, Human Anatomy and Physiology. (4th ed.) Benjamin/Cummings, Menlo Park, CA, 1998. [50] P. B. C. Matthews, Evidence that the secondary as well as the primary endings of the muscle spindles may be responsible for the tonic stretch-reflex of the decerebrate eat. J. Physiol. London 204 (1969), 365–93. [51] P. B. C. Matthews, Mammalian Muscle Receptors and Their Central Action. tWilliams & Wilkins, Baltimore, 1972. [52] D. W. Molavi, Neuroscience Tutorial. School of Medicine, Washington Univ., 1997.

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[53] T. R. Nichols, J. C. Houk, The improvement in linearity and the regulation of stiffness that results from the actions of the stretch–reflex. J. Neurophysiol. 39 (1976), 119–142. [54] H. Nijmeijer, A. J. van der Schaft, Nonlinear Dynamical Control Systems. Springer, New York, 1990. [55] J. Park, W. -K. Chung, Geometric Integration on Euclidean Group With Application to Articulated Multibody Systems. IEEE Trans. Rob. 21(5) (2005), 850–863. [56] A. Pellionisz, R. Llinas, Tensorial approach to the geometry of brain function: cerebellar coordination via a metric tensor. Neurosci. 5 (1980), 1125–1136. [57] A. Pellionisz, R. Llinas, Space-time representation in the brain. The cerebellum as a predictive space-time metric tensor. Neurosci. 7(12) (1982), 2949–2970. [58] A. Pellionisz, R. Llinas, Tensor network theory of the meta-organization of functional geometries in the central nervous system. Neurosci. 16(2) (1985), 245–273. [59] S. Schaal, C. G. Atkeson, Constructive incremental learning from only local information. Neural Comput. 10 (1998), 2047–2084. [60] R. S. Sutton, A. G. Barto, Reinforcement Learning: An Introduction. MIT Press, Cambridge, MA, 1998. [61] S. Schaal, Robot learning. In: M. Arbib (ed.), Handbook of Brain Theory and Neural Networks (2nd ed.), MIT Press, Cambridge, 1998. [62] S. Schaal, Is imitation learning the route to humanoid robots?. Trends Cogn. Sci. 3 (1999), 233–242. [63] P. van der Smagt, Cerebellar control of robot arms. Conn. Sci. 10 (1998), 301–320. [64] W. T. Thach, H. Goodkin, J. Keating, The cerebellum and the adaptive coordination of movement. Ann. Rev. Neurosci. (1992), 403–442. [65] M. Vukobratovic, B. Borovac, D. Surla, D. Stokic, Biped Locomotion: Dynamics, Stability, Control, and Applications. Springer, Berlin, 1990. [66] D. R. Wilkie, The mechanical properties of muscle. Brit. Med. Bull. 12 (1956), 177– 182.

In: Advances in Mathematics Research, Volume 10 ISBN 978-1-60876-265-1 c 2010 Nova Science Publishers, Inc. Editor: Albert R. Baswell, pp. 53-204

Chapter 3

T RAVELING WAVES A RISING FROM S YNAPTICALLY C OUPLED N EURONAL N ETWORKS Linghai Zhang∗ Department of Mathematics, Lehigh University 14 East Packer Avenue, Bethlehem, Pennsylvania 18015

Abstract We will investigate the traveling wave solutions of the following nonlinear singularly perturbed systems of integral differential equations arising from synaptically coupled neuronal networks: (I)

(II)

ut + f (u) + w = α

Z

K(x − y)H(u(y, t) − θ)dy, R

wt = ε(g(u) − γw). Z Z ut + f (u) + w = α K(x − y)H(u(y, t) − θ)dy+β K(x−y)H(u(y, t)−Θ)dy, R

R

wt = ε(g(u) − γw).     Z 1 (III) ut + f (u) + w = α K(x − y)H u y, t − |x − y| − θ dy, c R wt = ε(g(u) − γw).

These equations involve the convolution product of a synaptic coupling and the Heaviside step function. Some of the model equations involve spatial temporal delays as well, where the speed c of the action potential is finite. The main purpose of this chapter is to establish the existence and stability of traveling wave solutions to the nonlinear systems of integral differential equations. We will apply fundamental ideas, methods and techniques in differential equations and functional analysis to establish the existence of the traveling wave solutions and we will use linearized stability criterion and Evans functions (i.e complex analytic functions) to accomplish the stability. Let L(ε) represent a linear differential operator obtained by linearizing the integral differential equations with respect to a traveling wave solution, and let σ(L(ε)) denote the spectrum of the linear differential operator L(ε). The traveling wave solution of ∗

E-mail address: [email protected]

54

Linghai Zhang each nonlinear singularly perturbed system of integral differential equations is stable, if max{ Re λ : λ ∈ σ(L(ε)), λ 6= 0} ≤ c0 , and λ = 0 is algebraically simple, where c0 > 0 is a positive constant. This is a sufficient and necessary condition for the wave stability of the nonlinear systems of integral differential equations.

Keywords: synaptically coupled neuronal networks, nonlinear singularly perturbed systems of integral differential equations, traveling waves, speed index functions, existence and uniqueness, eigenvalue problems, Evans functions, nonlinear stability.

1.

Introduction to Integral Differential Equations

An excitable medium in biological, chemical or physical systems is a spatially distributed dynamical system that stays at a stable equilibrium, usually called the resting state, in the absence of external stimulus. However, a localized, super-threshold perturbation can trigger a local transition from the resting state to an excited state. Such a transition, called the excitation, can propagate away at constant speed and constant shape from the site of stimulation in the form of a traveling wave front, a traveling pulse solution, or other more complicated realistic stereo-typical forms such as target waves, spirals, and scrolls, depending on the local dynamics, the dimension of the medium, and the distribution of the global dynamics. In cardiobiology, excitable media have been encountered in many different systems including cardiac tissues, a field of aggregating slime moulds, the surface membrane of an excitable cell such as a nerve fiber, and the cytoplasm of a calcium excitable nerve cell. In chemistry, the Belousov-Zhabotinskii reaction modeled by the system ut = △u + u(1 − u − v), vt = △v − uv, is the best-known excitable medium that has been studied extensively in experiments and in mathematical analysis. In physics, inflammable materials is one of the best studied excitable medium. These excitable media are mostly modeled by nonlinear partial differential equations of the reaction-diffusion type, such as the scalar bistable equation ut = △u + u(1 − u)(u − a), where 0 < a < 1 is a constant; the well known FitzhughNagumo equations ut = △u + u(1 − u)(u − a) − w, wt = ε(u − γw), where 0 < a < 1, 0 < γ < 4/(1 − a)2 and 0 < ε ≪ 1 are constants; the diffusive predator-prey systems ut + u[(u − 2)(u − a) + v)] = ε2 △u, vt + (1 + v − u)v = △v, where 0 < ε ≪ 1 and 0 < a < 1 are constants; and the famous Hodgkin-Huxley equations CM ut = DM uxx − GNa m3 h(u − uNa ) − GK n4 (u − uK ) − GL (u − uL ), mt = αm (1 − m) − βm m,

nt = αn (1 − n) − βn n,

ht = αh (1 − h) − βh h,

where each term in the first equation represents a separate electrical current. Moreover   V + bm V   αm = am , βm = cm exp , V + bm dm −1 exp bm

Traveling Waves Arising from Synaptically Coupled Neuronal Networks   V + bn V   , αn = an βn = cn exp , V + bn dn exp −1 bn   V 1   αh = ah exp , βh = , V + ch bh exp +1 dh

55

where (CM , DM ), (GNa , GK , GL , uNa , uK , uL ), (am , bm , cm , dm ), (an , bn , cn , dn ), (ah , bh , ch , dh ) are five groups of parameters. In the models of calcium waves, aggregating slime moulds, and the Belousov-Zhabotinskii reaction, the n-dimensional Laplace operator △ = Pn 2 2 i=1 ∂ /∂xi naturally arises when the diffusion of some key molecules is included. In the model for traveling nerve pulses, the Laplace operator appears when describing the nerve fiber as a conducting cable. In modeling cardiac tissues, however, the Laplace operator occurs when taking the continuous limit of discrete muscle cells coupled only to the nearest neighbours. Nonlinear traveling waves have been observed ubiquitously in many fields of natural science (dispersive medium, nonlinear optics and plasma physics, etc). Solitary waves have been modeled very well with nonlinear dispersive wave equations, such as the Kortewegde Vries equation ut + ux + uxxx + uux = 0, the nonlinear cubic Schr¨odinger equation iut + uxx + |u|2 u + (|u|2 u)x = 0, and the Benjamin-Ono equation ut + Huxx + 2uux = 0, where H denotes the Hilbert transformation. The main feature of these equations is that each of them possesses infinitely-many conservation laws. The system with the richest variety of solitary waves that we observe is the nervous system. Traditional model equations (such as the nonlinear dispersive wave equations and the reaction-diffusion equations) are not good enough in modeling brain waves that are observed in nerve tissues which contain a macroscopic number of neurons such as a slice of the thalamus containing the thalamocortical and the reticular cells and a layer of the visual cortex. The main reason is that it is impossible that a neuron is only connected to its nearest neighbours. The extensive tree-structured dendritic and axonal processes enable each neuron to interact with many neurons that are not necessarily located in its vicinity. Thus neuron-to-neuron interaction is intrinsically nonlocal. The equations that describe the dynamic-kinetic behaviors of neural tissues consisting of a continuum of neurons that interact in a nonlocal manner are typically integral differential equations. An important aspect of mathematical neuroscience is to develop and solve mathematical model equations for neuronal activity patterns. These model equations are used to understand how various activity patterns are generated and how the patterns change as the model parameters are modulated. Patterns of neuronal waves underlie many activities of the nervous system. A neuronal network’s population behaviors result mainly from interactions between three separate components: the intrinsic properties of individual neurons, the synaptic coupling between neurons and the architecture of coupling. Neuronal networks may include several different kinds of neurons and different kinds of coupling. An important and challenging problem is to determine the role each component plays in shaping the emergent behaviors of the neuronal networks. It is appropriate to apply mathematical analysis to investigate biological mechanisms (described by mathematical parameters, synaptic couplings and nonlinear functions) that

56

Linghai Zhang

support traveling waves and other activity patterns in synaptically coupled neuronal networks. There have been some numerical simulations and rigorous analysis attempting to understand the mathematical systems that describe synaptically coupled neuronal networks. Rigorous mathematical analysis of neuronal network models is very important to make progress in understanding propagating activity patterns observed in neural tissues. These patterns of activity, often manifested as traveling wave fronts and pulses, arise both as natural physiological occurrence and in pathological situations. For example, the presentation of stimuli in turtle visual cortex has been shown to evoke propagating waves of activity [58]. Neurological disorders, such as cortical epilepsy [18] and migraine [52] in human beings, are also characterized by traveling waves of neural activity across the nervous system. The propagation of nerve impulses in synaptically coupled neuronal networks has some practical applications. Biologically and mathematically, stable wave solutions are the most interesting solutions. Consider the following nonlinear singularly perturbed systems of integral differential equations: (I) Z ut + f (u) + w = α K(x − y)H(u(y, t) − θ)dy, (1) R

wt = ε(g(u) − γw),

(2)

where α > 0, γ > 0, 0 < ε ≪ 1 and θ > 0 are constants, f and g are smooth functions of u, K is a synaptic coupling and H is the Heaviside step function. (II) Z Z K(x − y)H(u(y, t) − Θ)dy, ut + f (u) + w = α K(x − y)H(u(y, t) − θ)dy + β R

R

(3) wt = ε(g(u) − γw),

(4)

where α > 0, β > 0, γ > 0, 0 < ε ≪ 1, θ > 0 and Θ > 0 are constants, K is a synaptic coupling and H is the Heaviside step function. (III)     Z 1 ut + f (u) + w = α K(x − y)H u y, t − |x − y| − θ dy, (5) c R wt = ε(g(u) − γw), (6) where α > 0, γ > 0, 0 < ε ≪ 1, θ > 0 and c > 0 are constants, K is a synaptic coupling and H is the Heaviside step function. These equations often arise from mathematical neuroscience, see Atay and Hutt [5, page 645], Atay and Hutt [6, page 673], Bressloff and Folias [13, page 131], Bressloff, Folias, Prat and Li [14, page 178101-1], Coombes, Lord and Owen [20, page 222], Coombes and Owen [21, page 575], Folias and Bressloff [33, page 378], Folias and Bressloff [34, page 2067], Guo and Chow [36, page 218], Hutt and Atay [40, page 35], Kapitula, Kutz and Sandstede [46, page 1097], Laing [49, page 590], Laing and Troy [50, page 488], Laing, Troy, Gutkin and Ermentrout [51, page 62], Pinto and Ermentrout [55, page 207, page 212 and page 213], Pinto, Jackson and Wayne [56, page 955], Troy and Shusterman [71, page 264], Zhang [78, page 513], [79, page 163], [82, page 491], [83, page 285] and [84, page 600].

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

57

The following scalar integral differential equations may be obtained by setting either ε = 0 and w = 0 or ε = 0 and w = w0 (a positive constant) in the above systems: (I-1) Z (7) ut + f (u) = α K(x − y)H(u(y, t) − θ)dy, R

and (I-2)

Z

ut + f (u) + w0 = α

R

K(x − y)H(u(y, t) − θ)dy.

(8)

(II-1) Z

K(x − y)H(u(y, t) − θ)dy Z K(x − y)H(u(y, t) − Θ)dy, +β

ut + f (u) =α

R

(9)

R

and (II-2) ut + f (u) + w0 = α

Z

K(x − y)H(u(y, t) − θ)dy

RZ

+β

R

(III-1)

K(x − y)H(u(y, t) − Θ)dy.

(10)

    1 ut + f (u) = α K(x − y)H u y, t − |x − y| − θ dy, c R

(11)

    1 ut + f (u) + w0 = α K(x − y)H u y, t − |x − y| − θ dy. c R

(12)

Z

and (III-2) Z

The main goal of this chapter is to systematically investigate the existence and stability of traveling wave solutions of systems (1)–(2), (3)–(4) and (5)–(6). As well known, traveling wave solutions include homoclinic orbits and heteroclinic orbits. We will apply speed index functions to study the existence and we will use stability index functions (which are also called Evans functions) to study the stability. We will couple various ideas and techniques from linear algebra, real analysis, complex analysis, functional analysis, ordinary differential equations and partial differential equations to construct the speed index functions and the stability index functions. There are other kinds of integral differential equations as well, such as Z  Z t η(t − τ ) K(x − y)H(u(y, τ ) − θ)dy dτ, u(x, t) = α −∞ R Z     Z t  1 η(t − τ ) K(x − y)H u y, τ − |x − y| − θ dy dτ, u(x, t) = α c −∞ R and ut + u = α

Z

0

∞

ξ(c)

Z

     1 K(x − y)H u y, t − |x − y| − θ dy dc c R

58

Linghai Zhang +β

Z

0

∞

η(τ )

Z

R



W (x − y)H(u(y, t − τ ) − θ)dy dτ.

Moreover, there are similar integral differential equations from neuronal networks and other areas. For example Z 2 a ∈ (−1, 1), ut + u + (u − 1)(u − a) = α K(x − y)u(y, t)dy, R

and ut + δ(u − γ) = β(α − u)

Z

R

K(x − y)H(u(y, t) − θ)dy,

where δ and β represent sodium and synaptic conductances, γ and α are the sodium and synaptic reversal potentials. Here is another example Z ut + u = α K(x − y)H(ε, u(y, t) − θ)dy, R

where H(ε, u) is a smooth function, such as H(ε, u) =

1 u 1 + tanh , 2 ε

where ε > 0 is a constant. Note that, unlike traditional traveling wave problems in nonlinear dispersive wave equations such as the well-known Korteweg-de Vries equation ut + ux + uxxx + uux = 0 (see Benjamin [8] and Bona [9]-[10]) and reaction diffusion equations such as the scalar bistable equation ut = uxx +u(1−u)(u−a) (see Fife and McLeod [31]), these systems have neither conservation law nor maximum principle.

1.1.

Biological Backgrounds

In order to understand the mathematical model equations, let us review some basic neurobiology. Neurons are surrounded by a cell membrane, just like any other cells. They contain cytoplasm, mitochondria and other organelles and they have a nucleus that contain genes. Different from other cells, neurons have specialized extensions called dendrites and axons. Dendrites bring information to the cell body and axons take information away from the cell body. Neurons communicate with each other through an electrochemical process. Another difference is that neurons contain specialized structures (for example synapses) and chemicals (for example, neurotransmitters). The human brain consists of approximately a trillion interconnected neurons. Each neuron is a cell that uses biochemical or biophysical reactions to receive, integrate and conduct information. The dendrite is connected to about ten thousand neurons in its neighbourhood. When one neuron fires, a positive or negative input is received by one of the dendrites. The strengths of all the received inputs are added together through the integration process. Spatial summation occurs when several weak signals are converted into a strong one, while temporal summation converts a series of rapid weak pulses from one source into a large signal. The aggregated input is then conducted

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

59

to the cell body. The cell body and the enclosed nucleus play a less significant role in the processing of incoming and outgoing information. Their main purpose is to maintain the regular cellular functions such as protein synthesis and energy production. Biologically, the part of the cell body that does concern itself with the signal propagation is the axon hillock. The most common site of initiation of an action potential is the axon hillock where the highest concentration of voltage gated ion channels is found. If the aggregated input is greater than the threshold value of the axon hillock, then the neuron fires, and an output signal is transmitted down the axon. The strength of the output information is constant, regardless of whether the input is just above the threshold, or exceeds the threshold to a large scale. However, the output strength may be affected by the many divisions in the axon; it reaches each terminal button with weaker intensity than it has at the hillock. While the real interactions among neurons are very complicated, we can always take a reasonable approximation. If the membrane potential of a neuron is less than some threshold, then it makes no contribution to other neurons. If the membrane potential is larger than the threshold, then it makes contributions to other neurons. That is why the Heaviside step function is introduced to represent the gain function: H(u−θ) = 0 if u < θ, 1 H(0) = and H(u − θ) = 1 if u > θ. 2 Here is a little bit more biology. The membranes of neurons can be viewed as capacitors, rather than conductors. Note that passive channels across the cell membrane are always open and ions can flow through (this is the so-called leaking current). Whether the gates of other channels are open or closed is controlled by membrane potentials. In most real neuronal networks, due to the fast activation of the sodium channels and the slow activation of the potassium channels across the cell membrane, there must be at least two distinct timescales: the fast timescale and the slow timescale. Mathematically, this is reflected through the presence of the singular perturbation parameter 0 < ε ≪ 1 in a model system.

1.2.

Derivation of Model Equations and Nonlocal Terms

The model equations are derived by using Kirchhoff’s current law: The sum of voltage drops along a closed circuit is zero. We also used Ohm’s law: V (x, t) = I(x, t)R(x, t), where R(x, t) represents the resistance of the resistor, measured in ohms (Ω), and I(x, t) denotes the current passing through the resistor at position x and time t. To derive the nonlocal terms, first of all, we consider one-dimensional discrete array of neurons. Each neuron receives strong contributions of membrane potentials from some neurons and weak contributions from other neurons: [α − ωu(x, t)]K(x − yj )H(u(yj , t) − θ)△yj , where K(x − yj ) (representing a synaptic weight function) can be viewed as the probability that there is a synaptic connection between the neuron at yj and the neuron at x; H(u(yj , t) − θ) = 1 if u(yj , t) > θ and H(u(yj , t) − θ) = 0 if u(yj , t) < θ; α − ωu(x, t) denotes synaptic coupling strength; and △yj = yj − yj−1 . Note that synapses are the principal sites where neurons communicate with each other. Most neurons communicate with each other at chemical synapses in which messages are transmitted by the release of chemicals called neurotransmitters from the presynaptic neurons and detected by receptors

60

Linghai Zhang

on the post-synaptic neurons. This coupling is indeed of chemical synapse type. Summing up the contributions of membrane potentials from all other neurons to a particular neuron would yield the Riemann sum [α − ωu(x, t)]

N X j=1

K(x − yj )H(u(yj , t) − θ)△yj ,

where N denotes the total number of neurons (see [69, page 1581]). Secondly, note that there are about a trillion (1012 ) neurons and approximately a zillion 15 (10 ) synapses and a single neuron may receive inputs from about ten thousand (104 ) synapses. Therefore, the number of neurons and their tree-like dendritic structure in even a small piece of cortex is so large that a natural approach to take in modeling layers of cortex is to consider a continuum of neurons. That is to say, a cortical layer of nerve tissue can be regarded as a neuronal field that supports and propagates interacting patterns of excitation, see [12], [20] and [55]. Motivated by this fact, if we consider an ideal situation, namely, in the one-dimensional neural field, the neurons are continuously distributed (so that N → ∞), then we obtain a nonlocal term lim

N →∞

N o n X K(x − yj )H(u(yj , t) − θ)△yj [α − ωu(x, t)] j=1

= [α − ωu(x, t)]

Z

R

K(x − y)H(u(y, t) − θ)dy.

Thus, the equations (1)–(2) are of great interest in nonlocal neuronal networks. For simplicity, we consider the case ω = 0. See also [5], [6], [13]–[14], [19]–[23], [25]–[27] for similar models and more backgrounds of (1)–(2) in synaptically coupled neuronal networks. Modeling persistent patterns of propagating activity using a new characterization of the gain function in which there is a low persistent activity rate and a high saturating rate yields the two integrals in (3)–(4). See Guo and Chow [36, pages 218-219].

1.3.

Motivations to Find Exact Traveling Wave Solutions

Some nonlinear partial differential equations have exact traveling wave solutions. We list several classes of equations and solutions. (I) Nonlinear dispersive wave equations. (I-1) Consider the nonlinear Korteweg-de Vries equation ∂u ∂ 3 u ∂u + 3 + up = 0, ∂t ∂x ∂x where p ≥ 1 is a constant. It has a family of traveling wave solutions U (z) = U (x − c2 t) =

(

  2 )1/p 1 1 (p + 1)(p + 2) c sech cpz , 2 2

where z = x − c2 t and c2 is the wave speed.

Traveling Waves Arising from Synaptically Coupled Neuronal Networks (I-2) Consider the nonlinear Schr¨odinger equation i

∂2u ∂u + α 2 + β|u|2 u = γu, ∂t ∂x

where α > 0, β > 0 and γ > 0 are constants. It has the standing wave solution r r  2γ γ U (α, x) = sech x . β α (I-3) Consider the nonlinear cubic Schr¨odinger equation i

∂2u ∂u + α 2 + β|u|2 u = 0, ∂t ∂x

where α > 0 and β > 0 are constants. It has a family of solitary wave solutions s     2αc2 U (z) = exp i(ax + α(c2 − a2 )t + ω) sech c(x − 2αat) , β where a, c and ω ∈ R are real constants, such that c2 > a2 . (I-4) Consider the nonlinear Boussinesq equations ∂ ∂3w ∂u ∂w + + (uw) + ∂t ∂x ∂x ∂x3 ∂w ∂u ∂w + +w ∂t ∂x ∂x

= 0, = 0.

Let v(x, t) = 1 + u(x, t) and let v(x, t) = φ(x + ct),

w(x, t) = ψ(x + ct),

z = x + ct.

Then (c + ψ)φ + ψ ′′ = 0,   1 φ + c + ψ ψ = 0. 2 We get   1 ψ = ψ(c + ψ) c + ψ . 2 ′′

Let   1 ψ = −ψ c + ψ . 2 ′

Then   1 ψ ′′ = −(c + ψ)ψ ′ = ψ(c + ψ) c + ψ . 2

61

62

Linghai Zhang

Therefore ψ(z) = −

2c , 1 + 2 exp(cz)

and 2c2 exp(cz) φ(z) =  2 . 1 + 2 exp(cz)

It has another family of exact traveling wave solutions

u(x, t) = 2α2 sech2 (αx − 2α2 t) − 1,

w(x, t) = 2α + 2α tanh(αx − 2α2 t), where α is a real constant. (II) Nonlinear dissipative dispersive wave equations. (II-1) Consider the Korteweg-de Vries-Burgers equation ut +

∂3u ∂ 2 ∂2u + (u ) = α , ∂x3 ∂x ∂x2

where α > 0 is a constant. It has the traveling wave solutions    3α3 α 6α2 3α2 1 + tanh − x− t , u(x, t) = 25 50 10 125 3α2 sech2 U (z) = 50



   3α3 3α2 3α3 3α2 α α x− t − tanh x− t + , 10 125 25 10 125 25

3α2 U (z) = sech2 50



   3α3 3α2 3α3 3α2 α α x+ t − tanh x+ t − . 10 125 25 10 125 25

and

(II-2) Consider the modified Korteweg-de Vries-Burgers equation ut −

∂3u ∂ 3 ∂2u + (u ) = α 2 , 3 ∂x ∂x ∂x

where α > 0 is a constant. It has the traveling wave solutions √ α  √2 2 α tanh (x − ct) ∓ α. U (z) = ± 6 6 6 (II-3) Consider the generalized Korteweg-de Vries-Burgers equation ut −

∂3u ∂2u ∂ 2 3 + (u + u ) = α , ∂x3 ∂x ∂x2

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

63

where α > 0 is a constant. It has the traveling wave solutions √ α  √2 2 1 U (z) = ± α tanh (x − ct) ∓ α− . 6 6 6 3 (III) Nonlinear convection equations. (III-1) Consider the Burgers equation ut = αuxx + β(u2 )x , where α > 0 and β > 0 are constants. It has a family of traveling wave solutions cα  , U (z) = β + exp − c(x + cαt) where z = x + cαt. (III-2) Consider the n-dimensional Burgers equation ∂u = △u + 2(u · ∇)u, ∂t

in Rn × R+ .

It has a family of traveling wave solutions
u(x, t) = a + a tanh a · x + 2|a|2 t ,

where a = (a1 , a2 , · · · , an )T ∈ Rn is any constant vector. (IV) Nonlinear reaction diffusion equations. (IV-1) Consider the reaction-diffusion equation ∂u ∂2u = + u(1 − u)(u − a), ∂t ∂x2

1 0
It has a unique traveling wave front U (z) =

1 √ , 1 + exp(−z/ 2)

z = x + ν0 t,

ν0 =

1 − 2a √ . 2

(IV-2) Consider the generalized Fisher’s equation ∂u ∂2u = + u(1 − up )(up − a), ∂t ∂x2 where p > 0 is a constant, a is a real number. Let z = x + ct and u(x, t) = φ(x + ct). Let φ′ = Dφ(1 − φp ). We find D= √

1 , p+1

c=

1 − (p + 1)a √ . p+1

It has a family of explicit traveling wave solutions   1/p   p 1 − (p + 1)a     t x+ √  exp √  p+1 p+1    u(x, t) = .   1 − (p + 1)a p   1 + exp √  t  x+ √ p+1 p+1

64

Linghai Zhang

(IV-3) Consider the system of the Belousov-Zhabotinskii reaction-diffusion equations ∂u ∂t ∂v ∂t where a=

∂2u + u(1 − u − av), ∂x2 ∂2v − buv, ∂x2

= D =

−1 + 5D > 0, 1+D

b=

6 > 0, 1+D

1 D> , 5

 2  2 are constants. Let z = x + ct, u(x, t) = φ(z) and v(x, t) = 1 − φ(z) . Suppose that φ′ = cφ(1 − φ).

Then φ′′ = c2 φ(1 − φ)(1 − 2φ). It has one family of exact traveling wave solutions 1  2 ,   1 1 x+ √ t 1 + exp − √ 1+D 1+D    2  1 1     x+ √ t  exp − √  1 + D 1 + D    v(x, t) = .   1 1   1 + exp − √  t  x+ √ 1+D 1+D

u(x, t) =

(V) Nonlinear hyperbolic equations. (V-1) Consider the hyperbolic partial differential equation ∂2u ∂2u − 2 + a2 u2p+1 − b2 u = 0, ∂t2 ∂x where a > 0, b > 0 and p ≥ 1 are constants. It has a family of a traveling wave solutions φ(z) =



b2 (p + 1) sech2 a2



bp √ z c2 − 1

1/2p

where |c| > 1 is a constant. (V-2) Consider the Sine-Gordon equation ∂2u ∂2u − 2 + sin u = 0. ∂t2 ∂x It has the exact traveling wave solution −1

u(x, t) = 2 cos where a is a constant, |a| < 1.



tanh



x − at √ 1 − a2



,

,

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

65

(V-3) Consider the Landau-Lifschitz system ∂Z ∂2Z =Z× , ∂t ∂x2

in R × R+ ,

where Z = (Z1 , Z2 , Z3 )T and |Z(x, t)| = 1, for all (x, t) ∈ Rn × R+ . It has a family of traveling wave solutions   Z(x, t) = Z1 cos α + Z2 cos(cx − (c2 cos α)t) + Z3 sin(cx − (c2 cos α)t) sin α,

where Z1 , Z2 , Z3 ∈ R3 are constant vectors, such that Z1 × Z2 = Z3 and Z3 × Z1 = Z2 , and |Z1 | = |Z2 | = |Z3 | = 1, α and c are real constants. When trying to solve these differential equations for the exact traveling wave solutions, one would often use the following derivatives. (I) (sech x)′ = −sech x tanh x,

(tanh x)′ = (sech x)2 . (II)

(sech x)′′ = sech x(tanh x)2 − (sech x)3 ,

(tanh x)′′ = −2(sech x)2 tanh x. (III)

 ′ (sech x)2 = −2(sech x)2 tanh x,   ′ (tanh x)2 = 2 tanh x(sech x)2 . (IV)  ′′ (sech x)2 = −2(sech x)4 + 4(sech x)2 (tanh x)2 ,   ′′ (tanh x)2 = 2(sech x)4 − 4(tanh x)2 (sech x)2 . Motivated by the exact traveling wave solutions of these equations, we investigate explicit traveling wave solutions of these integral differential equations. Nevertheless, the nonlinear neuronal network model equations have neither conservation laws nor maximum principle.

1.4.

Assumptions

Suppose that the synaptic coupling K is at least piecewise continuous, satisfying the following conditions Z ∞ Z 0 Z 1 K(x)dx = , K(x)dx = K(x)dx = 1, (13) 2 0 −∞ R Z 0 Z |x|K(x)dx ≥ 0, (14) |K(x)|dx < ∞, R

−∞

66

Linghai Zhang |K(x)| ≤ C exp(−ρ|x|) on

R,

(15)

for some constants C > 0 and ρ > 0. We are concerned with the following three classes of synaptic couplings. (A) Pure excitations between neurons (representedrby nonnegative kernel functions). For
ρ ρ exp − ρ|x|2 may represent pure examples, K(x) = exp(−ρ|x|) and K(x) = 2 π excitations, where ρ > 0 is a constant. Here, ρ has a biological meaning. It indicates how the excitation of a synaptic coupling is distributed. Roughly speaking, if ρ is large, then a neuron is strongly coupled with neurons in a relatively small region and if ρ is small, then a neuron is strongly coupled with all neurons in a relatively large region. (B) Lateral inhibitions (represented by Mexican hat kernel functions, that is, each coupling satisfies K ≥ 0 on (−M, M ) and K ≤ 0 on (−∞, −M ) ∪ (M, ∞) for a positive constant M . This implies that neurons close to one another have excitatory connections and neurons far away have inhibitory connections.) For example, K(x) = A exp(−a|x|2 ) − B exp(−b|x|2 ) may represent a lateral inhibition, where A > B > 0 and a > b > 0 are positive constants, such that r r r A π π 1 B A ≥ , A −B = 1, M= ln . a b a b a−b B (C) Lateral excitations (represented by upside down Mexican hat kernel functions, that is, each coupling satisfies K ≤ 0 on (−M, M ) and K ≥ 0 on (−∞, −M )∪(M, ∞), for some positive constant M ). For example, K(x) = A exp(−a|x|) − B exp(−b|x|) may represent a lateral excitation, where 0 < A < B and 0 < a < b are positive constants, such that 1 A B − = , a b 2

A B ≥ 2, 2 a b

M=

1 B ln . b−a A

See Amari [2], Bressloff and Folias [13], Bressloff, Folias, Prat and Li [14], Coombes, Lord and Owen [20], Coombes and Owen [21], Ermentrout [25], Ermentrout and Terman [27], Laing [49], Laing and Troy [50], Pinto and Ermentrout [55], Terman, Ermentrout and Yew [69], and [77]–[85] for more information on various synaptic couplings represented by the synaptic couplings. Five Mexican hat kernel functions or upside down Mexican hat kernel functions, their Fourier transformations, together with some associated integrals, are given below (I)

(II)

K(x) = A exp(−a|x|) − B exp(−b|x|), Z Z 0 2A 2B A B K(x)dx = |x|K(x)dx = 2 − 2 , − and a b a b R −∞ 2aA 2bB b K(ξ) = 2 − . a + ξ 2 b2 + ξ 2 K(x) = A exp(−ax2 ) − B exp(−bx2 ), r r Z 0 Z A B π π |x|K(x)dx = −B and − , K(x)dx = A a b 2a 2b −∞ R r r     π π 1 1 b =A K(ξ) exp − ξ 2 − B exp − ξ 2 . a 4a b 4b

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

67

1.5

1

0.5

y

0

−0.5

−1

−1.5

−2

−1.5

−1

−0.5

0 x

0.5

1

1.5

2

Figure 1. Graphs of three kernel functions (A) Pure excitation (dotted curve). (B) Lateral inhibition (solid curve). (C) Lateral excitation (dash-dotted curve).

(III)

(IV)

(V)

K(x) = A|x| exp(−a|x|) − B|x| exp(−b|x|), Z Z 0 2A 2B 2A 2B K(x)dx = 2 − 2 and |x|K(x)dx = 3 − 3 , a b a b R −∞ 2 2 2 2 b −ξ a −ξ b − 2B 2 . K(ξ) = 2A 2 2 2 (a + ξ ) (b + ξ 2 )2 K(x) = Ax2 exp(−ax2 ) − Bx2 exp(−bx2 ), r r Z Z 0 A π A B π B K(x)dx = |x|K(x)dx = 2 − 2 , − and 2a a 2b b 2a 2b R −∞ r r     2 2 2a − ξ 2b − ξ 1 1 π π 2 2 b exp − ξ − B exp − ξ . =A K(ξ) a 4a2 4a b 4b2 4b

K(x) = A|x|4 exp(−ax2 ) − B|x|4 exp(−bx2 ), r r Z Z 0 3A π A 3B π B K(x)dx = 2 |x|K(x)dx = 3 − 3 , − 2 and 4a a 4b b a b R −∞ r   4 2 2 ξ − 12aξ + 12a π 1 2 b =A K(ξ) exp − ξ 16a4 a 4a

68

Linghai Zhang

1.2

1

0.8

y

0.6

0.4

0.2

0

−0.2 −3

−2

−1

0 x

1

2

3

Figure 2. Comparison of an on-center pure excitation and an off-center pure excitation. ξ 4 − 12bξ 2 + 12b2 −B 16b4

r

  π 1 2 exp − ξ . b 4b

Here, if the constants satisfy A > B > 0 and a > b > 0, then K represents a Mexican hat kernel function. If the constants satisfy 0 < A < B and 0 < a < b, then K represents an upside down Mexican hat kernel function. In each example, the function K satisfies Z

K(x)dx = 1 and R

Z

0 −∞

|x|K(x)dx ≥ 0.

These conditions imply that excitation exceeds inhibition in the neuronal networks. We also consider more complicated synaptic couplings where intervals of excitation are separated by intervals of inhibition (modeled by oscillatory synaptic couplings). For examples, K(x) = exp(−|x|) cos x and K(x) = exp(−|x|) sin(|x|). Note that the Fourier transformations of these synaptic couplings are, respectively 4 + 2ξ 2 b K(ξ) = , 4 + ξ4

4 − 2ξ 2 b K(ξ) = . 4 + ξ4

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

1.5.

69

Definition and Properties of Traveling Wave Solutions

Traveling wave solutions are non-constant, bounded, at least piecewise smooth functions, and they usually take the following form:

u(x, t), w(x, t) ≡ U (ε, x + ν(ε)t), W (ε, x + ν(ε)t) ,

(16)

where ν(ε) represents the wave speed and z = x + ν(ε)t stands for the moving coordinate. If (U (ε, z), W (ε, z)) is a traveling wave solution, then (U (ε, z + h), W (ε, z + h)) is also a traveling wave solution, for any real number h. This property is called the translation invariance. For our purposes, we will make a priori assumptions on the form of the traveling waves. Indeed, the knowledge of the excited intervals (where U > θ and/or U > Θ) of the traveling waves allows their exact computations, as the gain function depends only on the excited intervals. We restrict our analysis and results on the existence and uniqueness to the most common kinds of traveling waves: homoclinic orbits (also called traveling pulses or solitary pulses) where excitation exceed the thresholds on compact intervals, and heteroclinic orbits where excitation exceed the thresholds on unbounded open intervals. The waves also satisfy certain boundary conditions at plus/minus infinity, More precisely, the homoclinic orbits satisfy the homogeneous boundary conditions lim

z→±∞


b, W c ), U (ε, z), W (ε, z) = (U

lim

z→±∞


Uz (ε, z), Wz (ε, z) = (0, 0),

(17)

b, W c ) ∈ R2 is a fixed real vector, and the heteroclinic orbits satisfy the nonhomowhere (U geneous boundary conditions
U (ε, z), W (ε, z) = (U ∗ , W ∗ ), z→−∞
lim U (ε, z), W (ε, z) = (U # , W # ), z→∞
lim Uz (ε, z), Wz (ε, z) = (0, 0), lim

z→±∞

(18) (19) (20)

where the constant vectors satisfy (U ∗ , W ∗ ) 6= (U # , W # ). There may hold the decay estimate |U (ε, z)| + |W (ε, z)| ≤ C exp(−ρ|z|),

on

R,

for some of the traveling waves, where C > 0 and ρ > 0 are constants. When ν = 0, the traveling waves become standing waves. In general, it is much easier to study standing waves than traveling waves. For example, the mathematical analysis of the existence and stability of standing waves is simpler. There are other kinds of nonlinear waves in neuronal networks, such as lurching waves, see Coombes [19] and Terman, Ermentrout and Yew [69]. There also exists other sophisticated propagating activity patterns in large-scale inhibitory neuronal networks, see John Rinzel, David Terman, Xiao-Jing Wang and Bard Ermentrout [62]. Spiral waves have also been found by using biological experiments and numerical simulations, see [38] and [49].

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1.6.

Existence of Traveling Wave Solutions

When studying the existence and uniqueness of traveling waves of reaction diffusion equations, many mathematicians use the traditional geometric singular perturbation theory. They establish the transversal intersection of unstable manifold of a fixed point of the traveling wave equations and the stable manifold of another fixed point of the same traveling wave equations to prove the existence and uniqueness of the traveling wave. Many important results on existence, uniqueness and stability of traveling waves of reaction diffusion equations, nonlinear dispersive wave equations and integral differential equations have been obtained. We will use various ideas and techniques in functional analysis and ordinary differential equations. In particular, we will use fixed point theorem, implicit function theorem and the method of variation of parameters to prove the existence and uniqueness of the traveling wave of each system. The main results of each system or each scalar equation will be given in each section. Of course there are still many important open problems to be solved.

1.7.

Stability of Traveling Wave Solutions

There are many stability definitions in different senses, such as the L2 stability (for nonlinear dispersive wave equations [8]–[10]), the L∞ stability (for reaction diffusion equations and integral differential equations, where the essential spectrum of a linear differential operator lies to the left of the imaginary axis [1], [28], [31], [35], [44]), the weighted L2 or L∞ stability (for reaction diffusion equations, where the essential spectrum of a linear differential operator crosses the origin of the complex plane [45]). In this book chapter, we will focus on L∞ stability, without any weight, see [77]–[85]. Definition 1. The Banach space Y consists of all complex, vector-valued, bounded, uniformly continuous functions defined on R, and the Banach space X = {ψ ∈ Y : ψ ′ ∈ Y}. We define a linear differential operator L(ε) : X → Y by   f ′ (U ) 1 ψ L(ε)ψ = −ν(ε)ψ ′ − −εg ′ (U ) γε     α α K(z) 0 K(z − Z(ε)) 0 + ψ(0) − ψ(Z(ε)). 0 0 0 0 Uz (ε, 0) Uz (ε, Z(ε)) Here (U (ε, ·), W (ε, ·)) denotes the fast traveling pulse solution of system (1)–(2), ν(ε) represents the wave speed and z = x + ν(ε)t denotes the moving coordinate. Note that the operator L(ε) is not self-adjoint. Definition 2. If there exists a complex number λ0 and a complex vector-valued function ψ0 ∈ X, such that L(ε)ψ0 = λ0 ψ0 and limz→±∞ ψ0 (z) = 0 exponentially fast, then λ0 is an eigenvalue of L(ε) and ψ0 is a corresponding eigenfunction of L(ε) associated with the eigenvalue λ0 . Definition 3. If there exist three positive constants C, M and ρ, such that whenever the initial data (P0 , Q0 ) satisfy k(P0 , Q0 ) − (U, W )kL∞ (R) ≤ C, there is a nonzero constant

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h, with |h| ≤ M k(P0 , Q0 ) − (U, W )kL∞ (R) , and the global solution of the initial value problem Z Pt + ν(ε)Pz + f (P ) + Q = α K(z − y)H(P (y, t) − θ)dy, R

Qt + ν(ε)Qz = ε(g(P ) − γQ), P (z, 0) = P0 (z),

Q(z, 0) = Q0 (z).

satisfies the decay estimate k(P (·, t), Q(·, t)) − (U (· + h), W (· + h))kL∞ (R) ≤ M exp(−ρt)k(P0 , Q0 ) − (U, W )kL∞ (R) , for all t > 0, then the traveling wave solution (U, W ) is nonlinearly stable (also called exponentially stable). Definition 4. If there exist two positive constants M and ρ, such that for any initial data (p0 , q0 ), there exists a nonzero constant h with |h| ≤ M k(p0 , q0 )kL∞ (R) , and the global solution of the initial value problem αK(z) αK(z − Z(ε)) p(0, t) − p(Z(ε), t), Uz (ε, 0) Uz (ε, Z(ε)) qt + ν(ε)qz = ε(g ′ (U )p − γq),

pt + ν(ε)pz + f ′ (U )p + q =

p(z, 0) = p0 (z),

q(z, 0) = q0 (z).

satisfies the decay estimate k(p(·, t), q(·, t)) − h(Uz , Wz )kL∞ (R) ≤ M exp(−ρt)k(p0 , q0 )kL∞ (R) , for all t > 0, then the traveling wave solution is linearly stable. Definition 5. If there exists a positive constant c0 > 0, such that max{Re λ : λ ∈ σ(L(ε)), λ 6= 0} ≤ −c0 , and λ = 0 is a simple eigenvalue of L(ε), then the traveling wave solution is spectrally stable, where σ(L(ε)) represents the spectrum of L(ε). If there is an eigenvalue with positive real part or if the neutral eigenvalue λ = 0 is not simple, then the traveling wave solution is unstable. Theorem 0. The nonlinear stability and the linear stability of a traveling wave solution of systems (1)–(2), (3)–(4) and (5)–(6) are equivalent. The linear stability and the spectral stability of a traveling wave solution of these systems are also equivalent. For nonlinear reaction diffusion equations and nonlinear dispersive wave equations, it is not very difficult to establish the equivalence of the nonlinear stability, linear stability and spectral stability. The essential spectrum of linear differential operators arising from those equations are easy to calculate, see [1] and [53].

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1.8.

Linghai Zhang

The Evans Functions

Motivation of the Evans functions. Consider the n × n constant matrix A. It may be viewed as a bounded linear operator from the finite-dimensional Banach space Cn to Cn . The Evans function for this operator A is defined by E(λ) = det(λI − A) for all

λ ∈ C,

where I is the identity matrix and λ is the eigenvalue parameter. Properties of the Evans function. (1) The complex number λ0 is an eigenvalue of the operator A if and only if E(λ0 ) = 0. (2) The algebraic multiplicity of any eigenvalue λ0 of the operator A is equal to the order of λ0 as a zero of the Evans function. History on using Evans functions to study the stability of traveling waves. In 1975, John W. Evans [28] published a paper on the nerve axon equations ∂2u ∂u = M 2 + f (u), ∂t ∂x   1 0 ··· 0  0 0 ··· 0   where M =   · · · · · · · · · · · ·  is a real square matrix, and u = (u0 , u1 , · · · , un ) ∈ 0 0 ··· 0 Rn+1 is a real vector-valued function. The nerve axon equations contain the well known Hodgkin-Huxley equations and the Fitzhugh-Nagumo equations, to which there is no maximum principle to apply. John W. Evans constructed the very first Evans function. In 1984, Christopher K. R. T. Jones [44] applied the Evans function to establish the stability of the fast traveling pulse solution of the Fitzhugh-Nagumo equations ut = uxx + u(1 − u)(u − a) − w,

wt = ε(u − γw),

1 4 ,0 < γ < and 0 < ε ≪ 1 are constants. Jones is the first 2 (1 − a)2 mathematician to apply such a complex analytic function to accomplish the exponential stability of a traveling pulse solution of a nonlinear singularly perturbed system of partial differential equations. In 1990, Alexander, Gardner and Jones [1] published a paper on the nonlinear system of reaction-diffusion equations

where 0 < a <

∂u ∂2u = D 2 + f (u), ∂t ∂x   d1 0 ··· 0  0 d2 · · · 0  n  where D =   · · · · · · · · · · · · , all elements of D are positive constants, u ∈ R . 0 0 · · · dn They constructed and applied the Evans function.

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In 1992, Robert L. Pego and Michael I. Weinstein [53] published a paper on nonlinear dispersive wave equations, including the nonlinear Korteweg-de Vries equations ∂u ∂u ∂ 3 u ∂u + + = 0, + up ∂t ∂x ∂x3 ∂x where p = 1, 2, 3, 4, and the nonlinear Schr¨odinger equation i

∂u ∂2u + α 2 + β|u|2 u = 0. ∂t ∂x

Pego and Weinstein constructed and applied the Evans function to study the stability/instability of solitary waves of these equations in [53]–[54]. In this chapter, the author will construct complex analytic functions, usually called the Evans functions, also called the stability index functions, and study their properties, and then apply them to accomplish the stability of traveling waves. Evans functions have been constructed and proved to be a very powerful tool in stability analysis of traveling waves of reaction-diffusion equations ([28], [35] and [64]), nonlinear dispersive wave equations ([53]–[54]) and integral differential equations ([21], [46], [56], [66], [77]–[85]).

1.9.

Relationship between Integral Differential Equations and Partial Differential Equations

The systems of integral differential equations and the scalar integral differential equations contain many particular partial differential equations if one chooses the synaptic coupling appropriately. For concreteness, let us use equation (7) to illustrate the problem. If N  1 X δ(x + dk ) + δ(x − dk ) , K(x) = 2N k=1

where N ≥ 1 is an integer, 0 < d1 < d2 < · · · < dN < ∞, i.e. K is the linear combination of 2N Dirac delta impulse functions, then (7) becomes an ordinary differential equation N  α X ut + f (u) = H(u(x + dk , t) − θ) + H(u(x − dk , t) − θ) . 2N k=1

Define the Fourier transformation and the inverse Fourier transformation of the synaptic coupling K by Z Z 1 ˇ b K(ξ) exp(ixξ)dξ. K(x) exp(−ixξ)dx, K(x) = K(ξ) = 2π R R

Performing the Fourier transformation R to the differential equation (7), and noting that for the convolution product [f ∗ g](x) = R f (x − y)g(y)dy, the Fourier transformation enjoys the property f[ ∗ g(ξ) = fb(ξ)b g (ξ), we obtain \ b H(u u bt (ξ, t) + fd (u)(ξ, t) = αK(ξ) − θ)(ξ, t).

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Linghai Zhang

1 , where ω(ξ) is an even, polynomial or exponential function of ξ, then ω(ξ) we may multiply the above equation by ω(ξ) and apply the inverse Fourier transformation to obtain the equivalent partial differential equation. By equivalence, we mean that u is a solution of the integral differential equation (7), if and only if u is a solution of the partial differential equation. We offer a few examples below. ρ (I) Setting K(x) = exp(−ρ|x|), where ρ > 0 is a constant, then the Fourier transfor2 2 ρ b mation K(ξ) = 2 . Now (7) becomes a partial differential equation ρ + ξ2 1 1 ut − 2 uxxt + f (u) − 2 f (u)xx = αH(u − θ). ρ ρ   r
1 2 ρ 2 b exp −ρ|x| , then K(ξ) = exp − ξ and (7) is equiv(II) Letting K(x) = π 4ρ alent to the interesting equation ∞ X  (−1)m ∂ 2m  ut + f (u) = αH(u − θ). m 2m m!(4ρ) ∂x

b If K(ξ) =

m=0

1 sin(ρξ) b for |x| ≤ ρ and K(x) = 0 for |x| > ρ, then K(ξ) = 2ρ ρξ and (7) reduces to the differential-difference equation  α H(u(x + ρ, t) − θ) − H(u(x − ρ, t) − θ) . uxt + f (u)x = 2 The kernel functions in (I), (II) and (III) imply that the synaptic couplings between neurons in the neuronal networks are excitatory. Although it is not known what kind of kernels best describe the real synaptic couplings in human neuronal networks, we can pick up these kernels from mathematical point of view. Note that the classical reaction diffusion equation, see [30] 1 ut = uxx + u(1 − u)(u − a), 0
ut = uxx + u(1 − u)(u − a) − w,

wt = ε(u − γw), 4 1 and 0 < ε ≪ 1 are constants, are more or less where 0 < a < , 0 < γ < 2 (1 − a)2 similar to ut − uxxt + f (u) − f (u)xx = αH(u − θ), and ut − uxxt + f (u) − f (u)xx + w − wxx = αH(u − θ), wt = ε(u − γw),

respectively. These partial differential equations are equivalent to integral differential equations. Therefore, integral differential equations are much more general than reaction diffusion equations.

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

2.

75

A Nonlinear Singularly Perturbed System of Integral Differential Equations

In this section, we investigate the nonlinear singularly perturbed system of integral differential equations arising from synaptically coupled neuronal networks Z
(21) ut + f (u) + w =α K(x − y)H u(y, t) − θ dy, R

wt =εg(u) − γεw,

in R × R+ .

(22)

We will be concerned with the existence and stability of traveling wave solutions of the system. In this model, α stands for the synaptic rate constant, γ denotes the decay rate of the recovery variable, and θ represents the threshold for excitation of the neuronal network. In equations (21)–(22), u(x, t) stands for the membrane potential of a neuron at position x and time t, and w represents a leaking current, a local slow recovery variable that brings each neuron back to rest after a period of electroactivity. The integral stands for nonlocal interactions between neurons. This system has been proposed to model the propagation of nerve impulses in neuroscience. See [12]–[14], [25], [26], [33], [34], [46], [50], [55], [56], [66], [70], [71] and [77]–[85] for the biological backgrounds of the model system. In particular, one can consult the paper by Pinto and Ermentrout [55], the work by Pinto, Jackson and Wayne [56], and the paper by Terman, Ermentrout and Yew [69] for detailed background on the functions f , g and K, as well as the backgrounds on the nonlocal term. For more connections of the model equations to neuroscience, see [18], [24], [38], [52], [58], [59], [68], [72]–[74]. The functions f and g are continuously differentiable, f (0) = g(0) = 0, f ′ (0) > 0 and g ′ (·) > 0 on R. We assume that f ′ (0) = g ′ (0) = 1. The functions w = −f (u) and 1 w = g(u) intersect only once, at (u, w) = (0, 0). It is simple to see that (u, w) = (0, 0) γ is a stable constant solution of system (21)–(22). This model system includes several classes of functions.
(I) The first class consists of linear functions such as f (u), g(u) = (u, u). In this case, the system (21)–(22) reduces to Z ut + u + w =α K(x − y)H(u(y, t) − θ)dy, R

wt =ε(u − γw), see Bressloff [12], Bressloff and Folias [13], Bressloff, Folias, Prat and Li [14], Pinto and Ermentrout [55], Pinto, Jackson and Wayne [56], Sandstede [66], and the author [78], [82]. This class is a linear approximation of the real model functions. For this class only, the model is mathematically analytically tractable. (II) The
second class consists of nonlinear increasing functions and a typical example is f (u), g(u) = (sinh u, tanh u), see Terman, Ermentrout and Yew [69]. This is a nonlinear approximation of the real model functions. (III) The third class involves
cubic polynomial function and hyperbolic trigonometric functions, such as f (u), g(u) = (u(u − 1)(Au − 1), tanh u), where A > 1 is a constant,

76

Linghai Zhang

see Kapitula, Kutz and Sandstede [46] and Zhang [79]. This cubic function is basically the same as the nonlinear function in the Fitzhugh-Nagumo equations, see [27], [44] and [61]. Suppose that α is appropriately large, such that f (u) = α has a unique solution, say at some number β, such that β > θ and f ′ (β) > 0. Moreover 2f (θ) < α ≤ f (θ) +

1 g(θ). γ

When f (u) = u(u − 1)(Au − 1), we assume that   p p 1  1  A + 1 − A2 − A + 1 < θ < A + 1 + A2 − A + 1 , 3A 3A    √ √ 1 1  where A + 1 − A2 − A + 1 and A + 1 + A2 − A + 1 are the knees of 3A 3A the cubic function at which it attains a local maximum and a local minimum, respectively.

2.5

2

1.5

1

0.5

0

−0.5

0

0.5

1

1.5

2

Figure 3. Graphs of four sodium current functions: f+ (u) = sinh u (dotted curve), f0 (u) = u (solid curve), f− (u) = tanh u (dash-dotted curve) and fc (u) = u(u−1)(3u−1) (dashed curve). Note that if the parameter ε = 0, then the w variable reduces to a constant. Without loss of generality, let w = 0. Thus, we also study the scalar integral differential equation Z (23) ut + f (u) = α K(x − y)H(u(y, t) − θ)dy. R

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77

The difficult point of the problems (21)–(22) and (23) is that the derivative of the Heaviside step function is equal to the Dirac delta impulse function. To keep the analysis of this section clear, we will investigate the scalar integral differential equation (23) and the system (21)–(22) (where 0 < ε ≪ 1) separately. We provide detailed analysis for the case ε = 0 and the case 0 < ε ≪ 1. We will explain in great detail how to construct the Evasn functions and use the properties of the Evans functions to explore the eigenvalues of an associated linear differential operator L(ε) and to investigate the exponential stability of traveling waves. When studying the existence and uniqueness of each of the traveling wave solutions, we have to assume that the singular perturbation parameter ε is sufficiently small, see [12], [14], [33], [34], [39]–[42], [46], [55], [56], [66] and [77]–[85]. The existence of the traveling waves remains open for large ε. Although the formal derivation of the Evans function does not necessarily depend on the smallness of the parameter ε, we must let it be very small when proving the properties because of certain technical requirements. For example, we 1 will use the fact that Z(ε) = O(1) as ε → 0, where Z(ε) is a point at which the fast ε traveling pulse solution crosses the threshold θ and (0, Z(ε)) is the
excited interval, that is, U (ε, ·) < θ on (−∞, 0) ∪ (Z(ε), ∞) and U (ε, ·) > θ on 0, Z(ε) .

2.1.

The Mathematical Analysis of the Traveling Wave Front

Theorem 1 (Existence and uniqueness). Suppose that all of the assumptions made earlier are satisfied. Then there exists a unique traveling wave front U = Ufront (·) to equation (23), such that U (0) = θ, U ′ (0) > 0, U < θ on (−∞, 0) and U > θ on (0, ∞). The front satisfies the equation Z (24) ν0 Uz + f (U ) = α K(z − y)H(U (y) − θ)dy, R

and the decay estimates |U (z)| ≤ C exp(−ρ|z|),

|U (z) − β| ≤ C exp(−ρ|z|), ′

|U (z)| ≤ C exp(−ρ|z|),

on

(−∞, 0),

on

(0, ∞),

on

R,

where ν0 = νfront > 0 is the unique positive wave speed and z = x + ν0 t, β > θ is the unique solution of the equation f (U ) = α such that f ′ (β) > 0, and C > 0 and ρ > 0 are positive constants. There hold the following speed estimates. (I) Let f (u) = u, then   Z 0 x α exp α K(x)dx = − θ. ν0 2 −∞ f (θ) u on (0, θ). Then there hold the estimates θ   Z 0 α x K(x)dx ≤ − θ, exp α ν 2 0 −∞

(II) Let f (u) satisfy u ≤ f (u) ≤

78

Linghai Zhang   Z 0 α f (θ) x K(x)dx. exp − f (θ) ≤ α 2 θ ν0 −∞ f (θ) u ≤ f (u) ≤ u on (0, θ). Then there hold the estimates θ   Z 0 α f (θ) x K(x)dx ≤ − f (θ), α exp θ ν0 2 −∞   Z 0 α x K(x)dx. exp −θ ≤α 2 ν0 −∞

(III) Let f (u) satisfy

(IV) Let f (u) = u(u − 1)(Au − 1), where A > 1 is a constant, satisfy u + f (θ) − θ ≤ f (u) ≤ u on (0, θ). Then there hold the estimates α −θ ≤α 2

Z

0

exp −∞



x ν0



K(x)dx ≤

α − f (θ). 2

See [26], [78], [79] and [85] for the proofs of these results. There also exists a unique standing wave to the scalar integral differential equation (23). Due to the presence of a positive eigenvalue of an associated linear operator, the standing wave is unstable. Theorem 2 (Stability). The traveling wave front U = Ufront (·) is stable relative to the following initial value problem for the integral differential equation Z Pt + ν0 Pz + f (P ) =α K(z − y)H(P (y, t) − θ)dy, R

P (z, 0) =P0 (z), that is, if kP0 − U kL∞ (R) ≤ C, for some constant C > 0, then kP (·, t) − U (· + h)kL∞ (R) ≤ M exp(−ρt)kP0 − U kL∞ (R) , for all t > 0, where h is a finite real number, such that |h| ≤ M kP0 − U kL∞ (R) . Proof. To investigate the stability of the front, we have to derive the eigenvalue problem and construct the Evans function. Let us make a change of variable. Set z = x + ν0 t and P (z, t) = u(x, t). Then (23) becomes Z Pt + ν0 Pz + f (P ) = α K(z − y)H(P (y, t) − θ)dy. R

The linearization of this equation with respect to the front is pt + ν0 pz + f ′ (U )p =

α K(z)p(0, t). U ′ (0)

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2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 −1

0

1

2

3

4

Figure 4. Graphs of three traveling wave fronts of equation (7). (A) A front (solid curve) of equation (7), where K represents a pure excitation. (B) A front (dotted curve) of equation (7), where K represents a lateral inhibition. (C) A front (dashed curve) of equation (7), where K represents a lateral excitation. If the synaptic coupling is in class (A), then the front U is nonnegative and nondecreasing, see [82]-[84]. If the synaptic coupling is in class (C), then the front U is always nonnegative, see [84]. If the synaptic coupling is in class (B), then the front U is negative in some unbounded open interval (−∞ − N ) and the front is positive in the interval (−N, ∞), where N > 0 is a constant, depending on K, see [82]-[84]. The shapes of the traveling wave fronts are based on our previous rigorous mathematical analysis of the integral differential equations in [82], [83] and [84]. Suppose that p(z, t) = exp(λt)ψ(λ, z) is a solution of this equation. Then we obtain the eigenvalue problem ν0 ψ ′ + [λ + f ′ (U )]ψ =

α U ′ (0)

K(z)ψ(λ, 0).

Obviously ψ(λ, z) ≡ 0 if ψ(λ, 0) = 0, for all λ ∈ C. Thus, without loss of generality, let ψ(λ, 0) 6= 0. Let us define the linear differential operator L by Lψ = −ν0 ψ ′ − f ′ (U )ψ +

α K(z)ψ(0). U ′ (0)

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With the definition of the operator, the eigenvalue problem can be written as Lψ = λψ. The essential spectrum of L consists of two vertical lines situated in the left half plane σessential (L) = {λ ∈ C : λ = −f ′ (0) + iν0 ξ

or

λ = −f ′ (β) + iν0 ξ, ξ ∈ R}.

To study the simplicity of the neutral eigenvalue λ = 0 and to investigate where the inverse operator (λI − L)−1 exists and is bounded, we study a more general eigenvalue problem Lψ = λψ + ϕ, where the function ϕ ∈ C 0 (R) ∩ L∞ (R), with exponential decay at plus/minus infinity. This problem is merely a first order, linear, ordinary differential equation and it is not very difficult to solve. We will use two simple steps. First step: It is easy to check that     Z ∞  ′  λ + f ′ (β) 1 ′ ψ1 (λ, z) = exp − f (U (x)) − f (β) dx , z exp ν0 ν0 z     Z z  ′  λ + f ′ (0) 1 ′ ψ2 (λ, z) = exp + f (U (x)) − f (0) dx , z exp ν0 ν0 −∞ are solutions of the homogeneous differential equations ν0 ψ ′ + [λ + f ′ (U )]ψ =0, ν0 ψ ′ − [λ + f ′ (U )]ψ =0, respectively, for all λ ∈ C. Note that ψ1 (λ, z)ψ2 (λ, z) = ψ1 (λ, 0)ψ2 (λ, 0) = ψ1 (0, 0)ψ2 (0, 0), for all z ∈ R and all λ ∈ C. It is not difficult to establish the following asymptotic behaviors   λ + f ′ (0) z O(1), as z → −∞, ψ1 (λ, z) = exp − ν0   λ + f ′ (β) ψ1 (λ, z) = exp − z O(1), as z → ∞, ν0   λ + f ′ (0) ψ2 (λ, z) = exp + z O(1), as z → −∞, ν0   λ + f ′ (β) z O(1), as z → ∞. ψ2 (λ, z) = exp + ν0 The general solution of the eigenvalue problem Lψ = λψ + ϕ is given by ψ(λ, z) = C(λ)ψ1 (λ, z) Z z  αψ(λ, 0) + K(x)ψ2 (λ, x)dx ψ1 (λ, z) ν0 U ′ (0)ψ1 (λ, 0)ψ2 (λ, 0) −∞ Z z  1 ϕ(x)ψ2 (λ, x)dx ψ1 (λ, z), − ν0 ψ1 (λ, 0)ψ2 (λ, 0) −∞ where Reλ > −ρ−f ′ (0) and Reλ > −ρ−f ′ (β), C(λ) is a complex function of λ to be determined later. However, these solutions are not necessarily compatible for arbitrary C(λ),

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81

namely, when setting z = 0, the value ψ(λ, 0) on the left hand side is not necessarily the same as the value ψ(λ, 0) on the right hand side. Below we will find compatible solutions by choosing particular function C = C(λ). Second step: Letting z = 0, we find   Z 0 ψ(λ, 0) α K(x)ψ2 (λ, x)dx C(λ) = 1− ψ1 (λ, 0) ν0 U ′ (0)ψ2 (λ, 0) −∞ Z 0  1 + ϕ(x)ψ2 (λ, x)dx , ν0 ψ1 (λ, 0)ψ2 (λ, 0) −∞ where  Z ∞  ′  1 ′ f (U (x)) − f (β) dx > 0, ψ1 (λ, 0) = exp ν0 0  Z 0   ′  1 ψ2 (λ, 0) = exp f (U (x)) − f ′ (0) dx > 0. ν0 −∞ 

Substituting C(λ) back into the general solution yields the compatible solution, for all λ. Note that if |K(x)| + |ϕ(x)| ≤ C exp(−ρ|x|), on R, then Z z  ≤C exp(−ρ|z|), K(x)ψ (λ, x)dx ψ (λ, z) 2 1 −∞ Z z  ϕ(x)ψ2 (λ, x)dx ψ1 (λ, z) ≤C exp(−ρ|z|), −∞

for all z ∈ R and for all complex numbers λ with Reλ > −ρ−f ′ (0) and Reλ > −ρ−f ′ (β). To find the eigenvalues of L, we have to let ϕ = 0. It is not difficult to verify that the compatible solution is bounded on R if and only if C(λ) = 0, that is α 1− ν0 U ′ (0)ψ2 (λ, 0)

Z

0

K(x)ψ2 (λ, x)dx = 0.

−∞

This is the very motivation to construct the Evans function E(λ) in this way. The Evans function is defined in the open region to the right of the essential spectrum because we will explore all eigenvalues which may threat the stability of the wave. We summarize these results below. Lemma 1. (I) The compatible solutions of the general eigenvalue problem Lψ = λψ + ϕ are given by Z 0  1 ψ(λ, 0) ϕ(x)ψ2 (λ, x)dx ψ1 (λ, z) E(λ)ψ1 (λ, z) + ψ(λ, z) = ψ1 (0, 0) ν0 ψ1 (0, 0)ψ2 (0, 0) −∞ Z z  αψ(λ, 0) K(x)ψ (λ, x)dx ψ1 (λ, z) + 2 ν0 U ′ (0)ψ1 (0, 0)ψ2 (0, 0) −∞

82

Linghai Zhang 1 − ν0 ψ1 (0, 0)ψ2 (0, 0)

Z

z −∞

 ϕ(x)ψ2 (λ, x)dx ψ1 (λ, z).

(25)

Let ϕ = 0, then ψ(λ, z) 6= 0 if ψ(λ, 0) 6= 0 and ψ(λ, z) = 0 if ψ(λ, 0) = 0, for all λ ∈ C. Furthermore, ψ(λ, z) is unbounded on R if E(λ) 6= 0 and ψ(λ0 , z) is bounded on R if E(λ0 ) = 0. (II) The Evans function Z 0 α E(λ) = 1 − K(x)ψ2 (λ, x)dx ν0 U ′ (0)ψ2 (0, 0) −∞ Z 0  Z 0  =1− K(x)ψ2 (λ, x)dx / K(x)ψ2 (0, x)dx . −∞

−∞

The domain of the Evans function is Ω = {λ ∈ C : Reλ > −ρ − f ′ (0) and (III) If

Reλ > −ρ − f ′ (β)}.

   sup ρ > 0 : sup exp(ρ|x|)|K(x)| < ∞ = ∞, R

then Ω = C.

Proof. By the method of variation of parameters, matching technique and Lebesgue’s dominated convergence theorem, we can complete the proof. Since the details are not difficult, they are omitted. See [77]–[79] for a complete derivation of the solution ψ and the Evans function E(λ). Lemma 2. The following statements are correct. (I) The Evans function is complex analytic and it is real-valued on the real line. (II) The eigenvalues of the linear differential operator L have a one-to-one correspondence with the zeros of the Evans function. (III) The algebraic multiplicity of any eigenvalue λ0 of L is equal to the order of λ0 as a zero of the Evans function. (IV) Furthermore, E(0) = 0, and lim E(λ) = 1 in Ω.

|λ|→∞

Proof. The proofs of (I), (II) and (IV) are standard. See [66] for the proof of (III). Lemma 3. Suppose that f is continuously differentiable on R, such that f (0) = 0 and f ′ (0) > 0. Furthermore, suppose that the equation f (u) = α has a unique solution β > θ such that f ′ (β) > 0. Then the essential spectrum of the linear differential operator L is given by σessential (L) = {λ ∈ C : λ = −f ′ (0) + iν0 ξ or √ where i = −1. The normal spectrum is given by

λ = −f ′ (β) + iν0 ξ, ξ ∈ R},

σnormal (L) = {λ ∈ C : λ = 0}.

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83

Proof. It is simple and omitted. Lemma 4. The neutral eigenvalue λ = 0 is algebraically simple. Proof. In the compatible solution representation, let λ = 0 and ϕ = Uz , we find   Z 0 α ψ(0, 0) K(x)ψ2 (0, x)dx 1− C(0) = ψ1 (0, 0) ν0 U ′ (0)ψ2 (0, 0) −∞ Z 0  1 ′ + U (x)ψ2 (0, x)dx ν0 ψ1 (0, 0)ψ2 (0, 0) −∞ Z 0  1 ′ U (x)ψ2 (0, x)dx 6= 0. = ν0 ψ1 (0, 0)ψ2 (0, 0) −∞ Therefore, the neutral eigenvalue λ = 0 is algebraically simple. The proof is completed. By coupling Lemma 1, Lemma 2, Lemma 3 and Lemma 4 together, we finish the proof of Theorem 2 immediately.

2.2.

The Existence and Uniqueness of the Fast Traveling Pulse Solution

Theorem 3 (Existence and uniqueness). Suppose that the functions f , g, K and H and the model parameters α, γ and θ satisfy all of the assumptions made before. Then, there is a sufficiently small positive number 0 < ε0 ≪ 1, such
that for any ε ∈ (0, ε0 ), there
exists a unique fast traveling pulse solution U (ε, ·), W (ε, ·) = Ufast (ε, ·), Wfast (ε, ·) to system (21)–(22), such that Z ν(ε)Uz + f (U ) + W =α K(z − y)H(U (y) − θ)dy, R

ν(ε)Wz =εg(U ) − γεW,

|U (ε, z)| + |W (ε, z)| ≤C exp(−ρ1 |z|),

|U (ε, z)| + |W (ε, z)| ≤C exp(−ρ2 |z|),

on

(−∞, 0),

on

(0, ∞),

where ν(ε) = νfast (ε) > 0 represents the unique fast speed and z = x + ν(ε)t stands for the moving coordinate. Additionally, ρ1 = O(1) and ρ2 = O(ε) as ε → 0. Moreover 1 (I) U (ε, 0) = U (ε, Z(ε)) = θ, for some number Z = Zfast (ε) = O(1). ε (II) Uz (ε, 0) > 0 and Uz (ε, Z(ε)) < 0. (III) U < θ on (−∞, 0) ∪ (Z(ε), ∞) and U > θ on (0, Z(ε)). (IV) lim ν(ε) = νfront

lim Z(ε) = ∞,

lim Ufast (ε, z), Wfast (ε, z) = Ufront (z), 0 .

ε→0

and

ε→0

ε→0


There also exists a
unique slow traveling pulse solution U (ε, ·), W (ε, ·) = Uslow (ε, ·), Wslow (ε, ·) , together with a slow speed ν(ε) = νslow (ε) > 0 to system (21)– (22), for all ε ∈ (0, ε0 ). The slow pulse is close to a standing wave of (23) and lim νslow (ε) = 0.

ε→0

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Linghai Zhang

Remark 1. The existence of the traveling pulse solutions is open for large ε. Remark 2. Define ω1 (ε) =

1 + γε +

p

(1 − γε)2 − 4ε , 2

ω2 (ε) =

1 + γε −

p

(1 − γε)2 − 4ε , 2

and

     1 1 − ω2 (ε) 1 − ω1 (ε) ω1 (ε) ω2 (ε) exp x − exp x , C(ν, ε, x) = ω1 (ε) − ω2 (ε) ω1 (ε) ν(ε) ω2 (ε) ν(ε)      1 ω1 (ε) 1 ω2 (ε) 1 − exp x + exp x . D(ν, ε, x) = ω1 (ε) − ω2 (ε) ω1 (ε) ν(ε) ω2 (ε) ν(ε)

To find the speeds of the pulses, we may construct speed index functions
and use them to solve some equations for the speeds. For the special case f (u), g(u) = (u, u), the equations are Z 0 Z 0 γ θ C(ν, ε, x)[K(x) − K(Z(ε))]dx = K(x)dx − , (26) 1 + γ α −∞ −Z Z Z Z 0 θ γ K(x)dx − . (27) C(ν, ε, x)[K(x + Z(ε)) − K(x)]dx = 1 + γ α 0 −∞ The representation for the traveling pulse solutions is 1  × ν(ε) ω1 (ε) − ω2 (ε)     Z z  ω1 (ε) ω2 (ε) × [1 − ω2 (ε)] exp (x − z) − [1 − ω1 (ε)] exp (x − z) ν(ε) ν(ε) −∞ (" Z # ) x ε   × α K(ξ)dξ + U (ε, x) − f (U (ε, x)) dx + ν(ε) ω1 (ε) − ω2 (ε) x−Z(ε)     Z z   ω1 (ε) ω2 (ε) × exp U (ε, x) − g(U (ε, x)) dx, (x − z) − exp (x − z) ν(ε) ν(ε) −∞     Z z  ω2 (ε) ε ω1 (ε)   (x − z) + exp (x − z) W (ε, z) = − exp ν(ε) ν(ε) ν(ε) ω1 (ε) − ω2 (ε) −∞ (" Z # ) x ε   × α K(ξ)dξ + U (ε, x) − f (U (ε, x)) dx + ν(ε) ω (ε) − ω2 (ε) 1 x−Z(ε)     Z z  ω1 (ε) ω2 (ε) × [ω1 (ε) − 1] exp (x − z) − [ω2 (ε) − 1] exp (x − z) ν(ε) ν(ε) −∞  × U (ε, x) − g(U (ε, x)) dx. U (ε, z) =

In particular, if f (u) = u and g(u) = u, then the representation for the traveling pulse solutions is Z z Z z   αγ U (ε, z) = C(ν, ε, x − z) K(x) − K(Z(ε)) dx, K(x)dx − α 1 + γ z−Z(ε) −∞ Z z Z z   α D(ν, ε, x − z) K(x) − K(Z(ε)) dx. K(x)dx − α W (ε, z) = 1 + γ z−Z(ε) −∞

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85

0.5 0.4

w

0.3

back

0.2

right

left

0.1 0

front

−0.1 0 0.5 1

u

−0.2

−0.1

0

0.1

0.2

0.3

v

Figure 5. Phase space portrait of a singular traveling pulse solution of system (1)-(2), where ε = 0. This is a graph in three-dimensional space consisting of several curves rather than surfaces. The real fast traveling pulse solution of system (1)-(2) (where 0 < ε ≪ 1) is very close to the singular solution. Figure 5. Phase space portrait of a singular traveling pulse solution of system (1)-(2), where ε = 0. This is a graph in three-dimensional space consisting of several curves rather than surfaces. The real fast traveling pulse solution of system (1)-(2) (where 0 < ε ≪ 1) is very close to the singular solution. Graph of a singular homoclinic orbit of equations (21)–(22), where ε = 0, f (u) = u(u − 1)(3u
− 1) and g(u) = tanh u. The real fast traveling pulse solution Ufast (ε, ·), Wfast (ε, ·) is very close to the singular homoclinic orbit. Here are the main ideas in the rigorous mathematical analysis of the existence and uniqueness of the fast traveling pulse solution. Let ν > 0 represent the wave speed parameter and let Z > 0 denote another parameter. We will have to assume that the desired pulse crosses the threshold θ on certain excited interval, say (0, Z), so that we may reduce the traveling wave equations (integral differential equations) to ordinary differential equations. Then we have to solve these equations to find the formal solutions. But these are not necessarily real traveling wave solutions. We must make sure that the formal solution really crosses the threshold on the prescribed points and exceeds the threshold on the prescribed interval: U (ε, 0) = U (ε, Z) = θ and U (ε, z) > θ on (0, Z). This can be guaranteed by

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Linghai Zhang

using the implicit function theorem to determine the speed and the parameter Z appearing in the excited interval (0, Z). Meanwhile, we have to demonstrate that the formal solutions are close to a singular homoclinic orbit, which can be constructed by using a traveling wave front and a traveling wave back. This will guarantee that the traveling pulse solution exceeds the threshold only on the excited interval. This may also yield valuable estimates on the speed and the wave shape. The estimate on Z plays a crucial role in stability analysis. We may also have to justify that the solutions are uniformly bounded with respect to ε and other parameters in case we have to apply the Leray-Schauder’s fixed point principle to prove the existence. It is worth of mentioning that the existence, uniqueness and stability of the fast traveling pulse solution have been established for (21)–(22) with any nonnegative synaptic couplings. But, in the sense of rigorous mathematical analysis, these results are still open for (21)–(22) with lateral inhibition and lateral excitation kinds of kernel functions. To keep the chapter from too long, we are not going to provide the rigorous mathematical analysis of the existence and uniqueness of the fast pulse of (21)–(22) with lateral inhibition or lateral excitation kinds of synaptic couplings. We simply assume that the existence and uniqueness are true. On the other hand, the existence and uniqueness of the traveling wave front and the traveling wave back, respectively, which are used to construct the singular homoclinic orbit (together with some other pieces of smooth curves), can be proved for all synaptic couplings in the three classes, see Zhang [84] for detailed analysis. The real homoclinic orbit may be established by applying the singular homoclinic orbit and the well-known geometric singular perturbation technique. 1 Let us prove that Zfast (ε) = O(1), as ε → 0. Set τ = εz. By using the traveling wave ε equation ν(ε)Wz = εg(U )−γεW , we obtain the new equation ν(ε)Wτ = g(U )−γW . By assumption, g(U ) − γW > 0 on an appropriate segment of S ≡ {(U, W ) : U > θ, f (U ) + W = α, f ′ (U ) > 0}. Without loss of generality, let the compact interval [0, εZ(ε)] be such that g(U (ε, ·)) + γW (ε, ·) > 0. By a mean value theorem, there exists a number ξ ∈ (0, εZ(ε)), such that 1 εZ(ε)

Z

εZ(ε)

0

[g(U (ε, τ )) − γW (ε, τ )]dτ = g(U (ε, ξ)) − γW (ε, ξ) ≥ min[g(U ) − γW ] > 0. S

Now we have ν(ε)[W (ε, εZ(ε)) − W (ε, 0)] = ν(ε)

Z

εZ(ε) 0

Wτ (ε, τ )dτ = εZ(ε)[g(U (ε, ξ)) − γW (ε, ξ)].

Finally, we obtain the estimate εZ(ε) = ν(ε)

2.3.

W (ε, εZ(ε)) − W (ε, 0) = O(1) > 0. g(U (ε, ξ)) − γW (ε, ξ)

The Exponential Stability of the Fast Traveling Pulse Solution

To keep the chapter from too long, we will focus on the investigation of the exponential stability of the fast traveling pulse solution.

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

87

Theorem 4 (Stability). Let ε ∈ (0, ε0 ).
Then the fast traveling pulse solution
U (ε, ·), W (ε, ·) = Ufast (ε, ·), Wfast (ε, ·) of system (21)–(22) is exponentially stable relative to the system Z Pt + ν(ε)Pz + f (P ) + Q = α K(z − y)H(P (y, t) − θ)dy, R

Qt + ν(ε)Qz = ε(g(P ) − γQ), P (z, 0) = P0 (z),

Q(z, 0) = Q0 (z),

that is, if k(P0 , Q0 ) − (Ufast (ε, ·), Wfast (ε, ·))kL∞ (R) ≤ C, for some constant C > 0, then k(P (·, t), Q(·, t)) − (Ufast (ε, · + h), Wfast (ε, · + h))kL∞ (R) ≤ M exp(−ρt)k(P0 , Q0 ) − (Ufast , Wfast )kL∞ (R) ,

where h is a finite real number, such that |h| ≤ M k(P0 , Q0 ) − (Ufast (ε, ·), Wfast (ε, ·))kL∞ (R) . The constants C and M may depend on the parameters α, γ and ε, they may also depend on the nonlinear functions (f, g) and (K, H).

2.4.

The Eigenvalue Problem and the Evans Function



Let ν(ε) = νfast (ε), z = x + ν(ε)t and P (z, t), Q(z, t) = u(x, t), w(x, t) . Then we get the following integral differential equations from system (21)–(22) Z Pt + ν(ε)Pz + f (P ) + Q = α K(z − y)H(P (y, t) − θ)dy, R

Qt + ν(ε)Qz = εg(P ) − γεQ.

The fast pulse U (ε, ·), W (ε, ·) = Ufast (ε, ·), Wfast (ε, ·) is a stationary solution of these equations. The linearization with respect to the fast pulse is given by α α K(z)p(0, t) − K(z − Z)p(Z, t), Uz (ε, 0) Uz (ε, Z) qt + ν(ε)qz = εg ′ (U )p − γεq.

pt + ν(ε)pz + f ′ (U )p + q =

The right hand
side of the first equation comes from the nonlocal term. Suppose that
p(z, t), q(z, t) = exp(λt) ξ(λ, z), η(λ, z) is a solution of this linear system, where λ is a complex number. Here the temporal variable t and the spatial variable z are separated. Thus we obtain the following linear system of ordinary differential equations, that is, the eigenvalue problem   λ + f ′ (U ) 1 ′ ψ ν(ε)ψ + −εg ′ (U ) λ + γε

88

Linghai Zhang 



  α α K(z) 0 K(z − Z) 0 = ψ(λ, ε, 0) − ψ(λ, ε, Z), (28) 0 0 0 0 Uz (ε, 0) Uz (ε, Z)   ξ . The eigenvalue problem is associated with the linear differential operwhere ψ = η ator L(ε) : X → Y. Recall that the operator is defined by   f ′ (U ) 1 ′ ψ L(ε)ψ = −ν(ε)ψ − −εg ′ (U ) γε     α α K(z) 0 K(z − Z) 0 + ψ(0) − ψ(Z), (29) 0 0 0 0 Uz (ε, 0) Uz (ε, Z) Y is a Banach space consisting of all complex, vector-valued, bounded, uniformly continuous functions defined on R, and X = {ψ ∈ Y : ψ ′ ∈ Y}. Note that the operator L(ε) is not self-adjoint. Below we plan to solve the eigenvalue problem L(ε)ψ = λψ for bounded compatible solutions in X. A simple observation reveals that the solution is identically equal to zero if and only if ξ(λ, ε, 0)
= ξ(λ, ε, Z) = 0. Without loss of generality, we consider the case ξ(λ, ε, 0), ξ(λ, ε, Z) 6= (0, 0). To study the simplicity of the neutral eigenvalue λ = 0 and to investigate where the inverse operator (λI − L(ε))−1 exists and is bounded, let us investigate compatible solutions of the general eigenvalue problem L(ε)ψ = λψ + ϕ, where ϕ ∈ C 0 (R) ∩ L∞ (R). First of all p p 1 + γε + (1 − γε)2 − 4ε 1 + γε − (1 − γε)2 − 4ε ω1 (ε) = , ω2 (ε) = , 2 2     1 1 1 are the eigenvalues of the coefficient matrix and ; ω1 (ε) − 1 −ε γε   1 are the corresponding eigenvectors. The homogeneous system with conω2 (ε) − 1 stant coefficients     0 λ+1 1 , φ= ν(ε)φ′ + 0 −ε λ + γε possesses the solutions    λ + ω1 (ε) 1 φ1 (λ, ε, z) = exp − , z ω1 (ε) − 1 ν(ε)    λ + ω2 (ε) 1 . z φ2 (λ, ε, z) = exp − ω2 (ε) − 1 ν(ε) Obviously,


Φ(λ, ε, z) ≡ φ1 (λ, ε, z), φ2 (λ, ε, z)

is a fundamental matrix of the homogeneous system with constant coefficients. The inverse matrix
−1 Φ−1 (λ, ε, z) = φ1 (λ, ε, z), φ2 (λ, ε, z)

Traveling Waves Arising from Synaptically Coupled Neuronal Networks 





λ + ω1 (ε) z  [1 − ω2 (ε)] exp ν(ε) 1  =    ω1 (ε) − ω2 (ε)  λ + ω2 (ε) [ω1 (ε) − 1] exp z ν(ε)

+ exp



λ + ω1 (ε) z ν(ε)

89  

   .  λ + ω2 (ε) − exp z ν(ε) 

Moreover

−1 Φ(λ, ε, z)Φ−1 (λ, ε, x) = φ1 (λ, ε, z), φ2 (λ, ε, z) φ1 (λ, ε, x), φ2 (λ, ε, x)      λ+ω2 (ε) λ+ω1 (ε) (x − z) −[1−ω1 (ε)] exp (x − z)  [1−ω2 (ε)] exp ν(ε) ν(ε) 1  =      ω1 (ε)−ω2 (ε)  λ + ω1 (ε) λ + ω2 (ε) −ε exp (x − z) + ε exp (x − z) ν(ε) ν(ε)      λ + ω1 (ε) λ + ω2 (ε) exp (x − z) − exp (x − z)  ν(ε) ν(ε)      .  λ + ω1 (ε) λ + ω2 (ε) [ω1 (ε) − 1] exp (x − z) − [ω2 (ε) − 1] exp (x − z) ν(ε) ν(ε)

When (f, g) are nonlinear functions, by using Picard iteration technique [17], we can establish the existence and uniqueness of each of the following two solutions. We omit
the tedious details. Suppose that Ψ(λ, ε, z) ≡ ψ1 (λ, ε, z), ψ2 (λ, ε, z) is a fundamental matrix, where     ξ2 (λ, ε, z) ξ1 (λ, ε, z) , , ψ2 (λ, ε, z) = ψ1 (λ, ε, z) = η2 (λ, ε, z) η1 (λ, ε, z) are the solutions of the homogeneous equations     0 λ + f ′ (U ) 1 ′ , ψ= ν(ε)ψ + ′ 0 −εg (U ) λ + γε such that Ψ(λ, ε, 0) = Φ(λ, ε, 0) =



1 1 ω1 (ε) − 1 ω2 (ε) − 1



.

Actually, Ψ satisfies Ψ(λ, ε, z) = Φ(λ, ε, z)   Z z 1 1 − f ′ (U (ε, x)) 0 Ψ(λ, ε, x)dx. Φ(λ, ε, z) Φ−1 (λ, ε, x) + εg ′ (U (ε, x)) − ε 0 ν(ε) 0 Then, one can easily establish the following asymptotic behaviors     0 0 0 0 −1 , , lim Ψ (λ, ε, z) = lim Ψ(λ, ε, z) = z→∞ z→−∞ 0 0 0 0 for all λ with Reλ > −ω2 (ε). These solutions depend analytically on λ and ε, they converge exponentially fast to zero. In the left half plane {λ ∈ C: Reλ < −ω1 (ε)}, the asymptotic behaviors are reversed     0 0 0 0 −1 . , lim Ψ (λ, ε, z) = lim Ψ(λ, ε, z) = z→∞ z→−∞ 0 0 0 0

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Linghai Zhang

By using the method of variation of parameters [17], we may solve the general eigenvalue problem L(ε)ψ = λψ + ϕ to find the solutions   a(λ, ε) ψ(λ, ε, z) = Ψ(λ, ε, z) b(λ, ε)   Z z 1 1 0 −1 + Ψ (λ, ε, x) Ψ(λ, ε, z) 0 0 ν(ε) −∞   α α K(x)ψ(λ, ε, 0) − K(x − Z)ψ(λ, ε, Z) dx × Uz (ε, 0) Uz (ε, Z) Z z 1 − Ψ−1 (λ, ε, x)ϕ(x)dx, Ψ(λ, ε, z) ν(ε) −∞   a(λ, ε) is a complex vector-valued function of (λ, ε) where Reλ > −ρ − ω2 (ε), and b(λ, ε) to be determined later. Lemma 5. Suppose that |K(x)| + |ϕ(x)| ≤ C exp(−ρ|x|),

on

R,

for some positive constants C and ρ. Then   Z z K(x) 0 −1 Ψ(λ, ε, z) dx Ψ (λ, ε, x) 0 0 −∞   Z z K(x − Z) 0 −1 dx + Ψ(λ, ε, z) Ψ (λ, ε, x) 0 0 −∞ Z z + Ψ(λ, ε, z) Ψ−1 (λ, ε, x)ϕ(x)dx ≤ C exp(−ρ|z|),

on

R.

−∞


The solution ψ(λ, ε, z) is unbounded on R if a(λ,
ε), b(λ, ε) 6= (0, 0) and the solution ψ(λ0 , ε, z) is bounded on R if a(λ0 , ε), b(λ0 , ε) = (0, 0).

Proof. By using the asymptotic behaviors of Ψ(λ, ε, z) and Ψ−1 (λ, ε, z), the conditions on K and ϕ, we find that the results are true.

2.5.

The Compatible Solutions of the Eigenvalue Problem

To find the compatible solutions of the general eigenvalue problem, we have to determine
the vector a(λ, ε), b(λ, ε) . By setting z = 0 and z = Z(ε) respectively in the solution representation, we get   a(λ, ε) ψ(λ, ε, 0) = Ψ(λ, ε, 0) b(λ, ε)     Z 0 α K(x) 0 + dx ψ(λ, ε, 0) Ψ−1 (λ, ε, x) Ψ(λ, ε, 0) 0 0 ν(ε)Uz (ε, 0) −∞     Z 0 α K(x − Z) 0 −1 dx ψ(λ, ε, Z) Ψ (λ, ε, x) Ψ(λ, ε, 0) − 0 0 ν(ε)Uz (ε, Z) −∞

Traveling Waves Arising from Synaptically Coupled Neuronal Networks Z 0 1 − Ψ−1 (λ, ε, x)ϕ(x)dx, Ψ(λ, ε, 0) ν(ε) −∞

91

and  a(λ, ε) ψ(λ, ε, Z) = Ψ(λ, ε, Z) b(λ, ε)     Z Z α K(x) 0 −1 dx ψ(λ, ε, 0) Ψ(λ, ε, Z) Ψ (λ, ε, x) + 0 0 ν(ε)Uz (ε, 0) −∞     Z Z α K(x − Z) 0 −1 − dx ψ(λ, ε, Z) Ψ (λ, ε, x) Ψ(λ, ε, Z) 0 0 ν(ε)Uz (ε, Z) −∞ Z Z 1 Ψ−1 (λ, ε, x)ϕ(x)dx. Ψ(λ, ε, Z) − ν(ε) −∞ 

Rearranging terms produces the following equations   a(λ, ε) Ψ(λ, ε, 0) b(λ, ε)   Z 0 α K(x) −1 = I− Ψ (λ, ε, x) Ψ(λ, ε, 0) 0 ν(ε)Uz (ε, 0) −∞   Z 0 α K(x − Z) + Ψ−1 (λ, ε, x) Ψ(λ, ε, 0) 0 ν(ε)Uz (ε, Z) −∞ Z 0 1 Ψ−1 (λ, ε, x)ϕ(x)dx, Ψ(λ, ε, 0) + ν(ε) −∞

 dx ψ(λ, ε, 0)   0 dx ψ(λ, ε, Z) 0

0 0



and Ψ(λ, ε, Z)



a(λ, ε) b(λ, ε)



   Z Z α K(x − Z) 0 −1 dx ψ(λ, ε, Z) Ψ (λ, ε, x) Ψ(λ, ε, Z) = I+ 0 0 ν(ε)Uz (ε, Z) −∞     Z Z α K(x) 0 −1 − dx ψ(λ, ε, 0) Ψ (λ, ε, x) Ψ(λ, ε, Z) 0 0 ν(ε)Uz (ε, 0) −∞ Z Z 1 Ψ−1 (λ, ε, x)ϕ(x)dx, Ψ(λ, ε, Z) + ν(ε) −∞ 

where I denotes the identity matrix. Note that
the second component of ψ does not really get involved when we solve a(λ, ε), b(λ,
ε) . Thus, we will use the first component of ψ. Let R(λ, ε, z) = ξ1 (λ, ε, z), ξ2 (λ, ε, z) , for all λ, ε and z. We have R(λ, ε, 0)  = 1−



a(λ, ε) b(λ, ε)



α R(λ, ε, 0) ν(ε)Uz (ε, 0)

Z

0 −1

Ψ −∞

(λ, ε, x)



1 0



 K(x)dx ξ(λ, ε, 0)

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Linghai Zhang 

   0 α 1 −1 + K(x − Z)dx ξ(λ, ε, Z) Ψ (λ, ε, x) R(λ, ε, 0) 0 ν(ε)Uz (ε, Z) −∞ Z 0 1 Ψ−1 (λ, ε, x)ϕ(x)dx R(λ, ε, 0) + ν(ε) −∞ ≡ [1 − M11 (λ, ε)]ξ(λ, ε, 0) − M12 (λ, ε)ξ(λ, ε, Z) + N1 (λ, ε),   a(λ, ε) R(λ, ε, Z) b(λ, ε)     Z Z α 1 −1 = − K(x)dx ξ(λ, ε, 0) Ψ (λ, ε, x) R(λ, ε, Z) 0 ν(ε)Uz (ε, 0) −∞     Z Z α 1 −1 K(x − Z)dx ξ(λ, ε, Z) Ψ (λ, ε, x) R(λ, ε, Z) 1+ 0 ν(ε)Uz (ε, Z) −∞ Z Z 1 R(λ, ε, Z) Ψ−1 (λ, ε, x)ϕ(x)dx + ν(ε) −∞ ≡ −M21 (λ, ε)ξ(λ, ε, Z) + [1 − M22 (λ, ε)]ξ(λ, ε, Z) + N2 (λ, ε). Z

Combining them together appropriately, we have        ξ(λ, ε, 0) a(λ, ε) R(λ, ε, 0) + N (λ, ε), = I − M(λ, ε) ξ(λ, ε, Z) b(λ, ε) R(λ, ε, Z)   M11 (λ, ε) M12 (λ, ε) is a 2 × 2 complex analytic matrix of λ, the where M(λ, ε) = M21 (λ, ε) M22 (λ, ε) four elements are given by   Z 0 α K(x) −1 M11 (λ, ε) = dx, Ψ (λ, ε, x) R(λ, ε, 0) 0 ν(ε)Uz (ε, 0) −∞   Z 0 −α K(x − Z) −1 dx, Ψ (λ, ε, x) R(λ, ε, 0) M12 (λ, ε) = 0 ν(ε)Uz (ε, Z) −∞   Z Z α K(x) −1 M21 (λ, ε) = dx, Ψ (λ, ε, x) R(λ, ε, Z) 0 ν(ε)Uz (ε, 0) −∞   Z Z −α K(x − Z) −1 dx. Ψ (λ, ε, x) R(λ, ε, Z) M22 (λ, ε) = 0 ν(ε)Uz (ε, Z) −∞   N1 (λ, ε) is a complex analytic function of (λ, ε), where The N (λ, ε) = N2 (λ, ε) Z 0 1 N1 (λ, ε) = Ψ−1 (λ, ε, x)ϕ(x)dx, R(λ, ε, 0) ν(ε) −∞ Z Z 1 N2 (λ, ε) = Ψ−1 (λ, ε, x)ϕ(x)dx. R(λ, ε, Z) ν(ε) −∞ Note that the inverse matrix Ψ−1 (λ, ε, x) =

1 exp det Ψ(λ, ε, 0)



1 ν(ε)

Z

0

x

 f ′ (U (ε, ξ)) + 2λ + γε dξ



Traveling Waves Arising from Synaptically Coupled Neuronal Networks   +η2 (λ, ε, x) −ξ2 (λ, ε, x) . × −η1 (λ, ε, x) +ξ1 (λ, ε, x)

93

It is not difficult to show that for all complex numbers λ ∈ C, 0 < ε < ε0 and z ∈ R, there holds   λz Ψ(λ, ε, z) = exp − Ψ(0, ε, z). ν(ε) Therefore, we have       λZ R(0, ε, 0) R(λ, ε, 0) det 6= 0. = exp − det R(0, ε, Z) R(λ, ε, Z) ν(ε) Actually, if ξ1 (λ, ε, 0) = ξ2 (λ, ε, 0) and ξ1 (λ, ε, Z) = ξ2 (λ, ε, Z) for some (λ, ε), then ψ1 (λ, ε, z) = ψ2 (λ, ε, z) for all z ∈ R. This is impossible because ψ1 (λ, ε, 0) 6= ψ2 (λ, ε, 0). Therefore, the above determinant is not equal to zero. Now one can easily solve the above equations to obtain    −1      ξ(λ, ε, 0) R(λ, ε, 0) a(λ, ε) + N (λ, ε) , [I − M(λ, ε) = ξ(λ, ε, Z) R(λ, ε, Z) b(λ, ε)
provided that the complex vector ξ(λ, ε, 0), ξ(λ, ε, Z) is given. Because
we are investigating analytic solutions of λ, we would choose ξ(λ, ε, 0), ξ(λ,
ε, Z) to be analytic functions of λ and smooth functions of ε. Hence a(λ, ε), b(λ, ε) would also be analytic. This in turn shows that ψ(λ, ε, z) is complex analytic in λ, for all fixed z ∈ R and ε > 0. The compatible solutions are given by   −1     ξ(λ, ε, 0) + N (λ, ε) × I −M(λ, ε) ψ(λ, ε, z) = Ψ(λ, ε, z) ξ(λ, ε, Z)   Z z 1 1 0 + Ψ(λ, ε, z) Ψ−1 (λ, ε, x) 0 0 ν(ε) −∞   α α × K(x)ψ(λ, ε, 0) − K(x − Z)ψ(λ, ε, Z) dx Uz (ε, 0) Uz (ε, Z) Z z 1 Ψ(λ, ε, z) Ψ−1 (λ, ε, x)ϕ(x)dx, − ν(ε) −∞ 

where Reλ > −ρ − ω2 (ε).

R(λ, ε, 0) R(λ, ε, Z)



0 0





ξ(λ, ε, 0) ξ(λ, ε, Z)





0 0



and 6= if Let ϕ = 0. Then the solution ψ(λ, ε, z) 6=       0 ξ(λ, ε, 0) 0 , for all λ ∈ C and for all ε ∈ (0, ε0 ). = if ψ(λ, ε, z) = 0 ξ(λ, ε, Z) 0 Lemma 6. The dimension of the compatible solution space of the eigenvalue problem L(ε)ψ = λψ is equal to two.
Proof. In the compatible solution representation, letting ξ(λ, ε, 0), ξ(λ, ε, Z) = (1, 0)
and ξ(λ, ε, 0), ξ(λ, ε, Z) = (0, 1), respectively, we obtain two linearly independent solutions ψe1 (λ, ε, z) and ψe2 (λ, ε, z) of the eigenvalue problem L(ε)ψ = λψ. Any other choice

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Linghai Zhang
of ξ(λ, ε, 0), ξ(λ, ε, Z) would also give a well defined solution of the eigenvalue problem, because it is essentially a linear system. Actually, all of the solutions of the eigenvalue problem L(ε)ψ = λψ can be written in the form ψ(λ, ε, z) = c(λ, ε)ψe1 (λ, ε, z) + d(λ, ε)ψe2 (λ, ε, z),

for some appropriate functions c = c(λ, ε) and d = d(λ, ε).

Defined the unbounded, simply connected open region ( ) p 1 + γε − (1 − γε)2 − 4ε Ω(ε) = λ ∈ C : Reλ > −ρ − . 2 The Evans function is defined by   E(λ, ε) = det I − M(λ, ε) for λ ∈ Ω(ε),

0 < ε < ε0 .

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The domain of the Evans function can be extended to the entire complex plane C if |K(x)| ≤ C exp(−ρ|x|2 ) on R, for some positive constants C and ρ. Actually, all of the improper integrals in M(λ, ε) would be convergent for any complex number λ ∈ C. Roughly speaking, when ε → 0, 1 − M11 (λ, ε) becomes the Evans function of the traveling wave front, 1 − M22 (λ, ε) becomes the Evans function of the traveling wave back, M12 (λ, ε) is due to the right, and M21 (λ, ε) is due to the left in the singular homoclinic orbit. If we make a simple change of variable, then we obtain the nice expressions    λx K(x) dx, Ψ−1 (0, ε, x) 0 ν(ε) −∞     Z 0 −α λx K(x − Z) M12 (λ, ε) = dx, R(0, ε, 0) Ψ−1 (0, ε, x) exp 0 ν(ε)Uz (ε, Z) ν(ε) −∞     Z 0 α λx K(x + Z) −1 M21 (λ, ε) = dx, R(0, ε, Z) Ψ (0, ε, x + Z) exp 0 ν(ε)Uz (ε, 0) ν(ε) −∞     Z 0 λx −α K(x) dx. R(0, ε, Z) Ψ−1 (0, ε, x + Z) exp M22 (λ, ε) = 0 ν(ε)Uz (ε, Z) ν(ε) −∞ M11 (λ, ε) =

2.6.

α R(0, ε, 0) ν(ε)Uz (ε, 0)

Z

0

exp



The Properties of the Evans Function

Let us investigate the properties of the Evans function. Fix ε ∈ (0, ε0 ). Lemma 7. (I) The Evans function is a complex analytic function in the right half plane Ω(ε) and it is a real-valued function if the eigenvalue parameter λ is in R ∩ Ω(ε). (II) The complex number λ0 is an eigenvalue of the associated linear differential operator L(ε) if and only if E(λ0 , ε) = 0. (III) The algebraic multiplicity of any eigenvalue λ0 of L(ε) is equal to the order of λ0 as a zero of the Evans function.

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

95

(IV) Inside Ω(ε), there holds the following limit lim E(λ, ε) = 1.

|λ|→∞

Proof. (I) Each element Mij (λ, ε) is a complex analytic function of λ and ε, and it is realvalued if the eigenvalue parameter λ is in R ∩ Ω(ε), for all i = 1, 2 and j = 1, 2. Thus E(λ, ε) shares the same property. (II) If the complex number λ0 ∈ Ω(ε) is a solution of the equation E(λ0 , ε) = 0, then there must be some nonzero complex numbers ξ(λ0 , ε, 0) and ξ(λ0 , ε, Z), such that     ξ(λ0 , ε, 0) = 0, I − M(λ0 , ε) ξ(λ0 , ε, Z) and then



R(λ0 , ε, 0) R(λ0 , ε, Z)



a(λ0 , ε) b(λ0 , ε)



= 0.


Hence a(λ0 , ε), b(λ0 , ε) = (0, 0) and λ0 is an eigenvalue of L(ε). On the other hand, if
λ0 is an eigenvalue of L(ε), then a(λ0 , ε), b(λ0 , ε) = (0, 0) and        0 a(λ0 , ε) R(λ0 , ε, 0) ξ(λ0 , ε, 0) . = = [I − M(λ0 , ε)] 0 b(λ0 , ε) R(λ0 , ε, Z) ξ(λ0 , ε, Z)
This implies that E(λ0 , ε) = 0, since ξ(λ0 , ε, 0), ξ(λ0 , ε, Z) 6= (0, 0), for nontrivial solutions. The above analysis shows that λ0 ∈ Ω(ε) is an eigenvalue of L(ε) if and only if E(λ0 , ε) = 0. See [66] for the proof of (III). (IV) is easy to prove. Lemma 8. The spectrum of the linear differential operator L(ε) consists of two parts. (I) The essential spectrum is given by σessential (L(ε)) = {λ ∈ C : λ = iν(ε)ξ − ω1 (ε) or

λ = iν(ε)ξ − ω2 (ε), ξ ∈ R} ,

where ω1 (ε) =

1 + γε +

p

(1 − γε)2 − 4ε , 2

ω2 (ε) =

1 + γε −

p

(1 − γε)2 − 4ε . 2

(II) The normal spectrum of L(ε) consists of isolated eigenvalues with finite algebraic multiplicities. It is given by    C , σnormal (L(ε)) = λ ∈ C : λ0 = 0, λstable = − exp − ε where both eigenvalues λ0 = 0 and λstable are algebraically simple. In particular, λstable ′ (ε) > 0 on (0, ε0 ). Proof. The essential spectrum is standard to find by following the original ideas of John Evans, see [28]. The isolated eigenvalues are found by using the Evans function and by applying the one-to-one correspondence between the eigenvalues of L(ε) and the zeros of the Evans function E(λ, ε). It is not difficult to show that E(λ, ε) 6= 0 in the right half plane

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Linghai Zhang

Reλ ≥ 0, except for
λ = 0. Let λ(ε) denote the nonzero eigenvalue of L(ε). From the equation E λ(ε), ε = 0, we find λ′ (ε)


∂E
∂E λ(ε), ε + λ(ε), ε = 0. ∂λ ∂ε

Note that for any real number λ ∈ Ω(ε), 0 < ε < ε0 , we have ∂E (λ, ε) > 0, ∂λ

∂E (λ, ε) < 0. ∂ε

From this above equation, we obtain the derivative    ∂E ∂E (λ(ε), ε) / (λ(ε), ε) = O(1) > 0. λ (ε) = − ∂ε ∂λ 

′

The proof of Lemma 8 is finished. For all complex numbers λ 6= 0, if Reλ ≥ λstable , then the inverse operator [λI − L(ε)]−1 exists and is bounded.

2.7.

The Simplicity of the Neutral Eigenvalue λ = 0

In the compatible solution representation, let λ = 0 and ϕ = (Uz , Wz )T , we find that 

a(0, ε) b(0, ε)

   −1   ξ(0, ε, 0) R(0, ε, 0) + N (0, ε) [I − M(0, ε) = ξ(0, ε, Z) R(0, ε, Z)   −1  0 R(0, ε, 0) . N (0, ε) 6= = 0 R(0, ε, Z) 



Therefore the neutral eigenvalue λ = 0 is algebraically simple. Proof of Theorem 4. By using the properties of the Evans function, we find that E(λ, ε) 6= 0, for all complex numbers λ ∈ Ω(ε), except for λ = 0 and λstable . Therefore, there exists no nonzero eigenvalue of L(ε) in {λ ∈ C : Reλ ≥ 0}. Because E(λ, ε) is complex analytic and lim|λ|→∞ E(λ, ε) = 1 in Ω(ε), we conclude that max{Reλ : λ ∈ σ(L(ε)), λ 6= 0} ≤ λstable

  C = − exp − . ε

By the linearized stability criterion, see [21], [28], [78]-[79], the exponential stability of the fast traveling pulse solution is proved. Remark 3. The linear operator corresponding to the slow traveling pulse solution has a positive eigenvalue: λunstable = exp (−Cε) > 0. That is why the slow pulse is exponentially unstable relative to (21)–(22).

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

2.8.

97

The Traveling Wave Front versus the Traveling Wave Back

When constructing the singular homoclinic orbit, we have to use the traveling wave front and the traveling wave back. Sometimes it is possible to use symmetry to obtain the traveling wave back from the traveling wave front. The traveling wave front and the traveling wave back have the same speed and they satisfy very similar differential equations - the only difference is a constant. Below we will show several examples. (I) Consider the integral differential equations Z ut + u = α K(x − y)H(u(y, t) − θ)dy, R

and ut + u + α − 2θ = α

Z

R

K(x − y)H(u(y, t) − θ)dy.

Let z = x + ν0 t. It is easy to show that Z Z z K(x)dx − α Ufront (z) = α

z

exp

−∞

−∞



x−z ν0



K(x)dx

is a front of the former and Uback (z) ≡ 2θ − Ufront (z) is a traveling wave back of the latter. This is basically due to the identity: H(x) + H(−x) = 1, for all x ∈ R. (II) Consider the integral differential equations Z ut + u(u − 1)(Au − 1) = α K(x − y)H(u(y, t) − θ)dy, R

and 2(A + 1)(A − 2)(2A − 1) ut + u(u − 1)(Au − 1) + α + 27A2 Z = α K(x − y)H(u(y, t) − θ)dy. R

1+a . It is easy to see that if U = Ufront (·) is the 3 2A + 2 − Ufront (·) is the traveling wave traveling wave front of the former, then Uback ≡ 3A back of the latter. Let z = x + ν0 t. Suppose that θ =

2.9.

A Concrete Example

The scalar integral differential equation Z ut + u = α K(x − y)H(u(y, t) − θ)dy, R

possesses the exact traveling wave front Z Z z K(x)dx − α U (z) = α −∞

z

−∞

exp



x−z ν0



K(x)dx.

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Linghai Zhang

The speed index function and the stability index function are defined by Z 0 x exp φ(ν) = α K(x)dx, ν −∞   ν0 1 . φ E(λ) = 1 − φ(ν0 ) λ+1 ρ Let K(x) = exp(−ρ|x|), where ρ > 0 is a constant. We find that the speed of the front 2 α − 2θ > 0. The speed index function and the stability index function are given is ν0 = 2ρθ by αρν , φ(ν) = 2(1 + ρν) 2θλ E(λ) = . α + 2θλ The speed index function is defined in the open domain (0, ∞) and the stability index function is define in Ω = {λ ∈ C: Reλ > −1}. Clearly s (Reλ)2 + (Imλ)2 , |E(λ)| = (Reλ + α/2θ)2 + (Imλ)2 and (α/2θ)(Reλ) + (Reλ)2 + (Imλ)2 , (Reλ + α/2θ)2 + (Imλ)2 αImλ . ImE(λ) =  2θ (Reλ + α/2θ)2 + (Imλ)2

ReE(λ) =

For this Evans function, we can directly prove the properties mentioned earlier. It is straight2αθ 2θλ and E ′ (λ) = to verify that the front is expoforward to use E(λ) = α + 2θλ (α + 2θλ)2 nentially table.

3.

Another Nonlinear Singularly Perturbed System of Integral Differential Equations

In the brain, persistently active neurons fire at rates far below their saturated maximum. How does a neuronal network maintain persistent activity at low firing rates has not been completely understood. It is worth of mentioning that the problem of persistent activity at low firing rates cannot be solved with a single Heaviside step function. To circumvent this limitation, Guo and Chow [36] suggested the gain function G(u) = [α+β(u−θ)]H(u−θ), where α > 0, β > 0 and θ > 0 are constants, H stands for the Heaviside step function 1 H(u−θ) = 0 for all u < θ, H(0) = and H(u−θ) = 1 for all u > θ. They established the 2 existence and stability of standing pulse solutions of the scalar integral differential equation Z  ∂u τ K0 (x − y) α + β[u(y, t) − θ] H(u(y, t) − θ)dy, +u= ∂t R

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

99

where one may set the synaptic decay time constant τ = 1. The synaptic coupling K0 is a symmetric Mexican hat kernel function. Typically, K0 (x) = A exp(−a|x|) − exp(−|x|), where A > 1 and a > 1 are constants. The gain function is piecewise linear and unbounded. Note that if β = 0, then G(u) = αH(u − θ), this is the usual Heaviside step function. However, if β > 0, then G becomes arbitrarily large when u goes to infinity. We will avoid using this kind of unbounded gain function - we use a new piecewise constant function with two jumps. Here we will provide a mathematical justification and explanation why we choose a new gain function. Probably the most common gain function  is the continuous and monotone sigmoid firing rate function H(Λ, u) = exp[Λ(u − θ)]/ 1 + exp[Λ(u − θ)] , with the positive gain parameter Λ and the threshold value θ, which saturates to a constant for large u. Note that lim H(Λ, u) = H(u − θ). Λ→∞

The nonlinear transfer function H represents the ensemble pulse activity generated by the effective membrane potential and is monotone increasing. See Bressloff and Folias [13] and Pinto and Ermentrout [55]. Note that the continuous monotone gain function H(Λ, u) can be approximated by piecewise constant functions, such as GN (u) =

N X j=1

βj H(u − θj ),

where N ≥ 2 is an integer, β1 > 0, β2 > 0, · · · , βN > 0 and 0 < θ1 < θ2 < · · · < θN < ∞ are constants. Our new gain function, G2 (u) = αH(u − θ) + βH(u − Θ), is probably the simplest of this form. In this section, we will investigate the nonlinear singularly perturbed system of integral differential equations (3)–(4) arising from synaptically coupled neuronal networks. For simplicity, we focus on the case f (u) = g(u) = u. Thus, we consider the system Z ∂u + u + w = α K(x − y)H(u(y, t) − θ)dy ∂t R Z K(x − y)H(u(y, t) − Θ)dy, (31) +β R

∂w = ε(u − γw), ∂t

in R × R+ .

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The variable u represents the membrane potential of a neuron at positive x and time t, w denotes the leaking current. The parameters α and β stand for synaptic rate constants, γ denotes the decay rate, ε controls the fast/slow activation of chemical ion channels, and 0 < ε ≪ 1, θ and Θ are the threshold constants for excitation. H is the Heaviside step function and K is the synaptic coupling. Both K ∗ [H(u − θ)] and K ∗ [H(u − Θ)] stand for nonlocal interactions between neurons. We incorporate a recovery mechanism in the equations, i.e. the addition of a slow, local negative feedback recovery variable w. This negative feedback could represent spike frequency adaptation, synaptic depression or some other slow process that controls the excitation of the neuronal networks. It has been established in [55] that there exists a direct connection between a similar model and

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Linghai Zhang

w

α +β

α

0

0 θ

u

Θ

Figure 6. Graph of the gain function with two thresholds.

experimental investigations of wave propagation in cortical slices where synaptic inhibition is pharmacologically blocked. Due to the presence of the singular perturbation parameter ε, we may use some ideas in geometric singular perturbation theory, but we will not follow all the sophisticated ideas, techniques, and analysis. After all, this is an almost linear system except for the nonlinearity due to the Heaviside step function. Based on the particular form of the model (31)–(32), by virtue of certain assumptions of the traveling wave solutions, by the reduction of the integral differential equations to ordinary differential equations, and by certain standard techniques in differential equations and dynamical systems, we can always obtain the formal traveling wave solutions. Then by compatibility conditions and implicit function theorem, we can verify the existence and uniqueness of each of the traveling waves. By compatibility, we mean that if we assume one wave crosses the thresholds at certain points and exceeds the thresholds on some open intervals, then after rigorously proving the existence, we must verify that the wave really crosses the threshold at these points and exceeds the thresholds on the desired open intervals.

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

3.1.

101

The Traveling Wave Fronts

First of all, we will consider the scalar integral differential equation (9) with f (u) = u. 1 Theorem 5. Suppose that 0 < 2θ < α < Θ < (α + β) and α(Θ − α) < βθ. Then there 2 exist exactly three exponentially stable traveling wave fronts to the equation Z Z K(x − y)H(u(y, t) − Θ)dy. ut + u = α K(x − y)H(u(y, t) − θ)dy + β R

R

(I) The first traveling wave front U = Ufront−1 (·) crosses the small threshold θ only once. It does not cross the large threshold Θ. The front satisfies   Z z Z z x−z K(x)dx, exp K(x)dx − α Ufront−1 (z) = α ν1 −∞ −∞   Z x−z α z d K(x)dx, exp Ufront−1 (z) = dz ν1 −∞ ν1 d lim Ufront−1 (z) = 0, lim Ufront−1 (z) = α, lim Ufront−1 (z) = 0. z→∞ z→−∞ z→±∞ dz The first wave speed ν1 is determined by the equation   Z 0 Z 0 x α K(x)dx − α exp K(x)dx = θ. ν1 −∞ −∞ (II) The second traveling wave front U = Ufront−2 (·) crosses the large threshold Θ only once. It does not cross the small threshold θ. The front satisfies   Z z Z z x−z K(x)dx, exp K(x)dx − β Ufront−2 (z) = α + β ν2 −∞ −∞   Z x−z β z d exp K(x)dx, Ufront−2 (z) = dz ν2 −∞ ν2 d lim Ufront−2 (z) = α, lim Ufront−2 (z) = α + β, lim Ufront−2 (z) = 0. z→∞ z→−∞ z→±∞ dz The second wave speed ν2 is determined by the equation   Z 0 Z 0 x K(x)dx = Θ. exp K(x)dx − β α+β ν 2 −∞ −∞ (III) The third traveling wave front U = Ufront−3 (·) crosses the small threshold θ once and the large threshold Θ once. The front satisfies Z

z

Z

z−Z

K(x)dx K(x)dx + β −∞ −∞   Z z x−z exp − [αK(x) + βK(x − Z)]dx, ν3 −∞

Ufront−3 (z) = α

102

Linghai Zhang   d x−z 1 z exp [αK(x) + βK(x − Z)]dx, Ufront−3 (z) = dz ν3 −∞ ν3 d Ufront−3 (z) = 0. lim Ufront−3 (z) = 0, lim Ufront−3 (z) = α + β, lim z→∞ z→−∞ z→±∞ dz Z

The third wave speed ν3 together with the positive number Z = Zfront (it is at z = Z that U crosses Θ) are determined by the equations α

Z

0

K(x)dx + β

Z

−Z

−∞

−∞

K(x)dx −

Z

0

exp −∞



x ν3



[αK(x) + βK(x − Z)]dx = θ

and α

Z

Z

K(x)dx + β

Z

0

−∞

−∞

K(x)dx −

Z

0

exp

−∞



x ν3



[αK(x + Z) + βK(x)]dx = Θ.

(IV) There hold the estimates ν1 < ν3 < ν2 .

W

θ

Θ α

U

α+β

Figure 7A. Different portraits of the three traveling wave fronts of the integral differential equation (9).

Proof. First of all, we find that U0 ≡ 0, U1 ≡ α and U2 ≡ α + β are the stable constant solutions of the equation. Let us investigate the existence and uniqueness of the first traveling wave front, satisfying the traveling wave equation Z ′ νU + U = α K(z − y)H(U (y) − θ)dy. R

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

103

U=α+β

U=Θ

U=α

U

U=θ

U=0 Z

Figure 7B. Different portraits of the three traveling wave fronts of the integral differential equation (9). Due to translation invariance of the traveling wave, we may assume that U (0) = θ. Moreover, assume that U < θ on (−∞, 0) and U > θ on (0, ∞). Therefore Z z Z K(x)dx. K(z − y)H(U (y) − θ)dy = −∞

R

Now the traveling wave equation reduces to the nonhomogeneous first order linear ordinary differential equation Z νU ′ + U = α

z

K(x)dx,

−∞

where ν is a real parameter. Solving this equation, we obtain the following results   Z x  Z α z x−z U (z) = exp K(ξ)dξ dx ν −∞ ν −∞   Z z Z z x−z K(x)dx, exp K(x)dx − α =α ν −∞ −∞ lim U (z) = 0,

z→−∞

lim U (z) = α,

z→∞

lim U ′ (z) = 0.

z→±∞

Furthermore, we find the derivatives   Z α z x−z U ′ (z) = exp K(x)dx, ν −∞ ν   Z x−z α z α ′′ exp K(x)dx. U (z) = K(z) − 2 ν ν −∞ ν

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Linghai Zhang

Note that Z 0 x α U (0) = − α exp K(x)dx, 2 ν −∞ Z x α 0 exp K(x)dx, U ′ (0) = ν −∞ ν Z x α α 0 U ′′ (0) = K(0) − 2 exp K(x)dx. ν ν −∞ ν We construct the speed index function φ on R+ by Z 0 x K(x)dx. exp φ(ν) = α ν −∞

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Then φ′ (ν) =

α ν2

Z

0 −∞

|x| exp

x ν

K(x)dx.

Obviously, we have the limits lim φ(ν) = 0 <

ν→0+

α α − θ < = lim φ(ν). ν→∞ 2 2

Moreover φ′ (ν) > 0, if K ≥ 0 everywhere. For any Mexican hat kernel function, there is a constant M > 0, such that K ≤ 0 on (−∞, −M ) and K ≥ 0 on (−M, 0). Now it is easy to verify that   x M 0 ≥ |x| exp |x|K(x), K(x) ≥ exp − ν ν on (−∞, −M ), and also

  M |x|K(x) ≥ 0, K(x) ≥ exp − |x| exp ν ν x

on (−M, 0). Therefore Z 0  α M φ (ν) > 2 exp − |x|K(x)dx ≥ 0. ν ν −∞ ′

Hence, there exists a unique positive number ν1 , such that φ(ν1 ) = U (0) =

α − θ. Hence 2

α − φ(ν1 ) = θ. 2

Obviously, U ′ (0) > 0 if the kernel function is nonnegative. For the Mexican hat kernel function   Z x α 0 exp K(x)dx U ′ (0) = ν1 −∞ ν1

Traveling Waves Arising from Synaptically Coupled Neuronal Networks Z 0  α M > K(x)dx ≥ 0. exp − ν1 ν1 −∞

105

Additionally, U ′ ≥ 0 if K ≥ 0 on R. For the Mexican hat kernel function, this is no longer true. Note that   Z −M x exp K(x)dx < 0, ν1 −∞ Z 0    Z 0 M x K(x)dx ≥ 0. K(x)dx > exp − exp ν1 ν1 −∞ −∞ Thus, there exists a negative number N ∈ (−M, 0), such that   Z z x−z K(x)dx ≤ 0, on (−∞, N ), exp ν1 −∞   Z z x−z exp K(x)dx ≥ 0, on (N, ∞). ν1 −∞ In another word, the first traveling wave front is a decreasing function on (−∞, N ) and it is an increasing function on (N, ∞). It attains a global minimum at z = Z. Additionally, U < θ on (−∞, 0) and U > θ on (0, ∞). Taking the boundary conditions lim U (z) = 0,

z→−∞

lim U (z) = α,

z→∞

into account, the proof of the existence and uniqueness of the wave speed ν1 and the first traveling wave front Ufront−1 are completed. Similarly, the proof of the existence and uniqueness of the wave speed ν2 and the second traveling wave front   Z z Z z x−z K(x)dx, exp K(x)dx − β Ufront−2 (z) = α + β ν2 −∞   −∞ Z z x−z β exp K(x)dx, Ufront−2 ′ (z) = ν2 −∞ ν2   Z z β x−z β ′′ Ufront−2 (z) = K(z) − 2 exp K(x)dx, ν2 ν2 −∞ ν2 are also completed. The Evans functions for the first two traveling wave fronts are defined in Ω = {λ ∈ C: Reλ > −1 − ρ}. Explicitly     νi 1 νi νi 1 ′ ′ , Ei (λ) = , (34) φ φ Ei (λ) = 1 − φ(νi ) λ+1 φ(νi ) (λ + 1)2 λ+1 where i = 1, 2. It can be established that all of these Evans functions are complex analytic functions. We recall that Z 0 Z x x α 0 ′ exp φ(ν) = α |x| exp K(x)dx, φ (ν) = 2 K(x)dx > 0, ν ν −∞ ν −∞ for any synaptic coupling. Now |Ei ′ (λ)| > 0,

i = 1, 2,

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Linghai Zhang

for all complex numbers λ ∈ Ω. There exists a trivial, simple solution λ = 0 to the equation Ei (λ) = 0, and there exists no nonzero solution to that equation in Ω. The exponential stability of the first two traveling wave fronts is proved. Now let us investigate the existence and uniqueness of the third traveling wave front connecting U0 = 0 at z = −∞ to U2 = α + β at z = ∞. Due to translation invariance of the traveling wave front, we assume that U (0) = θ, U < θ on (−∞, 0) and U > θ on (0, ∞). Similarly, let U (Z) = Θ, U < Θ on (−∞, Z) and U > Θ on (Z, ∞), for some constant Z > 0. Under these conditions, the traveling wave equation becomes ′

νU + U = α

Z

z

K(x)dx + β

−∞

Z

z−Z

K(x)dx. −∞

Solving the differential equation, we obtain the traveling wave solution U (z) =α

Z z−Z K(x)dx K(x)dx + β −∞ −∞   Z z  x−z  exp − αK(x) + βK(x − Z) dx, ν −∞

Z

lim U (z) =0,

z

lim U (z) = α + β,

z→∞

z→−∞

lim U ′ (z) = 0.

z→±∞

We also obtain the first order derivative   Z  1 z x−z  ′ U (z) = exp αK(x) + βK(x − Z) dx, ν −∞ ν and the second order derivative U ′′ (z) =

 1 αK(z) + βK(z − Z) ν   Z z  1 x−z  − 2 exp αK(x) + βK(x − Z) dx. ν −∞ ν

In particular Z

x   αK(x) + βK(x − Z) dx > 0, ν −∞ Z 0   1 x  U ′ (Z) = exp αK(x + Z) + βK(x) dx > 0. ν −∞ ν U ′ (0) =

1 ν

0

exp

We will use certain compatibility conditions to explore the existence and uniqueness of the wave speed ν and the positive number Z. For this purpose, we construct the following auxiliary functions Z −Z Z 0 x   α +β K(x)dx − exp αK(x) + βK(x − Z) dx, 2 ν −∞ −∞ Z 0 Z Z   x  β exp αK(x + Z) + βK(x) dx. K(x)dx + − g(ν, Z) = α 2 ν −∞ −∞ f (ν, Z) =

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To apply the implicit function theorem to establish the existence and uniqueness, let us compute the first order partial derivatives Z 0 x   ∂f 1 |x| exp =− 2 αK(x) + βK(x − Z) dx, ∂ν ν −∞ ν Z 0   ∂f β x exp K(x − Z)dx, =− ∂Z ν −∞ ν Z 0 x   1 ∂g |x| exp =− 2 αK(x + Z) + βK(x) dx, ∂ν ν −∞ ν Z 0   x α ∂g exp = K(x + Z)dx. ∂Z ν −∞ ν For nonnegative kernel functions and for all ν > 0 and Z > 0, it is not difficult to find that ∂f ∂f ∂g ∂g < 0, < 0, < 0, > 0. ∂ν ∂Z ∂ν ∂Z These estimates are also true for Mexican hat kernel functions and upside down Mexican hat kinds of kernel functions. Therefore, by the equations f (ν, Z) = θ and g(ν, Z) = Θ, respectively, we conclude that the functions ν ≡ A(Z) and ν ≡ B(Z) are well defined on R+ , such that f (A(Z), Z) = θ and g(B(Z), Z) = Θ. Set A0 = lim A(Z),

A+ = lim A(Z),

B0 = lim B(Z),

B+ = lim B(Z).

Z→0

Z→0

Z→∞

Z→∞

The existence and uniqueness of each of these limits A0 , A+ , B0 , B+ are guaranteed by the conditions on the positive parameters α, β, θ and Θ. The detailed analysis is very similar to the existence and uniqueness of the first wave speed ν1 and is omitted. The numbers A0 , A+ , B0 and B+ satisfy the following equations   Z 0 x α+β K(x)dx = θ, − (α + β) exp 2 A 0 −∞   Z 0 α x K(x)dx = θ, exp −α 2 A+ −∞   Z 0 x α+β K(x)dx = Θ, exp − (α + β) 2 B0 −∞   Z 0 β x α+ −β K(x)dx = Θ, exp 2 B+ −∞ respectively. Differentiating f (A(Z), Z) = θ and g(B(Z), Z) = Θ with respect to Z, we obtain ∂f ∂g ∂g ∂f (A(Z), Z)A′ (Z) + (A(Z), Z) = 0, (B(Z), Z)B′ (Z) + (B(Z), Z) = 0. ∂ν ∂Z ∂ν ∂Z Hence A′ (Z) = −

∂Z f (A(Z), Z) < 0, ∂ν f (A(Z), Z)

B ′ (Z) = −

∂Z g(B(Z), Z) > 0. ∂ν g(B(Z), Z)

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Linghai Zhang

Therefore, A is strictly decreasing and B is strictly increasing on R+ . Note that A0 > B0 and A+ < B+ due to the condition α(Θ − α) < βθ. Thus, A = A(Z) and B = B(Z) intersect only once in the (ν, Z) plane. Now we obtain a unique pair (ν(Z0 ), Z0 ), such that f (ν(Z0 ), Z0 ) = θ and g(ν(Z0 ), Z0 ) = Θ. The existence and uniqueness of the third front are proved.

2.5

ν

2

B∞

A0

1.5

B0

1

0.5

0

A∞

0.5

1 Z

1.5

2

Figure 8. Graphs of the two monotone curves ν = A(Z) and ν = B(Z). Let ν3 = ν(Z0 ), z = x + ν3 t and P (z, t) = u(x, t). The given integral differential equation is equivalent to the following integral differential equation Z Pt + ν3 Pz + P = α K(z − y)H(P (y, t) − θ)dy R Z K(z − y)H(P (y, t) − Θ)dy. +β R

The third traveling wave front is a stationary solution, we linearize it to get β α K(z − Z)p(Z, t). pt + ν3 pz + p = ′ K(z)p(0, t) + ′ U (0) U (Z) For convenience, we define a linear differential operator L from the Banach space C 1 (R) ∩ L∞ (R) to the Banach space C 0 (R) ∩ L∞ (R) by Lψ = −ν3 ψ ′ − ψ +

α β K(z)ψ(0) + ′ K(z − Z)ψ(Z). U ′ (0) U (Z)

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The essential spectrum of L is easy to calculate. It is a vertical line {λ ∈ C: Reλ = −1} in the left half plane, presenting no threat to the stability. It remains to investigate if the eigenvalues of L are also safe. Substituting solutions of the form p(z, t) = exp(λt)ψ(z) into the differential equation, where λ ∈ C is a complex number and ψ ∈ C 1 (R) ∩ L∞ (R), we obtain the following eigenvalue problem ν3 ψ ′ + (λ + 1)ψ =

α U ′ (0)

K(z)ψ(0) +

β U ′ (Z)

K(z − Z)ψ(Z).

Let us study a general eigenvalue problem ν3 ψ ′ + (λ + 1)ψ + ϕ =

α U ′ (0)

K(z)ψ(0) +

β U ′ (Z)

K(z − Z)ψ(Z),

where ϕ ∈ C 0 (R)∩L∞ (R). It is a nonhomogeneous, first order, linear differential equation. The general solution is     Z z λ+1 λ+1 αψ(0) ψ(z) = C(λ) exp − exp − z + (z − x) K(x)dx ν3 ν3 U ′ (0) −∞ ν3   Z z λ+1 βψ(Z) exp − (z − x) K(x − Z)dx + ν3 U ′ (Z) −∞ ν3   Z z λ+1 1 exp − (z − x) ϕ(x)dx, − ν3 −∞ ν3 where C(λ) is a complex function of λ to be determined later. It is simple to show that ψ is bounded on R if and only if C(λ) = 0. To determine the constant C(λ), we let z = 0 and z = Z, respectively, to get     Z 0 λ+1 α exp x K(x)dx ψ(0) C(λ) = 1 − ν3 U ′ (0) −∞ ν3     Z 0 β λ+1 − exp x K(x − Z)dx ψ(Z) ν3 U ′ (Z) −∞ ν3   Z λ+1 1 0 exp x ϕ(x)dx, + ν3 −∞ ν3 and       Z 0 λ+1 λ+1 α C(λ) exp − exp Z =− x K(x + Z)dx ψ(0) ν3 ν3 U ′ (0) −∞ ν3     Z 0 λ+1 β exp x K(x)dx ψ(Z) + 1− ν3 U ′ (Z) −∞ ν3   Z λ+1 1 0 exp x ϕ(x + Z)dx. + ν3 −∞ ν3 We consider the cases ψ(0) 6= 0 or ψ(Z) 6= 0. By eliminating ψ(Z), we get 

β C(λ) 1 − ν3 U ′ (Z)

Z

0

−∞

exp



 λ+1 x K(x)dx ν3

110

Linghai Zhang    0 λ+1 λ+1 β x K(x − Z)dx exp − Z exp + ν3 U ′ (Z) −∞ ν3 ν3     Z 0 α λ+1 = 1− x K(x)dx exp ′ ν3 U (0) −∞ ν3     Z 0 β λ+1 × 1− x K(x)dx ψ(0) exp ν3 U ′ (Z) −∞ ν3     Z 0 λ+1 α x K(x + Z)dx exp − ν3 U ′ (0) −∞ ν3     Z 0 β λ+1 × x K(x − Z)dx ψ(0) exp ν3 U ′ (Z) −∞ ν3    Z 0     Z 0 β λ+1 1 λ+1 + 1− x K(x)dx exp x ϕ(x)dx exp ν3 U ′ (Z) −∞ ν3 ν3 −∞ ν3    Z 0     Z 0 β λ+1 1 λ+1 + x K(x − Z)dx x ϕ(x + Z)dx . exp exp ν3 U ′ (Z) −∞ ν3 ν3 −∞ ν3 Z



Note that   Z 0 β λ+1 F(λ) ≡ 1 − exp x K(x)dx ν3 U ′ (Z) −∞ ν3     Z 0 λ+1 λ+1 β exp x K(x − Z)dx exp − Z 6= 0. + ν3 U ′ (Z) −∞ ν3 ν3 Define the open, unbounded domain Ω = {λ ∈ C: Reλ > −1 − ρ} and construct the Evans function     Z 0 α λ+1 E3 (λ) = 1 − exp x K(x)dx ν3 U ′ (0) −∞ ν3     Z 0 λ+1 β exp x K(x)dx × 1− ν3 U ′ (Z) −∞ ν3     Z 0 λ+1 α exp x K(x + Z)dx − ν3 U ′ (0) −∞ ν3     Z 0 β λ+1 × exp x K(x − Z)dx . (35) ν3 U ′ (Z) −∞ ν3 Then E3 ′ (λ)        Z 0 Z 0 β λ+1 λ+1 α x K(x)dx x K(x)dx 1 − |x| exp exp = ν3 2 U ′ (0) −∞ ν3 ν3 U ′ (Z) −∞ ν3        Z 0 Z 0 α β λ+1 λ+1 + 1− x K(x)dx x K(x)dx exp |x| exp ν3 U ′ (0) −∞ ν3 ν3 2 U ′ (Z) −∞ ν3        Z 0 Z 0 α β λ+1 λ+1 − x K(x + Z)dx x K(x−Z)dx |x| exp exp ν3 2 U ′ (0) −∞ ν3 ν3 U ′ (Z) −∞ ν3        Z 0 Z 0 α β λ+1 λ+1 − x K(x+Z)dx x K(x−Z)dx . exp |x| exp ν3 U ′ (0) −∞ ν3 ν3 2 U ′ (Z) −∞ ν3

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

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Note that E3 (λ) is also well defined when λ belongs to the essential spectrum. This may be the main difference between the Evans functions of integral differential equations and the Evans functions of partial differential equations. Furthermore, if ϕ = 0, then the complex number λ0 ∈ C is an eigenvalue of the linear differential operator L if and only if it is a zero of the Evans function because C(λ)F(λ) = E3 (λ)ψ(λ, 0). The algebraic multiplicity of any eigenvalue λ0 of L is equal to the order of λ0 as a zero of E3 (λ). Note that lim E3 (λ) = 1.

|λ|→∞

1 , for all 2 |λ| > M . In another word, any complex number λ with Reλ ≥ 0 and |λ| > M is not an eigenvalue of L. As well known, the number λ = 0 is a neutral eigenvalue of L, due to translation invariance of the traveling wave front. To investigate if λ = 0 is simple, let us consider the variational equation Therefore, there exists a sufficiently large constant M > 0, such that |E(λ)| >

ν3 ψ ′ + ψ + U ′ =

α U ′ (0)

K(z)ψ(0) +

β U ′ (Z)

K(z − Z)ψ(Z).

The general solution of this equation is     Z z x−z 1 z ψ(z) = C exp − exp − U ′ (x)dx ν3 ν3 −∞ ν3   Z z x−z αψ(0) K(x)dx exp + ν3 U ′ (0) −∞ ν3   Z z x−z βψ(Z) exp K(x − Z)dx. + ′ ν3 U (Z) −∞ ν3 The solution of the variational equation is unbounded because     Z 0    Z 0 x 1 β x 1 ′ exp 1− K(x)dx U (x)dx exp F(0) ν3 U ′ (0) −∞ ν3 ν3 −∞ ν3     Z 0    Z 0 1 x β 1 x ′ + exp K(x − Z)dx U (x + Z)dx . exp F(0) ν3 U ′ (Z) −∞ ν3 ν3 −∞ ν3

C=

This means that λ = 0 is a simple eigenvalue of L. Based on Theorem 0, the stability of the third front is proved.

3.2.

The Traveling Wave Backs

Next we will consider the scalar integral differential equation (10) with an appropriate constant. 1 Theorem 6. Suppose that 0 < 2θ < α < Θ < (α + β) and α(Θ − α) < βθ. There exist 2 exactly three stable traveling wave solutions Uback−1 , Uback−2 and Uback−3 to the integral differential equation Z ∂u + u + w = α K(x − y)H(u(y, t) − θ)dy ∂t R

112

Linghai Zhang Z +β K(x − y)H(u(y, t) − Θ)dy R

with the same speeds as the corresponding fronts Ufront−1 , Ufront−2 and Ufront−3 , respectively. Moreover   Z z Z ∞ x−z K(x)dx − α + 2θ, exp K(x)dx + α Uback−1 (z) = α ν1 −∞ z   Z z Z ∞ x−z exp K(x)dx + β Uback−2 (z) = β K(x)dx − α − β + 2Θ, ν2 −∞ z Z ∞ Z ∞ K(x)dx K(x)dx + β Uback−3 (z) = α z−Z z   Z z x−z [αK(x) + βK(x − Z)]dx − W, exp + ν3 −∞ where W = Wback and Z = Zback . These traveling wave solutions cross their thresholds in the following way. (I) The first traveling wave solution Uback−1 crosses the small threshold θ only once. It does not cross the large threshold Θ. (II) The second traveling wave solution Uback−2 crosses the large threshold Θ only once. It does not cross the small threshold θ. (III) The third traveling wave solution Uback−3 crosses the small threshold θ once and the large threshold Θ once. Proof. The traveling wave equations for the first two traveling wave solutions are Z ∞ Z ′ K(x)dx, ν1 U + U + α − 2θ = α K(z − y)H(U (y) − θ)dy = α z

R

and ′

ν2 U + U + α + β − 2Θ = β

Z

R

K(z − y)H(U (y) − Θ)dy = β

Z

∞

K(x)dx.

z

Therefore, the first two traveling wave solutions are given by   Z ∞ Z z x−z Uback−1 (z) = α K(x)dx + α exp K(x)dx − α + 2θ, ν1 z −∞   Z α z x−z ′ Uback−1 (z) = − exp K(x)dx, ν1 −∞ ν1   Z z x−z α α exp K(x)dx, Uback−1 ′′ (z) = − K(z) + 2 ν1 ν1 −∞ ν1   Z 0 x α K(x)dx − α + 2θ = θ, exp Uback−1 (0) = + α 2 ν 1 −∞

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113

w

α+β

w=u/γ α back−3

back−2

back−1

O

α+β

α

v u+w=0

u+w=α

u

u+w=α+β

Figure 9. Phase space portrait of the three traveling wave solutions of the integral differential equation (10).

Uback−1

′

α (0) = − ν1

Z

0

−∞

exp



x ν1



K(x)dx = −

 1 α −θ , ν1 2

and Z

∞

Z

z



x−z ν2



K(x)dx − α − β + 2Θ, exp K(x)dx + β −∞   Z x−z β z exp K(x)dx, Uback−2 ′ (z) = − ν2 −∞ ν2   Z z x−z β β ′′ K(x)dx, exp Uback−2 (z) = − K(z) + 2 ν2 ν2 −∞ ν2   Z 0 x β K(x)dx − α − β + 2Θ = Θ, exp Uback−2 (0) = + β 2 ν2 −∞     Z β 0 x 1 β ′ Uback−2 (0) = − exp K(x)dx = − α+ −Θ . ν2 −∞ ν2 ν2 2 Uback−2 (z) = β

z

Let us investigate the existence and uniqueness of the third traveling wave solution Uback−3 . As before, we assume that U (0) = θ, U > θ on (−∞, 0) and U < θ on (0, ∞); similarly

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Linghai Zhang

U (Z) = Θ, U > Θ on (−∞, Z) and U < Θ on (Z, ∞), where Z = Zback < 0 and W = Wback > 0 are constants to be determined later. Then we have the traveling wave equation Z ∞ Z ∞ ′ ν3 U + U + W = α K(x)dx + β K(x)dx. z

z−Z

Solving the equation, we find the solution Z ∞ Z ∞ K(x)dx − W K(x)dx + β U (z) = α z−Z z   Z z x−z exp + [αK(x) + βK(x − Z)]dx. ν3 −∞ The derivatives are 1 U (z) = − ν3 ′

Z

z

exp

−∞





x−z ν3

[αK(x) + βK(x − Z)]dx

and 1 1 U (z) = − [αK(z) + βK(z − Z)] + 2 ν3 ν3 ′′

Z

z

exp

−∞



x−z ν3



[αK(x) + βK(x − Z)]dx.

The compatible conditions are α α

Z

Z

∞

K(x)dx + β

0 ∞

K(x)dx + β

Z

∞

K(x)dx +

−Z Z ∞

K(x)dx +

0

Z

Z

0

−∞ Z 0

exp exp

−∞





x ν3 x ν3

 

[αK(x) + βK(x − Z)]dx − W = θ, [αK(x + Z) + βK(x)]dx − W = Θ.


To establish the existence and uniqueness of (W, Z) = Wback , Zback , let us define the following functions f (W, Z) = α

Z

∞

K(x)dx + β

0

Z

∞

K(x)dx +

−Z

Z

exp



x ν3



[αK(x) + βK(x − Z)]dx − W,

exp



x ν3



[αK(x + Z) + βK(x)]dx − W.

0

−∞

and g(W, Z) = α

Z

∞

Z

K(x)dx + β

Z

0

∞

K(x)dx +

Z

0

−∞

on (0, ∞) × (−∞, 0). The first order partial derivatives of these functions are ∂f = −1, ∂W   Z x β 0 ∂f exp K(x − Z)dx, = ∂Z ν3 −∞ ν3 ∂g = −1, ∂W   Z ∂g x α 0 exp K(x + Z)dx. =− ∂Z ν3 −∞ ν3

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Therefore, by using the equations f (W, Z) = θ and g(W, Z) = Θ and the implicit function theorem, we obtain two well defined functions W ≡ A(Z) and W ≡ B(Z), such that f (A(Z), Z) = θ and g(B(Z), Z) = Θ. Clearly, there exist the limits A0 = lim A(Z),

A− = lim A(Z),

B0 = lim B(Z),

B− = lim B(Z),

Z→0

Z→0

Z→−∞

Z→−∞

and they satisfy   Z 0 α+β x K(x)dx − A0 = θ, exp + (α + β) 2 ν3 −∞   Z 0 x α K(x)dx − A− = θ, +α exp 2 ν3 −∞   Z 0 α+β x K(x)dx − B0 = Θ, exp + (α + β) 2 ν3 −∞   Z 0 x β K(x)dx − B− = Θ. exp α+ +β 2 ν3 −∞ Differentiating the equations f (A(Z), Z) = θ and g(B(Z), Z) = Θ with respect to Z, we obtain ∂f ∂f (A(Z), Z)A′ (Z) + (A(Z), Z) = 0, ∂W ∂Z ∂g ∂g (B(Z), Z)B′ (Z) + (B(Z), Z) = 0. ∂W ∂Z Therefore A′ (Z) = −

∂Z f (A(Z), Z) , ∂W f (A(Z), Z)

B ′ (Z) = −

∂Z g(B(Z), Z) . ∂W g(B(Z), Z)

It is easy to see that A is a strictly increasing function of Z and B is a strictly decreasing function of Z on (−∞, 0). Note that A− < B− and A0 > B0 . Therefore the two functions A = A(Z) and B = B(Z) intersect exactly once. Finally, we obtain a unique solution (W, Z) = (Wback , Zback ) to the equations f (W, Z) = θ and g(W, Z) = Θ. The existence and uniqueness of the third traveling wave back are proved. The stability of the third back is very similar to that of the third front and is omitted. 1 Remark 4. The conditions 0 < 2θ < α < Θ < (α + β) guarantee that the existence, 2 uniqueness and stability of each of the first two traveling wave fronts as well as the first two traveling wave backs are true; and the conditions 0 < α(Θ − α) < βθ guarantee that the existence, uniqueness and stability of the third traveling wave front and the third traveling wave back are true.

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Linghai Zhang

2.5

W

2

A

B

0

−∞

1.5

A−∞ 1

0.5 −2

B0

−1.5

−1 Z

−0.5

0

Figure 10. Graphs of the two monotone curves W = A(Z) and W = B(Z).

3.3. The Traveling Pulse Solutions There are three fixed points to the model equations (31)–(32):     αγ α (α + β)γ α + β (U0 , W0 ) = (0, 0), (U1 , W1 ) = , (U2 , W2 ) = . , , 1+γ 1+γ 1+γ 1+γ The second fixed point exists if αγ > (1 + γ)θ and the third fixed point
exists if (α + β)γ > (1 + γ)Θ. We are interested in traveling waves U (ε, ·), W (ε, ·) connecting one fixed point (Ui , Wi ) at z = −∞ to another fixed point (Uj , Wj ) at z = ∞, where i = 0, 1 and j = 0, 1, 2. Note that i = j for homoclinic orbits and i < j for heteroclinic orbits. Theorem 7. Suppose that the positive parameters satisfy the conditions 1 0 < 2θ < α < Θ < (α + β), α(Θ − α) < βθ, 2 (α + β)γ αγ 6= θ, 6= Θ, 0 < ε ≪ 1. 1+γ 1+γ
There exist exactly three exponentially stable, fast homoclinic orbits U (ε, ·), W (ε, ·) =
Ufast−pulse (ε, ·), Wfast−pulse (ε, ·) to system (31)–(32). They satisfy the traveling wave

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

117

equations ′

Z

K(z − y)H(U (y) − θ)dy Z K(z − y)H(U (y) − Θ)dy, +β

ν(ε)U + U + W = α

R

R

′

ν(ε)W = ε(U − γW ), and the following boundary conditions, respectively: (I) lim

z→±∞


Upulse−1 (ε, z), Wpulse−1 (ε, z) = (0, 0), if

αγ <θ 1+γ

and

(α + β)γ 6= Θ. 1+γ

(II) lim

z→±∞


Upulse−2 (ε, z), Wpulse−2 (ε, z) =

if

αγ >θ 1+γ

and



α αγ , 1+γ 1+γ



,

(α + β)γ < Θ. 1+γ

(III) lim

z→±∞


Upulse−3 (ε, z), Wpulse−3 (ε, z) = (0, 0), if

αγ 6= θ 1+γ

and

(α + β)γ < Θ. 1+γ

These homoclinic orbits cross their thresholds in the following way.
(I) The first homoclinic orbit Upulse−1 (ε, ·), Wpulse−1 (ε, ·) crosses the small threshold θ twice but does not cross the large threshold Θ.
(II) The second homoclinic orbit Upulse−2 (ε, ·), Wpulse−2 (ε, ·) crosses the large threshold Θ twice but does not cross the small threshold θ.
(III) The third homoclinic orbit Upulse−3 (ε, ·), Wpulse−3 (ε, ·) crosses the small threshold θ twice and the large threshold Θ twice. There are precisely three exponentially unstable,
slow homoclinic orbits
U (ε, ·), W (ε, ·) = Uslow−pulse (ε, ·), Wslow−pulse (ε, ·) with the same boundary conditions and crossings as above. Overall, there exist three fast stable pulses and three slow unstable pulses. The phase plane portraits of the three fast homoclinic orbits. Proof. We are going to present the rigorous mathematical analysis for the existence and uniqueness of all of the traveling pulse solutions. The rigorous analysis and results will play

118

Linghai Zhang 14

w

12 α+β 10 8

w=u/γ

6 α 4 back

2

v

left

right α

front

α+β

u

0

O u+w=0 −2

−2

0

u+w=α+β

u+w=α

2

4

6

8

10

12

Figure 11A. Phase space portrait of the three traveling pulse solutions of the system of integral differential equations (3)-(4). a very important role in applications to mathematical neuroscience. Suppose that α > 0, β > 0, γ > 0, θ > 0 and Θ > 0 are given constants. Let ε > 0, ν > 0, Γ > 0, Λ > 0 and Z > 0 be parameters. First of all, let us derive the traveling wave equations and the pulse solution representations for all of the three homoclinic orbits. (I) Let 0 < 2θ < α < Θ <

α+β , 2

α(Θ − α) < βθ,

and αγ < θ, 1+γ

0 < ε ≪ 1.

We assume that the first traveling pulse solution crosses its threshold θ in the following way: U (0) = U (Z) = θ, U < θ on (−∞, 0) ∪ (Z, ∞) and U > θ on (0, Z), for some constant Z = Zpulse−1 > 0. Moreover, this pulse does not cross the large threshold Θ, namely, U < Θ on R. Under these hypotheses, we have the simpler traveling wave equations Z z K(x)dx, νU ′ + U + W = α z−Z

νW ′ = ε(U − γW ). Solving the differential equations, we get  Z Z z A(ν, ε, x − z) α U (ε, z) = −∞

x

x−Z

 K(ξ)dξ dx,

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119

w

α+β

w=u/γ

back

α

right

left front

v

O u+w=0

α

α+β

u+w=α

u+w=α+β

u

Figure 11B. Phase space portrait of the three traveling pulse solutions of the system of integral differential equations (3)-(4). W (ε, z) =

Z

z

−∞

 Z B(ν, ε, x − z) α

x

x−Z



K(ξ)dξ dx,

where

     ω1 (ε) ω2 (ε) 1 [1 − ω2 (ε)] exp x − [1 − ω1 (ε)] exp x , A(ν, ε, x) = ν[ω1 (ε) − ω2 (ε)] ν ν      ε ω1 (ε) ω2 (ε) B(ν, ε, x) = − exp x + exp x . ν[ω1 (ε) − ω2 (ε)] ν ν If we use integration by parts, then we obtain Z z Z z αγ C(ν, ε, x − z)[K(x) − K(x − Z)]dx, K(x)dx − α U (ε, z) = 1 + γ z−Z −∞ Z z Z z α W (ε, z) = D(ν, ε, x − z)[K(x) − K(x − Z)]dx, K(x)dx − α 1 + γ z−Z −∞ where

     1 1 − ω2 (ε) 1 − ω1 (ε) ω1 (ε) ω2 (ε) C(ν, ε, x) = exp x − exp x , ω1 (ε) − ω2 (ε) ω1 (ε) ν ω2 (ε) ν      1 1 ω1 (ε) ω2 (ε) ε − exp x + exp x . D(ν, ε, x) = ω1 (ε) − ω2 (ε) ω1 (ε) ν ω2 (ε) ν

Note that A(ν, ε, x) =

∂ C(ν, ε, x), ∂x

120

Linghai Zhang

w

α+β

w=u/γ α

back

left

right

v

front α+β

α

O

u+w=0

u+w=α

u

u+w=α+β

Figure 11C. Phase space portrait of the three traveling pulse solutions of the system of integral differential equations (3)-(4).

B(ν, ε, x) =

∂ D(ν, ε, x). ∂x

The wave speed ν = νpulse−1 (ε) and the positive number Z = Zpulse−1 (ε) satisfy Z

Z 0   αγ C(ν, ε, x) K(x) − K(x − Z) dx = α K(x)dx − θ, 1 + γ −Z −∞ Z Z Z 0   αγ K(x)dx − θ. C(ν, ε, x) K(x + Z) − K(x) dx = α 1+γ 0 −∞ 0

As before, by using the implicit function theorem,
we can establish the existence and uniqueness of (ν, Z) = νpulse−1 (ε), Zpulse−1 (ε) , such that U (ε, 0) = U (ε, Z) = θ, for all ε ∈ (0, ε0 ), where 0 < ε0 ≪ 1 is a number. Moreover, we can prove that U < θ on (−∞, 0) ∪ (Z, ∞) and U > θ on (0, Z). Therefore,
the existence and uniqueness of the first homoclinic orbit Upulse−1 (ε, ·), Wpulse−1 (ε, ·) are established. (II) Let 0 < 2θ < α < Θ <

α+β , 2

α(Θ − α) < βθ,

and αγ > θ, 1+γ

(α + β)γ < Θ, 1+γ

0 < ε ≪ 1.

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

121

We assume that the second traveling pulse solution crosses its threshold Θ in the following way: U (0) = U (Z) = Θ, U < Θ on (−∞, 0) ∪ (Z, ∞) and U > Θ on (0, Z), for some constant Z = Zpulse−2 > 0. Moreover, this pulse does not cross the small threshold θ, namely, U > θ on R. Under these hypotheses, we have the new traveling wave equations Z z ′ K(x)dx, νU + U + W = α + β z−Z

′

νW = ε(U − γW ). Solving the differential equations, we get   Z x Z z K(ξ)dξ dx, A(ν, ε, x − z) α + β U (ε, z) = x−Z −∞   Z x Z z K(ξ)dξ dx. B(ν, ε, x − z) α + β W (ε, z) = x−Z

−∞

If we use integration by parts, then we obtain   Z z Z z γ α+β K(x)dx − β C(ν, ε, x − z)[K(x) − K(x − Z)]dx, 1+γ z−Z −∞   Z z Z z 1 α+β K(x)dx − β D(ν, ε, x − z)[K(x) − K(x − Z)]dx. W (ε, z) = 1+γ z−Z −∞

U (ε, z) =

The wave speed ν = νpulse−2 (ε) and the positive number Z = Zpulse−2 (ε) satisfy

  Z 0 γ β K(x)dx − Θ, α+β 1+γ −∞ −Z   Z Z Z 0   γ K(x)dx − Θ. α+β C(ν, ε, x) K(x + Z) − K(x) dx = β 1+γ 0 −∞ Z

0

  C(ν, ε, x) K(x) − K(x − Z) dx =

As before, by using the implicit function theorem,
we can establish the existence and uniqueness of (ν, Z) = νpulse−2 (ε), Zpulse−2 (ε) , such that U (ε, 0) = U (ε, Z) = Θ, for all ε ∈ (0, ε0 ), where 0 < ε0 ≪ 1 is a number. Moreover, we can prove that U < Θ on (−∞, 0) ∪ (Z, ∞) and U > Θ on (0, Z). Therefore,
the existence and uniqueness of the second homoclinic orbit Upulse−2 (ε, ·), Wpulse−2 (ε, ·) are established. (III) Let 0 < 2θ < α < Θ <

α+β , 2

α(Θ − α) < βθ,

and (α + β)γ < Θ, 1+γ

0 < ε ≪ 1.

We assume that the third traveling pulse solution crosses its thresholds in the following way: U (0) = U (Z) = θ, U < θ on (−∞, 0) ∪ (Z, ∞) and U > θ on (0, Z); similarly, U (Γ) = U (Λ) = Θ, U < Θ on (−∞, Γ) ∪ (Λ, ∞) and U > Θ on (Γ, Λ), where 0 < Γ < Λ < Z,

122

Linghai Zhang

Z = Zpulse−3 , are real numbers to be determined later. Under these hypotheses, we have the new traveling wave equations Z z−Γ Z z K(x)dx, K(x)dx + β νU ′ + U + W = α z−Λ

z−Z

νW ′ = ε(U − γW ). By applying standard techniques in differential equations, it is easy to solve the following boundary value problems   ′   U 1 1 U + ν W −ε γε W    Z z Z z−Γ 1 , K(x)dx K(x)dx + β = α 0 z−Λ z−Z     0 U (z) , = lim z→−∞ 0 W (z) to obtain the solution Z U (ε, z) = W (ε, z) =

z

−∞ Z z

 Z A(ν, ε, x − z) α

−∞



B(ν, ε, x − z) α

x

K(ξ)dξ + β

x−Z Z x

x−Z

Z

K(ξ)dξ + β

x−Γ



K(ξ)dξ dx,

x−Λ Z x−Γ x−Λ



K(ξ)dξ dx.

By using integration by parts, we obtain the solution representation   Z z Z z−Γ γ U (ε, z) = K(x)dx K(x)dx + β α 1+γ z−Λ z−Z Z z  − C(ν, ε, x − z) α[K(x) − K(x − Z)] + β[K(x − Γ) − K(x − Λ)] dx, −∞

  Z z Z z−Γ 1 K(x)dx K(x)dx + β W (ε, z) = α 1+γ z−Λ z−Z Z z  D(ν, ε, x − z) α[K(x) − K(x − Z)] + β[K(x − Γ) − K(x − Λ)] dx. − −∞

The solution converges to zero exponentially fast, as z → ±∞. We have made the assumption that the third homoclinic orbit crosses the small threshold θ at z = 0 and z = Z and it crosses the large threshold Θ at z = Γ and z = Λ. Now we have to make sure that it really crosses the thresholds at the desired points. To establish the existence of the wave speed ν and the numbers Γ, Λ and Z, let us define the following auxiliary functions by using the four values U (ε, 0), U (ε, Γ), U (ε, Λ) and U (ε, Z). Define ( Z ) Z −Γ 0 γ α K(x)dx + β K(x)dx f1 (ν, Γ, Λ, Z) = 1+γ −Z −Λ Z 0  − C(ν, ε, x) α[K(x) − K(x − Z)] + β[K(x − Γ) − K(x − Λ)] dx, −∞

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

123

( Z ) Z 0 Γ γ f2 (ν, Γ, Λ, Z) = α K(x)dx + β K(x)dx 1+γ Γ−Z Γ−Λ Z 0  − C(ν, ε, x) α[K(x + Γ) − K(x + Γ − Z)] + β[K(x) − K(x + Γ − Λ)] dx, −∞

( Z ) Z Λ−Γ Λ γ f3 (ν, Γ, Λ, Z) = α K(x)dx + β K(x)dx 1+γ Λ−Z 0 Z 0  − C(ν, ε, x) α[K(x + Λ) − K(x + Λ − Z)] + β[K(x + Λ − Γ) − K(x)] dx, −∞

) ( Z Z Z−Γ Z γ f4 (ν, Γ, Λ, Z) = K(x)dx α K(x)dx + β 1+γ Z−Λ 0 Z 0  − C(ν, ε, x) α[K(x + Z) − K(x)] + β[K(x + Z − Γ) − K(x + Z − Λ)] dx. −∞

As before, by using the implicit function theorem, we find
that there exists a unique vector-valued solution (ν, Γ, Λ, Z) = ν(ε), Γ(ε), Λ(ε), Z(ε) , such that f1 (ν, Γ, Λ, Z) = f4 (ν, Γ, Λ, Z) = θ and f2 (ν, Γ, Λ, Z) = f3 (ν, Γ, Λ, Z) = Θ, that is to say, U (ε, 0) = U (ε, Z(ε)) = θ and U (ε, Γ(ε)) = U (ε, Λ(ε)) = Θ, for all ε ∈ (0, ε0 ), where 0 < ε0 ≪ 1 is a number. It is not difficult to see that (ν, Γ, Λ, Z) is a smooth function of ε. See the next few subsections for the proof of the exponential stability. The proof of Theorem 7 is finished.

3.4.

The Other Traveling Wave Solutions

Theorem 8. Suppose that 1 0 < 2θ < α < Θ < (α + β), 2 αγ (α + β)γ θ< <Θ< , 1+γ 1+γ

α(Θ − α) < βθ, 0 < ε ≪ 1.


There are precisely three
exponentially stable heteroclinic orbits U (ε, ·), W (ε, ·) = Uwave (ε, ·), Wwave (ε, ·) to system (31)–(32). They satisfy the traveling wave equations Z ′ ν(ε)U + U + W = α K(z − y)H(U (y) − θ)dy R Z K(z − y)H(U (y) − Θ)dy, +β R

ν(ε)W ′ = ε(U − γW ), and the following boundary conditions, respectively: (I) lim

z→−∞


Uwave−1 (ε, z), Wwave−1 (ε, z) = (0, 0),

124

Linghai Zhang  
αγ α lim Uwave−1 (ε, z), Wwave−1 (ε, z) = , , z→∞ 1+γ 1+γ   d d Uwave−1 (ε, z), Wwave−1 (ε, z) = (0, 0). lim z→±∞ dz dz

(II)  
αγ α lim Uwave−2 (ε, z), Wwave−2 (ε, z) = , , z→−∞ 1+γ 1+γ  
(α + β)γ α + β lim Uwave−2 (ε, z), Wwave−2 (ε, z) = , , z→∞ 1+γ 1+γ   d d lim Uwave−2 (ε, z), Wwave−2 (ε, z) = (0, 0). z→±∞ dz dz (III)
Uwave−3 (ε, z), Wwave−3 (ε, z) = (0, 0), z→−∞  
(α + β)γ α + β , , lim Uwave−3 (ε, z), Wwave−3 (ε, z) = z→∞ 1+γ 1+γ   d d lim Uwave−3 (ε, z), Wwave−3 (ε, z) = (0, 0). z→±∞ dz dz lim

Furthermore, the second component of each of the heteroclinic orbits is increasing on R. The heteroclinic orbits cross their thresholds in the following way.
(I) The first heteroclinic orbit Uwave−1 (ε, ·), Wwave−1 (ε, ·) crosses the small threshold θ only once and it does not cross the large threshold Θ.
(II) The second heteroclinic orbit Uwave−2 (ε, ·), Wwave−2 (ε, ·) crosses the large threshold Θ only once and it does not cross the small threshold θ.
(III) The third heteroclinic orbit Uwave−3 (ε, ·), Wwave−3 (ε, ·) crosses the small threshold θ only once and the large threshold Θ only once. The phase plane portraits of the three heteroclinic orbits. Proof. We will use the same notations as before. Suppose that 0 < 2θ < α < Θ <

α+β , 2

α(Θ − α) < βθ,

and θ<

αγ (α + β)γ <Θ< , 1+γ 1+γ

0 < ε ≪ 1.

Let us derive the traveling wave equations and the traveling wave solution representations for all of the three heteroclinic orbits.
(I) We assume that the first heteroclinic orbit Uwave−1 (ε, ·), Wwave−1 (ε, ·) crosses its threshold θ in the following way: U (0) = θ, U < θ on (−∞, 0) and U > θ on (0, ∞).

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

125

w

α+β

w=u/γ

α

v

O u+w=0

α

α+β

u+w=α

u+w=α+β

u

Figure 12A. Phase space portrait of the three traveling wave solutions of the system of integral differential equations (3)-(4). Moreover, this orbit does not cross the large threshold Θ, namely, U < Θ on R. Under these hypotheses, we have the simpler traveling wave equations Z z ′ νU + U + W = α K(x)dx, −∞

′

νW = ε(U − γW ). By applying standard techniques in differential equations, we find the solution   Z x Z z K(ξ)dξ dx, A(ν, ε, x − z) α U (ε, z) = −∞ −∞   Z x Z z K(ξ)dξ dx. B(ν, ε, x − z) α W (ε, z) = −∞

−∞

By using integration by parts, we obtain Z z Z z αγ C(ν, ε, x − z)K(x)dx, K(x)dx − α U (ν, ε, z) = 1 + γ −∞ −∞ Z z Z z α W (ν, ε, z) = D(ν, ε, x − z)K(x)dx. K(x)dx − α 1 + γ −∞ −∞ The wave speed ν = νwave−1 (ε) satisfies Z 0 Z 0 αγ K(x)dx − θ. C(ν, ε, x)K(x)dx = α 1 + γ −∞ −∞

126

Linghai Zhang

w

α+β

w=u/γ

α

v

O u+w=0

α

α+β

u+w=α

u+w=α+β

u

Figure 12B. Phase space portrait of the three traveling wave solutions of the system of integral differential equations (3)-(4). As before, we can establish the existence and uniqueness of ν = νwave−1 (ε), such that U (ε, 0) = θ, for all ε ∈ (0, ε0 ), where 0 < ε0 ≪ 1 is a number. Moreover, we can prove that U < θ on (−∞, 0) and U > θ on (0, ∞). Therefore,
the existence and uniqueness of the first heteroclinic orbit Uwave−1 (ε, ·), Wwave−1 (ε, ·) are established.
(II) We assume that the second heteroclinic orbit Uwave−2 (ε, ·), Wwave−2 (ε, ·) crosses its threshold Θ in the following way: U (0) = Θ, U < Θ on (−∞, 0) and U > Θ on (0, ∞). Moreover, this orbit does not cross the small threshold θ, namely, U > θ on R. Under these hypotheses, we have the reduced traveling wave equations Z z νU ′ + U + W = α + β K(x)dx, −∞

νW ′ = ε(U − γW ). The solution of the system is   Z x Z z K(ξ)dξ dx, A(ν, ε, x − z) α + β U (ε, z) = −∞ −∞   Z z Z x W (ε, z) = B(ν, ε, x − z) α + β K(ξ)dξ dx. −∞

−∞

By using integration by parts, we obtain   Z z Z z γ U (ν, ε, z) = C(ν, ε, x − z)K(x)dx, K(x)dx − β α+β 1+γ −∞ −∞

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

127

w

α+β

w=u/γ

α

v

O u+w=0

α

α+β

u+w=α

u+w=α+β

u

Figure 12C. Phase space portrait of the three traveling wave solutions of the system of integral differential equations (3)-(4).   Z z Z z 1 W (ν, ε, z) = D(ν, ε, x − z)K(x)dx. K(x)dx − β α+β 1+γ −∞ −∞ The wave speed ν = νwave−2 (ε) satisfies   Z 0 Z 0 γ K(x)dx − Θ. α+β C(ν, ε, x)K(x)dx = β 1+γ −∞ −∞ As before, we can establish the existence and uniqueness of ν = νwave−2 (ε), such that U (ε, 0) = Θ, for all ε ∈ (0, ε0 ), where 0 < ε0 ≪ 1 is a number. Moreover, we can prove that U < Θ on (−∞, 0) and U > Θ on (0, ∞). Therefore,
the existence and uniqueness of the second heteroclinic orbit Uwave−2 (ε, ·), Wwave−2 (ε, ·) are established.
(III) We assume that the third heteroclinic orbit Uwave−3 (ε, ·), Wwave−3 (ε, ·) crosses its thresholds in the following way: U (0) = θ, U < θ on (−∞, 0) and U > θ on (0, ∞); similarly, U (Z) = Θ, U < Θ on (−∞, Z) and U > Θ on (Z, ∞), where Z = Zwave−3 > 0 is a positive number to be determined later. Under these hypotheses, we have the traveling wave equations Z z−Z Z z ′ K(x)dx, K(x)dx + β νU + U + W = α −∞

−∞

′

νW = ε(U − γW ). Solving the traveling wave equations, we find the solution  Z x Z Z z K(ξ)dξ + β A(ν, ε, x − z) α U (ε, z) = −∞

−∞

x−Z



K(ξ)dξ dx, −∞

128 W (ε, z) =

Z

z

−∞

Linghai Zhang  Z x Z K(ξ)dξ + β B(ν, ε, x − z) α

x−Z

K(ξ)dξ dx.

−∞

−∞



By using integration by parts, we obtain the better expressions   Z z Z z−Z γ K(x)dx K(x)dx + β α U (ε, z) = 1+γ −∞ −∞ Z z C(ν, ε, x − z)[αK(x) + βK(x − Z)]dx, − −∞

  Z z Z z−Z 1 W (ε, z) = K(x)dx K(x)dx + β α 1+γ −∞ −∞ Z z − D(ν, ε, x − z)[αK(x) + βK(x − Z)]dx. −∞

To establish the existence and uniqueness of the wave speed ν and the number Z, let us define the following auxiliary functions by using U (ε, 0) and U (ε, Z). Define  Z 0  Z −Z γ α K(x)dx + β K(x)dx g1 (ν, Z) = 1+γ −∞ −∞ Z 0 C(ν, ε, x)[αK(x) + βK(x − Z)]dx, − −∞

g2 (ν, Z) = −

Z

γ 1+γ

 Z α

Z

K(x)dx + β

−∞

Z

0 −∞

 K(x)dx

0

−∞

C(ν, ε, x)[αK(x + Z) + βK(x)]dx.

Now by applying the implicit function theorem, there is a positive number ε0 , 0 < ε0 ≪ 1,

for all ε ∈ (0, ε0 ), there exists a unique pair ν(ε), Z(ε) = νwave−3 (ε), Zwave−3 (ε) , such that

U ε, ν(ε), 0 = θ, U ε, ν(ε), Z = Θ.

Based on the closeness of the heteroclinic orbit to a singular heteroclinic orbit, we can easily conclude that Uz (ε, ν(ε), 0) > 0, Uz (ε, ν(ε), Z) > 0,

U (ε, ν(ε), z) < θ

on (−∞, 0),

U (ε, ν(ε), z) > θ

on

U (ε, ν(ε), z) < Θ on U (ε, ν(ε), z) > Θ on

(0, ∞),

(−∞, Z) , (Z, ∞) .

It is not difficult to understand that

νwave−3 (ε), Zwave−3 (ε) = ν3 − C1 (ε)ε, Zfront − C2 (ε)ε

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

129

is the only choice such that
g1 ν(ε), Z = θ

and


g2 ν(ε), Z = Θ.

Moreover, we have 0 < sup0<ε≤ε0 Ci (ε) < ∞. For each of the three heteroclinic orbits, U (ε, ·) > γW (ε, ·) on R, hence we claim that Wz (ε, ·) > 0 on R. This completes the proof of the existence and uniqueness of the third heteroclinic orbit. See the next few subsections for the proof of the exponential stability. Remark 5. Given the constants α > 0, β > 0 and γ > 0, Theorem 7 and Theorem 8 tell us that many different kinds of stable waves may coexist (ignore the unstable pulses temporarily), depending on the positions of the thresholds θ and Θ. More precisely, (I) αγ there exist one large homoclinic orbit and one small homoclinic orbit if < θ and 1+γ (α + β)γ < Θ. (II) There exist one large homoclinic orbit, one small homoclinic orbit and 1+γ αγ (α + β)γ one small heteroclinic orbit if > θ and < Θ. (III) There exist one large 1+γ 1+γ (α + β)γ αγ < θ and > Θ. (IV) heteroclinic orbit and one small homoclinic orbit if 1+γ 1+γ αγ There exist one large heteroclinic orbit and two small heteroclinic orbits if > θ and 1+γ (α + β)γ > Θ. 1+γ

3.5. The Eigenvalue Problem and the Evans Function
Denote by U (ε, ·), W (ε, ·) any of the homoclinic orbits or any of the heteroclinic orbits of the nonlinear singularly perturbed system of integral differential equations (31)–(32). Setting z = x + ν(ε)t and (P (z, t), Q(z, t)) = (u(x, t), w(x, t)), where ν = ν(ε) represents the wave speed of any of the homoclinic orbits or the wave speed of any of the heteroclinic orbits. We will focus on the stability analysis of the third homoclinic orbit, because this is the most challenging problem. Any other problem is relatively easier and simpler to analyze. We are going to drop the subscripts and superscripts to simplify the notations. The system (31)–(32) becomes Z Pt + ν(ε)Pz + P + Q = α K(z − y)H(P (y, t) − θ)dy R Z K(z − y)H(P (y, t) − Θ)dy, +β R

Qt + ν(ε)Qz = ε(P − γQ).
The traveling wave solution U (ε, ·), W (ε, ·)
is a stationary solution of this nonlinear system. The linearization about U (ε, ·), W (ε, ·) is given by pt + ν(ε)pz + p + q = aK(z)p(0, t) + bK(z − Z(ε))p(Z(ε), t)

+ cK(z − Γ(ε))p(Γ(ε), t) + dK(z − Λ(ε))p(Λ(ε), t),

130

Linghai Zhang qt + ν(ε)qz = ε(p − γq),

where α α , b= , |Uz (ε, 0)| |Uz (ε, Z(ε))| β β , d= . c= |Uz (ε, Γ(ε))| |Uz (ε, Λ(ε))| a=

 ξ and we define the linear differential operator L(ε) from the Banach Setting ψ = η space C 1 (R) ∩ L∞ (R) to the Banach space C 0 (R) ∩ L∞ (R) by   1 1 ′ ψ L(ε)ψ = − ν(ε)ψ − −ε γε   1 0  aK(z)ψ(0) + bK(z − Z(ε))ψ(Z(ε)) + 0 0  + cK(z − Γ(ε))ψ(Γ(ε)) + dK(z − Λ(ε))ψ(Λ(ε)) . (36)

Suppose that p(z, t), q(z, t) = eλt ξ(z), η(z) is a solution of the linear system, where λ ∈ C and (ξ, η) ∈ C 0 (R) ∩ L∞ (R). Then we obtain the eigenvalue problem 

ν(ε)ξz + (λ + 1)ξ + η = aK(z)ξ(λ, ε, 0) + bK(z − Z(ε))ξ(λ, ε, Z(ε))

+ cK(z − Γ(ε))ξ(λ, ε, Γ(ε)) + dK(z − Λ(ε))ξ(λ, ε, Λ(ε)),

ν(ε)ηz + λη = ε(ξ − γη). Let Φ(λ, ε, z) ≡



(37) (38)

      λ + ω1 (ε) λ + ω2 (ε) 1 1 exp − , exp − . z z ω1 (ε) − 1 ω2 (ε) − 1 ν(ε) ν(ε)

Recall that ω1 (ε) =

1 + γε +

p (1 − γε)2 − 4ε , 2

ω2 (ε) =

1 + γε −

p

(1 − γε)2 − 4ε . 2

Let ϕ ∈ C 0 (R) ∩ L∞ (R) and limz→±∞ ϕ(z) = 0. The solution of the general eigenvalue problem L(ε)ψ = λψ + ϕ subject to the homogeneous boundary condition limz→−∞ (ξ(λ, ε, z), η(λ, ε, z)) = (0, 0) is     Z z 1 1 0 a(λ, ε) −1 Φ (λ, ε, x) Φ(λ, ε, z) + ψ(λ, ε, z) = Φ(λ, ε, z) 0 0 b(λ, ε) ν(ε) −∞  × aK(x)ψ(λ, ε, 0) + bK(x − Z(ε))ψ(λ, ε, Z(ε)) +cK(x − Γ(ε))ψ(λ, ε, Γ(ε)) + dK(x − Λ(ε))ψ(λ, ε, Λ(ε)) dx Z z 1 Φ−1 (λ, ε, x)ϕ(x)dx, Φ(λ, ε, z) − ν(ε) −∞   a(λ, ε) is a complex vector-valued function of (λ, ε) to be determined later by where b(λ, ε) some compatibility conditions.

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

3.6.

131

The Compatible Solutions of the Eigenvalue Problem

Using the first component of the solution representation and setting z = 0, z = Γ(ε), z = Λ(ε) and z = Z(ε), respectively, we obtain the following equations   a(λ, ε) ξ(λ, ε, 0) = R(λ, ε, 0) b(λ, ε)     Z 0 a 1 −1 + K(x)dx ξ(λ, ε, 0) R(λ, ε, 0) Φ (λ, ε, x) 0 ν(ε) −∞     Z 0 b 1 −1 K(x − Z(ε))dx ξ(λ, ε, Z(ε)) R(λ, ε, 0) Φ (λ, ε, x) + 0 ν(ε) −∞     Z 0 c 1 −1 K(x − Γ(ε))dx ξ(λ, ε, Γ(ε)) Φ (λ, ε, x) R(λ, ε, 0) + 0 ν(ε) −∞     Z 0 d 1 −1 + K(x − Λ(ε))dx ξ(λ, ε, Λ(ε)) R(λ, ε, 0) Φ (λ, ε, x) 0 ν(ε) −∞   Z 0 1 −1 R(λ, ε, 0) Φ (λ, ε, x)ϕ(x)dx − ν(ε) −∞   a(λ, ε) + M11 (λ, ε)ξ(λ, ε, 0) + M12 (λ, ε)ξ(λ, ε, Γ(ε)) ≡ R(λ, ε, 0) b(λ, ε) +M13 (λ, ε)ξ(λ, ε, Λ(ε)) + M14 (λ, ε)ξ(λ, ε, Z(ε)) − N1 (λ, ε),   a(λ, ε) ξ(λ, ε, Γ(ε)) = R(λ, ε, Γ(ε)) b(λ, ε) # "   Z Γ(ε) a 1 −1 + K(x)dx ξ(λ, ε, 0) Φ (λ, ε, x) R(λ, ε, Γ(ε)) 0 ν(ε) −∞ # "   Z Γ(ε) b 1 K(x − Z(ε))dx ξ(λ, ε, Z(ε)) Φ−1 (λ, ε, x) R(λ, ε, Γ(ε)) + 0 ν(ε) −∞ # "   Z Γ(ε) c 1 K(x − Γ(ε))dx ξ(λ, ε, Γ(ε)) R(λ, ε, Γ(ε)) Φ−1 (λ, ε, x) + 0 ν(ε) −∞ # "   Z Γ(ε) d 1 K(x − Λ(ε))dx ξ(λ, ε, Λ(ε)) R(λ, ε, Γ(ε)) Φ−1 (λ, ε, x) + 0 ν(ε) −∞ " # Z Γ(ε) 1 −1 R(λ, ε, Γ(ε)) Φ (λ, ε, x)ϕ(x)dx − ν(ε) −∞   a(λ, ε) + M21 (λ, ε)ξ(λ, ε, 0) + M22 (λ, ε)ξ(λ, ε, Γ(ε)) ≡ R(λ, ε, Γ(ε)) b(λ, ε) +M23 (λ, ε)ξ(λ, ε, Λ(ε)) + M24 (λ, ε)ξ(λ, ε, Z(ε)) − N2 (λ, ε),   a(λ, ε) ξ(λ, ε, Λ(ε)) = R(λ, ε, Λ(ε)) b(λ, ε) # "   Z Λ(ε) a 1 + K(x)dx ξ(λ, ε, 0) R(λ, ε, Λ(ε)) Φ−1 (λ, ε, x) 0 ν(ε) −∞

132

Linghai Zhang #   Z Λ(ε) b 1 −1 K(x − Z(ε))dx ξ(λ, ε, Z(ε)) Φ (λ, ε, x) R(λ, ε, Λ(ε)) + 0 ν(ε) −∞ # "   Z Λ(ε) c 1 + K(x − Γ(ε))dx ξ(λ, ε, Γ(ε)) Φ−1 (λ, ε, x) R(λ, ε, Λ(ε)) 0 ν(ε) −∞ # "   Z Λ(ε) d 1 K(x − Λ(ε))dx ξ(λ, ε, Λ(ε)) Φ−1 (λ, ε, x) R(λ, ε, Λ(ε)) + 0 ν(ε) −∞ " # Z Λ(ε) 1 R(λ, ε, Λ(ε)) Φ−1 (λ, ε, x)ϕ(x)dx − ν(ε) −∞   a(λ, ε) + M31 (λ, ε)ξ(λ, ε, 0) + M32 (λ, ε)ξ(λ, ε, Γ(ε)) ≡ R(λ, ε, Λ(ε)) b(λ, ε) +M33 (λ, ε)ξ(λ, ε, Λ(ε)) + M34 (λ, ε)ξ(λ, ε, Z(ε)) − N3 (λ, ε),   a(λ, ε) ξ(λ, ε, Z(ε)) = R(λ, ε, Z(ε)) b(λ, ε) # "   Z Z(ε) a 1 −1 = K(x)dx ψ(λ, ε, 0) Φ (λ, ε, x) R(λ, ε, Z(ε)) 0 ν(ε) −∞ # "   Z Z(ε) b 1 −1 K(x − Z(ε))dx ξ(λ, ε, Z(ε)) Φ (λ, ε, x) R(λ, ε, Z(ε)) + 0 ν(ε) −∞ # "   Z Z(ε) c 1 K(x − Γ(ε))dx ξ(λ, ε, Γ(ε)) R(λ, ε, Z(ε)) Φ−1 (λ, ε, x) + 0 ν(ε) −∞ # "   Z Z(ε) d 1 K(x − Λ(ε))dx ξ(λ, ε, Λ(ε)) R(λ, ε, Z(ε)) Φ−1 (λ, ε, x) + 0 ν(ε) −∞ " # Z Z(ε) 1 Φ−1 (λ, ε, x)ϕ(x)dx R(λ, ε, Z(ε)) − ν(ε) −∞   a(λ, ε) + M41 (λ, ε)ξ(λ, ε, 0) + M42 (λ, ε)ξ(λ, ε, Γ(ε)) ≡ R(λ, ε, Z(ε)) b(λ, ε) +M43 (λ, ε)ξ(λ, ε, Λ(ε)) + M44 (λ, ε)ξ(λ, ε, Z(ε)) − N4 (λ, ε). "

We may rewrite these equations in the following form    R(λ, ε, 0) ξ(λ, ε, 0)    ξ(λ, ε, Γ(ε))   R(λ, ε, Γ(ε))  a(λ, ε)  =   ξ(λ, ε, Λ(ε))   R(λ, ε, Λ(ε))  b(λ, ε) R(λ, ε, Z(ε)) ξ(λ, ε, Z(ε))   ξ(λ, ε, 0)  ξ(λ, ε, Γ(ε))   +M(λ, ε)   ξ(λ, ε, Λ(ε))  − N (λ, ε), ξ(λ, ε, Z(ε)) 

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

133

where M(λ, ε) = (Mij (λ, ε)) is a four-by-four matrix and N (λ, ε) = (Ni (λ, ε)) is a complex vector. We have the simple equation     R(λ, ε, 0) ξ(λ, ε, 0)    R(λ, ε, Γ(ε))  a(λ, ε)    ξ(λ, ε, Γ(ε))      . + N (λ, ε) =  I − M(λ, ε)  R(λ, ε, Λ(ε))  b(λ, ε) ξ(λ, ε, Λ(ε))  R(λ, ε, Z(ε)) ξ(λ, ε, Z(ε)) Therefore, we obtain1 

a(λ, ε) b(λ, ε)



 −1  ξ(λ, ε, 0) R(λ, ε, 0)     R(λ, ε, Γ(ε))    ξ(λ, ε, Γ(ε))   =  R(λ, ε, Λ(ε))  ×  I − M(λ, ε)  ξ(λ, ε, Λ(ε))   ξ(λ, ε, Z(ε)) R(λ, ε, Z(ε)) 

Defined the open unbounded domain ( Ω(ε) =

where

λ ∈ C : Reλ > −ρmax −

1 + γε −

      + N (λ, ε) .     

) p (1 − γε)2 − 4ε , 2

    ρmax = sup ρ > 0 : sup exp(ρx)K(x) < ∞ , R

and we construct the Evans function by   E(λ, ε) = det I − M(λ, ε) .

(39)

The compatible solution of the general eigenvalue problem L(ε)ψ = λψ + ϕ is −1 R(λ, ε, 0)  R(λ, ε, Γ(ε))   ψ(λ, ε, z) = Φ(λ, ε, z)   R(λ, ε, Λ(ε))  R(λ, ε, Z(ε))     ξ(λ, ε, 0)         ξ(λ, ε, Γ(ε))    + N (λ, ε) × I − M(λ, ε)  ξ(λ, ε, Λ(ε))        ξ(λ, ε, Z(ε))   Z z a 1 0 −1 + ψ(λ, ε, 0)dx Φ (λ, ε, x) Φ(λ, ε, z) 0 0 ν(ε) −∞   Z z b 1 0 −1 + ψ(λ, ε, Z(ε))dx Φ(λ, ε, z) Φ (λ, ε, x) 0 0 ν(ε) −∞   Z z c 1 0 −1 ψ(λ, ε, Γ(ε))dx Φ (λ, ε, x) Φ(λ, ε, z) + 0 0 ν(ε) −∞ 

1

The inverse matrix of the 4 × 2 matrix is understood in generalized sense, not in classical sense. The inverse matrix is a 2 × 4 matrix.

134

Linghai Zhang   z d 1 0 −1 + ψ(λ, ε, Λ(ε))dx Φ (λ, ε, x) Φ(λ, ε, z) 0 0 ν(ε) −∞ Z z 1 − Φ−1 (λ, ε, x)ϕ(x)dx. Φ(λ, ε, z) ν(ε) −∞ Z

(40)

For all complex numbers λ 6= 0, if Reλ > −c0 , where c0 > 0 is a positive constant, then the inverse operator [λI − L(ε)]−1 exists and is bounded.

3.7.

The Properties of the Evans Functions

Lemma 9. The following results are correct relative to the Banach space C 0 (R) ∩ L∞ (R). The Evans function E(λ, ε) is complex analytic in λ and it is real-valued if λ is real. E(0, ε) = 0. The complex number λ0 is an eigenvalue of the linear differential operator L(ε) if and only if E(λ0 , ε) = 0. (This is the one-to-one correspondence.) The algebraic multiplicity of any eigenvalue λ0 of L(ε) is equal to the order of λ0 as a zero of the Evans function. The Evans function enjoys a limit lim E(λ, ε) = 1,

|λ|→∞

in the right half plane {λ ∈ C: Reλ ≥ 0}. There exists no nonzero eigenvalue of L(ε) in the right half plane {λ ∈ C: Reλ ≥ 0} for the fast homoclinic orbit and for the heteroclinic orbit, but there exists a positive eigenvalue for the slow homoclinic orbit.

3.8.

The Simplicity of the Neutral Eigenvalue λ = 0

Lemma 10. The neutral eigenvalue λ = 0 is algebraically simple. Proof. Let λ = 0 and ϕ(z) =



Uz (ε, z) Wz (ε, z)



in the general eigenvalue problem. Then −1 R(0, ε, 0)    R(0, ε, Γ(ε))  a(0, ε)  =  R(0, ε, Λ(ε))  b(0, ε) R(0, ε, Z(ε))     ξ(0, ε, 0)         ξ(0, ε, Γ(ε))    + N (0, ε) × I − M(0, ε)  ξ(0, ε, 0(ε))        ξ(0, ε, Z(ε)) −1  R(0, ε, 0)    R(0, ε, Γ(ε))  0   . N (0, ε) 6= =  R(0, ε, 0(ε)) 0 R(0, ε, Z(ε)) 

The lemma is proved.

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

135

This section extends the results of nonlocal models with a simple piecewise constant gain function jumping up at a single threshold θ to models with a different characterization of the gain function in which there is a low persistent activity rate θ and a high saturating rate Θ.

4.

A Nonlinear Singularly Perturbed System of Integral Differential Equations with Spatial Temporal Delay

Consider the nonlinear singularly perturbed system of integral differential equations (5)– (6). For simplicity, we focus on the case f (u) = g(u) = u. Thus, we consider the system     Z 1 ut + u + w = α K(x − y)H u y, t − |x − y| − θ dy, (41) c R wt = ε(u − γw). (42) Coombes, Lord and Owen [20], Coombes and Owen [21] also studied the existence and stability of the fast traveling pulse solution of this system. Consider the scalar integral differential equation with spatial temporal delay     Z 1 ut + u = α K(x − y)H u y, t − |x − y| − θ dy. (43) c R The model equation (43) has two constant solutions U0 ≡ 0 and U1 ≡ α. The constant solutions exist only when 0 < θ < α. They play a fundamental role for the existence and uniqueness of the traveling wave front. We will establish the existence and stability of a traveling wave front of equation (43) by applying fundamental ideas in differential equations and functional analysis (see [17], [28], [75]) and by using the method of linearization. In particular, we will introduce two concepts: the speed index function and the stability index function. We will derive formulas for the speed in terms of the given data, namely, the synaptic coupling, as well as the parameters α, θ and c. We will investigate the asymptotic behaviors of the wave and the speed as the synaptic coupling and the parameters tend to θ θ 1 certain critical cases, such as c → 0, c → ∞, → 0 and → , etc. We will also α α 2 exhibit valuable relation between the speed index function and the stability index function (i.e. complex analytic function). This is extremely important for studying the stability of the traveling wave front. Note that there have been many nice work using the so-called Evans function (i.e. the stability index function) to investigate stability of traveling waves.

4.1.

The Existence and Uniqueness of the Wave Speed of the Front

Define the speed index function φ by φ(µ) = α on the open domain (0, c).

Z

0

exp −∞



 c−µ ξ K(ξ)dξ, cµ

(44)

136

Linghai Zhang

Lemma 11. There exists a unique positive number µ0 ∈ (0, c), such that   Z 0 c − µ0 α exp α ξ K(ξ)dξ = − θ. cµ0 2 −∞ Moreover lim µ0 (c, α, θ) = 0,

lim µ0 (c, α, θ) = ν0 (α, θ),

c→∞

c→0

lim µ0 (c, α, θ) = c,

lim µ0 (c, α, θ) = 0.

θ →0 α

θ → 12 α

where ν0 (α, θ) is the solution of the equation   Z 0 α x K(x)dx = − θ. exp α ν0 2 −∞ Proof. The speed index function φ is defined by   Z 0 c−µ ξ K(ξ)dξ, exp φ(µ) = α cµ −∞ on (0, c). Then α φ (µ) = 2 µ ′

Z

0

−∞

|ξ| exp



 c−µ ξ K(ξ)dξ. cµ

Clearly we have lim φ(µ) = 0 <

µ→0

α α − θ < = lim φ(µ). µ→c 2 2

Therefore, the existence of a positive number µ0 in (0, c) is obviously true. It suffices to establish the uniqueness. We will show that φ′ > 0 for all synaptic couplings in classes (A) and (B). Also we will demonstrate that φ′ < 0 on (0, µ# ), φ′ (µ# ) = 0 and φ′ > 0 on (µ# , c), where the positive number µ# depends on the kernel function K, for all kernel α functions in class (C). Overall, if 0 < 2θ < α, then the graph of φ crosses the line φ = −θ 2 only once. The speed µ0 of the front is the unique solution of the equation φ(µ) =

α − θ. 2

The rest of the proof is divided into three parts. (A) For all nonnegative kernel functions, clearly

R0

−∞ K(x)dx

=

1 , so we must have 2

that φ′ (µ) > 0 on (0, c). (B) For each Mexican hat kernel function, recall that there is a constant M > 0, such that K ≥ 0 on (−M, 0) and K ≤ 0 on (−∞, −M ). Therefore, there hold the estimates     c−µ c−µ |ξ| exp ξ K(ξ) ≥ exp − M |ξ|K(ξ) ≥ 0, cµ cµ

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

137

on (−M, 0), and 0 ≥ |ξ| exp



   c−µ c−µ ξ K(ξ) ≥ exp − M |ξ|K(ξ), cµ cµ

on (−∞, −M ). There exists an nonempty interval (ξ1 , ξ2 ) ⊂ (−∞, 0), such that there R0 1 holds the strict inequality > on (ξ1 , ξ2 ), because −∞ K(x)dx = . Hence for all µ ∈ 2 (0, c), we have φ′ (µ) > 0, because Z 0  α c−µ ′ φ (µ) > 2 exp − M |ξ|K(ξ)dξ ≥ 0. µ cµ −∞ (C) Define a sequence of nonlinear smooth functions {φn (µ)} on (0, c), where   Z 0 c−µ n ξ K(ξ)dξ, |ξ| exp φn (µ) = α cµ −∞ for all integers n ≥ 1. Then lim φn (µ) = 0,

lim φn (µ) = α

µ→c

µ→0

Z

0

−∞

|ξ|n K(ξ)dξ > 0,

and Z

α φn (µ) = 2 µ ′

0

−∞

|ξ|

n+1

exp



 c−µ φn+1 (µ) ξ K(ξ)dξ = . cµ µ2

For each upside down Mexican hat kernel function, there exists a constant M > 0, such R0 1 that K ≤ 0 on (−M, 0) and K ≥ 0 on (−∞, −M ) and that −∞ K(x)dx = . Therefore, 2 we can pick up two positive numbers ∆1 and ∆2 , with ∆2 > ∆1 > 0, such that Z

0

K(ξ)dξ +

−M

Fix µ ∈ (0, c). Now we have Z



Z

−M (1+∆1 )

K(ξ)dξ > 0. −M (1+∆2 )

 c−µ |ξ| exp ξ K(ξ)dξ cµ −∞   Z 0 c−µ n+2 n+1 M x K(M x)dx =M |x| exp cµ −∞   Z 0 c−µ n+2 n+1 M x K(M x)dx =M |x| exp cµ −1   Z −1 c−µ |x|n+1 exp +M n+2 M x K(M x)dx cµ −∞   Z 0 c−µ M x K(M x)dx |x|n+1 exp ≥ M n+2 cµ −1 0

n+1

138

Linghai Zhang   −1−∆1 c−µ n+2 n+1 +M |x| exp M x K(M x)dx cµ −1−∆2 Z 0 Z −1−∆1 n+2 n+2 K(M x)dx ≥M K(M x)dx + M Z

−1 0

= M n+1

Z

K(ξ)dξ + M n+1

Z

−1−∆2 −M (1+∆1 )

K(ξ)dξ > 0,

−M (1+∆2 )

−M

where we have applied the following estimates   c−µ n+1 |x| exp M x ≤ 1 and K(M x) ≤ 0, on (−1, 0), for all n ≥ 0, cµ   c−µ |x|n+1 exp M x ≥ 1 and K(M x) ≥ 0, on (−1 − ∆2 , −1 − ∆1 ), cµ provided that the integer n is sufficiently large, namely, n ≥ N , where   c − µ M (1 + ∆2 ) . N =1+ cµ ln(1 + ∆1 ) Here [[x]] represents the greatest integer function of x. Therefore we obtain the estimates " # Z −M (1+∆1 ) n+1 α Z 0 M 1 K(ξ)dξ + K(ξ)dξ > 0. φn ′ (µ) = 2 φn+1 (µ) > µ µ2 −M −M (1+∆2 ) The results are also true on the open interval (µ, c). Similarly, we have Z

0 n

−∞

|ξ| K(ξ)dξ = M

=M

n+1

≥ M n+1 ≥M

n+1

=M

n

Z

Z

n+1

Z

0

−∞

|x|n K(M x)dx

0 n

−1 Z 0

−1 Z 0

|x| K(M x)dx + M

n+1

|x|n K(M x)dx + M n+1 K(M x)dx + M

−1

0

K(ξ)dξ + M

n

−M

Z

n+1

Z

Z

−1

|x|n K(M x)dx

−∞ Z −1−∆1

−1−∆2 −1−∆1

|x|n K(M x)dx

K(M x)dx

−1−∆2 −M (1+∆1 )

K(ξ)dξ > 0, −M (1+∆2 )

for all integers n ≥ 1. If φn (µ) = 0 at some number µ ∈ (0, c), then for the same number µ, we have φn+k (µ) > 0, because 

 c−µ ξ K(ξ)dξ cµ −∞   Z 0 c−µ M x K(M x)dx |x|n+k exp = αM n+k+1 cµ −∞

φn+k (µ) = α

Z

0

|ξ|n+k exp

Traveling Waves Arising from Synaptically Coupled Neuronal Networks   Z 0 c−µ n+k+1 n+k = αM |x| exp M x K(M x)dx cµ −1   Z −1 c−µ n+k+1 n+k M x K(M x)dx + αM |x| exp cµ −∞   Z 0 c−µ n n+k+1 |x| exp > αM M x K(M x)dx cµ −1   Z −1 c−µ n n+k+1 M x K(M x)dx |x| exp + αM cµ −∞   Z 0 c−µ M x K(M x)dx |x|n exp = αM n+k+1 cµ −∞   Z 0 c−µ k n = αM |ξ| exp ξ K(ξ)dξ = 0, cµ −∞

139

at the same number µ, for all integers k ≥ 1. On the other hand, by fixing the integer n = N and by making the change of variable ξ = µx, where µ ∈ (0, c), we have     Z 0 Z 0 c−µ c−µ n+2 n+1 n+1 ξ K(ξ)dξ = µ |x| exp x K(µx)dx, |ξ| exp cµ c −∞ −∞ and lim

Z

0

µ→0 −∞

n+1

|x|

= K(0)

Z

exp



 c−µ x K(µx)dx c

0 −∞

|x|n+1 exp(x)dx = (n + 1)!K(0) < 0,

if K(0) < 0. Therefore, there exists a small number µ = µn+1 > 0, such that   Z 0 c−µ n+1 |ξ| exp φn+1 (µ) = α ξ K(ξ)dξ cµ −∞   Z 0 c−µ n+1 n+2 x K(µx)dx < 0, |x| exp = αµ c −∞ for all µ with 0 < µ < µn+1 . This result is also true if K(0) = 0, K ≥ 0 on (−∞, −M ) and K ≤ 0 on (−M, 0), for some positive number M > 0. Set A∗ = inf{µ ≤ c : φn+1 (µ) > 0} and A∗ = sup{µ ≥ 0 : φn+1 (µ) < 0}. We will see that A∗ = A∗ . In fact, if φn+1 (µ# ) = 0 for some number µ# ∈ (0, c), then using similar analysis to the above, we see φn+2 (µ# ) = µ# 2 φn+1 ′ (µ# ) > 0. Thus the graph of the smooth function φn+1 (µ) crosses the µ-axis exactly once. Without any difficulty, we may conclude that A∗ = A∗ . Now φn+1 (A∗ ) = 0. Below we will use mathematical induction method. Note that lim φn (µ) = 0.

µ→0

Now it is easy to derive φn (µ) < 0 on (0, µn ) and φn (µ) > 0 on (µn , c), for some number µn ∈ (0, c). Recall that we have φn−1 ′ (µ) =

φn (µ) µ2

and

lim φn−1 (µ) = 0.

µ→0

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Linghai Zhang

Therefore, it is easy to conclude that φn−1 (µ) < 0 on (0, µn−1 ) and φn−1 (µ) > 0 on (µn−1 , c), for some number µn−1 ∈ (0, c), and so on. Finally, we get limµ→0 φ1 (µ) = 0, φ1 (µ) < 0 on (0, µ1 ) and φ1 (µ) > 0 on (µ1 , c), for some number µ1 ∈ (0, c). By mathematical induction method, we find that there exists some constant µm = µ(m), such that φm (µ) < 0 if 0 < µ < µm , φm (µ) = 0 if µ = µm and φm (µ) > 0 if µm < µ < c. Note that 0 < µm+1 < µm < c. Therefore, for all synaptic couplings in classes (A), (B) and (C), there exists a unique α speed µ0 = µ0 (c, α, θ, K) > 0, such that µ0 ∈ (0, c), φ(µ0 ) = − θ and U (0) = θ. 2 1 θ Note that the speed is a smooth function of β, where β ≡ − . To see if the speed is 2 α increasing with respect to β, differentiating the following equation with respect to β:   Z 0 θ 1 c − µ0 ξ K(ξ)dξ = β = − , exp cµ0 2 α −∞ we find



1 µ0 2

Z

0 −∞

|ξ| exp



  c − µ0 ∂µ0 ξ K(ξ)dξ = 1. cµ0 ∂β

α ∂µ0 = ′ > 0. Hence µ0 is an increasing ∂β φ (µ0 ) function of β. Moreover, it is not difficult to show that

Therefore, based on the above analysis,

lim µ0 (β) = 0,

β→0

lim µ0 (β) = c. 1 β→ 2

Similarly   Z 0 ∂µ0 α c − µ0 2 |ξ| exp ξ K(ξ)dξ = − ′ = −µ0 / < 0, ∂(θ/α) cµ0 φ (µ0 ) −∞  Z 0    2  c − µ0 1 ∂µ0 ∂µ0 φ′ (µ0 ) µ0 1 |ξ| exp ξ K(ξ)dξ = − − = 0. c2 µ0 2 ∂c cµ0 c2 ∂c α −∞ The proof of Lemma 11 is finished. ρ Example 1. Let K(x) = exp(−ρ|x|), where ρ > 0 is a constant. The speed index 2 function and the speed of the front satisfy cαρµ αρµ
→ φ(µ) = , as c → ∞, 2(1 + ρµ) 2 cρµ + c − µ (α − 2θ)c α − 2θ µ0 = → , as c → ∞. α − 2θ + 2cρθ 2ρθ Obviously, 0 < µ0 < c and 0 < µ0 <

α − 2θ . There hold the following limits 2ρθ

lim µ0 (c, α, θ, ρ) = 0,

ρ→∞

lim µ0 (c, α, θ, ρ) =

c→∞

α − 2θ , 2ρθ

lim µ0 (c, α, θ, ρ) = c,

ρ→0

lim µ0 (c, α, θ, ρ) = 0.

c→0

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

4.2.

141

The Existence and Uniqueness of the Traveling Wave Front

Theorem 9 (Existence and uniqueness of the traveling wave front). There exists a unique traveling wave front U = Ufront (·) to the following traveling wave equation Z     µ0 ′ µ0 U + U = α K(z − y)H U y − |z − y| − θ dy. (45) c R The front satisfies U (0) = θ, U < θ on (−∞, 0) and U > θ on (0, ∞), and the following boundary conditions lim U (z) = 0,

lim U ′ (z) = 0.

lim U (z) = α,

z→∞

z→−∞

z→±∞

Moreover lim µ0 (c, α, θ) = 0, lim µ0 (c, α, θ) = ν0 (α, θ), c→∞ # " Z z/2 α K(x)dx H(z), +α lim U (z) = c→0 2 0   Z z Z z x−z K(x)dx, lim U (z) = α K(x)dx − α exp c→∞ ν0 −∞ −∞ c→0

where µ0 (c, α, θ) and ν0 (α, θ) satisfy   Z 0 c − µ0 (c, α, θ) α exp α ξ K(ξ)dξ = − θ, cµ0 (c, α, θ) 2 −∞ and α

Z

0

exp

−∞



 α 1 ξ K(ξ)dξ = − θ, ν0 (α, θ) 2

respectively. (III) Define the function U by U(z) = 0 on (−∞, 0), U(0) = α U(z) = + α 2

Z

0

z/2

α K(x)dx − 2

Z

z

exp

0



x−z c



K

x 2

α , and 2

dx,

on (0, ∞), then lim µ0 (c, α, θ) = c,

θ →0 α

lim U (z) = U(z),

θ →0 α

lim µ0 (c, α, θ) = 0,

θ → 21 α

lim U (z) = α

θ → 12 α

Z

z

K(x)dx. −∞

Proof. Because of translation invariance of the traveling wave front, we may assume that U (0) = θ, U < θ on (−∞, 0) and U > θ on (0, ∞). First of all, let us consider the case µ ∈ (0, c). Let s(x) = −1 for all x < 0, s(0) = 0 and s(x) = 1 for all x > 0. To simplify the convolution product, we make the following change of variable η=y−

µ |z − y|. c

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Linghai Zhang

Then z−y =

c (z − η), c + s(z − η)µ

and dy =

c cµ(z − η) dη − s′ (z − η)dη. c + s(z − η)µ [c + s(z − η)µ]2

Clearly, z > η if z > y and z < η if z < y. Moreover, for each fixed number z, η is a strictly increasing function of y and s(z − y) = s(z − η). Now the traveling wave equation reduces to the ordinary differential equation   Z c c ′ µU + U = α K (z − η) H(U (η) − θ)dη c + s(z − η)µ R c + s(z − η)µ   Z ∞ c c =α K (z − η) dη c + s(z − η)µ c + s(z − η)µ 0 Z cz/(c+s(z)µ) K(x)dx, =α −∞

where x=

c c cµ(z − η) (z − η), dx = − dη + s′ (z − η)dη. c + s(z − η)µ c + s(z − η)µ [c + s(z − η)µ]2

We obtain the traveling wave solution representation Z

cz/(c+s(z)µ)

K(x)dx     c c x−z K x dx. exp µ c + s(x)µ c + s(x)µ −∞

U (z) = α Z −α

−∞ z

Moreover, we have 

   x−z c c exp K x dx, µ c + s(x)µ c + s(x)µ −∞   cz cα K U ′′ (z) = [c + s(z)µ]µ c + s(z)µ     Z z α x−z c c − 2 exp K x dx. µ −∞ µ c + s(x)µ c + s(x)µ

α U (z) = µ ′

Z

z

There hold the following limits lim U (z) = 0,

z→−∞

lim U (z) = α,

z→∞

lim U ′ (z) = 0.

z→±∞

Below we will investigate if the traveling wave solution really crosses the threshold as expected. To do that, we have to study on which interval the solution is increasing or decreasing, and whether it attains a global maximum or a global minimum somewhere.

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

143

We have fixed the phase condition U (0) = θ, that is α −α 2

    x c c exp K x dx = θ, µ c−µ c−µ −∞

Z

0

for some number µ ∈ (0, c). If we can find the unique speed µ0 ∈ (0, c) and demonstrate that the solution U (z) really crosses the threshold θ at z = 0, and it really lies below and above the threshold θ on (−∞, 0) and (0, ∞), respectively, then we can finish the proof immediately. The existence and uniqueness of the positive speed µ0 ∈ (0, c) has been proved in Lemma 11. The crossing of U through the threshold θ can be guaranteed by the assumptions on K.  1 α First of all, if U (0) = θ then U ′ (0) = − θ > 0. Clearly, if the synaptic µ0 2 coupling represents a pure excitation, then the front U is increasing, namely, if K ≥ 0, then U ′ ≥ 0. In particular, if K is positive almost everywhere, then U is strictly increasing. Therefore, synaptic couplings corresponding to pure excitation yields monotone solutions, thus crossing the threshold θ as desired. Let us look at the traveling wave solution of (43) with Mexican hat or upside down Mexican hat kinds of kernel functions. For all z ≤ 0, we find     Z α z x−z c c ′ U (z) = exp K x dx µ −∞ µ c−µ c−µ   Z cz/(c−µ)   α c−µ z = exp − exp ξ K(ξ)dξ, µ µ cµ −∞ and for all z > 0, we find     Z x−z c c α 0 exp K x dx U (z) = µ −∞ µ c−µ c−µ     Z x−z c c α z exp K x dx + µ 0 µ c+µ c+µ  Z 0   α c−µ z = exp − exp ξ K(ξ)dξ µ µ cµ −∞   Z cz/(c+µ)   α c+µ z + exp − exp ξ K(ξ)dξ µ µ cµ 0   Z 0   α c−µ z = exp − exp ξ K(ξ)dξ µ µ cµ −∞ )   Z cz/(c+µ) c+µ ξ K(ξ)dξ . exp + cµ 0 ′

Moreover, if µ = µ0 , then U ′ (z) =

 Z cz/(c−µ0 )    α c − µ0 z exp exp − ξ K(ξ)dξ, µ0 µ0 cµ0 −∞

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Linghai Zhang

for all z ≤ 0 and

) (    Z cz/(c+µ0 ) 1 α c + µ0 z U (z) = exp −θ+α exp − ξ K(ξ)dξ , µ0 µ0 2 cµ0 0 ′

for all z ≥ 0. Let the synaptic coupling represent a lateral inhibition. Then there exists a constant M > 0, such that K ≥ 0 on (−M, M ) and K ≤ 0 on (−∞, −M ) ∪ (M, ∞). Obviously U ′ ≤ 0 on (−∞, −M ), in particular, U ′ (−M ) < 0, and U ′ (0) > 0. Therefore, there exists a negative number z0 ∈ (−M, 0), such that U ′ (z0 ) = 0, and U ′ ≤ 0 on (−∞, z0 ) and U ′ ≥ 0 on (z0 , ∞), i.e. the front is decreasing on (−∞, z0 ) and it is increasing on (z0 , M ). Therefore the front U attains a global minimum at z0 ∈ (−∞, 0). Let us investigate the behavior of U on the right half line (M, ∞). Differentiating the traveling wave equation with respect to z, we have   cα c µ0 U ′′ + U ′ = K z ≤ 0. c + s(z)µ0 c + s(z)µ0 If we multiply this inequality by the integrating factor exp(z/µ0 ), we obtain    ′   z z ′ µ0 exp U (z) = exp (µ0 U ′′ + U ′ ) ≤ 0. µ0 µ0 If we integrate this inequality with respect to z over (z, ∞), we have the estimate       z z U ′ (z) ≤ exp U ′ (z). 0 < lim exp z→∞ µ0 µ0 Therefore, we find U ′ (z) > 0, for all z > M . Overall, U (z) is decreasing on (−∞, z0 ) and it is increasing on (z0 , ∞), and U (z0 ) is a global minimum. Let the synaptic coupling represent a lateral excitation. Then there exists a constant M > 0, such that K ≥ 0 on (−∞, −M ) ∪ (M, ∞) and K ≤ 0 on (−M, M ). Clearly U ′ ≥ 0 on (−∞, −M ). The fact U ′ (0) > 0 implies that U ′ ≥ 0 on (−∞, 0). For each upside down Mexican hat kernel function, it is easy to show that   Z cz/(c+µ0 ) c + µ0 exp ξ K(ξ)dξ > 0, cµ0 0 for all sufficiently large z > 0. Let us find the point zN where the synaptic coupling makes the most negative conczN c + µ0 tribution. Set the upper limit = M , thus zN = M . Additionally, if c + µ0 c   c + µ0 M ≥ 0, then U ′ ≥ 0 on R. In this case the front is a monotone increasing U′ c   c + µ0 M < 0, curve and crosses the threshold exactly once at z = 0, as expected. If U ′ c

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

145

then U ′ must change sign from positive to negative positive,hence there exist  and then to   c + µ0 c + µ0 a unique pair of positive numbers (z1 , z2 ): z1 ∈ 0, M , z2 ∈ M, ∞ , c c such that U ′ (z1 ) = 0 and U ′ (z2 ) = 0, U ′ ≥ 0 on (−∞, z1 ) ∪ (z2 , ∞) and U ′ ≤ 0 on (z1 , z2 ). The front attains a local maximum at z = z1 and a local minimum at z = z2 . We have to investigate if the traveling wave solution crosses the threshold on the right half line. It is easy to show that U (z1 ) = α

Z

cz1 /(c+s(z1 )µ0 )

K(x)dx

−∞

=α

Z

0

K(x)dx + α

−∞

Z

cz1 /(c+s(z1 )µ0 )

K(x)dx 0

Z cz1 /(c+µ0 ) α K(x)dx < α, = +α 2 0 Z cz2 /(c+s(z2 )µ0 ) Z M α U (z2 ) = + α K(x)dx K(x)dx + α 2 N 0 Z M α K(x)dx ≥ θ. > +α 2 0 Thus the local maximum U (z1 ) < α and the local minimum U (z2 ) > θ. Therefore U crosses the threshold θ only once. Overall, for any synaptic coupling in the three classes, the front crosses the threshold only once. Second, let us prove the existence and uniqueness of the front for the case µ = c. The traveling wave equation is Z
′ cU + U = α K(z − y)H U (y − |z − y|) − θ dy R Z z
K(z − y)H U (y − |z − y|) − θ dy =α Z −∞ ∞
K(z − y)H U (y − |z − y|) − θ dy +α   Zz α z z−y α = K H(U (y) − θ)dy + H(U (z) − θ). 2 −∞ 2 2 That is cU ′ + U = 0, cU ′ + U = α

Z

on

(−∞, 0),

z/2

K(x)dx + 0

α , 2

Solving this equation, we obtain the following solution U (z) = 0 for z < 0,

U (0) =

α , 2

on

(0, ∞).

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Linghai Zhang Z

z/2 α K(x)dx +α 2 0     Z α z x−z x − exp K dx for z > 0, 2 0 c 2     Z z α x−z x U ′ (z) = exp K dx for z > 0, 2c 0 c 2 lim U (z) = α.

U (z) =

z→∞

This solution is discontinuous at z = 0 but is continuous everywhere else. The mathematical analysis of the existence and uniqueness of the front is very similar to the case µ ∈ (0, c) and we can demonstrate that the wave crosses the threshold exactly once.

4.3.

The Exponential Stability of the Traveling Wave Front

Theorem 10. The traveling wave front U = Ufront (·) is exponentially stable. Proof. Let z = x + µ0 t and P (z, t) = u(x, t). Then ut = Pt + µ0 Pz . Now the equation (43) becomes     Z µ0 1 Pt + µ0 Pz + P = α K(z − y)H P y − |z − y|, t − |z − y| − θ dy. c c R Making the change of variable η = y −

c µ0 (z − η) |z − y|, then z − y = c c + s(z − η)µ0

and the nonlocal term becomes     Z µ0 1 K(z − y)H P y − |z − y|, t − |z − y| − θ dy c c R   Z c c K (z − η) = c + s(z − η)µ0 R c + s(z − η)µ0     1 ×H P η, t − |z − η| − θ dη. c + s(z − η)µ0

The traveling wave front is a stationary solution of this equation. We linearize it with respect to the traveling wave front to get     α c |z| c pt + µ0 pz + p = ′ . · ·K z · p 0, t − U (0) c + s(z)µ0 c + s(z)µ0 c + s(z)µ0

4.4.

The Eigenvalue Problem and the Evans Function

Suppose that p(z, t) = exp(λt)ψ(z) is a solution of the linear equation. We obtain the eigenvalue problem     c λ|z| c α ψ(λ, 0). · ·K z · exp − µ0 ψ ′ + (λ + 1)ψ = ′ U (0) c + s(z)µ0 c + s(z)µ0 c + s(z)µ0

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

147

Define the linear operator Lψ = −µ0 ψ ′ − ψ, where ψ ∈ C 1 (R) ∩ L∞ (R). Definition 6. The complex number λ0 is called an eigenvalue of the linear differential operator L if there exists a bounded, uniformly continuous function ψ = ψ(λ0 , ·), such that     α c λ|z| c ′ µ0 ψ + (λ + 1)ψ = ′ ψ(λ, 0). (46) K z exp − U (0) c + s(z)µ0 c + s(z)µ0 c + s(z)µ0 The function ψ is called an eigenfunction of the operator L corresponding to the eigenvalue λ0 . This is a nonlinear (in the eigenvalue parameter λ) eigenvalue problem. The nonlinearity comes from the spatial temporal delay. If there is no delay, then the eigenvalue problem α becomes linear, that is, ν0 ψ ′ + (λ + 1)ψ = ′ K(z)ψ(λ, 0). Note that the eigenvalue U (0) problem cannot be represented as Lψ = λψ, so we can not apply the general theory established in [28] and [67]. We are not sure whether the linearized stability criterion is true. Moreover, we are not sure whether the spectral stability, linear stability and nonlinear stability of the front are equivalent to each other. To study the stability of the traveling wave front of the integral differential equation (43), we are going to compute the essential spectrum and the eigenvalues of the linear differential operator L relative to the Banach space C 0 (R)∩L∞ (R). The essential spectrum σessential (L) is relatively easy to find: σessential (L) = {λ ∈ C: Reλ = −1}. We will construct an Evans function, defined in the right half plane Ω = {λ ∈ C: Reλ > −1}, to locate all of the eigenvalues of L, namely, σnormal (L). The equation µ0 ψ ′ + (λ + 1)ψ = 0 has a solution ψ1 (λ, z) =   differential λ+1 z , for all complex numbers λ. Define the function exp − µ0 χ(z) =

2cα . (α − 2θ)[c + s(z)µ0 ]

Let ϕ ∈ C 0 (R) ∩ L∞ (R). The solution of the general eigenvalue problem     c λ|z| c α ′ ψ(λ, 0) K z exp − µ0 ψ + (λ + 1)ψ + ϕ = ′ U (0) c + s(z)µ0 c + s(z)µ0 c + s(z)µ0 is  λ+1 z ψ(λ, z) = C(λ) exp − µ0       Z z ψ(λ, 0) c λ+1 λ|x| + χ(x)K x exp (x − z) exp − dx µ0 c + s(x)µ0 µ0 c + s(x)µ0 −∞   Z z 1 λ+1 − exp (x − z) ϕ(x)dx, µ0 −∞ µ0 

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Linghai Zhang

where C(λ) is a complex function of λ to be specified later. For each fixed λ with Reλ > −1, the solution ψ is bounded on R if and only if C(λ) = 0. Letting z = 0 in this solution, we have         Z 1 0 c λ+1 λ|x| C(λ) = 1− χ(x)K x exp x exp − dx ψ(λ, 0) µ0 −∞ c+s(x)µ0 µ0 c+s(x)µ0   Z 0 1 λ+1 + exp x ϕ(x)dx. µ0 −∞ µ0

Note that Z

     λ|x| λ+1 c dx x exp x exp − χ(x)K c + s(x)µ0 µ0 c + s(x)µ0 −∞     Z 0 2α c − µ0 λ = exp (λ + 1) ξ K(ξ)dξ ξ exp α − 2θ −∞ cµ0 c      Z 0 λ+1 1 1 µ0 2α exp ξ K(ξ)dξ = . − φ = α − 2θ −∞ µ0 c φ(µ0 ) λ+1 0



We construct the Evans function 1 E(λ) = 1 − φ φ(µ0 )



µ0 λ+1



,

(47)

in the open domain Ω = {λ ∈ C: Reλ > −1}, where we recall that the speed index function φ(µ) is defined by   Z 0 c−µ ξ K(ξ)dξ. exp φ(µ) = α cµ −∞ ρ Example 2. Let ρ > 0 be a constant and define the kernel function K(x) = exp(−ρ|x|). 2 Then the Evans function is given by E(λ) =

2θλ λ → , λ + 1 + ρµ0 − µ0 /c α + 2θλ

as c → ∞, where the wave speed

µ0 = Clearly

c(α − 2θ) . α − 2θ + 2cρθ

 λ+1   ξ K(ξ)dξ −∞ exp 1 ν0 ν0   lim E(λ) = 1 − =1− , φ R0 c→∞ ξ φ(ν0 ) λ+1 exp K(ξ)dξ −∞ ν0 R0



where ν0 = ν0 (α, θ) is the speed of the front of (43) where there is no delay. Note that if ρ > 0 and when K(x) is replaced with ρK(ρx), then the wave speed and the Evans function are functions of ρ. For 0 < c < ∞, we have lim µ0 (c, α, θ, ρ) = c,

ρ→0

lim

ρ→0

c[c − µ0 (c, α, θ, 1)] c − µ0 (c, α, θ, ρ) = , ρ µ0 (c, α, θ, 1)

Traveling Waves Arising from Synaptically Coupled Neuronal Networks lim

ρ→0

1 c − µ0 (c, α, θ, ρ) c − µ0 (c, α, θ, 1) = , ρ cµ0 (c, α, θ, ρ) cµ0 (c, α, θ, 1)

lim µ0 (c, α, θ, ρ) = 0,

lim ρµ0 (c, α, θ, ρ) =

ρ→∞

lim

ρ→∞

ρ→∞

c − µ0 (c, α, θ, 1) 1 c − µ0 (c, α, θ, ρ) = . ρ cµ0 (c, α, θ, ρ) cµ0 (α, θ, 1)

We also have

cµ0 (c, α, θ, 1) , c − µ0 (α, θ, 1)



  λ+1 1 ξ ρK(ρξ)dξ − µ0 (c, α, θ, ρ) c    lim E(λ, ρ) = 1 − lim ρ→0 ρ→0 R 0 1 1 ξ ρK(ρξ)dξ − −∞ exp µ0 (c, α, θ, ρ) c     R0 1 λ+1 1 x K(x)dx − −∞ exp ρ µ0 (c, α, θ, ρ) c     = 1 − lim ρ→0 R 0 1 1 1 x K(x)dx − −∞ exp ρ µ0 (c, α, θ, ρ) c = 1, for all λ 6= 0 with Reλ ≥ 0, if 0 < c < ∞,   R0 λ+1 exp x K(x)dx −∞ ν0 (α, θ, 1)   =1− , if c = ∞, R0 1 −∞ exp ν (α, θ, 1) x K(x)dx 0 R0

−∞ exp

for all nonzero complex numbers with Reλ ≥ 0 except for λ = 0,    R0 λ+1 1 exp ξ ρK(ρξ)dξ − −∞ µ0 (c, α, θ, ρ) c    lim E(λ, ρ) = 1 − lim ρ→∞ ρ→∞ R 0 1 1 ξ ρK(ρξ)dξ − −∞ exp µ0 (c, α, θ, ρ) c   R0 c − µ0 (c, α, θ, 1) −∞ exp (λ + 1) cµ (c, α, θ, 1) x K(x)dx 0   , =1− R0 c − µ0 (c, α, θ, 1) x K(x)dx −∞ exp cµ0 (c, α, θ, 1) for all λ ∈ Ω. Recall that lim µ0 (c, α, θ) = c,

lim µ0 (c, α, θ) = 0.

θ → 12 α

θ →0 α

Therefore lim E(λ) =

θ →0 α

and

  

1−2

R0

−∞ exp

 0

lim E(λ) = 1 −

θ → 21 α

 λ ξ K(ξ)dξ, c

1 λ = , λ+1 λ+1

if

c < ∞,

if

c = ∞.

149

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Linghai Zhang

if K(0) 6= 0. If ψ(λ, 0) = 0, then the eigenvalue problem only has the trivial solution ψ(λ, ·) ≡ 0 on R. It turns out that for nontrivial solutions, there must hold ψ(λ, 0) 6= 0.

4.5.

Preliminary Speed Analysis

Theorem 11 (Preliminary Speed Analysis - I). Suppose that 0 < 2θ < α. Then the speeds of the traveling wave fronts of (43) satisfy 1 1 1 = + , µ0 (c, α, θ) c ν0 (α, θ) 0 < µ0 (c, α, θ) < c, 0 < µ0 (c, α, θ) < ν0 (α, θ), µ0 (c, α, θ) lim µ0 (c, α, θ) = ν0 (α, θ), lim µ0 (c, α, θ) = 0, lim = 1. c→∞ c→0 c→0 c Proof. The speeds µ0 (c, α, θ) and ν0 (α, θ) satisfy the equations   Z 0 c − µ0 (c, α, θ) α exp α x K(x)dx = − θ, cµ0 (c, α, θ) 2 −∞ and  α x K(x)dx = − θ, α exp ν0 (α, θ) 2 −∞ Z

0



respectively. By uniqueness, we obtain the relationship 1 c − µ0 (c, α, θ) = , cµ0 (c, α, θ) ν0 (α, θ) that is 1 1 1 = + . µ0 (c, α, θ) c ν0 (α, θ) The estimates and limits follow immediately from this formula. The proof of Theorem 11 is completed. The results are elegant because the formula does not depend on α, θ and K directly. Theorem 12 (Preliminary Speed Analysis - II). Let 0 < 2θ < α. The wave speed µ = µ(c, α, θ) of the traveling wave front is an increasing function of c and it is an increasing function of α, but it is a decreasing function of θ. More precisely, when c = ∞, we have ∂ν0 ∂ν0 (α, θ) > 0, (α, θ) < 0, ∂α ∂θ lim ν0 (α, θ) = ∞, lim ν0 (α, θ) = ∞, α→∞ θ→0 Z 1 0 ν0 (α, θ) |ξ|K(ξ)dξ, lim = α→∞ α θ −∞

Traveling Waves Arising from Synaptically Coupled Neuronal Networks Z 0   |ξ|K(ξ)dξ, lim θν0 (α, θ) = α θ→0

−∞

lim ν0 (α, θ) = ∞,

θ →0 α

θν0 (α, θ) lim = θ α →0 α lim

θ → 12 α

Z

lim ν0 (α, θ) = 0,

θ → 12 α

0 −∞

|ξ|K(ξ)dξ,

2α 1 ν0 (α, θ) = , if K(0) 6= 0. α − 2θ K(0)

When 0 < c < ∞, we have

∂µ0 ∂µ0 ∂µ0 (c, α, θ) > 0, (c, α, θ) > 0, (c, α, θ) < 0, ∂c ∂α ∂θ lim µ0 (c, α, θ) = c, lim µ0 (c, α, θ) = c, lim µ0 (c, α, θ) = 0,

α→∞

θ→0

  2 lim c[ν0 (α, θ) − µ0 (c, α, θ)] = ν0 (α, θ) , c→∞ Z 0  2 |ξ|K(ξ)dξ, lim α[c − µ0 (c, α, θ)] = c θ/

α→∞

θ → 21 α

−∞

Z c2 0 c − µ0 (c, α, θ) |ξ|K(ξ)dξ, = lim θ→0 θ α −∞ Z 0 o nα 2 [c − µ0 (c, α, θ)] = c |ξ|K(ξ)dξ, lim θ →0 θ −∞ α lim µ0 (c, α, θ)

θ → 12 α

1 2α = if K(0) 6= 0. α − 2θ K(0)

Proof. First of all, differentiating the equation 1 1 1 = + µ0 (c, α, θ) c ν0 (α, θ) with respect to c, noting that ν0 (α, θ) is independent of c, we have

Now we find that

∂µ0 1 1 (c, α, θ) = − 2 . − 2 c µ0 (c, α, θ) ∂c  2 µ0 (c, α, θ) ∂µ0 (c, α, θ) = > 0. ∂c c2

Secondly, differentiating the equation 1 1 1 = + µ0 (c, α, θ) c ν0 (α, θ) with respect to α, we obtain ∂µ0 ∂ν0 1 1 − (c, α, θ) = −  (α, θ). 2 2 ∂α µ0 (c, α, θ) ν0 (α, θ) ∂α

151

152

Linghai Zhang

That is  2 µ0 (c, α, θ) ∂ν0 ∂µ0 (c, α, θ) =  (α, θ). 2 ∂α ∂α ν0 (α, θ)

Now upon differentiating the equation   Z 0 ξ α − 2θ exp K(ξ)dξ = ν0 (α, θ) 2α −∞ with respect to α, we get ∂ν0 (α, θ)  2 ν0 (α, θ) ∂α 1

Therefore

Z

 ξ θ |ξ| exp K(ξ)dξ = 2 . ν0 (α, θ) α −∞ 0



  Z 2 . 0 θ 1 ξ θ  ∂ν0 |ξ| exp K(ξ)dξ = (α, θ) = 2 ν0 (α, θ) > 0, ∂α α ν (α, θ) α φ′ (ν0 ) 0 −∞   Z 2 . 0 θ  ξ ∂µ0 K(ξ)dξ > 0. |ξ| exp (c, α, θ) = 2 µ0 (c, α, θ) ∂α α ν0 (α, θ) −∞ We can obtain the following partial derivatives with respect to θ:  2 µ0 (c, α, θ) ∂ν0 ∂µ0 (c, α, θ) =  (α, θ), 2 ∂θ ∂θ ν0 (α, θ)  2 Z   ν0 (α, θ) . 0 ∂ν0 1 ξ K(ξ)dξ = − ′ (α, θ) = − < 0, |ξ| exp ∂θ α ν (α, θ) φ (ν 0 0) −∞  2 Z   µ0 (c, α, θ) . 0 ∂µ0 ξ K(ξ)dξ < 0. (c, α, θ) = − |ξ| exp ∂θ α ν0 (α, θ) −∞ Recall that   Z 0 α − 2θ c − µ0 (c, α, θ) ξ K(ξ)dξ = = β, exp cµ (c, α, θ) 2α 0 −∞   Z  2 . 0 c − µ0 (c, α, θ) α ∂µ0 |ξ| exp (c, α, θ) = µ0 (c, α, θ) ξ K(ξ)dξ = ′ > 0, ∂β cµ (c, α, θ) φ (µ 0 0) −∞   Z  2 . 0 ∂µ0 c−µ0 (c, α, θ) α |ξ| exp (c, α, θ) = − µ0 (c, α, θ) ξ K(ξ)dξ = − ′ < 0, ∂(θ/α) cµ (c, α, θ) φ (µ 0 0) −∞ and similarly 

 1 α − 2θ ξ K(ξ)dξ = = β, ν (α, θ) 2α 0 −∞   Z  2 . 0 1 ∂ν0 |ξ| exp (α, θ) = ν0 (α, θ) ξ K(ξ)dξ ∂β ν0 (α, θ) −∞

Z

0

exp

Traveling Waves Arising from Synaptically Coupled Neuronal Networks .Z 0 =1 |x| exp(x)K(ν0 (α, θ)x)dx,

153

−∞

where     Z 0 Z 0 1 c − µ0 (c, α, θ) |ξ| exp ξ K(ξ)dξ > 0 and ξ K(ξ)dξ > 0. |ξ| exp cµ0 (c, α, θ) ν0 (α, θ) −∞ −∞ As a function of α, ν0 (α, θ) is increasing and positive, hence 0 < l := lim ν0 (α, θ) ≤ ∞. α→∞

Suppose that the limit is finite, 0 < l < ∞. In the speed equation   Z 0 ξ 1 θ exp K(ξ)dξ = − , ν0 (α, θ) 2 α −∞ setting α → ∞, we find that   Z 0 Z 0 1 ξ 1 K(ξ)dξ = . = K(ξ)dξ > exp 2 l 2 −∞ −∞ This is a contradiction. Therefore l = lim ν0 (α, θ) = ∞. α→∞

We have ν0 (α, θ) ∂ν0 L := lim = lim (α, θ) α→∞ α→∞ ∂α α      .Z 0  ν0 (α, θ) 2 ξ = lim lim θ K(ξ)dξ |ξ| exp α→∞ α→∞ α ν0 (α, θ) −∞  .Z 0  = L2 θ |ξ|K(ξ)dξ . −∞

Thus 1 ν0 (α, θ) = α→∞ α θ

L = lim

Z

0

−∞

|ξ|K(ξ)dξ.

Following this idea, all other limits are straightforward to establish. Recall that β = α − 2θ . Now we have 2α 2α ν0 (α, θ) ν0 (α, θ) = lim β→0 β→0 α − 2θ β   Z  2 . 0 ξ ∂ν0 (α, θ) |ξ| exp K(ξ)dξ = lim ν0 (α, θ) = lim β→0 β→0 ∂β ν0 (α, θ) −∞ .Z 0 1 . |x| exp(x)K(ν0 (α, θ)x)dx = = lim 1 β→0 K(0) −∞ lim

154

Linghai Zhang

Now let us investigate the limits involving the delay. Obviously     2  lim c ν0 (α, θ) − µ0 (c, α, θ) = lim µ0 (c, α, θ)ν0 (α, θ) = ν0 (α, θ) . c→∞

c→∞

Moreover cν0 (α, θ) = c, α→∞ α→∞ c + ν0 (α, θ) .Z 0   cαµ0 (c, α, θ) 2 |ξ|K(ξ)dξ. =c θ lim α c − µ0 (c, α, θ) = lim α→∞ α→∞ ν0 (α, θ) −∞ lim µ0 (c, α, θ) = lim

Additionally cν0 (α, θ) = c, θ→0 θ→0 c + ν0 (α, θ) Z c2 . 0 cµ0 (c, α, θ) c − µ0 (c, α, θ) = lim = |ξ|K(ξ)dξ. lim θ→0 θν0 (α, θ) θ→0 θ α −∞ lim µ0 (c, α, θ) = lim

Finally cν0 (α, θ) = c, θ →0 c + ν0 (α, θ) α

lim µ0 (c, α, θ) = lim

θ →0 α

.  α cαµ0 (c, α, θ) lim c − µ0 (c, α, θ) = lim = c2 θ θ →0 θ →0 θν0 (α, θ) α α

Z

0

−∞

|ξ|K(ξ)dξ.

The proof of Theorem 12 is completed.

4.6.

The Existence and Uniqueness of the Traveling Wave Back

Consider the scalar integral differential equation with the parameter W     Z 1 ut + u + W = α K(x − y)H u y, t − |x − y| − θ dy. c R

(48)

Lemma 12. Suppose that W = α − 2θ > 0. There exists a unique stable traveling wave solution to (49) with the same speed µ0 as the front of (43). Proof. Let 0 < 2θ < α. If there exists a traveling wave solution with the speed µ0 , then we must have Z     µ0 µ0 U ′ + U + W = α K(z − y)H U y − |z − y| − θ dy, c R where z = x + µ0 t. We are seeking a decreasing traveling wave solution satisfying U (0) = θ, U ′ (0) < 0, and U > θ on (−∞, 0) and U < θ on (0, ∞). Additionally, lim U (z) = α,

z→−∞

lim U (z) = 0,

z→∞

lim U ′ (z) = 0.

z→±∞

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

155

µ0 As before, by making the change of variable η = y − |z −y|, the traveling wave equation c becomes Z ∞ ′ K(x)dx. µ0 U + U + W = α cz/(c+s(z)µ0 )

By using integrating factor idea, we find the solution Z ∞ K(x)dx U (z) = α cz/(c+s(z)µ0 )     Z z x−z c cx exp +α dx − W. K µ0 c + s(x)µ0 c + s(x)µ0 −∞ Setting z = 0 yields     Z 0 Z ∞ c cx x dx − W K exp K(x)dx + α U (0) = α µ0 c + s(x)µ0 c + s(x)µ0 −∞ 0   Z 0 c − µ0 α exp ξ K(ξ)dξ − W. = +α 2 cµ0 −∞ By comparing this with α

Z

0

exp −∞



 c − µ0 α ξ K(ξ)dξ = − θ, cµ0 2

we find that only when W = α − 2θ, there holds U (0) = θ. The stability analysis is very similar to that of the front and is omitted. Lemma 13. Suppose that 0 < θ < α. Let u = u(x, t) be a solution of the integral equation Z      Z t 1 η(t − τ ) K(x − y)H u y, τ − |x − y| − θ dy + I(x, τ ) dτ, u(x, t) = α c −∞ R or     1 ut + u = α K(x − y)H u y, t − |x − y| − θ dy + I(x, t), c R Z

respectively, where Z Define u e=u e(x, t) by Then u e is a solution of αθ u e(x, t) = α−θ

Z

∞

η(t)dt = 1

K(x)dx = 1.

R

0

u e(x, t) − θ =

t

−∞

and

Z

η(t − τ )

θ [θ − u(x, t)]. α−θ

(49)

     1 θ I(x, τ ) dτ, K(x−y)H u e y, τ − |x − y| −θ dy− c α−θ R

Z

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Linghai Zhang

or αθ u et + u e= α−θ

respectively.

    1 θ K(x − y)H u e y, t − |x − y| − θ dy − I(x, t), c α−θ R

Z

Proof. For any real number x, there holds H(x) + H(−x) = 1. Thus         1 1 H u y, τ − |x − y| − θ + H u e y, τ − |x − y| − θ = 1. c c

Note that

Z

∞

η(t)dt = 1

and

Z

K(x)dx = 1.

R

0

The rest of the proof is omitted because it is straightforward. Remark 6. The two solutions u and u e are complementary in the sense that excited intervals αθ of u correspond to non-excited intervals of u e and vice versa. Furthermore, let α e= . α−θ If 0 < 2θ < α and U is the traveling wave front with speed µ0 found in Theorem 9, then e is the traveling wave front of (43) with the same speed µ0 . If α = 2θ, θ<α e < 2θ and U then the complementary solution u e coincides with the solution u, because α = α e = 2θ.

4.7.

The Existence and Stability of the Fast Traveling Pulse Solution

We may piece together the traveling wave front, the traveling wave back and two additional curves to form a singular homoclinic orbit. Here are some challenging problems. Can we apply the classical geometric singular perturbation theory to the nonlinear singularly perturbed systems of integral differential equations with spatial temporal delay? Is the real fast traveling pulse solution close to the singular homoclinic orbit? How to find the wave speed µ = µ(ε) and the number Z = Z(ε), such that U (ε, 0) = U (ε, Z) = θ? Can we prove that the traveling pulse solution really satisfies U (ε, ·) < θ on (−∞, 0) ∪ (Z, ∞) and U (ε, ·) > θ on (0, Z)? Denote by µ = µ(ε) the wave speed of the traveling pulse solutions and
let z = x + µ(ε)t. The traveling pulse solutions (U, W ) = Upulse (ε, ·), Wpulse (ε, ·) satisfy the traveling wave equations Z     µ µU ′ + U + W = α K(z − y)H U y − |z − y| − θ dy, c R ′ µW = ε(U − γW ), and the boundary conditions lim z→±∞


Upulse (ε, z), Wpulse (ε, z) = (0, 0),

lim z→±∞



 ∂ ∂ Upulse (ε, z), Wpulse (ε, z) = (0, 0). ∂z ∂z

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

157

c µ |z − y|. Then z − y = (z − η). Suppose that U (ε, 0) = c c + s(z − η)µ U (ε, Z) = θ, U (ε, ·) < θ on (−∞, 0) ∪ (Z, ∞) and U (ε, ·) > θ on (0, Z), for some constant Z = Zpulse (ε) > 0. Then the traveling wave equations become Z cz/(c+s(z)µ) ′ µU + U + W = α K(x)dx, Let η = y −

c(z−Z)/(c+s(z−Z)µ)

′

µW = ε(U − γW ). Solving the differential equations, we obtain the pulse solution representation # " Z Z cx/(c+s(x)µ)

z

U (ε, z) =

−∞

W (ε, z) =

Z

A(µ, ε, x − z) α

z −∞

K(ξ)dξ dx,

c(x−Z)/(c+s(x−Z)µ)

" Z B(µ, ε, x − z) α

cx/(c+s(x)µ)

#

K(ξ)dξ dx.

c(x−Z)/(c+s(x−Z)µ)

By using integration by parts, we have the slightly better expression Z cz/(c+s(z)µ) αγ U (ε, z) = K(x)dx 1 + γ c(z−Z)/(c+s(z−Z)µ)    Z z c c C(µ, ε, x − z) −α K x c + s(x)µ c + s(x)µ −∞   c c K (x − Z) dx, − c + s(x − Z)µ c + s(x − Z)µ Z cz/(c+s(z)µ) α K(x)dx W (ε, z) = 1 + γ c(z−Z)/(c+s(z−Z)µ)    Z z c c D(µ, ε, x − z) −α K x c + s(x)µ c + s(x)µ −∞   c c K (x − Z) dx. − c + s(x − Z)µ c + s(x − Z)µ The wave speed µ and the number Z satisfy the equations 

  c c K x c + s(x)µ c + s(x)µ −∞   Z 0 c c αγ − K (x − Z) dx = K(x)dx − θ, c + s(x − Z)µ c + s(x − Z)µ 1 + γ −cZ/(c−µ)    Z 0 c c α K (x + Z) C(µ, ε, x) c + s(x + Z)µ c + s(x + Z)µ −∞   Z cZ/(c+µ) c αγ c K x dx = K(x)dx − θ. − c + s(x)µ c + s(x)µ 1+γ 0

α

Z

0

C(µ, ε, x)

That is Z

0



c α C(µ, ε, x) K c − µ −∞



   c c c x − K (x − Z) dx c−µ c−µ c−µ

158

Linghai Zhang = α

Z

αγ 1+γ

Z

0

−cZ/(c−µ)

0

C(µ, ε, x)

−∞

αγ = 1+γ

Z



K(x)dx − θ,

c K c + s(x + Z)µ

cZ/(c+µ)

0



   c c c (x + Z) − K x dx c + s(x + Z)µ c−µ c−µ

K(x)dx − θ.

By using
the implicit function theorem, we can find a unique solution (µ, Z) = µ(ε), Z(ε) , such that U (ε, 0) = U (ε, Z) = θ. Moreover, we find that U (ε, ·) < θ on (−∞, 0) ∪ (Z, ∞) and U (ε, ·) > θ on (0, Z). Let z = x + µ(ε)t and (P (z, t), Q(z, t)) = (u(x, t), w(x, t)). Then (ut , wt ) = (Pt + µ(ε)Pz , Qt + µ(ε)Qz ). Then system (41)–(42) becomes     Z µ(ε) 1 Pt + µ(ε)Pz + P + Q = α K(z − y)H P y − |z − y|, t − |z − y| − θ dy, c c R Qt + µ(ε)Qz = ε(P − γQ). Making the change of variable η = y − and the integral

µ(ε) c |z − y|, then z − y = (z − η) c c + s(z − η)µ(ε)

    µ(ε) 1 K(z − y)H P y − |z − y|, t − |z − y| − θ dy c c R   Z c c K (z − η) = c + s(z − η)µ(ε) R c + s(z − η)µ(ε)     1 ×H P η, t − |z − η| − θ dη. c + s(z − η)µ(ε)

Z

We may linearize
this new system with
respect to the traveling pulse solution U (ε, ·), W (ε, ·) = Upulse (ε, ·), Upulse (ε, ·) to obtain     α c |z| c pt + µ(ε)pz + p + q = K z p 0, t − Uz (ε, 0) c + s(z)µ(ε) c + s(z)µ(ε) c + s(z)µ(ε)     c |z − Z| c α K (z−Z) p Z, t− , − Uz (ε, Z) c+s(z−Z)µ(ε) c+s(z−Z)µ(ε) c+s(z−Z)µ(ε) qt + µ(ε)qz = ε(p − γq).

Suppose that (p(z, t), q(z, t)) = exp(λt)ψ(λ, z) is a solution of the linear system, where λ is a complex number. Then we obtain the following eigenvalue problem   λ+1 1 ′ ψ µ(ε)ψ + −ε λ + γε       α c λ|z| c 1 0 = ψ(λ, ε, 0) K z exp − 0 0 Uz (ε, 0) c+s(z)µ(ε) c+s(z)µ(ε) c+s(z)µ(ε)   c c α K (z − Z) − Uz (ε, Z) c + s(z − Z)µ(ε) c + s(z − Z)µ(ε)

Traveling Waves Arising from Synaptically Coupled Neuronal Networks    λ|z − Z| × exp − ψ(λ, ε, Z) . c + s(z − Z)µ(ε)

159

Consider the general eigenvalue problem  λ+1 1 ψ+ϕ µ(ε)ψ + −ε λ + γε       c λ|z| c α 1 0 K z exp − ψ(λ, ε, 0) = 0 0 Uz (ε, 0) c + s(z)µ(ε) c+s(z)µ(ε) c+s(z)µ(ε)   α c c − K (z − Z) Uz (ε, Z) c + s(z − Z)µ(ε) c + s(z − Z)µ(ε)    λ|z − Z| × exp − ψ(λ, ε, Z) , c + s(z − Z)µ(ε) 

′

where ϕ ∈ C 0 (R) ∩ L∞ (R). The solution of the general eigenvalue problem is 

a(λ, ε) b(λ, ε)



ψ(λ, ε, z) = Φ(λ, ε, z)   Z z 1 1 0 −1 + Φ(λ, ε, z) Φ (λ, ε, x) 0 0 µ(ε) −∞      α c λ|x| c K x exp − ψ(λ, ε, 0) Uz (ε, 0) c + s(x)µ(ε) c + s(x)µ(ε) c + s(x)µ(ε)   c α c K (x − Z) − Uz (ε, Z) c + s(x − Z)µ(ε) c + s(x − Z)µ(ε)    λ|x − Z| ψ(λ, ε, Z) dx × exp − c + s(x − Z)µ(ε) Z z 1 − Φ(λ, ε, z) Φ−1 (λ, ε, x)ϕ(x)dx. µ(ε) −∞

In the first component, letting z = 0 and z = Z, respectively, we have R(λ, ε, 0)



 = 1−

a(λ, ε) b(λ, ε)



  Z 0 α 1 −1 R(λ, ε, 0) Φ (λ, ε, x) 0 µ(ε)Uz (ε, 0) −∞      c λ|x| c K x exp − dx ξ(λ, ε, 0) × c + s(x)µ(ε) c + s(x)µ(ε) c + s(x)µ(ε)    Z 0 α 1 + R(λ, ε, 0) Φ−1 (λ, ε, x) 0 µ(ε)Uz (ε, Z) −∞      c λ|x−Z| c K (x−Z) exp − dx ξ(λ, ε, Z) × c+s(x−Z)µ(ε) c+s(x−Z)µ(ε) c+s(x−Z)µ(ε) Z 0 1 R(λ, ε, 0) Φ−1 (λ, ε, x)ϕ(x)dx + µ(ε) −∞ ≡ [1 − M11 (λ, ε)]ξ(λ, ε, 0) − M12 (λ, ε)ξ(λ, ε, Z) + N1 (λ, ε),   a(λ, ε) R(λ, ε, Z) b(λ, ε)

160

Linghai Zhang "

  Z Z α 1 R(λ, ε, Z) Φ−1 (λ, ε, x) 0 µ(ε)Uz (ε, 0) −∞      c c λ|x| × K x exp − dx ξ(λ, ε, 0) c + s(x)µ(ε) c + s(x)µ(ε) c + s(x)µ(ε) "   Z Z α 1 + 1+ R(λ, ε, Z) Φ−1 (λ, ε, x) 0 µ(ε)Uz (ε, Z) −∞      c c λ|x−Z| × K (x−Z) exp − dx ξ(λ, ε, Z) c+s(x−Z)µ(ε) c+s(x−Z)µ(ε) c+s(x−Z)µ(ε) Z Z 1 R(λ, ε, Z) Φ−1 (λ, ε, x)ϕ(x)dx + µ(ε) −∞ = −

≡ −M21 (λ, ε)]ξ(λ, ε, 0) + [1 − M22 (λ, ε)]ξ(λ, ε, Z) + N2 (λ, ε).

Combine them together appropriately, we get        N1 (λ, ε) ξ(λ, ε, 0) a(λ, ε) R(λ, ε, 0) . + = [I − M(λ, ε)] N2 (λ, ε) ξ(λ, ε, Z) b(λ, ε) R(λ, ε, Z) Solving this equation, we find    −1      N1 (λ, ε) ξ(λ, ε, 0) R(λ, ε, 0) a(λ, ε) . + [I − M(λ, ε)] = N2 (λ, ε) ξ(λ, ε, Z) R(λ, ε, Z) b(λ, ε) The Evans function is E(λ, ε) = det[I − M(λ, ε)]. It enjoys the same properties as in Lemma 9. If λ = 0 and ϕ(z) = 

a(0, ε) b(0, ε)



Uz (ε, z) Wz (ε, z)

 , then

   −1   N1 (0, ε) ξ(0, ε, 0) + [I − M(0, ε)] N2 (0, ε) ξ(0, ε, Z)    −1   0 N1 (0, ε) R(0, ε, 0) . 6= = 0 N2 (0, ε) R(0, ε, Z)



=



R(0, ε, 0) R(0, ε, Z)

The compatible solution of the general eigenvalue problem L(ε)ψ = λψ + ϕ is

  −1     ξ(λ, ε, 0) + N (λ, ε) I − M(λ, ε) ψ(λ, ε, z) = Φ(λ, ε, z) ξ(λ, ε, Z)   Z z 1 1 0 + Φ(λ, ε, z) Φ−1 (λ, ε, x) 0 0 µ(ε) −∞      c λ|x| c α K x exp − ψ(λ, ε, 0) × Uz (ε, 0) c + s(x)µ(ε) c + s(x)µ(ε) c + s(x)µ(ε)   c α c − K (x − Z) Uz (ε, Z) c + s(x − Z)µ(ε) c + s(x − Z)µ(ε)    λ|x − Z| × exp − ψ(λ, ε, Z) dx c + s(x − Z)µ(ε) Z z 1 − Φ(λ, ε, z) Φ−1 (λ, ε, x)ϕ(x)dx. µ(ε) −∞ 

R(λ, ε, 0) R(λ, ε, Z)

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

5.

161

The Mathematical Analysis of Wave Speeds

Consider the scalar integral differential equations     Z 1 ut + f (u) = α K(x − y)H u y, t − |x − y| − θ dy, c ZR ut + f (u) = α K(x − y)H(u(y, t) − θ)dy,

(50) (51)

R

where 0 < c < ∞ is a constant. We are going to investigate how various neurobiological mechanisms (in particular, synaptic couplings, threshold and synaptic rate constant) influence traveling wave speed. We will be concerned with three kinds of synaptic couplings: pure excitation (modeled with nonnegative kernel function), lateral inhibition (modeled with Mexican hat kernel function), lateral excitation (modeled with upside down Mexican hat kernel function). We are concerned with asymptotic behaviors of the speed as various parameters approach certain numbers or infinity. Speeds play a very important role in the study of traveling waves of nonlinear integral differential equations. Indeed, once the speed is found, the traveling wave solution is easy to solve by using techniques in differential equations. Moreover, the speeds are closely related to the stability of traveling waves. Intuitively, stable waves are the most important solutions. θ u Making changes of variables: w = , Θ = , if f (u) = u, then equations (51) and α α (52) may be written as Z K(x − y)H(w(y, t) − Θ)dy, (51∗) wt + w = R     Z 1 wt + w = K(x − y)H w y, t − |x − y| − Θ dy, (52∗) c R where 0 < Θ < 1 is a constant. However, to investigate clearly how the wave speed depends on the biological mechanisms α, θ and K, we will work with equations (51) and (52) rather than equations (51*) and (52*).

5.1.

The Influence of Synaptic Rate Constant and Threshold on Wave Speeds

Theorem 13. Consider (51) with two pairs of parameters (α1 , θ1 ) and (α2 , θ2 ), such that 0<

θ1 θ2 < . α1 α2

We have the estimate µ2 (α2 , θ2 ) < µ1 (α1 , θ1 ). In particular, if θ1 < θ2 and α1 = α2 , then µ2 (α2 , θ2 ) < µ1 (α1 , θ1 ). If α1 > α2 and θ1 = θ2 , then µ2 (α2 , θ2 ) < µ1 (α1 , θ1 ). Proof. The wave speeds of (51) satisfy the following equations 1 − 2

Z

0

−∞

exp



 c − µ1 θ1 x K(x)dx = , cµ1 α1

162

Linghai Zhang

and 1 − 2

Z

0

exp

−∞



 c − µ2 θ2 x K(x)dx = . cµ2 α2

Since       Z 0 Z 0 c−µ c−µ 1 d 1 |x| exp exp − x K(x)dx = − 2 x K(x)dx < 0, dµ 2 cµ µ −∞ cµ −∞ see [84], the function 1 − 2

Z

0

exp −∞



 c−µ x K(x)dx cµ

is a strictly decreasing function of µ. Thus, we find that µ2 (α2 , θ2 ) < µ1 (α1 , θ1 ),

if

θ2 θ1 < . α1 α2

The proof of Theorem 13 is finished.

5.2.

The Influence of Sodium Currents on Wave Speeds

Theorem 14. Let f and g be two functions, such that f (u) ≤ u ≤ g(u),

on (0, θ).

Let u(x, t) = Uf (x + µf t) and u(x, t) = Ug (x + µg t) be the traveling wave fronts of     Z 1 ut + f (u) = α K(x − y)H u y, t − |x − y| − θ dy, c R and

    1 ut + g(u) = α K(x − y)H u y, t − |x − y| − θ dy, c R Z

respectively, such that Uf (0) = Ug (0) = θ, Uf < θ and Ug < θ on (−∞, 0), Uf > θ and Ug > θ on (0, ∞), and there exist the limits lim Uf (z) = 0,

z→−∞

lim Ug (z) = 0.

z→−∞

Then µf ≥ µg . In particular, if additionally f 6= g, then µf > µg . Proof. First of all, the traveling wave fronts satisfy the traveling wave equations Z     µf ′ |z − y| − θ dy, µf Uf + f (Uf ) = α K(z − y)H Uf y − c R d , and dz Z     µg ′ µg Ug + g(Ug ) = α K(z − y)H Ug y − |z − y| − θ dy, c R

where z = x + µf t and ′ =

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

163

where z = x + µg t. Using the assumptions f (u) ≤ u ≤ g(u),

on (0, θ),

and a series of change of variables, see [84], we get Z cz/(c+s(z)µf ) ′ K(x)dx, µf Uf + Uf ≥ α Z

µg Ug ′ + Ug ≤ α

−∞ cz/(c+s(z)µg )

K(x)dx,

−∞

on (−∞, 0). Using integrating factor technique, we find Uf (z) ≥ α Ug (z) ≤ α

Z

cz/(c+s(z)µf )

−∞ Z cz/(c+s(z)µg ) −∞

K(x)dx − α K(x)dx − α

on (−∞, 0). Let z = 0, recall that

R0

z

exp

−∞ Z z

exp

−∞

−∞ K(x)dx

α −α 2

Z

α θ = Ug (0) ≤ − α 2

Z

θ = Uf (0) ≥

Z





x−z µf x−z µg





c K c + s(x)µf c K c + s(x)µg



 c − µf x K(x)dx. cµf

exp



 c − µg x K(x)dx. cµg

−∞



cx c + s(x)µf

cx c + s(x)µg





dx,

dx,

= 1/2, we have

exp

0



Similarly, we have 0

−∞

Note that       Z Z 0 d α c−µ c−µ α 0 |x| exp exp −α x K(x)dx = − 2 x K(x)dx < 0. dµ 2 cµ µ −∞ cµ −∞ Therefore, it follows from the above inequalities that µf ≥ µg . Moreover, if f 6= g, then µf > µg . The proof of Theorem 14 is finished.

5.3.

A Technical Lemma

We study how the speed µ0 depends on the synaptic coupling K. We compare wave speeds of (51) corresponding to different synaptic couplings. Lemma 14. Assume that f (u) = u. Suppose that K1 and K2 satisfy (13)–(15). Let µ1 and µ2 be the speeds of fronts of (51) with K1 and K2 , respectively. (I) If there exists a positive number M > 0 such that K1 ≤ K2

K1 ≥ K2

on (−∞, −M ), on

(−M, 0),

then µ1 ≤ µ2 . Here µ1 = µ2 holds if and only if K1 = K2 almost everywhere on (−∞, 0).

164

Linghai Zhang

(II) Let n be an integer, let M0 = 0 and M2n+2 = ∞. Suppose that there exist 2n + 1 positive numbers 0 < M1 < M2 < M3 < · · · < M2n+1 < ∞, such that Z −M2k Z −M2k K1 (x)dx ≥ K2 (x)dx, −M2k+2

−M2k+2

for all k = 0, 1, · · · , n, and that K1 ≤ K2

K1 ≥ K2

on (−∞, −M2n+1 ) ∪ · · · ∪ (−M4 , −M3 ) ∪ (−M2 , −M1 ),

on (−M2n+1 , −M2n ) ∪ · · · ∪ (−M3 , −M2 ) ∪ (−M1 , 0).

Then µ1 ≤ µ2 . Here µ1 = µ2 holds if and only if K1 = K2 almost everywhere on (−∞, 0).

2 1.8 K

1

1.6 1.4

y

1.2 1 0.8 K

2

0.6 0.4 0.2 0 −2

−1.8

−1.6

−1.4

−1.2

−1 x

−0.8

−0.6

−0.4

−0.2

Figure 13. Illustration of the intersection of two synaptic couplings. Proof. We have defined the following speed index function   Z 0 c−µ x K(x)dx. exp φ(µ) = α cµ −∞ (I) By the assumptions on Ki , we have the estimates α

Z

0

−∞

exp



   Z 0 c−µ c−µ x K1 (x)dx − α x K2 (x)dx exp cµ cµ −∞

0

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

165

 0   c−µ x K1 (x) − K2 (x) dx =α exp cµ −∞     Z 0 Z −M     c−µ c−µ x K1 (x) − K2 (x) dx + α x K1 (x) − K2 (x) dx exp =α exp cµ cµ −M −∞     Z −M Z 0     c−µ c−µ ≥α exp − M K1 (x)−K2 (x) dx+α M K1 (x)−K2 (x) dx exp − cµ cµ −∞ −M  Z 0   c−µ = α exp − M (52) K1 (x) − K2 (x) dx = 0, cµ −∞ Z



and the equality holds if and only if K1 = K2 almost everywhere on (−∞, 0). Since α φ(µ1 , K1 ) = φ(µ2 , K2 ) = − θ and φ(µ) is strictly increasing in µ, we obtain µ1 ≤ µ2 . 2 (II) We have    Z 0 c−µ c−µ exp x K1 (x)dx − α x K2 (x)dx exp α cµ cµ −∞ −∞   Z 0   c−µ exp =α x K1 (x) − K2 (x) dx cµ −∞   Z n X −M2k   c−µ x K1 (x) − K2 (x) dx =α exp cµ k=0 −M2k+2  Z 0   c−µ ≥ α exp − K1 (x) − K2 (x) dx M1 cµ −M2  Z −M2    c−µ K1 (x) − K2 (x) dx + · · · M3 +α exp − cµ −M4  Z −M2n    c−µ M2n+1 K1 (x) − K2 (x) dx +α exp − cµ −∞  Z 0   c−µ ≥ α exp − K1 (x) − K2 (x) dx = 0. M2n+1 cµ −∞ Z

0



(53)

α Therefore, from φ(µ1 , K1 ) = φ(µ2 , K2 ) = − θ, we get the estimate µ1 ≤ µ2 . The proof 2 of Lemma 14 is completed.

5.4.

The Influence of Synaptic Coupling on Wave Speeds

Intuitively, Lemma 14 implies that if the excitation in one neuronal network is more widely distributed than the excitation in a second neuronal network, then the wave speed in the first network is faster than the wave speed in the second network. Biologically, this makes sense: the more excitatory couplings one neuron has with distant neurons, the faster the wave propagates. Mathematically, this can be seen by comparing K(x) and Kρ (x) ≡ ρK(ρx). It is easy to prove that the speed of the traveling wave front of (51) with Kρ (x) is given by cµ(1) µ(1) µ(ρ) = . In particular, if c = ∞, then µ(ρ) = , where µ(1) is the µ(1) + ρ[c − µ(1)] ρ speed of the traveling wave front of (51) with K1 (x) = K(x). Note that Kρ (x) is more

166

Linghai Zhang

widely distributed than K(x), for all ρ < 1. Needless to say, this consideration is related to the transformation x 7→ ρx. Taking these considerations into account, we would like to introduce a quantity which may be invariant under the transformation x 7→ ρx. We define the moment of degree one by Z |x|K(x)dx, (54) m= R

which measures the first absolute moment if the synaptic coupling K represents a probability density function. For Kρ (x) = ρK(ρx), we have Z Z Z   1 m |x| ρK(ρx) dx = |x|Kρ (x)dx = m(ρ) = |x|K(x)dx = . ρ R ρ R R µ(ρ) µ0 = is invariant under the transformation x 7→ ρx. m(ρ) m µ0 is very important. Mathematically, the quantity m Let A > 0 and m > 0 be positive parameters. For any synaptic coupling K (pure excitation, lateral inhibition, or lateral excitation), we define the following functions Z x K(ξ)dξ, on (−∞, 0), κ(x) = Therefore, if c = ∞, then

−∞

κm (x) = 0 on (−∞, −m),

κm (x) = A on (−m, 0).

Lemma 15. Suppose that the positive numbers A and m are such that Aα > θ, κ − κm ∈ L1 (−∞, 0), and κ(x) − κm (x) ≥ 0 on

(−∞, −m),

κ(x) − κm (x) ≤ 0 on (−m, 0), Z 0   κ(x) − κm (x) dx ≤ 0. −∞

Then there holds the estimate      Z c−µ 0 c−µ c−µ exp x κ(x)dx ≤ A 1 − exp − m , cµ −∞ cµ cµ for every number µ ∈ (0, c), where the equality holds if and only if κ = κm on (−∞, 0). Proof. First of all, we have 

   c−µ exp x κ(x) − κm (x) dx cµ −∞     Z −m Z 0     c−µ c−µ = exp x κ(x) − κm (x) dx + x κ(x) − κm (x) dx exp cµ cµ −∞ −m   Z −m  Z 0     c−µ ≤ exp − m κ(x) − κm (x) dx + κ(x) − κm (x) dx cµ −∞ −m

Z

0

Traveling Waves Arising from Synaptically Coupled Neuronal Networks 

= exp −

c−µ m cµ

Z

0

−∞

  κ(x) − κm (x) dx ≤ 0,

167 (55)

and the equality holds if and only if κ = κm almost everywhere on (−∞, 0). Therefore     Z 0 Z 0 c−µ c−µ exp exp x κ(x)dx ≤ x κm (x)dx. cµ cµ −∞ −∞ Note that    Z c−µ c−µ c−µ 0 exp exp x κm (x)dx = A x dx cµ cµ −m cµ  −∞  c−µ m . = A 1 − exp − cµ

c−µ cµ

Z

0



The proof of Lemma 15 is completed. Theorem 15. Suppose that f (u) ≥ u for all u. (I) If the synaptic coupling K represents a pure excitation, then the speed satisfies the estimates     Z Z α |x|K(x)dx . (56) + 0 < µ0 < c |x|K(x)dx / c ln α − 2θ R R If K(x) =

 1 δ(x + d) + δ(x − d) , 2

where d > 0 is a parameter, then 1 1 1 α = + ln . µ0 c d α − 2θ (II) If the synaptic coupling K represents a lateral inhibition, then    Z  Z α 0 < µ0 < c |x||K(x)|dx / c ln + |x||K(x)|dx . α − 2θ R R

(57)

If K(x) =

 1 2δ(x + d1 ) + 2δ(x − d1 ) − δ(x + d2 ) − δ(x − d2 ) , 2

where d2 > d1 > 0 are parameters, then     c − µ0 α − 2θ c − µ0 d1 − exp − d2 = . 2 exp − cµ0 cµ0 α (III) If the synaptic coupling K represents a lateral excitation, then    Z  Z αA 0 < µ0 < c |x|K(x)dx / c ln |x|K(x)dx , + αA − θ R R

(58)

168

Linghai Zhang R −M

where A = −∞ K(x)dx, M > 0 is a constant such that K ≥ 0 on (−∞, −M ) and K ≤ 0 on (−M, 0). If K(x) =

 1 2δ(x + d2 ) + 2δ(x − d2 ) − δ(x + d1 ) − δ(x − d1 ) , 2

where d2 > d1 > 0 are parameters, then     c − µ0 c − µ0 α − 2θ 2 exp − . d2 − exp − d1 = cµ0 cµ0 α

Proof. Based on Theorem 14, to derive the upper bound for the wave speed, we only need to consider the case fR(u) = u in equation (51). Let K be any R x synaptic coupling, and define its moment by m = R |x|K(x)dx. Recall that κ(x) = −∞ K(ξ)dξ. It is easy to check that xκ(x) → 0 as x → −∞. Since m=

Z

R

|x|K(x)dx = −2

Z

0



 0 xκ (x)dx = −2 xκ(x) −∞ + 2 ′

−∞

we find that

Z

0

κ(x)dx =

−∞

Z

0

κ(x)dx = 2 −∞

Z

0

κ(x)dx, −∞

m . 2

On the other hand, we have that     Z 0 Z 0 c−µ c−µ exp exp x K(x)dx = x κ′ (x)dx cµ cµ −∞ −∞    0    Z c−µ c−µ 0 c−µ exp x κ(x) x κ(x)dx − = exp cµ cµ −∞ cµ −∞   Z c−µ 1 c−µ 0 exp x κ(x)dx. = − 2 cµ −∞ cµ Therefore, φ(µ) satisfies     α c−µ c−µ x κ(x)dx ≥ − αA 1 − exp − m cµ 2 cµ −∞   α c−µ = − αA + αA exp − m , 2 cµ

φ(µ) =

c−µ α −α 2 cµ

Z

0

exp



where we have used the following estimate of Lemma 15      Z c−µ 0 c−µ c−µ exp x κ(x)dx ≤ A 1 − exp − m . cµ −∞ cµ cµ If µ0 is the speed of the front, then we obtain the estimate   c − µ0 α − 2θ α − αA + αA exp − m < φ(µ0 ) = < ∞. 2 cµ0 2 This implies that αA < exp αA − θ



 c − µ0 m , cµ0

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

169

or equivalently ln

αA c − µ0 < m. αA − θ cµ0

Finally, we obtain the desired estimates    Z 0 < µ0 < c |x|K(x)dx / c ln R

αA + αA − θ

Z

R

 |x|K(x)dx .

If K is in class (A), then we may simply let Z

Z

0

1 K(x)dx = , A= 2 −∞

m=

R

|x|K(x)dx,

so that Z

αA − θ > 0,

0 −∞

  κ(x) − κm (x) dx = 0,

κ − κm ≥ 0 on (−∞, −m) and κ − κm ≤ 0 on

(−m, 0).

If K is in class (C), then we may let A=

Z

−M

K(x)dx,

m=

−∞

Z

R

|x|K(x)dx,

so that αA − θ > 0,

Z

0 −∞

  κ(x) − κm (x) dx ≤ 0,

κ − κm ≥ 0 on (−∞, −m) and κ − κm ≤ 0 on (−m, 0), where M > 0 is a number such that K ≥ 0 on (−∞, −M ) and K ≤ 0 on (−M, 0). Given any Mexican hat kernel function K, there exists a positive number M , such that e in K ≤ 0 on (−∞, −M ) and K ≥ 0 on (−M, 0). Define a nonnegative kernel function K e e e the following way: K = 0 on (−∞, −M ), K = 0 on (−M, −N ), K = K on (−N, 0), e e and K(−x) = K(x) on R, where N is a positive constant determined by the following condition Z 0 Z 0 e K(x)dx = K(x)dx. −∞

−∞

Let µ0 and m denote the speed and the moment associated with K, let µ e0 and m e denote the e speed and the moment associated with K. By Lemma 14, µ0 < µ e0 . Moreover m=2

Z

0

|x|K(x)dx = 2 −∞ "Z −M

<m e + 2M

Z

−M

|x|K(x)dx + 2 −∞ # Z

−N

−M

−N

K(x)dx +

−∞

Z

−M

K(x)dx = m. e

|x|K(x)dx + 2

Z

0

−N

|x|K(x)dx

170

Linghai Zhang

Overall µ0 < µ e0 and m < m. e Moreover, based on (I), we get    Z  Z α e e / c ln µ e0 < c |x|K(x)dx |x|K(x)dx . + α − 2θ R R

Therefore, we obtain the estimates    Z 0 < µ0 < µ e0 < c |x||K(x)|dx / c ln R

α + α − 2θ

Z

R

 |x||K(x)|dx .

Given any upside down Mexican hat kernel function K, there exists a positive constant M , such that K ≥ 0 on (−∞, −M ) and K ≤ 0 on (−M, 0). Let Z x Z −M K(ξ)dξ. K(x)dx, κ(x) = A= −∞

−∞

Then κ ≥ 0 on (−∞, −m) and κ ≤ A on (−m, 0), for any m > 0. Define a nonnegative e K e = K on (−∞, −M −N ), K e = 0 on (−M −N, 0), K(−x) e e kernel function K: = K(x) on R, where N is a positive constant determined by the following condition Z 0 Z 0 e K(x)dx = K(x)dx. −∞

−∞

Let µ0 and m denote the speed and the moment associated with K, let µ e0 and m e denote the e By Lemma 14, µ speed and the moment associated with K. e0 < µ0 . Moreover Z 0 |x|K(x)dx m=2 =2

Z

−∞ −M −N

|x|K(x)dx + 2

−∞

>m e + 2M

Z

−M

Z

−M

−M −N Z 0

K(x)dx +

−M −N

−M

Overall, µ e0 < µ0 and m e < m. Moreover    Z e 0<µ e0 < c |x|K(x)dx / c ln R

|x|K(x)dx + 2

Z

 K(x)dx = m. e

α + α − 2θ

Z

R

The proof of Theorem 15 is finished.

0

−M

|x|K(x)dx

 e |x|K(x)dx .

Theorem 16. Estimates (56) are equivalent to the following  Z   Z α |x|K(x)dx . (59) + 0 < µ0 / |x|K(x)dx < c/ c ln α − 2θ R R R  The first inequality µ0 / R |x|K(x)dx > 0 is optimal in the senseRthat for any positive  number ε > 0, there exists a synaptic coupling Kε such that hR µ0 (Kε )/ iR |x|Kε (x)dx < ε e e such that µ0 (K)/ e and there is no synaptic coupling K, R |x|K(x)dx = 0. The estimate µ0 /

Z

R

  |x|K(x)dx < c/ c ln

α + α − 2θ

Z

R

 |x|K(x)dx

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

171

is also optimal in the sense that for any ε > 0, there exists a synaptic coupling Kε , such that   Z  Z α µ0 (Kε )/ |x|Kε (x)dx + ε > c/ c ln |x|Kε (x)dx , + α − 2θ R R

e such that but there is no synaptic coupling K Z   e e µ0 (K)/ |x|K(x)dx = c/ c ln R

α + α − 2θ

Z

R

 e |x|K(x)dx .

Proof. Let us show that there exists a sequence of positive synaptic couplings {Ki }, such µ(i) that the corresponding speeds satisfy → 0 as i → ∞. Define the synaptic couplings m(i) Ki by Ki (x) =

1 ai exp(−ai |x|) + bi exp(−bi |x|) , 4

where ai > 0 and bi > 0 are constants to be determined later. Then it is easy to see that Z Ki (x)dx = 1, R

and Z

1 |x|Ki (x)dx = m(i) = 2 R



1 1 + ai bi



.

Now    ai cµ α bi cµ c−µ x Ki (x)dx = + exp φi (µ) = α cµ 4 ai cµ + c − µ bi cµ + c − µ  −∞  α ai µ bi µ > . + 4 ai µ + 1 bi µ + 1 Z

0



1 and bi = i, then m(i) → ∞ as i → ∞. On the other hand, i 1 0 < µ(i) < c < ∞ if 0 < c < ∞. For the case c = ∞, let ζ = . Then µi   2θ 1 α − 4θ ζ− = 0. ζ2 + i + i 2α − 4θ α − 2θ

Here, if we choose ai =

By using quadratic formula, we find   s r  2  2   8θ  1 4θ − α 1 4θ − α 1 1 θ + + . > i+ ζ= i+   2 i 2α − 4θ i 2α − 4θ α − 2θ 4 α − 2θ Therefore, we obtain the uniform estimate r µi < 4

α − 2θ , 2θ

172

Linghai Zhang

for all integers i ≥ 1. Overall, there hold the estimates r

0 < µ(i) < 4

This proves that optimal.

α − 2θ . 2θ

R  µ(i) → 0 as i → ∞. Hence the estimate µ0 / R |x|K(x)dx > 0 is m(i)

1 Finally, given any K ≥ 0, define κm (x) = 0 on (−∞, −m) and κm (x) = on 2 R (−m, 0), where m = R |x|K(x)dx. There exists a sequence of positive functions {κj } in C ∞ (−∞, 0), such that κj (x) → κm (x), as j → ∞, in L1 -sense. For example, the sequence of functions 1 κj (x) = √ π

Z

R



exp − τ 2 κm x − τ /j dτ

is good, where j = 1, 2, 3, · · · . It is easy to prove that Z

0

|κj (x) − κm (x)|dx Z 0 Z   1 2 exp(−τ ) κm (x − τ /j) − κm (x) dτ dx =√ π −∞ R Z Z 2 m2 0 2 τ ≤√ exp(−τ ) 2 dτ dx. j π −m R

−∞

As j → ∞, we have

  Z α c−µ c−µ 0 φj (µ) = − α exp x κj (x)dx 2 cµ −∞ cµ   Z c−µ c−µ 0 α exp x κm (x)dx → −α 2 cµ −∞ cµ   c−µ α m , = exp − 2 cµ cµj corresponding to the function κj c − µj , as j → ∞. This shows that the estimate

uniformly in µ. This implies that the quantity converges to

m α ln α − 2θ µ0 /

Z

R

  |x|K(x)dx < c/ c ln

α + α − 2θ

is optimal. The proof of Theorem 16 is completed.

Z

R

 |x|K(x)dx

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

5.5.

173

Some Examples

Example 3. Let j ≥ 0 be any nonnegative integer and let ρ > 0 be a constant. We define Kj (x) = Certainly

1 j+1 j ρ |x| exp(−ρ|x|). 2j! Z

Kj (x)dx = 1, R

for all integers j ≥ 0. The synaptic coupling K0 is on-center and the synaptic coupling Kj j+1 is off-center for all j ≥ 1. Note that the maximum of Kj+1 is attained at xj+1 = − ρ j and the maximum of Kj is attained at xj = − , so we may say Kj+1 is more off-center ρ than Kj , for all j ≥ 1. Let us investigate the wave speed of (51) where c = ∞. The corresponding speed is given by µj =

1 (α − 2θ)1/(j+1) 1 1 · 1/(j+1) = , 1/(j+1) 1/(j+1) ρ α ργ − (α − 2θ) −1

α . Obviously, there hold the following estimates for the speeds of the α − 2θ fronts of (51) where γ =

α − 2θ = µ0 < µ1 < µ2 < · · · < µj < µj+1 < · · · , 2ρθ lim µj = ∞.

0<

j→∞

Next, let us consider as

µj , where mj is the moment corresponding to Kj , which is computed mj

Z Z 1 |x| ρj+1 |x|j exp(−ρ|x|)dx |x|Kj (x)dx = mj = 2j! R Z R 1 j+1 j+1 ρ |x| exp(−ρ|x|)dx = 2j! R Z j+1 1 = ρj+2 |x|j+1 exp(−ρ|x|)dx ρ 2(j + 1)! ZR j+1 j+1 Kj+1 (x)dx = . = ρ ρ R

Hence µj 1 (α − 2θ)1/(j+1) 1 1 = · 1/(j+1) · 1/(j+1) = . 1/(j+1) mj j+1 α j + 1 − (α − 2θ) γ −1 µj Note that is independent of ρ. Furthermore, we get mj µj+1 µj j + 1 γ 1/(j+1) − 1 γ 1/(j+1) − 1 γ 1/(j+2) − 1 / = · 1/(j+2) / , = 1 1 mj+1 mj j+2 γ −1 j+1 j+2

174 where γ =

Linghai Zhang α γx − 1 > 1. Here is an increasing function on (0, ∞). Note that α − 2θ x 0<

1 1 < . j+2 j+1

Thus, we obtain µj+1 µj < mj mj+1 µj is increasing in j. Even though the quantity mj µj is independent of ρ, it depends on j. In another word, as the synaptic coupling becomes mj µj more and more off-centered, the quantity becomes larger and larger. Furthermore, we mj get

for every j = 0, 1, 2, · · · . This shows that

lim

j→∞

µj+1 1 = , mj+1 ln γ

lim

j→∞

µj+1 µj / = 1. mj+1 mj

Example 4. Let A > 0, B > 0, a > 0, b > 0 and ρ > 0 be constants, such that A > B > 0, A B − = 1, a b

a > b > 0, A B ≥ 2. a2 b

We define the following synaptic couplings K(x) = and

ρ exp(−ρ|x|) 2

Aρ Bρ e K(x) = exp(−aρ|x|) − exp(−bρ|x|). 2 2

e It is easy to see that K(x) = AK(ax) − BK(bx). Let µ and µ e represent the wave speeds of e respectively. By Lemma 14, we have equation (51) with the synaptic couplings K and K, the estimate µ e < µ if A − B > 1 and K(0) > 0. We also find that Aρ ce µ ce µ cµ Bρ ρ − = . 2 ace µρ + c − µ e 2 bce µρ + c − µ e 2 cµρ + c − µ

Noting that a > b > 0, we get

Therefore, µ e < µ if a ≥ 1.

cµ ace µ < . acρe µ+c−µ e cρµ + c − µ

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

5.6.

175

Several Applications

Application 1. Synaptic coupling with high peak versus synaptic coupling with low peak. Given the synaptic coupling K, define Kρ (x) = ρK(ρx), for any constant ρ > 0. Note that Z Z Kρ (x)dx = 1. K(x)dx = R

R

The wave speeds µ and µ(ρ) of the traveling wave fronts of (51) corresponding to the synaptic couplings K and Kρ satisfy α

Z

0

−∞

exp



   Z 0 c−µ α c − µ(ρ) exp x K(x)dx = − θ = α x Kρ (x)dx. cµ 2 cµ(ρ) −∞

Making a change of variable, we find that   Z 0 α c − µ(ρ) x K(x)dx = − θ. exp α cρµ(ρ) 2 −∞ cµ . Therefore, if ρ > 0 is µ + ρ(c − µ) very small, then Kρ is very flat and the speed µ(ρ) is very big. On the other hand, if ρ > 0 is very large, then Kρ has a high peak (if K(0) > 0), and the speed µ(ρ) is very small.

Due to the uniqueness of the speed, we have µ(ρ) =

Application 2. On-center pure excitation versus off-center pure excitation. Given a nonnegative on-center kernel function K, we may generate the off-center synaptic couplings Kj (x) = Cj x2j K(x), for all integers j ≥ 1, where Cj > 0 is a normalization constant, so that Z Kj (x)dx = 1. R

Since Kj+1 (x) Cj+1 2 = x < 1, Kj (x) Cj

if

|x| <

s

Cj , Cj+1

|x| >

s

Cj , Cj+1

and Cj+1 2 Kj+1 (x) = x > 1, Kj (x) Cj

if

we may apply Lemma 14 to conclude that µj+1 > µj for every integer j = 1, 2, · · · . This implies that the traveling wave front corresponding to off-center synaptic couplings has a larger speed than traveling wave front corresponding to on-center synaptic couplings. The speed increases when j becomes larger and larger, that is, as the synaptic coupling Kj becomes more and more off-center. Application 3. Pure excitation versus lateral inhibition

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2

1.5

y

1

0.5

0

−0.5

−1 −2

−1.5

−1

−0.5

0 x

0.5

1

1.5

2

Figure 14. Comparison of a pure excitation and a lateral inhibition. Case 3A. Given any nonnegative kernel function K and given any constants A > B > 0 and a > b > 0, such that A B A B − = 1, ≥ 2, 2 a b a b we may generate a Mexican hat kernel function

Then

Z

R

e K(x) = AK(ax) − BK(bx). e K(x)dx = 1,

Z

0 −∞

e |x|K(x)dx ≥ 0.

e and their corresponding speeds µ and µ The synaptic couplings K and K e satisfy     Z 0 Z 0 c−µ α c−µ e e exp x K(x)dx = − θ = α x K(x)dx. exp α ce µ 2 cµ −∞ −∞ More explicitly, we have     Z Z Aα 0 c−µ e c−µ e Bα 0 exp exp x K(x)dx − x K(x)dx a −∞ ace µ b −∞ bce µ

Traveling Waves Arising from Synaptically Coupled Neuronal Networks   Z 0 c−µ exp =α x K(x)dx. cµ −∞

177

(I) If A − B > 1, K ′ ≤ 0 on (0, ∞) and K(0) > 0, then µ e < µ. (II) Noting that a > b > 0 and K ≥ 0, we have the estimate     Z 0 Z 0 c−µ e c−µ e exp x K(x)dx < x K(x)dx. exp bce µ ace µ −∞ −∞ Now we have Z

0

−∞

exp



   Z 0 c−µ e c−µ exp x K(x)dx < x K(x)dx. ace µ cµ −∞

Then c−µ e c−µ < , cµ ace µ

µ e<

cµ . µ + a(c − µ)

e < µ. Therefore, if a ≥ 1, then µ Case 3B. Let K1 and K2 be nonnegative synaptic couplings, satisfying the conditions 0≤

K2 (0) K2 (x) < 1 < lim = ∞. K1 (0) |x|→∞ K1 (x)

Suppose that K1 (x) is decreasing in |x| and K2 (x)/K1 (x) is increasing in |x| and there holds Z 0 Z 0 (1 + ρ) |x|K1 (x)dx ≥ ρ |x|K2 (x)dx, −∞

−∞

for all ρ ≥ 0. Set K(ρ, x) = (1 + ρ)K1 (x) − ρK2 (x), R where ρ ≥ 0 is a parameter. For any ρ > 0, we have R K(ρ, x)dx = 1, K(ρ, x) → 0 as |x| → ∞, and   > 0 for |x| < ξ(ρ), = 0 for |x| = ξ(ρ), K(ρ, x)  < 0 for |x| > ξ(ρ), where ξ(ρ) > 0 is a decreasing function of ρ. Hence K(ρ, x) is a Mexican hat kernel function for each fixed ρ > 0. Moreover, Z 0 Z 0 Z 0 |x|K2 (x)dx ≥ 0. |x|K1 (x)dx − ρ |x|K(x, ρ)dx = (1 + ρ) −∞

−∞

−∞

By definition, if ρ1 < ρ2 , then  e     > 0 for |x| < Z, e K(ρ2 , x) − K(ρ1 , x) = (ρ2 − ρ1 ) K1 (x) − K2 (x) = 0 for |x| = Z   < 0 for |x| > Z, e

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e > 0. Hence by Lemma 14, we have µ(ρ1 ) > µ(ρ2 ) if ρ1 < ρ2 , that for some constant Z is, µ(ρ) is strictly decreasing in ρ. This implies that the speed with the Mexican hat kernel function is smaller than the speed with the nonnegative kernel function. Application 4. Pure excitation versus lateral excitation

1.5

1

y

0.5

0

−0.5

−1 −2

−1.5

−1

−0.5

0 x

0.5

1

1.5

2

Figure 15. Comparison of a pure excitation and a lateral excitation. Case 4A. Given any nonnegative kernel function K and given any constants 0 < A < B and 0 < a < b, such that A B − = 1, a b we define an upside down Mexican hat kernel function

Note that

e K(x) = AK(ax) − BK(bx). Z

R

e K(x)dx = 1.

e and their corresponding speeds µ and µ The synaptic couplings K and K e satisfy     Z 0 Z 0 c−µ e c−µ α e exp α exp x K(x)dx x K(x)dx. = −θ =α ce µ 2 cµ −∞ −∞

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More explicitly, we have     Z Z Aα 0 c−µ e c−µ e Bα 0 exp exp x K(x)dx − x K(x)dx a −∞ ace µ b −∞ bce µ   Z 0 c−µ exp =α x K(x)dx. cµ −∞ e. (I) If 0 < A < B, K ′ ≤ 0 on (0, ∞) and K(0) > 0, then µ < µ (II) There holds the estimate     Z 0 Z 0 c−µ e c−µ e exp x K(x)dx < x K(x)dx, exp ace µ bce µ −∞ −∞ so that

Z

0

−∞

Thus

exp



   Z 0 c−µ c−µ e exp x K(x)dx < x K(x)dx. cµ ace µ −∞

c−µ c−µ e < , ace µ cµ

cµ <µ e. µ + a(c − µ)

e if a ≤ 1. Hence µ < µ Case 4B. Let K1 (x) and K2 (x) be nonnegative synaptic couplings. Suppose that both K1 (x) and K2 (x)/K1 (x) are decreasing in |x|. Let ρ∗ ≡ K1 (0)/{K2 (0) − K1 (0)} > 0. Define K(ρ, x) = (1 + ρ)K1 (x) − ρK2 (x), R where ρ ≥ 0 is a parameter. Then R K(ρ, x)dx = 1 for all ρ > 0. Moreover, K(ρ, x) is a nonnegative kernel function if 0 ≤ ρ ≤ ρ∗ . If ρ > ρ∗ , then K(ρ, x) → 0 as |x| → ∞, and  for |x| < η(ρ),  <0 =0 for |x| = η(ρ), K(ρ, x)  >0 for |x| > η(ρ), where η(ρ) > 0 is an increasing function of ρ > ρ∗ . Hence K(ρ, x) is an upside down Mexican hat kernel function for each ρ > ρ∗ . By definition, if ρ1 < ρ2 , then  e   > 0 for |x| < Z, e K(ρ1 , x) − K(ρ2 , x) = (ρ2 − ρ1 )[K2 (x) − K1 (x)] = 0 for |x| = Z   < 0 for |x| > Z, e

e > 0. We denote by µ(ρ) the speed of the traveling wave front for for some constant Z K(ρ, x). By Lemma 14, we have µ(ρ1 ) < µ(ρ2 ) if ρ1 < ρ2 , that is, µ(ρ) is strictly increasing in ρ. This implies that the speed of (51) with the upside down Mexican hat kernel function is larger than the speed of (51) with the nonnegative kernel function. Application 5. Lateral inhibition versus lateral excitation. Let K1 and K2 represent lateral inhibition and lateral excitation between neurons in neuronal networks. Then, by Lemma 14, the speeds corresponding to K1 and K2 satisfy µ1 < µ2 .

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Linghai Zhang

1 0.8 0.6 0.4

y

0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −2

−1.5

−1

−0.5

0 x

0.5

1

1.5

2

Figure 16. Comparison of a lateral inhibition and a lateral excitation.

6. Standing Waves 6.1.

Existence of Standing Waves

Let us investigate the standing waves of the scalar homogeneous integral differential equation Z ut + u = α K(x − y)H(u(y, t) − θ)dy. (60) R

Let K be a symmetric kernel function on R. (I) Let α = 2θ. Then there are two monotone standing waves Z ∞ Z x K(ξ)dξ. K(ξ)dξ, U− (x) = α U+ (x) = α −∞

x

The monotone standing waves are symmetric about the point (0, α/2). Indeed, we have Z 0 Z x Z 0 Z x α U+ (x) − K(ξ)dξ K(ξ)dξ = α K(ξ)dξ = α K(ξ)dξ − α =α 2 −x 0 −∞ −∞ Z −x Z 0 α K(ξ)dξ = − U+ (x). K(ξ)dξ − α =α 2 −∞ −∞

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181

(II) Let α and θ satisfy α

Z

Z

K(x)dx = θ, 0

for some constant Z > 0. Then there exists a standing wave Z x K(ξ)dξ. U (x) = α x−Z

(III) Let α and θ satisfy α−α

Z

Z

K(x)dx = θ, 0

for some constant Z > 0. Then there exists a standing wave Z x K(ξ)dξ. U (x) = α − α x−Z

Let us consider the scalar nonhomogeneous integral differential equation Z ut + u = α K(x − y)H(u(y, t) − θ)dy + I(x, t),

(61)

R

and the system ut + u + w = α

Z

R

K(x − y)H(u(y, t) − θ)dy + I(x, t),

wt = ε(u − γw).

(62) (63)

Let I(x, t) = I(x) be independent of t. Standing waves (U, W ) are independent of time, that is, Ut = Wt = 0 in (63)–(64). Thus we have the system Z U + W = α K(x − y)H(U (y) − θ)dy + I(x), R

U = γW. The system can be decoupled by eliminating W . In fact, we have Z αγ γ U= K(x − y)H(U (y) − θ)dy + I(x). 1+γ R 1+γ Without loss of generality, we will investigate the standing waves of the scalar integral differential equation (62). The standing waves satisfy the equation Z U (x) = α K(x − y)H(U (y) − θ)dy + I(x). R

To make the problem simple, we can assume that U (0) = U (Z) = θ, where 0 < Z < ∞ or Z = ∞. If U < θ on (−∞, 0) ∪ (Z, ∞) and U > θ on (0, Z), then we have Z x K(ξ)dξ + I(x). U (x) = α x−Z

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If U < θ on (0, Z) and U > θ on (−∞, 0) ∪ (Z, ∞), then we have Z x K(ξ)dξ + I(x). U (x) = α − α x−Z

Theorem 17. For any constants α and θ, 0 < θ < α, and for any synaptic coupling K, there exists at least a nonhomogeneous function I, such that (62) has a standing wave.

1.1

1

0.9

y

0.8

0.7

0.6

0.5

0.4 −0.5

0

0.5

1 x

1.5

2

2.5

Figure 17. Phase plane portrait of a standing wave solution of equation (9). Proof. We will consider three different cases. (I) Let α = 2θ, and let I be either always increasing or always decreasing and be suitably bounded on R, such that I(0) = 0, for example, the function I1 (x) = ε tanh x is increasing and the function I2 (x) = −ε tanh x is decreasing, both are bounded on R, then Z x K(ξ)dξ + I(x), if I is increasing on R, U1 (x) = α Z−∞ ∞ K(ξ)dξ + I(x), if I is decreasing on R, U2 (x) = α x

are the monotone standing waves of (62), such that U1 (0) = U2 (0) = θ, U1 < θ on (−∞, 0) and U1 > θ on (0, ∞); U2 > θ on (−∞, 0) and U2 < θ on (0, ∞).

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183

2.6

2.5

2.4

y

2.3

2.2

2.1

2

1.9 −0.5

0

0.5

1 x

1.5

2

2.5

Figure 18. Phase plane portrait of a standing wave solution of equation (9). (II) Let 0 < 2θ < α. By using intermediate value theorem, it is straightforward to show that there exists a number Z∗ > 0, such that Z Z∗ K(x)dx = θ. α 0

Choose the function I such that I(0) = I(Z∗ ) = 0, I < 0 on (−∞, 0) ∪ (Z∗ , ∞) and I > 0 on (0, Z∗ ). For example, the function I(x) = ε tanh[x(Z∗ − x)] satisfies the conditions, where 0 < ε ≪ 1. The standing wave solution is given by Z Z∗ K(x − y)dy + I(x). U∗ (x) = α 0

It satisfies the conditions U (0) = U (Z∗ ) = θ, U < θ on (−∞, 0) ∪ (Z∗ , ∞) and U > θ on (0, Z∗ ). (III) Let 2θ > α, then there exists a positive number Z ∗ > 0 such that Z Z∗ α−α K(x)dx = θ. 0

Choose the function I such that I(0) = I(Z ∗ ) = 0, I < 0 on (0, Z ∗ ) and I > 0 on (−∞, 0) ∪ (Z ∗ , ∞). For example, the function I(x) = ε tanh[x(x − Z ∗ )] satisfies the

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Linghai Zhang

conditions, where 0 < ε ≪ 1. The standing wave is given by U ∗ (x) = α − α

Z

0

Z∗

K(x − y)dy + I(x),

such that U (0) = U (Z ∗ ) = θ, U < θ on (0, Z ∗ ) and U > θ on (−∞, 0) ∪ (Z ∗ , ∞).

6.2.

Linear Stability of Standing Waves

Theorem 18. The monotone standing waves U1 and U2 are linearly stable. The nonmonotone standing waves U∗ and U ∗ are linearly unstable. Proof. First of all, let 0 < 2θ < α and Z = Z∗ . Then the linearization of (62) about the standing wave is given by Pt (x, t) + P (x, t) =

α α K(x)P (0, t) − ′ K(x − Z)P (Z, t). U ′ (0) U (Z)

If we substitute solutions of the form P (x, t) = exp(λt)ψ(x) into the differential equation, we obtain the eigenvalue problem (λ + 1)ψ(x) =

α U ′ (0)

K(x)ψ(0) −

α U ′ (Z)

K(x − Z)ψ(Z).

(64)

Define Ω = {λ ∈ C: Reλ > −1}. We will study the eigenvalue problem in Ω. If ψ(0) = ψ(Z) = 0, then ψ ≡ 0. If ψ(0) = 0 or ψ(Z) = 0, then simple analysis also shows that ψ ≡ 0. Without loss of generality, we assume that ψ(0) 6= 0 and ψ(Z) 6= 0. Now we define the linear operator L : C 0 (R) ∩ L∞ (R) → C 0 (R) ∩ L∞ (R) by Lψ = −ψ +

α α K(x)ψ(0) − ′ K(x − Z)ψ(Z). U ′ (0) U (Z)

(65)

The essential spectrum of L is empty. Due to translation invariance of the standing wave, λ = 0 is an eigenvalue of L and U ′ is the corresponding eigenfunction. To find other eigenvalues, setting x = 0 and x = Z in the eigenvalue problems, respectively, we get   α α λ + 1 − ′ K(0) ψ(0) = − ′ K(Z)ψ(Z), U (0) U (Z)   α α λ+1+ ′ K(0) ψ(Z) = + ′ K(Z)ψ(0). U (Z) U (0) Multiplying them together and canceling out ψ(0)ψ(Z) 6= 0, we get    α α [αK(Z)]2 λ + 1 − ′ K(0) λ + 1 + ′ K(0) = − ′ . U (0) U (Z) U (0)U ′ (Z) Solving this algebraic equation, noting that U ′ (0) = −U ′ (Z) > 0, we find that λ1 = α

K(0) − K(Z) − 1 = 0, U ′ (0)

λ2 = α

K(0) + K(Z) − 1 > 0. U ′ (0)

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185

Therefore, the standing wave U∗ is unstable. Similarly, the standing wave U ∗ for the case 2θ > α is also unstable. Suppose that α = 2θ. Then the linearization of equation (62) about the monotone standing wave is pt (x, t) + p(x, t) =

α U ′ (0)

p(0, t).

Suppose that p(x, t) = exp(λt)ψ(x) is a solution of this equation. Then we obtain the eigenvalue problem (λ + 1)ψ(x) =

α U ′ (0)

K(x)ψ(0).

Let x = 0 in this equation, we find (λ + 1)ψ(0) =

α U ′ (0)

K(0)ψ(0).

A nontrivial solution of the eigenvalue problem must satisfy ψ(0) 6= 0. Thus λ=

α U ′ (0)

K(0) − 1 = 0.

There exists no nonzero eigenvalue of the linear operator L in the right half plane {λ: Reλ > −1} and λ = 0 is a simple eigenvalue of L. Thus the monotone standing waves are stable relative to (62). Remark 7. If α 6= 2θ, then there are two eigenvalues, counting algebraic multiplicities, because the standing wave crosses the threshold θ twice. If α = 2θ, then there is only one eigenvalue, also counting algebraic multiplicity, because the standing wave crosses the threshold θ only once.

6.3.

Nonlinear Stability of Standing Waves

To study the nonlinear stability of the standing waves of the integral differential equation (62), we will investigate the initial value problems u(x, 0) = u0 (x). The following results demonstrate that the standing waves are nonlinear exponentially stable. Theorem 19. (I) Suppose that the initial data, the synaptic coupling and the nonhomogeneous function satisfy Z x K(ξ)dξ + I(x) ≤ θ on (−∞, 0), −∞ < u0 (x) ≤ θ and − ∞ < α −∞ Z x K(ξ)dξ + I(x) < ∞ on (0, ∞). θ ≤ u0 (x) < ∞ and θ ≤ α −∞

We are also assuming that −∞ < u0 (x) < θ

or

−∞<α

Z

x

K(ξ)dξ + I(x) < θ −∞

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Linghai Zhang

on some nontrivial subset Ω1 ⊂ (−∞, 0), where the Lebesgue measure of Ω1 is bigger than zero, or Z x

θ ≤ u0 (x) < ∞

θ≤α

or

−∞

K(ξ)dξ + I(x) < ∞

on some nontrivial subset Ω2 ⊂ (0, ∞), where the Lebesgue measure of Ω2 is bigger than zero. Other assumptions are understood in the same way. Then there exists a unique global solution   Z x −t −t K(ξ)dξ + I(x) , (66) u(x, t) = e u0 (x) + (1 − e ) α −∞

to the initial value problem for (62). Moreover, −∞ < u(x, t) ≤ θ on (−∞, 0) and θ ≤ u(x, t) < ∞ on (0, ∞). Additionally Z x K(ξ)dξ + I(x). lim u(x, t) = α t→∞

−∞

(II) Suppose that θ ≤ u0 (x) < ∞

and

x

K(ξ)dξ + I(x) < ∞ on (−∞, 0), Z x K(ξ)dξ + I(x) ≤ θ on (0, ∞). −∞<α

and θ ≤ α

−∞ < u0 (x) ≤ θ

Z

−∞

−∞

Then there exists a unique global solution  Z u(x, t) = e u0 (x) + (1 − e ) α −t

−t

x

∞

 K(ξ)dξ + I(x) ,

(67)

to the initial value problem for (62). Moreover −∞ < u(x, t) ≤ θ on (0, ∞) and θ ≤ u(x, t) < ∞ on (−∞, 0). Additionally Z ∞ K(ξ)dξ + I(x). lim u(x, t) = α t→∞

x

Proof. It is simple and omitted. Remark 8. Suppose that the initial data, the synaptic R x coupling and the nonhomogeneous function satisfy the conditions u0 (x) ≡ θ and α −∞ K(ξ)dξ + I(x) ≡ θ, for all x ∈ R, then there are at least two global solutions to the Cauchy problem u(x, 0) = u0 (x). The solutions are u(x, t) = e−t u0 (x) + (1 − e−t )I(x),

u(x, t) = e−t u0 (x) + (1 − e−t )[α + I(x)]. Suppose that the initial data, the synaptic R x coupling and the nonhomogeneous function satisfy the conditions u0 (x) ≡ Θ and α −∞ K(ξ)dξ + I(x) ≡ Θ, for all x ∈ R, then there are at least two global solutions to the Cauchy problem Z Z K(x − y)H(u(y, t) − Θ)dy, ut + u = α K(x − y)H(u(y, t) − θ)dy + β R

R

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

187

u(x, 0) = u0 (x). The solutions are u(x, t) = e−t u0 (x) + (1 − e−t )[α + I(x)],

u(x, t) = e−t u0 (x) + (1 − e−t )[α + β + I(x)]. Theorem 20. (I) Suppose that the initial data, the synaptic coupling and the nonhomogeneous function satisfy Z x K(ξ)dξ + I(x) ≤ θ on (−∞, 0) ∪ (Z∗ , ∞), −∞ < u0 (x) ≤ θ and − ∞ < α −∞ Z x K(ξ)dξ + I(x) < ∞ on (0, Z∗ ). θ ≤ u0 (x) < ∞ and θ ≤ α −∞

Then there exists a global solution  Z u(x, t) = e u0 (x) + (1 − e ) α −t

−t

x

x−Z∗

 K(ξ)dξ + I(x) ,

(68)

of the initial value problem for (62). Additionally, −∞ < u(x, t) ≤ θ on (−∞, 0)∪(Z∗ , ∞) and θ ≤ u(x, t) < ∞ on (0, Z∗ ). Moreover Z x K(ξ)dξ + I(x). lim u(x, t) = α t→∞

x−Z∗

(II) Suppose that θ ≤ u0 (x) < ∞

and

−∞ < u0 (x) ≤ θ

θ≤α

and

Z

x

K(ξ)dξ + I(x) < ∞ on −∞ Z x K(ξ)dξ + I(x) ≤ θ −∞<α

(−∞, 0) ∪ (Z ∗ , ∞), on

(0, Z ∗ ).

−∞

Then there exists a global solution  Z u(x, t) = e u0 (x) + (1 − e ) α − α −t

−t

x x−Z ∗

 K(ξ)dξ + I(x) ,

(69)

of the initial value problem for (62). Moreover −∞ < u(x, t) ≤ θ on (0, Z ∗ ) and θ ≤ u(x, t) < ∞ on (−∞, 0) ∪ (Z ∗ , ∞). Additionally Z x lim u(x, t) = α − α K(ξ)dξ + I(x). t→∞

x−Z ∗

Proof. It is simple and omitted. Consider the integral differential equation Z ut + u(u − 1)(Au − 1) = α K(x − y)H(u(y, t) − θ)dy, R

where A > 1 is a constant. We can also prove the existence and nonlinear stability (or linear instability) of several standing waves of this equation.

188

7.

Linghai Zhang

Open Problems and Concluding Remarks

7.1. Summary In this chapter, we investigated the existence and stability of traveling wave solutions of the following nonlinear singularly perturbed systems of integral differential equations: (I) Z ut + f (u) + w = α K(x − y)H(u(y, t) − θ)dy, R

wt = ε(g(u) − γw). (II) ut + f (u) + w = α +β

Z

Z

R

K(x − y)H(u(y, t) − θ)dy

R

K(x − y)H(u(y, t) − Θ)dy,

wt = ε(g(u) − γw). (III)     1 K(x − y)H u y, t − |x − y| − θ dy, c R wt = ε(g(u) − γw).

ut + f (u) + w = α

Z

We study traveling wave solutions of these nonlinear singularly perturbed systems of integral differential equations arising from synaptically coupled neuronal networks to model the propagation of nerve impulses. The integral differential equations generalize the classical Hodgkin-Huxley equations and the Fitzhugh-Nagumo equations. The stability of a traveling wave is completely determined by the structure of the spectrum of an associated linear differential operator L(ε), which is obtained by linearizing the nonlinear system about that traveling wave. We constructed Evans functions whose zeros coincide with the eigenvalues of the operator L(ε). We also have applications of the Evans functions. The study of the existence and stability of nonlinear traveling waves has important feedback in improving further modeling in mathematical neuroscience. The method we use can solve other similar integral differential equations as well. Remark 9. The Evans functions encode important information about The nonlinear functions The synaptic couplings The parameters The traveling waves Remark 10. The investigation of the existence and stability of the traveling waves can provide important feedback to the mathematical model equations

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

189

enhance our understanding of the nature of the propagating activities improve our further modeling in mathematical neuroscience enable us to study more realistic model equations arising from synaptically coupled neuronal networks

7.2.

Open Problems for System (1)–(2)

Consider the following nonlinear singularly perturbed system of integral differential equations Z ut + f (u) + w = α K(x − y)H(u(y, t) − θ)dy, R

wt = ε(g(u) − γw). Are the existence and stability of the fast traveling pulse solutions true for all ε ∈ (0, ∞)? What happens if ε ≫ 1? Are there any traveling waves if Z

0

−∞

|x|K(x)dx < 0.

If yes, how many? Which wave is stable and which wave is unstable? Can we construct Evans functions for this nonlocal model system with a smooth gain function, such as 1 H(ε, u) = , 1 + exp(−u/ε) where 0 < ε < ∞ is a constant. What is the influence of the cubic polynomial function f (u) = u(u − 1)(u − a) on the speed, the existence of the fast traveling pulse solution and the stability of the fast traveling pulse solution, where 0 < a < 1? Are there any traveling pulse solutions which cross the threshold θ for 2m times, where m > 1 is an integer? Are they exponentially stable or not? Are there any traveling periodic solutions which cross the threshold θ for infinitely many times? Are the traveling periodic solutions stable? In the Hodgkin-Huxley equations (see Section 1), if we replace the diffusion DM uxx by the integral αK ∗ [H(u − θ)], can we establish the existence and stability of a fast traveling pulse solution? Can we obtain bifurcation information by solving the equation E(λ) = 0? Note that if αγ < (1 + γ)θ, there exist two pulses. But if αγ > (1 + γ)θ, there exists only one wave. We may be able to calculate the limits lim

αγ/(1+γ)→θ

Efast−pulse (λ, ε),

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Linghai Zhang lim

Eslow−pulse (λ, ε),

lim

Eheteroclinic (λ, ε).

αγ/(1+γ)→θ αγ/(1+γ)→θ

We hope to obtain bifurcation and stability information through these limits.

7.3.

Open Problems for System (3)–(4)

1. The waves travel at different speeds. How do they interact with each other? 2. Can large complicated waves be represented as appropriate combinations of small simple waves? 3. Are the speeds related to each other through a mathematical equation? 4. Can speed index functions for large waves be constructed through speed index functions for small waves? 5. Can stability index functions for large waves be constructed through stability index functions for small waves? 6. For each large traveling wave, can the stability index function be constructed through its speed index function? 7. Can the speed index functions and stability index functions be used to investigate bifurcations of problem (1)–(2)?

7.4.

Open Problems for System (5)–(6)

1. Let f (u) = u. Then there exists a nice relationship between the wave speed of (7) and the wave speed of (11). We wonder if there is a similar relationship between the wave speed of (7) and the wave speed of (11), when f (u) is nonlinear, in particular, the cubic polynomial function f (u) = u(u − 1)(Au − 1). Biologically, the cubic polynomial function is usually used to describe sodium current in mathematical neuroscience. 2. Let f (u) = u and g(u) = u and let 0 < ε ≪ 1. Can we find a relationship between the speed of the unique fast traveling pulse solution of system (1)–(2) where there is no delay and the speed of the unique fast traveling pulse solution of the system (5)–(6) where there is a spatial temporal delay? This is definitely an interesting but difficult problem.

7.5.

Appendix One: Solutions of A Nonhomogeneous Linear Differential Equation

Let ν > 0, λ ∈ C and ψ(λ, 0) be given constants. Suppose that a, ϕ ∈ C 0 (R) ∩ L∞ (R) are real functions, such that |a(z)| + |ϕ(z)| ≤ C exp(−ρ|z|),

on

R,

for two positive constants C and ρ. Consider the nonhomogeneous linear differential equation νψ ′ + (λ + 1)ψ + ϕ(z) = a(z)ψ(λ, 0).

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

191

The general solution is     Z λ+1 λ+1 1 z exp − ψ(λ, z) = C(λ) exp − z − (z − x) ϕ(x)dx ν ν −∞ ν   Z ψ(λ, 0) z λ+1 + exp − (z − x) a(x)dx, ν ν −∞ where C(λ) is a complex function of λ to be determined later. Letting z = 0, we find       Z Z λ+1 λ+1 1 0 1 0 exp exp x a(x)dx ψ(λ, 0) + x ϕ(x)dx. C(λ) = 1 − ν −∞ ν ν −∞ ν Therefore, the compatible solution of the nonhomogeneous differential equation is given by       Z λ+1 λ+1 1 0 x a(x)dx exp − z exp ψ(λ, z) = ψ(λ, 0) 1 − ν −∞ ν ν  Z 0      1 λ+1 λ+1 + x ϕ(x)dx exp − z exp ν −∞ ν ν     Z z Z 1 z λ+1 λ+1 ψ(λ, 0) (z − x) a(x)dx − (z − x) ϕ(x)dx. exp − exp − + ν ν ν −∞ ν −∞

The compatible solution is bounded on R if and only if       Z Z λ+1 λ+1 1 0 1 0 exp x a(x)dx ψ(λ, 0) + x ϕ(x)dx = 0. 1− ν −∞ ν ν −∞ ν If λ = 0, C(λ) = 0, a = 0 and ϕ(z) = −α

Z

z

K(x)dx,

−∞

then Z

z

Z

z



x−z ν



K(x)dx, exp K(x)dx − α −∞   x−z α exp K(x)dx, ψ ′ (z) = ν −∞ ν   Z α x−z α z ′′ ψ (z) = K(z) − 2 exp K(x)dx. ν ν −∞ ν

ψ(z) = α

−∞ Z z

Moreover Z 0 x α K(x)dx, exp ψ(0) = − α 2 ν −∞ Z x α 0 ψ ′ (0) = exp K(x)dx, ν −∞ ν Z x α 0 α exp K(x)dx. ψ ′′ (0) = K(0) − 2 ν ν −∞ ν

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Linghai Zhang

If λ = 0, C(λ) = 0, a = 0 and ϕ(z) = −α

Z

∞

K(x)dx, z

then   Z z x−z K(x)dx exp K(x)dx + α ν −∞ z   Z α z x−z ′ ψ (z) = − exp K(x)dx, ν −∞ ν   Z x−z α z α ′′ exp K(x)dx. ψ (z) = − K(z) + 2 ν ν −∞ ν ψ(z) = α

Z

∞

Furthermore Z 0 x α exp K(x)dx, +α 2 ν −∞ Z x α 0 ψ ′ (0) = − exp K(x)dx, ν −∞ ν Z x α 0 α ′′ exp K(x)dx. ψ (0) = − K(0) + 2 ν ν −∞ ν ψ(0) =

7.6. Appendix Two: Solutions of A Nonhomogeneous Linear System of Differential Equations To facilitate the rigorous mathematical analysis of the existence and stability of traveling wave solutions of the nonlinear systems (1)–(2), (3)–(4) and (5)–(6), we study the solutions of a system of particular form. Let γ > 0, ε > 0, ν > 0, Z > 0, λ ∈ C be constants, let ψ(λ, ε, 0) and ψ(λ, ε, Z) be constant vectors. Suppose that a, b and ϕ ∈ C 0 (R) ∩ L∞ (R) are real functions, such that |a(z)| + |b(z)| + |ϕ(z)| ≤ C exp(−ρ|z|),

on

R,

for two positive constants C and ρ. Consider the nonhomogeneous linear system of differential equations      1 0  λ+1 1 ′ a(z)ψ(λ, ε, 0) + b(z)ψ(λ, ε, Z) . ψ + ϕ(z) = νψ + 0 0 −ε λ + γε   1 1 are given by (I) The eigenvalues and eigenvectors of the coefficient matrix −ε γε p p 1 + γε + (1 − γε)2 − 4ε 1 + γε − (1 − γε)2 − 4ε ω1 (ε) = , ω2 (ε) = , 2 2 and v1 (ε) =



1 ω1 (ε) − 1



,

v2 (ε) =



1 ω2 (ε) − 1



.

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

193

Note that ω1 (ε) + ω2 (ε) = 1 + γε > 0 and ω1 (ε)ω2 (ε) = (1 + γ)ε > 0. For all     √ 1 γ+2+2 γ+1 √ ε ∈ 0, , ∞ , ∪ γ2 γ+2+2 γ+1 the eigenvalues and the eigenvectors are real. Note that ω1 (0) = 1, ω2 (0) = 0 and

If

h p i ∂ω1 1 2 p γ (1 − γε) − 4ε + γ(γε − 1) − 4 < 0, (ε) = ∂ε 2 (1 − γε)2 − 4ε i h p 1 ∂ω2 (ε) = p γ (1 − γε)2 − 4ε − γ(γε − 1) + 4 > 0. ∂ε 2 (1 − γε)2 − 4ε ε∈



√  1 γ+2+2 γ+1 √ , , γ2 γ+2+2 γ+1

then the eigenvalues and eigenvectors are complex, but the real parts of the eigenvalues are positive. If √ 1 γ+2+2 γ+1 √ , ε= or ε = γ2 γ+2+2 γ+1 then there is only one eigenvalue √ γ+1− γ+1 ω(ε) = γ

or

ω(ε) =

√ γ+1+ γ+1 . γ

(II) The homogeneous system of differential equations     0 λ+1 1 ′ ψ= νψ + 0 −ε λ + γε possesses a fundamental matrix 

      λ+ω1 (ε) λ + ω2 (ε) 1 1 Φ(λ, ε, z) = exp − , exp − . z z ω1 (ε) − 1 ω2 (ε)−1 ν ν

The inverse matrix       λ+ω2 (ε) λ+ω1 (ε) 1 1 , exp − z z Φ−1 (λ, ε, z) = exp − ω1 (ε)−1 ω2 (ε)−1 ν ν      λ + ω1 (ε) λ + ω1 (ε) z + exp + z  [1 − ω2 (ε)] exp + ν ν 1  =      ω1 (ε) − ω2 (ε)  λ + ω2 (ε) λ + ω2 (ε) [ω1 (ε) − 1] exp + z − exp + z ν ν

−1 

  . 

When using the method of variation of parameters [17] to solve the system of differential equations for solutions, we have to compute the product of the fundamental matrix and its inverse matrix. After some tedious computations, we find that Φ(λ, ε, z)Φ−1 (λ, ε, x)

194

Linghai Zhang      λ + ω2 (ε) λ + ω1 (ε) z exp − z exp −   ν ν   =        λ + ω2 (ε) λ + ω1 (ε) z [ω2 (ε) − 1] exp − z [ω1 (ε) − 1] exp − ν ν       λ + ω1 (ε) λ + ω1 (ε) x + exp + x   [1 − ω2 (ε)] exp + ν ν   ×        λ + ω2 (ε) λ + ω2 (ε) x − exp + x [ω1 (ε) − 1] exp + ν ν      λ + ω1 (ε) λ + ω2 (ε) (x − z) − [1 − ω1 (ε)] exp (x − z)  [1 − ω2 (ε)] exp ν ν  =      λ + ω2 (ε) λ + ω1 (ε) (x − z) + ε exp (x − z) −ε exp ν ν      λ + ω2 (ε) λ + ω1 (ε) (x − z) − exp (x − z) exp  ν ν     .   λ + ω2 (ε) λ + ω1 (ε) (x − z) − [ω2 (ε) − 1] exp (x − z) [ω1 (ε) − 1] exp ν ν 

The general solution of the linear system is   a(λ, ε) ψ(λ, ε, z) = Φ(λ, ε, z) b(λ, ε)   Z z  1 1 0  −1 + Φ(λ, ε, z) a(x)ψ(λ, ε, 0) + b(x)ψ(λ, ε, Z) dx Φ (λ, ε, x) 0 0 ν Z−∞ z 1 Φ−1 (λ, ε, x)ϕ(x)dx. − Φ(λ, ε, z) ν −∞ Let R(λ, ε, z) =

     λ + ω1 (ε) λ + ω2 (ε) exp − z , exp − z . ν ν

Then 

a(λ, ε) b(λ, ε)



R(λ, ε, 0)     Z 0 1 a(x) −1 dx ξ(λ, ε, 0) Φ (λ, ε, x) R(λ, ε, 0) = 1− 0 ν(ε) −∞     Z 0 1 b(x) −1 dx ξ(λ, ε, Z) Φ (λ, ε, x) R(λ, ε, 0) + − 0 ν(ε) −∞   Z 0 1 −1 + Φ (λ, ε, x)ϕ(x)dx R(λ, ε, 0) ν(ε) −∞ = [1 − M11 (λ, ε)]ξ(λ, ε, 0) − M12 (λ, ε)ξ(λ, ε, Z) + N1 (λ, ε),

Traveling Waves Arising from Synaptically Coupled Neuronal Networks   a(λ, ε) R(λ, ε, Z) b(λ, ε)     Z Z 1 a(x) −1 = − dx ξ(λ, ε, 0) Φ (λ, ε, x) R(λ, ε, Z) 0 ν(ε) −∞     Z Z 1 b(x) −1 dx ξ(λ, ε, Z) R(λ, ε, Z) Φ (λ, ε, x) + 1− 0 ν(ε) −∞   Z Z 1 −1 + Φ (λ, ε, x)ϕ(x)dx R(λ, ε, Z) ν(ε) −∞ = −M21 (λ, ε)ξ(λ, ε, 0) + [1 − M22 (λ, ε)]ξ(λ, ε, Z) + N2 (λ, ε).

195

Let us couple them together appropriately. We have         ξ(λ, ε, 0) M11 (λ, ε) M12 (λ, ε) a(λ, ε) R(λ, ε, 0) = I− ξ(λ, ε, Z) M21 (λ, ε) M22 (λ, ε) b(λ, ε) R(λ, ε, Z)       ξ(λ, ε, 0) N1 (λ, ε) + N (λ, ε). = I − M(λ, ε) + ξ(λ, ε, Z) N2 (λ, ε) Therefore, the constant vector is given by   −1        ξ(λ, ε, 0) R(λ, ε, 0) a(λ, ε) + N (λ, ε) . I − M(λ, ε) = ξ(λ, ε, Z) R(λ, ε, Z) b(λ, ε) The compatible solution of the nonhomogeneous system is   −1      ξ(λ, ε, 0) R(λ, ε, 0) + N (λ, ε) I − M(λ, ε) ψ(λ, ε, z) = Φ(λ, ε, z) ξ(λ, ε, Z) R(λ, ε, Z)   Z z  1 1 0  −1 + Φ(λ, ε, z) a(x)ψ(λ, ε, 0) + b(x)ψ(λ, ε, Z) dx Φ (λ, ε, x) 0 0 ν Z−∞ z 1 Φ−1 (λ, ε, x)ϕ(x)dx. − Φ(λ, ε, z) ν −∞ Define    λ + ω1 (ε) 1  [1 − ω2 (ε)] exp x A(λ, ν, ε, x) =  ν ν ω1 (ε) − ω2 (ε)   λ + ω2 (ε) −[1 − ω1 (ε)] exp x , ν      1 λ + ω2 (ε) λ + ω1 (ε)   B(λ, ν, ε, x) = x + exp x . − exp ν ν ν ω1 (ε) − ω2 (ε)

Define

   1 − ω2 (ε) λ + ω1 (ε) 1 C(λ, ν, ε, x) = exp x ω1 (ε) − ω2 (ε) λ + ω1 (ε) ν   1 − ω1 (ε) λ + ω2 (ε) − exp x , λ + ω2 (ε) ν

196

Linghai Zhang    1 1 λ + ω1 (ε) D(λ, ν, ε, x) = − exp x ω1 (ε) − ω2 (ε) λ + ω1 (ε) ν   λ + ω2 (ε) 1 exp x . + λ + ω2 (ε) ν

It is easy to find that ∂ C(λ, ν, ε, x) = A(λ, ν, ε, x), ∂x ∂ D(λ, ν, ε, x) = B(λ, ν, ε, x), ∂x  ϕ1 (z) , then for all ν > 0, ε > 0 and x ∈ R. If a = 0, b = 0, λ = 0 and ϕ(z) = ϕ2 (z)   Z z 1 ξ(0, ε, z) −1 , Φ (0, ε, x)ϕ(x)dx = ψ(0, ε, z) = Φ(0, ε, z) η(0, ε, z) ν −∞ 

where   Z z  1 ω1 (ε)   ξ(0, ε, z) = [1 − ω2 (ε)] exp (x − z) ν ν ω1 (ε) − ω2 (ε) −∞   ω2 (ε) −[1 − ω1 (ε)] exp (x − z) ϕ1 (x)dx ν     Z z  ω1 (ε) 1 ω2 (ε)  exp +  (x − z) −exp (x − z) ϕ2 (x)dx, ν ν ν ω1 (ε)−ω2 (ε) −∞     Z z  ω1 (ε) 1 ω2 (ε)  −ε exp η(0, ε, z) =  (x−z) +ε exp (x−z) ϕ1 (x)dx ν ν ν ω1 (ε)−ω2 (ε) −∞   Z z  ω1 (ε) 1  [1 − ω1 (ε)] exp −  (x − z) ν ν ω1 (ε) − ω2 (ε) −∞   ω2 (ε) (x − z) ϕ2 (x)dx. −[1 − ω2 (ε)] exp ν

If a = 0, b = 0, λ = 0 and ϕ(z) = α

Z

1 0



,

K(ξ)dξ



1 0

z

K(x)dx

z−Z



then the solution is ψ(0, ε, z) =

1 Φ(0, ε, z) ν

Z

z

−∞

 Z Φ−1 (0, ε, x) α

x

x−Z



dx =



ξ(0, ε, z) η(0, ε, z)



,

where Z z Z z αγ K(x)dx − α C(0, ε, x − z)[K(x) − K(x − Z)]dx, 1 + γ z−Z −∞ Z z Z z γ η(0, ε, z) = K(x)dx − α D(0, ε, x − z)[K(x) − K(x − Z)]dx. 1 + γ z−Z −∞ ξ(0, ε, z) =

Traveling Waves Arising from Synaptically Coupled Neuronal Networks

7.7.

197

Appendix Three: The Implicit Function Theorem


Let −∞ ≤ a < b ≤ ∞, −∞ ≤ c < d ≤ ∞, and f ∈ C 1 (a, b) × (c, d) . Suppose that 0 < C1 ≤

∂f ≤ C2 , ∂y

for all (x, y) ∈ (a, b) × (c, d) and for two constants C2 > C1 > 0. Then the equation f (x, y) = 0 has a unique solution y = g(x) ∈ C 1 (a, b), such that f (x, g(x)) = 0.

7.8.

The Initial Value Problems

Consider the initial value problems ut + f (u) + w = α

Z

R

K(x − y)H(u(y, t) − θ)dy,

wt = ε(g(u) − γw),

u(x, 0) = u0 (x),

w(x, 0) = w0 (x).

Given an appropriate initial data, can we establish the existence and uniqueness of global strong solution of the initial value problem in some Banach space? Can we obtain decay estimate with sharp rate of the global solution as t → ∞? Some interesting cases are that the initial data is close to a traveling wave, or a standing wave, or a periodic wave, or a lurching wave, or a spiral wave, or the threshold θ. What happens to the global solution if the initial data is not close to any of those?

7.9.

The Sigmoid Gain Function

A reasonable gain function corresponding to experimental data is given by   exp Λ(u − θ)  , H(Λ, u) = 1 + exp Λ(u − θ)

where θ represents a threshold and Λ stands for a gain or steepness parameter. However, H is not derived from a biophysical model. The firing rate function H is monotonically increasing and nonlinear, saturating to a constant for sufficiently large u. Note that H is a solution of the nonlinear differential equation H′ = ΛH(1 − H), subject to the initial 1 condition H(θ) = . The existence and uniqueness of a traveling wave front together 2 with a positive speed of (7), where H is replaced with H, have been proved by using the homotopy ideas in Ermentrout and McLeod [26]. The speed index function and the stability index function have not been found. However, by using the speed index function defined in (33), we may provide nice estimates on the speed. Note also that lim H(Λ, u) = H(u − θ),

Λ→∞

where H is the well-known Heaviside step function, whose derivative is the famous Dirac delta impulse function. This may be a useful hint on how to construct the speed index function with large Λ > 0.

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Acknowledgments The author is very grateful to Professor Yulin Zhou and Professor Boling Guo for their valuable suggestions and comments, which made the presentation of this work better. Partially supported by a Faculty Research Grant at Lehigh University.

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[56] D. J. Pinto, R. K. Jackson, C. E. Wayne, Existence and stability of traveling pulses in a continuous neuronal network. SIAM Journal on Applied Dynamical Systems 4 (2005), 954–984. [57] D. J. Pinto, S. L. Patrick, W. C. Huang and B. W. Connors, Initiation, propagation, and termination of epileptiform activity in rodent neocortex in vitro involve distinct mechanisms. Journal of Neuroscience 25 (2005), 8131–8140. [58] J. C. Prechtl, L. B. Cohen, B. Pesaran, P. P. Mitra, D. Kleinfeld, Visual stimuli induce waves of electrical activity in turtle cortex. Proceedings of the National Academy of Sciences, USA 94 (1997), 7621–7626. [59] K. A. Richardson, S. J. Schiff and B. J. Gluckman, Control of traveling waves in the mammalian cortex. Physical Review Letters 94 (2005), 028103–1 to 028103–4. [60] J. Rinzel and J. B. Keller, Traveling wave solutions of a nerve conduction equation. Biophysics Journal 13 (1973), 1313–1337. [61] J. Rinzel and D. Terman, Propagation phenomena in a bistable reaction-diffusion system. SIAM Journal on Applied Mathematics 42 (1982), 1111–1137. [62] J. Rinzel, D. Terman, X.-J. Wang and G. B. Ermentrout, Propagating activity patterns in large-scale inhibitory neuronal networks. Science 279 (1998), 1351–1355. [63] J. E. Rubin and W. C. Troy, Sustained spatial patterns of activity in neuronal populations without recurrent excitation. SIAM Journal on Applied Mathematics 64 (2004), 1609–1635. [64] B. Sandstede, Stability of multiple-pulse solutions. Transactions of the American Mathematical Society 350 (1998), 429–472. [65] B. Sandstede, Stability of travelling waves. In: Handbook of dynamical systems, 2, 983–1055. North-Holland, Amsterdam, 2002. [66] B. Sandstede, Evans functions and nonlinear stability of travelling waves in neuronal network models. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. 17 (2007), 2693–2704. [67] D. H. Sattinger, On the stability of waves of nonlinear parabolic systems. Advances in Mathematics 22 (1976), 312–355. [68] G. Tamas, E. H. Buhl and P. Somogyi, Massive autaptic self-innervation of GABAergic neurons in cat visual cortex. Journal of Neuroscience 17 (1997), 6352–6364. [69] D. H. Terman, G. B. Ermentrout and A. C. Yew, Propagating activity patterns in thalamic neuronal networks. SIAM Journal on Applied Mathematics 61 (2001), 1578–1604. [70] W. C. Troy, Traveling waves and synchrony in an excitable large-scale neuronal network with asymmetric connections. SIAM Journal on Applied Dynamical Systems 7 (2008), 1247–1282.

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In: Advances in Mathematics Research, Volume 10 ISBN 978-1-60876-265-1 c 2010 Nova Science Publishers, Inc. Editor: Albert R. Baswell, pp. 205-247

Chapter 4

T HE I NITIAL -B OUNDARY VALUE P ROBLEMS FOR PARABOLIC E QUATIONS IN D OMAINS WITH C ONICAL P OINTS Nguyen Manh Hung1,∗ and Nguyen Thanh Anh2,† 1 Dean of Department of Mathematical Analysis Faculty of Mathematics and Informatics Hanoi National University of Education, Hanoi, Vietnam, 2 Faculty of Mathematics and Informatics Hanoi National University of Education, Hanoi, Vietnam

Abstract In this paper we consider a initial-boundary value problem for parabolic equation in a domain with conical points. We establish the weak well-posedness of the problem and the regularity of the solution. We also receive asymptotic formulas for the solution near the conical points.

Keywords: parabolic equation, initial-boundary value problem, nonsmooth domains, generalized solutions, regularity, asymptotics. 2000 MSC: 35D05, 35D10, 35K30, 35B40.

1.

Introduction

Elliptic boundary value problems in domains with point singularities were thoroughly investigated (see, e.g, [4,9] and the extensive bibliography therein). In this paper we deal with a initial-boundary value problem for parabolic equation in a domain with conical points. We investigate the unique solvability of the problem and asymptotics for solution near singularities of the boundary of the domain. Initial-boundary value problems for parabolic equations are studied widely on various aspects with differential approaches. For example, for parabolic equations of second order ∗ †

E-mail addresses: [email protected] E-mail addresses: [email protected]

206

Nguyen Manh Hung and Nguyen Thanh Anh

in a smooth domains, the solvability of the problems and the regularity of the solutions were established in both H¨older and Sobolev spaces in [15] by the method in which a regularizer was constructed and exact estimates of solutions in terms of the data were dealt with. Such ideas were also used in [6] with some modifications for the case of domains with edges. For the equation considered in [21], whose coefficients are independent of the time variable, the Fourier transform was used to reduce the problem to an elliptic one with a parameter. In the present paper, for a general higher order linear parabolic equation with coefficients depending on both spatial and time variable in a domain containing conical points we modify the approach suggested in [8, 11, 12]. First, in Section 3., we study the unique solvability of the problem and the regularity with respect to the time variable for the generalized solution by Galerkin’s approximate method. By modifying the arguments used in [11, 12], we can weaken the restrictions on the data at the initial time t = 0 imposed therein. These results are essential contained in the authors’ paper [13]. After that, in Section 4., we deal with asymptotics of the solution of the problem near singularities of the boundary. To this end, we take the term containing the derivative in time of the unknown function to the right-hand side of the equation such that the problem can be considered as an elliptic one depending parameter t. With some assumptions on the smoothness with respect to the variable t of eigenvalues, eigenvectors and generalized eigenvectors of the operator pencil generated by the problem we apply the results on elliptic boundary value problems in domains with conical points and our previous results on the regularity of the solutions to receive asymptotic formulas for the solution near singularities of the boundary as a sum of a linear combination of special singular functions and a regular function in which this functions and the coefficients of the linear combination are regular with respect to the variable t. Finally, in Section 5., with the help of the results on the analytic perturbation theory of linear operators ( [7, 20]) and the method of linearization of polynomial operator pencils ( [19]) we deal with the operator pencil generated by the problem. Our purpose here is to give a sufficient condition for the assumptions mentioned above to be valid.

2.

Notation and Formulation of the Problem

Let G be a bounded domain in Rn (n > 2) with the boundary ∂G. We suppose that S = ∂G \ {0} is a smooth manifold and G in a neighborhood U of the origin 0 coincides with the cone K = {x : x/|x| ∈ Ω}, where Ω is a smooth domain with the boundary Γ on the unit sphere S n−1 in Rn . Let T be a positive real number or T = +∞. For 0 6 t 6 T , we set Gt = G × (0, t), Ωt = Ω × (0, t), St = S × [0, t], Γt = Γ × [0, t]. For each multi-index α = (α1 , . . . , αn ) ∈ Nn , set |α| = α1 + · · · + αn , and ∂xα = ∂xα = ∂xα11 . . . ∂xαnn . For a nonnegative integer k we write utk instead of ∂tk u. Let us introduce some functional space used in this paper. Let l be a nonnegative integer. We denote by W2l (G) the usual Sobolev space of functions defined in G with the norm

kukW l (G) = 2

Z

X

G |α|6m

|∂xα u|2 dx

1 2

,

The Initial-Boundary Value Problems for Parabolic Equations... l− 21

and by W2

207

(S) the space of traces of functions from W2l (G) on S with the norm  kuk l− 12 = inf kvkW l (G) : v ∈ W2l (G), v|S = u . W2

2

(S)

l We denote by W2,loc (G) the space of all functions u(x) defined in G such that they and all their derivatives up to order l are square integrable on every compact subset of G, and by W2lm,l 2,loc (GT ) the space of all functions u(x, t) defined in GT with the derivatives

∂xα utk , |α| + 2mk 6 2lm, being square integrable on every compact subset of GT . l (K) (γ ∈ R) as the closure of C ∞ (K We define the weighted Sobolev space V2,γ \ {0}) 0 with respect to the norm kukV l

2,γ (K)

=

XZ |α|6l

r

2(γ+|α|−l)

K

|∂xα u|2 dx

1 2

,

(2.1)


1 Pn l− 1 2 2 where r = |x| = . If l > 1, then Vγ 2 (∂K) denote the space consisting of k=1 xk l (K) on the boundary ∂K with the norm traces of functions from V2,γ  l kuk l− 12 = inf kvkV l (K) : v ∈ V2,γ (K), v|∂K = u . (2.2) Vγ

2,γ

(∂K)

l− 1

l (G), V 2 The weighted spaces V2,γ (S) are defined similarly as in (2.1), (2.2) with K, ∂K γ replaced by G, S, respectively. l+k It is obvious from the definition that the space V2,γ+k (G) is continuously imbedded into l the space V2,γ (G) for an arbitrary nonnegative integer k. An analogous assertion holds for l− 1

the space V2,γ 2 (∂G). l (G) (γ ∈ R) as the set of all functions in G such that r γ ∂ α u ∈ L (G) We define W2,γ 2 x for |α| ≤ l with the norm kukW l

2,γ (G)

=

XZ |α|6l

G

r

2γ

|∂xα u|2 dx

1 2

.

l− 1

If l > 1, then W2,γ 2 (S) denote the space consisting of traces of functions from respective l (G) on the boundary S with the norm space W2,γ  l kuk l− 12 = inf kvkW l (G) : v ∈ W2,γ (G), v|S = u . W2,γ (S)

2,γ

Let X, Y be Banach spaces. Denote by B(X, Y ) the set of all continuous linear operators from X into Y . If Y = X, we write B(X) instead of B(X, Y ). We denote by L2 ((0, T ); X) the space consisting of all measurable functions u : (0, T ) → X with the norm 1 Z T 2 2 ku(t)kX dt , kukL2 ((0,T );X) = 0

208

Nguyen Manh Hung and Nguyen Thanh Anh

and by W21 ((0, T ); X, Y ) the space consisting of all functions u ∈ L2 ((0, T ); X) such that the generalized derivative ut = u′ exists and belongs to L2 ((0, T ); Y ). The norm in W21 ((0, T ); X, Y ) is defined by kukW21 ((0,T );X,Y ) = kuk2L2 ((0,T );X) + kut k2L2 ((0,T );Y )


1 2

.

Let h be a nonnegative integer. By W2h ((0, T ); X) we denote the Sobolev space of Xvalued functions defined on (0, T ) with the norm kf kW h ((0,T );X) = 2

X h Z k=0

T O

For shortness, we set

dk f (t) 2  21

< +∞.

dt dtk X

W2h ((0, T )) = W2h ((0, T ); C), W2l,h (ΩT ) = W2h ((0, T ); W2l (Ω)), l− 1 ,h

l− 1

l,h l V2,γ (GT ) = W2h ((0, T ); V2,γ (G)), V2,γ 2 (ST ) = W2h ((0, T ); V2,γ 2 (S)), l− 1 ,h

l− 1

l,h l W2,γ (GT ) = W2h ((0, T ); W2,γ (G)), W2,γ 2 (ST ) = W2h ((0, T ); W2,γ 2 (S)).

In this paper, we say that a function is smooth in a certain set if it is infinitely differentiable in that set. Recall that a X-valued function f (t) defined on [0, +∞) is said to be continuous (smooth, analytic, respectively) at t = +∞ if the function g(t) = f ( 1t ) is continuous (smooth, analytic, respectively) at t = 0 with a suitable definition of g(0) ∈ X. In these cases we can regard f (t) as a function defined on [0, +∞] with f (+∞) = g(0). Thus, if we say f is continuous (smooth, analytic, respectively) on [0, T ], then f is continuous (smooth, analytic, respectively) at all points of [0, T ] including the point +∞ in the case T = +∞. Denote by C a([0, T ]; X) the set of all X-valued functions defined and analytic on [0, T ]. Now we introduce a differential operator of order 2m L = L(x, t, ∂x ) =

m X

(−1)|α| ∂xα (aαβ (x, t) ∂xβ )

(2.3)

|α|,|β|=0

in GT with smooth coefficients in GT , aαβ = aβα for |α|, |β| 6 m. We introduce also a system of boundary operators X Bj = Bj (x, t, ∂x ) = bj,α (x, t)∂xα , j = 1, . . . , m, (2.4) |α|6µj

on ST with smooth coefficients in GT . Suppose that ordBj = µj 6 m − 1 for j = 1, . . . , χ, m 6 ordBj = µj 6 2m − 1 for j = χ + 1, . . . , m,

and coefficients of Bj are independent of t if ordBj < m. Suppose that {Bj (x, t, ∂x )}m j=1 is a normal system on S for all t ∈ [0, T ], i.e, the two following conditions are satisfied:

The Initial-Boundary Value Problems for Parabolic Equations...

209

(i) µj 6= µk for j 6= k, (ii) Bj◦ (x, t, ν(x)) 6= 0 for all (x, t) ∈ ST , j = 1, . . . , m. Here ν(x) is the unit outer normal to S at point x and Bj◦ (x, t, ∂x ) is the principal part of Bj (x, t, ∂x ), X Bj◦ = Bj◦ (x, t, ∂x ) = bj,α (x, t)∂xα , j = 1, . . . , m. |α|=µj

Furthermore, we assume that Bj◦ (0, t, ν(x)) 6= 0 for all x ∈ S sufficient enough the origin 0. To be able to reduce the problem considered to variational form we assume that it can be choose boundary operators Φj on ST , j = 1, . . . , m, such that B(t, u, v) =

Z

Luvdx +

G

χ Z X j=1

Φj uBj vds +

S

m Z X

Bj uΦj vds

(2.5)

j=χ+1 S

for all u, v ∈ C0∞ (G \ {0}) and a.e. t ∈ [0, T ]. Here B(t, u, v) =

Z m X

|α|,|β|=0 G

aαβ (., t) ∂xβ u∂xα vdx, t ∈ [0, T ]

is the bilinear form associated to the operator L(x, t, ∂x ). Of course, this is an essential restriction on the structure of the boundary operators in (2.4). However, if the system of boundary operators in (2.4) are a Dirichlet system (then all ord of Bj are less than m) or a generalized Neumann system (then m 6 ordBj 6 2m − 1 for all j = 1, . . . , m) then it is satisfied (2.5) for a suitable system Φj , j = 1, . . . , m (see [3, Sec. I.7.]). m (G)−elliptic uniformly with respect to We assume further that the form B(t, ., .) is HB t ∈ [0, T ], i.e., there exists a positive constant µ such that B(t, u, u) > µkuk2W m (G) 2

(2.6)

m (G) and all t ∈ [0, T ], where for all u ∈ HB  m HB (G) = u ∈ W2m (G) : Bj u = 0 on S for j = 1, . . . , χ

with the same norm in W2m (G). From this it follows that L(x, t, ∂x ) is positively elliptic, i.e., X a(x, t)ξ α ξ β > 0 (2.7) |α|=|β|=m

for all ξ ∈ Rn \ {0}, all (x, t) ∈ GT (see [1]). −m m (G). We write h., .i to denote the pairing By HB (G) we denote the dual space to HB −m m (G) and H between HB B (G), and (., .) to denote the inner product in L2 (G). By idenm (G) ֒→ L (G) ֒→ tifying L2 (G) with its dual, we have the continuous imbeddings HB 2 −m HB (G) with the equation hf, vi = (f, v)

−m m for f ∈ L2 (G) ⊂ HB (G), v ∈ HB (G).

210

Nguyen Manh Hung and Nguyen Thanh Anh In this paper we consider the following problem ut +Lu = f in GT ,

(2.8)

Bj u = 0, on ST , j = 1, . . . , m, u|t=0 = φ on G,

(2.9) (2.10)

where f : GT → C, φ : G → C are given functions. −m Let f ∈ L2 ((0, T ); HB (G)), φ ∈ L2 (G). A function u ∈ −m 1 m W2 ((0, T ); HB (G), HB (G)) is called a generalized solution of the problem (2.8)– (2.10) iff u(., 0) = φ and the equality hut , vi + B(t, u, v) = hf (t), vi

(2.11)

m (G). holds for a.e. t ∈ (0, +∞) and all v ∈ HB

3.

The Unique Solvability and the Regularity with Respect to the Time Variable

For simplicity in the following we will sometimes write v(t) instead of v(., t) for functions v(x, t) defined on GT . For integer k > 0, u, v ∈ H m,0 (GT ), t ∈ [0, T ] we set Btk (t, u, v) =

X

Z

|α|,|β|6m G

BtTk (u, v) =

Z

T

0

∂ k aαβ (x, t) β ∂x u(x, t)∂xα v(x, t)dx, ∂tk

Btk (t, u, v)dt, B T (u, v) = BtT0 (u, v).

m (G) × H m (G) satisfying Lemma 3.1. Let F (t, ., .) be a bilinear form on HB B

|F (t, v, w)| 6 CkvkHBm (G) kwkHBm (G) (C = const )

(3.12)

m (G), and F (., v, w) is measurable on [0, T ] for each for all t ∈ [0, T ] and all v, w ∈ HB m (G), H −m (G)) satisfies u(0) ≡ 0 m pair v, w ∈ HB (G). Assume that u ∈ W21 ((0, T ); HB B and Z t hut (., t), v(., t)i + B(t, u(., t), v(., t)) = F (θ, u(., θ), v(., t))dθ (3.13) 0

m (G)), τ is a arbitrary positive real number. for a.e. t ∈ [0, T ] and all v ∈ L2 ((0, τ ); HB Then u ≡ 0 on GT .

Proof. Substituting v := u into (3.13), then integrating both sides of the obtained equality with respect to t from 0 to b (b > 0), we arrive at Z

b 0

b

hut (., t), u(., t)idt + B (t, u(., t), u(., t)) =

Z bZ 0

t

F (θ, u(., θ), u(., t))dθdt. 0

The Initial-Boundary Value Problems for Parabolic Equations...

211

Using the assumptions (2.6) and (3.12), we get Z bZ t 1 ku(t)kHBm (G) ku(τ )kHBm (G) dτ dt ku(b)k2L2 (G) + µkuk2L2 ((0,b);H m (G)) 6 C B 2 0 0 Z bZ t 1 (ku(t)k2H m (G) + ku(τ )k2H m (G) )dτ dt 6 bCkuk2L2 ((0,b);H m (G)) . 6 C B B B 2 0 0
µ 1 Choosing b 6 ku(b)k2L2 (G) + µkuk2L2 ((0,b);H m (G)) 6 0. This implies , we have B 2 h 2C µ i . Repeating this argument we can show that u ≡ 0 on intervals u ≡ 0 on 0, 2C i h µ µ i h µ 3µ , , , , . . ., and, therefore, u ≡ 0 on GT . 2C C C 2C −m Theorem 3.2. If f ∈ L2 ((0, T ); HB (G)), φ ∈ L2 (G), then there exists a unique gen1 m (G), H −m (G)) of the problem (2.8)–(2.10) which eralized solution u ∈ W2 ((0, T ); HB B satisfies
kuk2W 1 ((0,T );H m (G),H −m (G)) 6 C kφk2L2 (G) + kf k2L ((0,T );H −m (G)) , (3.14) 2

B

2

B

B

where C is the constant independent of φ, f and u.

Proof. The uniqueness of the solution follows directly from Lemma 3.1. We will prove its existence. By the assumption aαβ = aβα for |α|, |β| 6 m, L is a formally self-adjoint operator. Moreover, W2m (G) is compact imbedded in L2 (G). Thus, the operator L(x, 0, ∂x ) possesses a set {ψk }∞ k=1 consisting of all its eigenfunctions, which is not only an orthogonal m basis of HB (G) but also an orthonormal basis of L2 (G). For each positive integer N , we P N N N consider the function uN (x, t) = N k=1 Ck (t)ψk (x), where {Ck (t)}k=1 is the solution of the ordinary differential system: N (uN t , ψl ) + B(t, u , ψl ) = hf, ψl i, l = 1, . . . , N,

CkN (0)

= Ck , k = 1, . . . , N.

(3.15) (3.16)

Here Ck = (φ, ψk ), k = 1, 2, . . . . After multiplying both sides of (3.15) by ClN (t), taking sum with respect to l from 1 to N , and integrating with respect to t from 0 to τ (τ > 0), we arrive at Z τ Z τ N N τ N N (3.17) hf, uN idt. (ut , u )dt + B (u , u ) = 0

0

Adding (3.17) with its complex conjugate, we obtain kuN (τ )k2L2 (G) + 2B τ (uN , uN ) = kuN (0)k2L2 (G) + 2Re

2

P

N (φ, ψk )ψk Noting that kuN (0)k2L2 (G) =

L2 (G)

k=1

Z τ Z N 2Re hf, u idt 6 2 0

0

τ

Z

τ 0

hf, uN idt.

6 kφk2L2 (G) and

kf kH −m (G) kuN kHBm (G) dt B

1 6 ǫkuN k2L2 ((0,τ );H m (G)) + kf k2L ((0,τ );H −m (G)) 2 B B ǫ

(3.18)

212

Nguyen Manh Hung and Nguyen Thanh Anh

(0 < ǫ < 2µ), and using the assumption (2.6), we have from (3.18) that kuN k2L2 ((0,τ );H m (G)) 6 C kφk2L2 (G) + kf k2L

−m 2 ((0,τ );HB (G))

B

Sending τ → T , we obtain kuN k2L2 ((0,T );H m (G)) 6 C kφk2L2 (G) + kf k2L




−m 2 ((0,T );HB (G))

B

.


.

(3.19)

m (G) with kvk m Now fix any v ∈ HB HB (G) 6 1, and write v = v1 + v2 , where v1 ∈ N span{ψl }l=1 , (v2 , ψl )L2 (G) = 0, l = 1, . . . , N. Since the functions {ψl }N l=1 are orthogonal m in HB (G), kv1 kHBm (G) 6 kvkHBm (G) 6 1. We obtain from (3.15) that N (uN t , v1 ) + B(t, u , v1 ) = hf, v1 i.

Therefore, N N N huN t , vi = (ut , v) = (ut , v1 ) = hf, v1 i − B(t, u , v1 ).

Hence, we get N m (G) |huN t , vi| 6 C kf kH −m (G) + ku kHB B

since kv1 k

m (G) HB

6 1. Thus,





N m (G) , kuN t kH −m (G) 6 C kf kH −m (G) + ku kHB B

B

and therefore, by (3.19), 2 kuN t kL

−m 2 ((0,T );HB (G))

+ kuN k2L2 ((0,T );H m (G)) B
2 2 6 C kφkL2 (G) + kf kL ((0,T );H −m (G)) .

6 C kf k2L

−m 2 ((0,T );HB (G)) 2




(3.20)

B

Combining (3.19) and (3.20), we get kuN k2W 1 ((0,T );H m (G),H −m (G)) 6 C kφk2L2 (G) + kf k2L 2

B

−m 2 ((0,T );HB (G))

B




,

(3.21)

where C is a constant independent of φ, f and N . This means that the sequence {uN } is m (G), H −m (G)). It has a subsequence, denoted by {uN } for bounded in W21 ((0, T ); HB B m (G), H −m (G)). We will show convenience, weakly converging to u in W21 ((0, T ); HB B below that u is a generalized solution of the problem (2.8)–(2.10). Fix a positive real number τ , τ 6 T and a positive integer h. Take a function η ∈ m (G)) in the form L2 ((0, τ ); HB η(x, t) =

h X

dl (t)ψl (x),

(3.22)

l=1

where dl (t) are smooth functions defined on [0, τ ]. Multiplying both sides of (3.15) with N > h by dl (t), taking sum with respect to l from 1 to h and integrating with respect to t from 0 to τ , we have Z τ Z τ N τ N hut , ηidt + B (t, u , η) = hf, ηidt. 0

0

The Initial-Boundary Value Problems for Parabolic Equations... From this equality sending N → +∞, we have Z τ Z τ τ hut , ηidt + B (t, u, η) = hf, ηidt. 0

213

(3.23)

0

m (G)), the equality Since the set of functions of the form (3.22) is dense in L2 ((0, τ ); HB m (3.23) holds for all η ∈ L2 ((0, τ ); HB (G)). This implies

hut , vi + B(t, u, v) = hf, vi.

(3.24)

m (G) and a.e. t ∈ [0, T ]. The inequality (3.14) follows from (3.21). for all v ∈ HB It remains to show that u(., 0) = φ. It follows from (3.23) that Z τ Z τ hu, η t idt − (u(., 0), η(., 0)) + B τ (t, u, η) = − hf, ηidt 0

(3.25)

0

m (G)) satisfying η(., τ ) = 0. Analogously, from (3.15) we holds for all η ∈ C 1 ([0, τ ], HB have Z τ Z τ N N τ N hu , η t idt − (u (., 0), η(., 0)) + B (t, u , η) = − hf, ηidt. 0

0

Passing to the limit as N → ∞ with noting that uN (., 0) → φ in L2 (G), we obtain Z τ Z τ hf, ηidt. (3.26) hu, η t idt − (φ, η(., 0)) + B τ (t, uN , η) = − 0

0

Compare (3.25) and (3.26) to find (u(., 0), η(., 0)) = (φ, η(., 0)). m (G) is arbitrary, then u(., 0) = φ. The proof is completed. Since η(., 0) ∈ HB

Now we improve slightly the regularity of the the generalized solution u by making the initial data φ and the right-hand side f more regularity. m (G) and f ∈ L (G ) or f ∈ W 1 ((0, T ); H −m (G)). Then the Lemma 3.3. Let φ ∈ HB 2 T 2 B m (G), H −m (G)) of the problem (2.8)–(2.10) in generalized solution u ∈ W21 ((0, T ); HB B m (G), L (G)) and the following estimate fact belongs to W21 ((0, T ); HB 2
(3.27) kuk2W 1 ((0,T );H m (G),L2 (G)) 6 C kφk2H m (G) + kf k2X 2

B

B

holds with the constant C independent of g, f, and u. −m W21 ((0, T ); HB (G)) which f belongs to.

Here X is L2 (GT ) or

Proof. (i) Let us consider first the case f ∈ L2 (GT ). Let uN be the functions defined as in the proof of Theorem 3.2 with Ck = (φ, ψk ) (k = 1, 2, . . .) replaced by m m m Ck = kψk k−2 H m (G) (φ, ψk )HB (G) , where (., .)HB (G) denotes the inner product in HB (G). B

Multiplying both sides of (3.15) by

dClN , then taking sum with respect to l from 1 to N , dt

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Nguyen Manh Hung and Nguyen Thanh Anh

after that integrating with respect to t from 0 to T (0 < T < +∞), and adding the attained equality with its complex conjugate, we arrive at 2 2kuN t kL2 (GT )

Z m X

+

|α|,|β|=0 GT

∂ β N α N
aαβ ∂ u ∂x u dx dt = 2Re ∂t x

Z

0

T

(f, uN t )dt.

By the integration by parts, we get 2 N N N N 2kuN t kL2 (GT ) + B(T, u , u ) = B(0, u , u )+ Z T T N N (f, uN +Bt (u , u ) + 2Re t )dt.

(3.28)

0

Since aαβ ,

∂aαβ are bounded on GT , using Cauchy’s inequality, we get ∂t 2 B(0, uN , uN ) 6 CkuN (0)k2 m H (G) 6 CkφkH m (G) , B

Z 2Re

T

0

T N N Bt (u , u ) 6 CkuN k2

(f, uN t )dt

B

m (G)) , L2 ((0,T );HB

2 6 ǫkuN t kL2 (GT ) +

1 kf k2L2 (GT ) (0 < ǫ < 2). 4ǫ

Hence, it follows from (3.19) and (3.28) that
2 2 2 kuN t kL2 (GT ) 6 C kφkH m (G) + kf kL2 (GT ) . B

Sending T → +∞, we obtain


2 2 2 kuN t kL2 (GT ) 6 kφkH m (G) + kf kL2 (GT ) .

(3.29)


kuN k2W 1 ((0,T );H m (G),L2 (G)) 6 C kφk2H m (G) + kf k2L2 (GT ) .

(3.30)

B

Combining (3.19) and (3.29), we have 2

B

B

This implies that the sequence {uN } contains a subsequence which weakly converges to m (G), L (G)). Passing to the limit of the subsequence, a function v ∈ W21 ((0, T ); HB 2 we can see that v is a generalized solution of the problem (2.8)–(2.10). Thus, u = v ∈ m (G), L (G)). The estimate (3.27) with X = L (G ) follows from (3.30). W21 ((0, T ); HB 2 2 T −m (ii) Now let f ∈ W21 ((0, T ); HB (G)). Then f is continuous on [0, T ] and has the Rt representation f (t) = f (s) + s ft (τ )dτ for all s, t ∈ [0, T ] (see [5, Sec. 5.9, Th. 2]). This implies Z (3.31) kf (t)k2H −m (G) 6 2kf (s)k2H −m (G) + 2 kft (τ )k2H −m (G) dτ, B

B

J

B

where J = [a, b] ⊂ [0, T ] such that a 6 s, t 6 b and b − a = 1. Integrating both sides of (3.31) with respect to s on J, we obtain kf (t)k2H −m (G) 6 2kf k2W 1 ((0,T );H −m (G)) (t ∈ [0, T ]). B

2

B

(3.32)

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215

Now by the same way to get (3.28), we have N N N N T N N 2kuN t kL2 (GT ) + B(T, u , u ) = B(0, u , u ) + Bt (u , u ) + 2Re

Z

T 0

hf, uN t idt. (3.33)

Noting that Z

0

RT 0

hf, uN t idt = −

RT 0

T hft , uN idt + hf, uN i 0 , and using (3.32), we obtain

T N hf, ut idt 6 kft kL2 ((0,T );H −m (G)) kuN kL2 ((0,T );HBm (G)) B

+ kf (T )kH −m (G) kuN (T )kHBm (G) + kf (0)kH −m (G) kuN (0)kHBm (G) B

B

6 C(ǫ)kf k2W 1 ((0,T );H −m (G)) + ǫ kuN k2L2 ((0,T );H m (G)) B 2 B
N 2 N 2 + ku (T )kW m (G) + ku (0)kH m (G) 2

B

(3.34)

Using (3.19), (2.6) and (3.34) for 0 < ǫ < µ, we get from (3.33) that
2 2 2 kuN t kL2 (GT ) 6 C kφkH m (G) + kf kW 1 ((0,T );H −m (G)) . B

2

B

Sending T → +∞, we can see


2 2 2 kuN t kL2 (GT ) 6 C kφkH m (G) + kf kW 1 ((0,T );H −m (G)) . B

2

(3.35)

B

Combining (3.19) and (3.35), we have
kuN k2W 1 ((0,T );H m (G),L2 (G)) 6 C kφk2H m (G) + kf k2W 1 ((0,T );H −m (G)) . 2

B

B

2

(3.36)

B

From this, by the same argument as in the part (i) above, we obtain the assertion of the −m lemma for the case f ∈ W21 ((0, T ); HB (G)). m (G) and Remark. It follows from the proof of Lemma 3.3 that if φ ∈ HB −m f = f1 + f2 , where f1 ∈ L2 (GT ), f2 ∈ W21 ((0, T ); HB (G)) then the general−m 1 m ized solution u ∈ W2 ((0, T ); HB (G), HB (G)) of the problem (2.8)–(2.10) belongs m (G), L (G)) and the estimate (3.27) holds with kf k2 replaced by to W21 ((0, T ); HB 2 X kf1 k2L2 (GT ) + kf2 k2W 1 ((0,T );H −m (G)) . 2

B

Now we investigate the regularity of the solution of the problem (2.8)–(2.10) with respect to the time variable. To this end, it is needed to impose the compatibility conditions on the known functions f and φ which we formulate as follow: (2h+1)m (G), f ∈ W2hm,h Let φ ∈ W2,loc 2,loc (GT ), where h is a nonnegative integer. We set φ0 :=φ,

φ1 := f (., 0) − L(x, 0, ∂x )φ0 ,  h−1  X h−1 Lth−1−k (x, 0, ∂x )φk . . . . , φh := fth−1 (., 0) − k k=0

(3.37)

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Nguyen Manh Hung and Nguyen Thanh Anh

where

m X

Ltk = Ltk (x, t, ∂x ) =

∂α

|α|,|β|=0

∂ k aαβ(x,t) β
∂ . ∂tk

We say that the hth -order compatibility conditions for problem (2.8)–(2.10) are fulfilled if φ0 , . . . , φh−1 belong to W22m (G) and s   X s (Bj )ts−k (x, 0, ∂x )φk |S = 0, s = 0, . . . , h − 1, j = 1, . . . , m. k

(3.38)

k=0

Theorem 3.4. Let h is a nonnegative integer.

(2h+1)m

Suppose φ ∈ W2,loc

(G), f ∈

m W2hm,h 2,loc (GT ) such that φk ∈ W2 (G), ftk ∈ L2 (GT ) for k = 0, . . . , h and, if h > 1, the hth -order compatibility conditions for problem (2.8)–(2.10) are fulfilled. Then the genm (G), H −m (G)) of the problem (2.8)–(2.10) satisfies eralized solution u ∈ W21 ((0, T ); HB B m utk ∈ W21 ((0, T ); HB (G), L2 (G)) for k = 0, . . . , h,

(3.39)

h X

(3.40)

and h X k=0

kutk k2W 1 ((0,T );H m (G),L2 (G)) 2 B

6C

k=0


kφk k2W m (G) + kftk k2L2 (GT ) , 2

where C is the constant independent of u, f, φ. Proof. We will show by induction on h that not only the assertions (3.39), (3.40) but also the following equalities hold: utk (0) = φk , k = 1, . . . , h,

(3.41)

and (uth+1 , η) +

h X k=0

h k


m Bth−k (t, utk , η) = (fth , η) for all η ∈ HB (G).

(3.42)

The case h = 0 follows from Lemmas 3.2, 3.3. Assuming now that they hold for h − 1, we will prove them for h (h > 1). We consider first the following problem: find a function m (G), H −m (G)) satisfying v(0) = φ and v ∈ W21 ((0, T ); HB h B hvt , ηi + B(t, v, η) = (fth , η) −

h−1 X k=0

h k




Bth−k (t, utk , η)

(3.43)

m (G) and a.e. t ∈ (0, +∞). for all η ∈ HB Let F (t), t ∈ [0, T ], be functionals defined by

hF (t), ηi = (fth , η) −

h−1 X k=0

h k




m Bth−k (t, utk , η), η ∈ HB (G).

(3.44)

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217

−m Then F ∈ L2 ((0, T ); HB (G)) by the induction hypothesis. Hence, according to Lemma m (G), H −m (G)). We put now 3.2, the problem (3.43) has a solution v ∈ W21 ((0, T ); HB B Z t w(x, t) = φh−1 (x) + v(x, τ )dτ, x ∈ G, t ∈ [0, T ]. 0

Then we have w(0) = φh−1 , wt = v, wt (0) = φh . It follows from (3.43) that hwtt , ηi +

∂ B(t, w, η) =(fth , η) + Bt (t, w − uth−1 , η) ∂t h−2 ∂ X h−1
− k Bth−1−k (t, utk , η). ∂t

(3.45)

k=0

It follows from equality (2.5) that Z

Lψηdx = B(t, ψ, η) +

G

J Z X

S

j=1

Bj ψBj′ ηds

m (G) and all t ∈ [0, T ]. Differentiating both sides of this for all ψ ∈ W22m (G), η ∈ HB equality with respect to t h − 1 − k times and taking ψ = φk (0 ≤ k ≤ h − 1), we have Z Lth−1−k φk ηdx =Bth−1−k (t, φk , η) G

+

m Z X j=1

S

h−1−k X

h−1−k s

s=0




(Bj )th−1−k−s φk (Bj′ )ts ηds.


Multiplying both sides of this equality with h−1 , taking sum in k from 0 to h − 1 and k



h−1 h−1−k h−1 h−1−s noting that k = s , we have s k Z h−1 X G k=0

h−1 k




Lth−1−k φk ηdx =

h−1 X k=0

+

h−1 k




Bth−1−k (t, φk , η)

m h−1 X X

h−1 s

j=1 s=0




Z

S

h−1−s X


(Bj )th−1−s−k φk (Bj′ )ts ηds.

h−1−s k

k=0

(3.46)

From this equality taking t = 0 together with (3.37) and (3.38) we obtain (φh , η) = (fth−1 (0), η) −

h−1 X k=0

h−1 k


Bth−1−k (0, φk , η).

(3.47)

Now integrating equality (3.45) with respect to t from 0 to t and using (3.47), we arrive at hwt , ηi + B(t, w, η) =(fth−1 , η) Z t h−1 X Bt (τ, w − uth−1 , η)dτ − + 0

k=0




h−1 k

Bth−1−k (t, utk , η). (3.48)

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Nguyen Manh Hung and Nguyen Thanh Anh

Put z = w − uth−1 . Then z(0) = 0 since u(0) = w(0) = φh−1 . It follows from the induction hypothesis (3.42) with h replaced by h − 1 and (3.48) that Z t hzt (t), ηi + B(t, z(t), η) = Bt (τ, z(.τ ), η)dτ. (3.49) 0

Applying Lemma 3.1, we can see from (3.49) that z ≡ 0 on GT . Therefore, uth = wt = m (G), H −m (G)). v ∈ W21 ((0, T ); HB B m (G), L (G)). We rewrite (3.43) in the Now we show that in fact uth ∈ W21 ((0, T ); HB 2 form hvt , ηi + B(t, v, η) = (fth , η) + hFb(t), ηi,

(3.50)

m (G) defined by where Fb(t), t ∈ [0, T ], are functionals on HB

hFb(t), ηi = −

h−1 X k=0

h k




m (G). Bth−k (t, utk , η), η ∈ HB

(3.51)

m (G)) for k = 0, . . . , h, we can see from (3.51) that F bt ∈ Since utk ∈ L2 ((0, T ), HB −m L2 ((0, T ); HB (G)) and

hFbt (t), ηi = −

h−1 X k=0

h+1 k




m (G). Bth+1−k (t, utk , η) − hBt (t, uth , η), η ∈ HB

Then, according to the remark below Lemma 3.3, we obtain from (3.50) that uth = m (G), L (G)). The inequality (3.40) holds since kf k v ∈ W21 ((0, T ); HB 2 th L2 (GT ) and b kFt kL2 ((0,T );H −m (G)) can be estimated by the right-hand side of it. The proof is comB pleted.

4.

Asymptotics of the Solution

Let r = |x| and ω be an arbitrary local coordinate system on S n−1 . Let δ be a positive real constant. A differential operator X P (x, t, ∂x ) = pα (x, t) ∂xα |α|6l

is called δ-admissible operator of order l near the the conical point 0 if the coefficients pα (x, t) are infinitely differentiable in GT and there are representation in a neighborhood U of the conical point 0 |α|−l+δ (1) pα (x, t) = r|α|−l p(0) pα (ω, r, t), α (ω, t) + r (0)

(1)

where pα ∈ C ∞ (ΩT ), and the functions pα are infinitely differentiable functions in Ω × R+ × [0, T ] such that |(r∂r )j ∂ωβ ∂tk p(1) α (ω, r, t)| < cj,k,β , ω ∈ Ω, r > 0, t ∈ [0, T ],

The Initial-Boundary Value Problems for Parabolic Equations...

219

for every multi-index β and every pair of nonnegative integers j, k. Here the constants cj,k,β do not depend on ω, r and t. The leading part P of the operator P at the point 0 is defined by X α r|α|−l p(0) P(x, t, ∂x ) = α (ω, t) ∂x . |α|6l

It can be directly verified that the derivative ∂xα has the form ∂xα

=r

−|α|

|α| X

pα,j (ω, ∂ω )(r∂r)j ,

(4.52)

p=0

where pα,j (ω, ∂ω ) are differential operators of order 6 |α| − j with smooth coefficients on Ω. Thus the operator P(x, t, ∂x ) can be represented as P(x, t, ∂x ) = r

−l

l X

pj (ω, t, ∂ω )(r∂r )j .

j=0

If P (x, t, ∂x ) is δ-admissible operator of order l near 0 and P is the leading part of P at the point 0, then there exists a constant such that k(P − P)ukV k−l,h (K 2,γ−δ

T)

6 ckukV k,h (K 2,γ

(4.53)

T)

k,h for all u ∈ V2,γ (KT ) with support in UT , k > l. Analogously, the δ-admissibility of a boundary differential operator on ∂G × [0, T ] is defined. An analogous assertion as (4.53) is valid for δ-admissible boundary differential operators on ∂G × [0, T ]. For convenience we rewrite the operator L(x, t, ∂x ) in the form X aα (x, t) ∂xα . L = L(x, t, ∂x ) = |α|62m

Let L = L(x, t, ∂x ), Bj = Bj (x, t, ∂x ) (j = 1, . . . , m) be the leading parts of L(x, t, ∂x ), Bj (x, t, ∂x ) at the point x = 0. Since the coefficients of the operators L(x, t, ∂x ) and Bj (x, t, ∂x ) are smooth, it is verified easily that L(x, t, ∂x ) =

X

|α|62m

aα (0, t) ∂xα , Bj (x, t, ∂x ) =

X

bj,α (0, t) ∂xα ,

|α|6µj

and the operators L(x, t, ∂x ) and Bj (x, t, ∂x ) are 1- admissible. Rewrite L(x, t, ∂x ), Bj (x, t, ∂x ) in the form L(x, t, ∂x ) = r−2m L (ω, t, ∂ω , r∂r ), Bj (x, t, ∂x ) = r−µj Bj (ω, t, ∂ω , r∂r ). We introduce the operator U (λ, t) = (L (ω, t, ∂ω , λ), B1 (ω, t, ∂ω , λ), . . . , Bm (ω, t, ∂ω , λ))

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Nguyen Manh Hung and Nguyen Thanh Anh

(λ ∈ C, t ∈ [0, T ]) of the parameter-depending elliptic boundary value problem L (ω, t, ∂ω , λ)u = f

in Ω,

Bj (ω, t, ∂ω , λ)u = gj

on ∂Ω,

j = 1, . . . , m.

This is a pencil of continuous operators from X≡

W2l (Ω)

into Y ≡

W2l−2m (Ω)

×

m Y

l−µj − 21

W2

(∂Ω) (l > 2m)

j=1

depending polynomially on λ ∈ C. We mention now some well-known definitions ( [9]). Let t0 ∈ [0, T ] fixed. If λ0 ∈ C, ϕ0 ∈ X such that ϕ0 6= 0, U (λ0 , t0 )ϕ0 = 0, then λ0 is called an eigenvalue of U (λ, t0 ) and ϕ0 ∈ X is called an eigenvector corresponding to λ0 . Λ = dim ker U (λ0 , t0 ) is called the geometric multiplicity of the eigenvalue λ0 . If the elements ϕ1 , . . . , ϕs of X satisfy the equations σ X 1 dq U (λ, t0 )|λ=λ0 ϕσ−q = 0 for σ = 1, . . . , s, q! dλq q=0

then the ordered collection ϕ0 , ϕ1 , . . . , ϕs is said to be a Jordan chain corresponding to the eigenvalue λ0 of the length s + 1. The rank of the eigenvector ϕ0 (rank ϕ0 ) is the maximal length of the Jordan chains corresponding to the eigenvector ϕ0 . A canonical system of eigenvectors of U (λ0 , t0 ) corresponding to the eigenvalue λ0 is a system of eigenvectors ϕ1,0 , . . . , ϕΛ,0 such that rank ϕ1,0 is maximal among the rank of all eigenvectors corresponding to λ0 and rank ϕj,0 is maximal among the rank of all eigenvectors in any direct complement in ker U (λ0 , t0 ) to the linear span of the vectors ϕ1,0 , . . . , ϕj−1,0 (j = 2, . . . , Λ). The number κj = rank ϕj,0 (j = 1, . . . , Λ) are called the partial multiplicities and the sum κ = κ1 + · · · + κΛ is called the algebraic multiplicity of the eigenvalue λ0 . The eigenvalue of λ0 is called simple if its algebraic multiplicity is equal to one. For each t ∈ [0, T ] fixed the set of all complex number λ such that U (λ, t) is not invertible is called the spectrum of U (λ, t). It is known that the spectrum of U (λ, t) is an enumerable set of its eigenvalues (see [9, Th. 5.2.1]). Moreover, there are constants δ, R such that U (λ, t) is invertible for all t ∈ [0, T ] and all λ in the set  D := λ ∈ C : | Re λ| 6 δ| Im λ|, |λ| > R

(4.54)

(see [9, Th. 3.6.1]). To receive asymptotic formulas of the solutions with the coefficients regular with respect to the variable t we require later that eigenvalues and eigenvectors of the pencil U (λ, t) satisfy the following assumption. Let l1 , l2 be nonnegative integers, β1 , β2 be real numbers such that l1 −β1 < l2 −β2 . We say that the assumption (H) for numbers l1 , l2 , β1 , β2 is fulfilled if the following conditions are satisfied:

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221

n (i) The lines Re λ = −βi + li − (i = 1, 2) do not contain eigenvalues of the pencil 2 n n U (λ, t), and the strip −β1 + l1 − < Re λ < −β2 + l2 − contains the eigenvalues 2 2 λµ (t), µ = 1, . . . , N , with the geometric multiplicities Λµ and the partial multiplicities κµ,k , µ = 1, . . . , N, k = 1, . . . , Λµ , not depending on t ∈ [0, T ]. These eigenvalues are smooth functions on [0, T ]. (ii) It is can be chosen a canonical system (µ)

ϕk,s (ω, t), k = 1, . . . , Λµ , s = 1, . . . , κµ,k , of Jordan chains of U (λ, t) corresponding to the eigenvalue λµ (t), µ ∈ {1, . . . , N }, consisting of functions smooth on t ∈ [0, T ] for all ω ∈ Ω. Now we introduce the formally adjoint pencil U+ (λ, t) of the pencil U(λ, t). Notice that the operators Bj (x, t, ∂x ), j = 1, . . . , m, has the same principal parts as the operators Bj (x, t, ∂x ), j = 1, . . . , m. Thus the system {Bj (x, t, ∂x )}m j=1 is normal on ∂K for all t ∈ [0, T ]. Therefore, there are boundary operators Bj (t, ∂x ), ordBj (x, t, ∂x ) = µj < 2m, j = m + 1, . . . 2m, such that the system {Bj (x, t, ∂x )}2m j=1 is a Dirichlet system of order 2m (for definition see [9], p. 63) on ∂K for all t ∈ [0, T ], and the following classical Green formula Z Z m Z m Z X X Bj+m uB′j vds (4.55) Luvdx + uL+ vdx + Bj uB′j+m vds = K

j=1

∂K

K

j=1

∂K

holds for u, v ∈ C0∞ (K \ {0}) and for all t ∈ [0, T ]. Here L+ = L+ (t, ∂x ) is the formal adjoint operator of L and B′j = B′j (t, ∂x ) are boundary operators of order µ′j = 2m − 1 − µj+m if j 6 m, and of order µ′j = 2m − 1 − µj−m if j > m + 1. The coefficients of B′j = B′j (t, ∂x ), j = 1, . . . , 2m, are dependent on t smoothly on [0, T ]. The operators L+ (t, ∂x ), B′j (t, ∂x ) can be written in the form L+ (t, ∂x ) = r−2m L + (ω, t, ∂ω , r∂r ), ′

B∗′j (t, ∂x ) = r−µj Bj′ (ω, t, ∂ω , r∂r ). From the Green formula (3.18) we get the following Green formula Z m Z X ′ L (t, λ)e uvedx + Bj (t, λ)e uBj+m (t, −λ + 2m − n)e v ds = Ω

Z

Ω

j=1

∂Ω

v dx + u eL + (t, −λ + 2m − n)e

m Z X j=1

∂Ω

v ds Bj+m (t, λ)e uBj′ (t, −λ + 2m − n)e

for u e, ve ∈ C ∞ (Ω) and for all t ∈ [0, T ] (see [9], p. 206). Here for the sake of brevity, we have omitted the arguments ω and ∂ω in the operators of this formula. We denote by U+ (λ, t) the operator of the boundary value problem L + (ω, t, ∂ω , −λ + 2m − n)v = f Bj+ (ω, t, ∂ω , −λ

+ 2m − n)v = gj

in

Ω,

on

∂Ω,

j = 1, . . . , m.

Repeat the arguments used to prove Lemma 2.1 in [16] and Lemma 4.1 in [17] we get

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Nguyen Manh Hung and Nguyen Thanh Anh

Lemma 4.1. Let λ0 (t) be a smooth function on [0, T ] such that λ0 (t) be an eigenvalue of U (λ, t) with the geometric multiplicity Λ(t) = Λ, the partial multiplicities κk (t) = κk , k = 1, . . . , Λ(t), not depending on t ∈ [0, T ]. Suppose that ϕk,s (ω, t), k = 1, . . . , Λ, s = 0, . . . , κk − 1, is a canonical system of Jordan chains of U (λ, t) corresponding to the eigenvalues λ0 (t) which can be chosen from C ∞ (ΩT ). Then 1) There exist functions ψk,s ∈ C ∞ (ΩT ), k = 1, . . . , Λ, s = 0, . . . , κk −1, such that for each t ∈ [0, T ] they form a canonical system of Jordan chains of U + (λ, t) corresponding to the eigenvalue λ0 (t) and the following biorthonormality condition holds: σ p+s+1 X X 1 p=0 q=p+1

+

m X

q!

(L (q) (λ0 (t), t)ϕk,p+s+1−q , ψl,σ−p )Ω (q)

Bj (λ0 (t), t)ϕk,p+s+1−q ,

σ−p X τ =0

j=1


 1 ′ (Bj+m )(τ ) (−λ0 (t) + 2m − n, t)ψl,σ−p−τ ∂Ω τ!

= δk,l δs,κk −1−σ

for k, l = 1, . . . , Λ, s = 0, . . . , κk − 1, σ = 0, . . . , κl − 1. 2) The inverse U −1 (λ, t) has the following representation U −1 (λ, t) =

Λ κX k −1 X k=1 s=0

Pk,s (t)
κ −s + P(λ, t) λ − λ0 (t) k

in the neighborhood of λ0 (t), where Pk,s (t) are pencils of finite-dimentional operators from Y into X defined by Pk,s (t)v =

s X

σ=0

hv, ψk,σ (., t)i ϕk,s−σ (., t), v ∈ Y,

and P(λ, t) is a pencil of continuous operators from Y into X depending analytically on λ ∈ C and smoothly on t ∈ [0, T ]. Here  m  k X X 1 ′ (q) hv, ψk,σ i = (v0 , ψk,σ )Ω + vj , (B ) (−λ0 (t) + 2m − n)ψk−q,σ q! j+m ∂Ω j=1

q=0

for v = (v0 , v1 , . . . , vm ) ∈ Y . Before giving results on asymptotics of the solution near the conical point 0, it is needed to establish its regularity in weighted Sobolev spaces. These are stated in three following lemmas. 0,h l+2m (GT ) be a solution of the Lemma 4.2. Let u ∈ W h ((0, T ); W2,loc (G \ {0})) ∩ V2,γ−l−2m problem

L(x, t, ∂x )u = f in GT , Bj (x, t, ∂x )u =gj on ST , j = 1, . . . , m,

(4.56) (4.57)

The Initial-Boundary Value Problems for Parabolic Equations... l+2m−µj − 12 ,h

l,h where f ∈ V2,γ (GT ), gj ∈ V2,γ l+2m,h V2,γ (GT )

and the following estimate

kuk2V l+2m,h (G ) T 2,γ

6C



kf k2V l,h (G ) T 2,γ

+

m X j=1

223

(ST ), l is a nonnegative integer. Then u ∈

kgj k2 l+2m−µj − 1 ,h 2

V2,γ

(ST )

+ kuk2V 0,h

2,γ−l−2m (GT )



(4.58) holds with the constant C independent of u, f, gj and t0 . Proof. Without generality we assume the domain G coincides with the cone K in the unit ball. Set G0 = {x ∈ G : |x| > 2−1 },  Gk = x|x ∈ G, 2−k 6 |x| 6 2−k+1 ,

k = 1, 2, . . . ,

and S k = ∂G ∩ ∂Gk , k = 0, 1 . . .. According to well-known results on the regularity of solutions of elliptic boundary problems in smooth domains (see, e.g., [2]), we have ku(., t)k2W l+2m (G2 ) 6C(kf (., t)k2W l (G1 ∪G2 ∪G3 ) 2

2

+ kgj (., t)k2

l+2m−µj − 1 2

W2

(S 1 ∪S 2 ∪S 3 )

+ ku(., t)k2L2 (G1 ∪G2 ∪G3 ) )

for a.e. t ∈ [0, T ] with the constant C independent of u, f, gj and t. By making change of variable x = 2−k x′ for a positive integer k, we get from (4.56), (4.57) that X

aα ∂xα′ u +

2(|α|−2m)k aα ∂xα′ u =2−2mk f in G,

(4.59)

|α|62m−1

|α|=2m

X

X

bα ∂xα′ u +

X

2(|α|−µj )k bα ∂xα′ u =2−µj k gj on S, j = 1, . . . , m.

(4.60)

|α|6µj −1

|α|=µj

Similarly as above, from (4.59), (4.60), we have ku(., t)k2W l+2m (G2 ) 6 C k2−2mk f (., t)k2W l (G1 ∪G2 ∪G3 ) 2

2

−µj k

+ k2

2

gj (., t)k

1 l+2m−µj − 2 (S 1 ∪S 2 ∪S 3 ) W2

+ ku(., t)k2L2 (G1 ∪G2 ∪G3 )




(4.61)

for a.e. t ∈ [0, T ] with the constant C independent of u, f, gj , t and k. Let g˜j (., t) ∈ l+2m−µj V2,γ (S) be arbitrary extensions of gj (., t) to G, j = 1, . . . , m. Then we have form (4.61) that ku(., t)k2W l+2m (G2 ) 6 C k2−2mk f (., t)k2W l (G1 ∪G2 ∪G3 ) 2

2

+ k2−µj k g˜j (., t)k2

l+2m−µj W2 (G1 ∪G2 ∪G3 )


+ ku(., t)k2L2 (G1 ∪G2 ∪G3 ) . (4.62)

224

Nguyen Manh Hung and Nguyen Thanh Anh

Returning to variable x with noting that, in Gk+2 , 2−k−2 6 r 6 2−k−1 , from (4.62) we have X X

r2(γ+|α|−l) f (., t) 2 k+1 k+2 k+3

r2(γ+|α|−l−2m) u(., t) 2 k+2 6 C L2 (G ∪G ∪G ) L2 (G ) |α|6l

|α|6l+2m

+

X

|α|6l+2m−µj

2(γ+|α|−l−2m+µ )

2 j

r gj (., t)

l+2m−µj

W2

(Gk+1 ∪Gk+2 ∪Gk+3 )

 . L2 (Gk+1 ∪Gk+2 ∪Gk+3 )

2 + r2(γ−l−2m) u(., t)

(4.63)

Taking sum both sides of these inequalities with respect to k from 1 to +∞, we have  ku(., t)k2V l+2m (G) 6C kf (., t)k2V l (G) 2,γ

2,γ

+

m X j=1

Here it is noted that

P

|α|6l+2m

k˜ gj (., t)k2 l+2m−µj V2,γ

(G)

+ ku(., t)k2V 0

2,γ−l−2m (G)



.

(4.64)

kr2(γ+|α|−l−2m) u(., t)k2L2 (G0 ) can be estimated by the right-

hand side of (4.64). It follows from (4.64) that  ku(., t)k2V l+2m (G) 6C kf (., t)k2V l (G) 2,γ

2,γ

+

m X j=1

kgj (., t)k2 l+2m−µj − 1 2

V2,γ

(S)

+ ku(., t)k2V 0

2,γ−l−2m (G)



with the constant C independent of u and t. Integrating this inequality with respect to t on [0, T ], we get kuk2V l+2m,0 (G ) T 2,γ

6C



kf k2V l,0 (G ) T 2,γ

+

m X j=1

kgj k2 l+2m−µj − 1 ,0 V2,γ

2

(ST )

+ kuk2V 0,0

2,γ−l−2m (GT )



(4.65) l+2m,0 Now assume by induction that utk ∈ V2,γ (GT ) and inequality (4.65) holds with u replaced by utk for k 6 h − 1 (h > 1). Differentiating equalities (4.56), (4.57) h times with respect to t, we have Luth Bj uth

h−1   X h L h−k utk in GT , = fh := fth − k t k=0 h−1   X h (Bj )th−k utk on ST , = gjh := (gj )th − k

(4.66)

j = 1, . . . , m.

(4.67)

k=0

l+2m−µj − 12 ,0

l,0 By the inductive assumption we see that fh ∈ V2,γ (GT ), gjh ∈ V2,γ

l+2m,0 V2,γ (GT )

1 . . . , m. Repeating the arguments above, we get uth ∈ holds with u replaced by uth . Consequently, the lemma is proved.

(ST ), j =

and inequality (4.65)

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225

Lemma 4.3. Let l, s be integers, l > 0, γ be a real number such that s− n2 < γ < s+1− n2 . P xα cα , cα ∈ C, be a polynomial of degree k. Then Let p = α! |α|6k l (G) if and only if c = 0 for all multi-index α, |α| 6 l − s − 1. (i) p ∈ V2,γ α l (G), then p ∈ V l+q (G) for all nonnegative integer q. (ii) If p ∈ V2,γ 2,γ+q l− 1

l+q− 1

(iii) If p|S ∈ V2,γ 2 (S), then p|S ∈ V2,γ+q 2 (S) for all nonnegative integer q. Proof. (i) First, consider a monomial pα = with |β| 6 l. We have

xα with |α| > l − s. Let β be a multi-index α!

xα−β 2 2 1 r2(γ+|α|−l) , r2(γ+|β|−l) ∂xβ pα = r2(γ+|β|−l) = (α − β)! [(α − β)!]2

1 = 0 if the multi-index α − β has a negative compo(α − β)! l (G). nent. Since γ +|α|−l > γ −s > − n2 , then r(γ+|β|−l) ∂xβ pα ∈ L2 (G). Hence, pα ∈ V2,γ P xα l (G). This also proves the converse statement of ∈ V2,γ cα This implies p1 := α! l−s6|α|6k assertion (i). l (G). Then we have Now let p ∈ V2,γ where we set by convention

p2 := p − p1 =

X

cα

|α|6l−s−1

xα l ∈ V2,γ (G). α!

From this we have X

∂xβ p2 =

cα+β

|α|6l−s−1−|β|

xα l−|β| 0 ∈ V2,γ (G) ⊂ V2,γ−s−1 α!

(4.68)

for |β| 6 l − s − 1. Since γ − s > − n2 , then all terms with |α| > 1 in the sum of (4.68) 0 0 belong to V2,γ−s−1 (G). This implies cβ ∈ V2,γ−s−1 (G). It is possible only when cβ = 0, n since γ − s − 1 < − 2 . Consequently, cβ = 0 for all |β| 6 l − s − 1. (ii) This part follows directly from the part (i). l− 1

(iii) Suppose p|S ∈ V2,γ 2 (S). Write p = p1 + p2 ≡

X

l−s6|α|6k

cα

xα + α!

X

|α|6l−s−1

cα

xα . α! l− 1

l (G) ∩ V l+q (G). Thus, p | ∈ V 2 It is known above that p1 ∈ V2,γ 1 S 2,γ (S), and therefore, 2,γ+q l− 1

p2 |S = p|S − p1 |S ∈ V2,γ 2 (S). This implies p2 |S = 0 according to [9, Le. 7.1.7]. l+q− 1

Consequently, p|S = p1 |S ∈ V2,γ+q 2 (S).

226

Nguyen Manh Hung and Nguyen Thanh Anh

m (G), L (G)) be the Lemma 4.4. Let h is a nonnegative integer and let u ∈ W21 ((0, T ); HB 2 generalized solution of the problem (2.8)–(2.10). Suppose that the assumptions of Theorem 2m,h 3.4 hold. Then u ∈ W2,m+ǫ (GT ) and

kukW 2m,h

2,m+ǫ (GT )

where 0 < ǫ < u, φ and f .

1 2

if m >

n 2

6C

h X k=0


kφk k2W m (G) + kftk k2L2 (GT ) , 2

(4.69)

and n is even, otherwise ǫ = 0, C is a constant independent of

Proof. First, we show by induction on h that 2m utk (., t) ∈ W2,loc (G \ {0}) for a.e. t ∈ [0, T ] and all k 6 h.

(4.70)

m (G), L (G)), k 6 h, and From Theorem 3.4 we have utk ∈ W21 ((0, T ); HB 2

B(t, u, η) = (f (., t) − ut (., t), η)

(4.71)

m (G) and a.e. t ∈ [0, T ]. Since f (., t) − u (., t) ∈ L (G) for a.e. t ∈ for all η ∈ HB t 2 [0, T ], according to results for elliptic boundary value problem in domains with smooth 2m (G \ {0}) for a.e. t ∈ [0, T ]. This boundaries, it follows from (4.71) that u(., t) ∈ W2,loc means (4.70) holds for h = 0. Assume now it holds for h − 1 (h > 1). Also from Theorem 3.4 we have h−1   X h Bth−k (t, utk , η) B(t, uth , η) = (fth (., t), η) − (uth+1 (., t), η) − k k=0

m (G), a.e. t ∈ [0, T ]. This yields for all η ∈ HB

B(t, uth , η) = (Fh (., t), η)

(4.72)

m (G) ∩ C ∞ (G \ {0}) and a.e. t ∈ [0, T ], where for all η ∈ HB 0

Fh = fth (., t) − uth+1 (., t) −

h−1   X h k=0

k

Lth−k (., t)utk ∈ L2,loc (G \ {0}).

2m (G \ {0}) for a.e. t ∈ [0, T ], Similarly as above, it follows from (4.72) that uth (., t) ∈ W2,loc and therefore, (4.70) holds for h. Now rewrite (2.8), (2.9) in the form

Lu = f1 := f − ut in GT ,

Bj u = 0

on ST , j = 1, . . . , m.

(4.73) (4.74)

m (G) = V m (G) by [9, Th. 7.1.1]. This Let us consider first the case m < n2 . In this case W2,0 2,0 m (G) ⊂ W m (G) = V m (G) ⊂ V 0 implies HB 2,0 2,0 2,−m (G). Therefore, the generalized solution 0,h 0,h 0,h u is an element of V2,−m (GT ). From this and that f1 ∈ V2,0 (GT ) ⊂ V2,−m (GT ) it follows

The Initial-Boundary Value Problems for Parabolic Equations...

227

2m,h 2m,h (GT ) according to Lemma 4.2. Consequently, the assertion of (GT ) ⊂ W2,m u ∈ V2,m the lemma holds. Now suppose m > n2 . Let ℓ = [ n2 ] be the greatest integer not exceeding n2 . Then we m,h m,h have ℓ − n2 < ǫ < ℓ + 1 − n2 . It follows from u ∈ W2m,h (GT ) ≡ W2,0 (GT ) ⊂ W2,ǫ (GT ) m (G) for all k 6 h and for a.e. t ∈ [0, T ]. Fix such t ∈ [0, T ]. According that utk (., t) ∈ W2,ǫ to [9, Th. 7.1.1], there is the representation X cα (t)xα + v(x, t), u(x, t) = |α|6m−ℓ−1

m (G), where v(., t) ∈ V2,ǫ

1 1 cα (t) = lim r→0 α! |Ω|

Z

∂xα u(ω, r, t)dω

Ω

(4.75)

with estimates m (G) 6 Cku(., t)kW m (G) , kvkV2,ǫ 2,ǫ m (G) , |α| 6 m − ℓ − 1. |cα (t)| 6 Cku(., t)kW2,ǫ

R Here |Ω| = Ω dω, and the constants C do not depend u and t. From these estimates it m,0 follows that v ∈ V2,ǫ (GT ), cα ∈ L2 ((0, T )) with kvkV m,0 (GT ) 6 CkukW m,0 (GT ) , 2,ǫ

2,ǫ

kcα kL2 (GT ) 6 CkukW m,0 (GT ) , |α| 6 m − ℓ − 1. 2,ǫ

Analogously, we have the representation X ut (x, t) =

α (1) c(1) α (t)x + v (x, t),

|α|6m−ℓ−1

m (G), where v (1) (., t) ∈ V2,ǫ

c(1) α (t)

1 1 lim = α! r→0 |Ω|

Z

Ω

∂xα ut (ω, r, t)dω

(4.76)

with estimates kv (1) kV m,0 (GT ) 6 Ckut kW m,0 (GT ) , 2,ǫ

2,ǫ

kc(1) α kL2 (GT ) 6 Ckut kW m,0 (GT ) , |α| 6 m − ℓ − 1. 2,ǫ

(1)

It following from (4.75), (4.76) that cα = (cα )t , |α| 6 m − ℓ − 1, and therefore, v (1) = vt . m,1 Thus, cα ∈ W21 ((0, T )), |α| 6 m − ℓ − 1, and v ∈ V2,ǫ (GT ). Repeating the arguments

m,h above we arrive at cα ∈ W2h ((0, T )), |α| 6 m − ℓ − 1, and v ∈ V2,ǫ (GT ). Moreover, we have (4.77) kwkW 2m,h (G ) + kvkV m,h (G ) 6 CkukW m,h (G ) , 2,m+ǫ

T

2,ǫ

T

2

T

228 where w(x, t) =

Nguyen Manh Hung and Nguyen Thanh Anh P cα (t)xα .

|α|6m−ℓ−1

Rewrite (2.8), (2.9) in the form

Lu = F in GT ,

(4.78)

Bj u = gj on ST , j = 1, . . . , m,

(4.79)

0,h 0,h where F = f − ut − Lw ∈ W2,m+ǫ (GT ) ≡ V2,m+ǫ (GT ), gj = −Bj w|ST , j = 1, . . . , m. We show now that 2m−µj − 21 ,h

gj ∈ V2,m+ǫ for j = 1, . . . , m. If µj > m −

n 2,

(ST )

(4.80) 2m−µ

then (4.80) holds, since in this case W2,m+ǫ j (G) 2m−µj − 1 ,h

2m−µ

2m−µj − 1 ,h

= V2,m+ǫ j (G) (see [9, Th. 7.1.1]), and therefore, W2,m+ǫ 2 (ST ) = V2,m+ǫ 2 (ST ). m (G), Now let us fix t ∈ [0, T ] and j ∈ {0, . . . , m} with µj 6 m − n2 . Since v(., t) ∈ V2,ǫ m−µj − 12

then −Bj w(., t)|S = Bj v(., t)|S ∈ V2,ǫ

(S). Since −Bj w(., t) is a polynomial with 2m−µj − 21

respect to x, then, according to Lemma 4.3, −Bj w(., t)|S ∈ V2,m+ǫ 2m−µj − 1 V2,m+ǫ 2 (S).

(S). This implies 2m−µj − 1 ,h

From this it follows gj ∈ V2,m+ǫ 2 (ST ). gj (., t) = −Bj w(., t)|S ∈ Consequently, we have (4.80). 2m,h Now we can apply Lemma 4.2 for problem (4.78), (4.79) to conclude v ∈ V2,m+ǫ (GT ).

2m,h Therefore, u = v + w ∈ W2,m+ǫ (GT ). The inequality (4.69) follows from estimates (4.58), (4.77) and (3.40).

Three following lemmas give asymptotic formulas for solutions of elliptic boundary problems depending on parameter t. They are essential to obtain the main result of this section. l1 ,h Lemma 4.5. Let u ∈ V2,β (GT ) be a solution of the problem 1

L(t, ∂x )u = f in GT , Bj (t, ∂x )u = gj on ST , j = 1, . . . , m,

(4.81) (4.82)

l2 −µj − 1 ,h

l2 −2m,h 2 (ST ), l1 , l2 > 2m, β1 − l1 > β2 − l2 . Suppose where f ∈ V2,β (GT ), gj ∈ V2,β2 2 that the assumption (H) is fulfilled. Then there exists a neighborhood V of the origin of Rn such that in VT the solution u has representation

u(x, t) =

Λµ κµ,k −1 N X X X

µ=1 k=1

cµ,k,s (t)r

λµ (t)

s=0

s X 1 (µ) (ln r)σ ϕk,s (ω, t) + w(x, t), σ! σ=0

(4.83)

l2 ,h where w ∈ V2,β (KT ), cµ,k,s (t) ∈ W2h ((0, T )). 2 (µ)

Proof. For each µ = 1, . . . , N , let ψk,s (ω, t), k = 1, . . . , Λµ , s = 1, . . . , κµ,k be a canonical system of Jordan chains of U + (λ, t) having the properties as in Lemma 4.1. We introduce functions s X 1 (µ) −λµ (t)+2m−n vµ,k,s (x, t) = r (− ln r)σ ψk,s−σ (ω, t) σ! σ=0

The Initial-Boundary Value Problems for Parabolic Equations...

229

for µ = 1, . . . , N , k = 1, . . . , Λµ , s = 0, . . . , κµ,k − 1. First, we assume that the function u has the support contained in UT , where U is a certain neighborhood of the origin 0 in which the domain G coincides with the cone K. By extension by zero to KT (respectively, ∂KT ) we can regard u, f (respectively, gj ) as functions defined in KT (respectively, ∂KT ). For each t ∈ [0, T ] fixed, according to results for elliptic boundary problem in a cone (see, e.g, [9, Th. 6.1.4, Th. 6.1.7]), the solution u(x, t) admits the representation (4.83) in K with Z 1 e w(x, t) = rλ U −1 (λ, t)F(ω, λ, t)dλ (4.84) 2πi Re λ=−β2 +l2 − n 2

and m X

′ cµ,k,s (t) = f (., t), vµ,k,κµ,k −1−s (., t) K + gj (., t), Bj+m vµ,k,κµ,k −1−s (., t) ∂K j=1


= f (., t), vµ,k,κµ,k −1−s (., t) G +

m X j=1


′ gj (., t), Bj+m vµ,k,κµ,k −1−s (., t) S (4.85)

2m f , r] µ1 g , . . . , r ^ µm g ). e = (r] for µ = 1, . . . , N , k = 1, . . . , Λµ , s = 1, . . . , κµ,k , where F 1 m Here ge(ω, λ), t denotes the Mellin transformation with respect to the variable r of g(ω, r, t), i.e, Z +∞

r−λ−1 g(ω, r, t)dr.

ge(ω, λ, t) =

0

l2 ,h We will prove below that w ∈ V2,β (KT ), cµ,k,s (t) ∈ W2h ((0, T )). 2 Now we make clear the first one. Since there are no eigenvalues of the operator pencil n U (λ, t) on the line Re λ = −β2 + l2 − , from the proof of [9, Th. 3.6.1] we have the 2 estimate m  X 2 e 2 l kU −1 (λ, t)Ψk 6 C ke η k + ke ηj k2 l−2m W (Ω,λ) W (Ω,λ) 2

2

l−µ − 1 W2 j 2 (∂Ω,λ)

j=1



(4.86)

n e = (e η , ηe1 , . . . , ηem ) for all λ on the line Re λ = −β2 + l2 − , t ∈ [0, T ], and all Ψ 2 1 Q l−µk − 2 e ∈ W2l−2m (Ω) × m (∂Ω), where the constant C is independent of λ, t and Ψ. k=1 W2 Here kukW l (Ω,λ) = kukW l (Ω) + |λ|l kukL2 (Ω) , 2

2

1

kukW l (∂Ω,λ) = kuk 2

l− 1 W2 2 (∂Ω)

+ |λ|l− 2 kukL2 (∂Ω) ,

l− 21

which are equivalent to the norms in W2l (Ω), W2 complex number λ.

(∂Ω), respectively, for arbitrary fixed

230

Nguyen Manh Hung and Nguyen Thanh Anh

We will prove by induction on h that m  X 2 e 2 l ke ηj k2 k(U −1 )th (λ, t)Ψk 6 C(h) ke η k + W (Ω,λ) W l−2m (Ω,λ) 2

2

l−µ − 1 W2 j 2 (∂Ω,λ)

j=1



. (4.87)

It holds for h = 0 by (4.86). Assume that it holds for h − 1. From the equality U (λ, t)U −1 (λ, t) = I,

differentiating both sides of it h (h ≥ 1) times with respect to t we obtain h−1 X

h−1 k

k=0


Uth−k (λ, t)(U −1 )tk (λ, t) + U (λ, t)(U −1 )th (λ, t) = 0.

Rewrite this equality in the form (U −1 )th (λ, t) = −U −1 (λ, t)

h−1 X




h−1 k

k=0

Uth−k (λ, t)(U −1 )tk (λ, t).

Then (4.87) follows from this equality and the inductive assumption. It is well-known (see [9, Le. 6.1.4]) that the norm (2.1) is equivalent to  1 Z 2 1 2 9u9V l (K) = ke u(., λ)kW l (Ω,λ) dλ , 2,β 2 2πi Re λ=−β+l− n 2

and the norm (2.2) is equivalent to  1 = 9u9 l− 21 2πi V2,β (∂K)

Z

Re λ=−β+l− n 2

1 2

2

ke u(., λ)k

l− 1 W2 2 (∂Ω,λ)

dλ

.

Using these with noting e λ, t) w(ω, e λ, t) = U −1 (λ, t)F(., (see [9, Le. 6.1.3]) and (4.86), we get from (4.84) that kw(., t)k2V l2

6

2,β2 (K)

=

6

C 2πi C 2πi

C 2πi Z

Z

kw(., e λ, t)k2W l2 (Ω,λ) dλ 2

Re λ=−β2 +l2 − n 2

e λ, t)k l2 kU −1 (λ, t)F(., dλ W (Ω,λ) 2

Re λ=−β2 +l2 − n 2

Z

Re λ=−β2 +l2 − n 2



^ kr2m f (., t)k2W l−2m (Ω,λ) +

 6 C kr2m f (., t)k2V l2 −2m

2,β2 −2m

2

+ (K)

 6 C kf (., t)k2V l2 −2m (K) + 2,β2

 = C kf (., t)k2V l2 −2m (G) + 2,β2

m X j=1

m X

j=1 m X j=1

m X j=1

j g (., t)k2 krµ^ j

l−µ − 1 W2 j 2

krµj gj (., t)k2 l2 −µj − 1 V2,β

2

2 −µj

kgj (., t)k2 l2 −µj − 1

(∂K)

kgj (., t)k2 l2 −µj − 1



V2,β

V2,β

2

2

2

2

(S)

(∂K)

(∂Ω,λ)



dλ



 (4.88)

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231

for all t ∈ [0, T ]. Here, and sometimes in this paper, for convenience, we denote different constants by the same symbol C. Integrating the last inequality with respect to t from 0 to l2 +∞, we obtain w ∈ V2,β (KT ) and 2 kwk2 l2 ,0

V2,β (KT ) 2

 6 C kf k2 l2 −2m,0 V2,β

2

(GT )

+

m X j=1

kgj k2 l2 −µj − 1 ,0 V2,β

2

2

(ST )



.

Differentiating (4.84) h times with respect to t we have wth (x, t) =

1 2πi

Z

Re λ=−β2 +l2 − n 2

rλ

h X

h k

k=0




e h−k (ω, λ, t)dλ. (U −1 )tk (λ, t)F t

Now using (4.87) and the same arguments as in (4.88) we arrive at kwth k2 l2 ,0

V2,β (KT )

6C

2

h  X kftk k2 l2 −2m,0 V2,β

k=0

2

(GT )

+

m X k=1

k(gj )tk k2 l2 −µj − 1 ,0 V2,β

2

2

(ST )



.

l2 ,h Therefore, w ∈ V2,β (KT ) and 2

kwk2 l2 ,h

V2,β (KT ) 2

 6 C kf k2 l2 −2m,h V2,β

2

(GT )

+

m X j=1

kgj k2 l2 −µj − 1 ,h V2,β

2

2

(ST )



.

Now we verify that cµ,k,s (t) ∈ W2h ((0, T )) for µ = 1, . . . , N , k = 1, . . . , Λµ , s = 1, . . . , κµ,k . For some such µ, k, s put (µ)

v(x, t) = r−λµ (t)+2m−n (ln r)σ ψk,s (ω, t). Using formula (4.52), we have α

∂ v=r

−|α|

|α| X p=0

=r


(µ) (r∂r)p r−λµ (t)+2m−n (ln r)σ Pα,p ψk,s

−|α|−λµ (t)+2m−n

|α| p   X X p (µ) (−λµ (t) + 2m − n)p−q (r∂r)q (ln r)σ Pα,p ψk,s q p=0 q=0

=r

−|α|−λµ (t)+2m−n

|α| p   X X p p=0 q=0

q

− λµ (t) + 2m − n


p−q

σ! (µ) (ln r)σ−q Pα,p ψk,s (σ − q)! (4.89)

n for all t ∈ [0, T ] and λµ (t) is smooth on [0, T ], then there 2 n is a real number ǫ > 0 such that Re λµ (t) 6 −β2 + l2 − − 2ǫ for all t ∈ [0, T ]. Thus, it 2 follows from (4.89) that Since Re λµ (t) < −β2 + l2 −

|r

−β2 +l2 −2m+|α| α

∂ v(x, t)| 6 Cr

+ǫ −n 2

|α| X p=0

(µ)

|Pα,p ψk,s (ω, t)|

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Nguyen Manh Hung and Nguyen Thanh Anh

l for all (x, t) ∈ GT and all multi-index α. This implies v(., t) ∈ V2,−β (G) and 2 +l2 −2m+l (µ)

kv(., t)kV l

2,−β2 +l2 −2m+l (G)

6 Ckψk,s (., t)kW l (Ω) 2

for an arbitrary integer l. Using Fa`a Di Bruno’s Formula for the higher order derivatives of composite functions (see, e.g, [14]), we have vtp =

p   X p q=0

=r

q

r−λµ (t)+2m−n

−λµ (t)+2m−n




(µ)

tp−q

p  X X p

q

q=0

(ψk,s )tq

n  (k) Y n! −λµ (t) mk (µ) m1 +···+mn (ln r) (ψk,s )tq , m1 ! . . . mn ! k! k=1

where the second sum is over all n−tuples (m1 , . . . , mn ) satisfying the condition m1 + 2m2 + · · · + nmn = n. Since λµ (t) is smooth on [0, T ], it together with its derivatives are bounded on [0, T ]. Repeating the arguments as above, we get kvtp (., t)kV l

2,−β2 +l2 −2m+l (G)

6C

p X q=0

(µ)

k(ψk,s )tq kW l (Ω) . 2

Thus, we have sup kvkV l,p

6C

2,−β2 +l2 −2m+l (G)

t∈[0,T ]

p X

(µ)

sup k(ψk,s )tq kW l (Ω) < +∞

(4.90)

2

q=0 t∈[0,T ]

for arbitrary nonnegative integers l, p. Set c(t) = (f (., t), v(., t))G . For p 6 h, using (4.90), we have p X |ctp (t)| = 2

p q

q=0

2
(ftp−q (., t), vtq (., t))G

6C

p X

6C

p X

q=0

q=0

krβ2 −l2 +2m ftq k2L2 (G)

kftq k2 l2 −2m V2,β

2

(G)

p  X q=0

kr−β2 +l2 −2m vtq k2L2 (G)



.

This implies c(t) ∈ W2h ((0, T )) and kckW h ((0,T )) 6 Ckf kV l2 −2m,h (G ) . 2

2,β2

T

(4.91)

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233

′ Now set cj (t) = (gj , Bj+m v)S , j = 1, . . . , m. Then also using (4.90), we have p X |(cj )tp (t)| =




p q

2

q=0

6C

p X

6C

p X

q=0

6C

q=0 p X q=0

2 ((gj )tp−q (., t), vtq (., t))S 1

krβ2 −l2 +µj + 2 (gj )tq k2L2 (G) k(gj )tq k2 l2 −µj − 1 V2,β

2

k(gj )tq k2 l2 −µj − 1 V2,β

(S)

2

2

2

(S)

p  X q=0

1

′ kr−β2 +l2 −µj − 2 (Bj+m v)tq k2L2 (G)

p  X . kvtq k2 2m−µj V2,−β

q=0

2 +l2 −µj

(G)





(p ≤ h).

This implies cj ∈ W2h ((0, T )) and kcj kW h ((0,T )) 6 Ckgj k 2

l2 −µj − 1 2 ,h (ST ) 2

V2,β

.

(4.92)

From (4.91) and (4.92), we can conclude that cµ,k,s (t) ∈ W2h ((0, T )) and 

kcµ,k,s kW h ((0,T )) 6 C kf kV l2 −2m,h (G 2

2,β2

T

+ )

m X j=1

kgj k

l2 −µ − 1 ,h V2,β j 2 (ST ) 2



.

l1 ,h (GT ) is arbitrary. Let η be an infinitely differential Now we consider the case u ∈ V2,β 1 function with support in U , equal to one in a neighborhood V of the origin. Let G′ , G′′ be domains with smooth boundary such that U \ V ⊂ G′ ⋐ G′′ ⊂ G. Denote by G the set of all subdomain G′ of G with the smooth boundary such that G ∩ U \ V ⊂ G′ . We will show that u ∈ W2l2 ,h (G′T ) for all G′ ∈ G. To this end, we will prove by induction on h that

utk ∈ W2l2 ,0 (G′T ) for all k = 0, . . . , h and for all G′ ∈ G.

(4.93)

According to the results on the regularity of solutions of elliptic boundary problems in smooth domains, we can conclude from (4.81), (4.82) that u(., t) ∈ W2l2 (G′ ) for each t ∈ [0, T ] and  ku(., t)kW l2 (G′ ) 6C ku(., t)kW l1 (G′′ ) + kf (., t)kW l2 −2m (G′′ ) 2

2

+

m X j=1

2

kgj (., t)k

1 l2 −µj − 2

W2

(S∩∂G′′ )



,

where G′′ ∈ G such that G′ ⊂ S ∪ G′′ and C is a constant independent of u, f , gj and t. Integrating this inequality with respect to t from 0 to T we get u ∈ W2l2 ,0 (G′T ). Thus (4.93) holds for h = 0. Assume that it holds for h − 1. Differentiating equalities (4.81), (4.82)

234

Nguyen Manh Hung and Nguyen Thanh Anh

with respect to t h times and using the inductive assumption, we have Luth Bj uth

h−1   X h L h−k utk ∈ W2l2 −2m (G′′T ), = fth − k t k=0 h−1   X h l2 −µj − 21 (Bj )th−k utk ∈ W2,β2 = (gj )th − (ST ∩ ∂G′′T ), k k=0

where G′ , G′′ ∈ G, G′ ⊂ S ∪ G′′ . Applying the arguments above for uth , we get uth ∈ W2l2 ,0 (G′T ). From (4.81) we have L(ηu) = ηf + [L, η]u in GT , (4.94) where [L, η] = Lη − ηL is the commutator of L and η. Noting that u ∈ W2l2 ,h (G′T ) for all G′ ∈ G and [L, η] is a differential expression (acting on u) of order 6 2m − 1 with l2 −2m,h coefficients having the supports contained in U \ V , we have [L, η]u is in W2,β (GT ). 2 So is the right-hand side of (4.94). Similarly, we have l2 −µj − 21

Bj (ηu) = ηgj + [Bj , η]u ∈ W2,β2

(ST ) (j = 1, . . . , m).

(4.95)

Applying the result above for the function ηu, we conclude from (4.94) and (4.95) that u admits the decomposition (4.83) in VT . The lemma is proved. Lemma 4.6. Suppose that the operators L(x, t, ∂x ) and Bj (x, t, ∂x ) are δ−admissible near l1 ,h the origin 0. Let u ∈ V2,β (GT ) be a solution of the problem 1 L(x, t, ∂x )u = f in GT ,

(4.96)

Bj (x, t, ∂x )u = gj on ST , j = 1, . . . , m,

(4.97)

l2 −µj − 1 ,h

l2 −2m,h 2 where f ∈ V2,β (GT ), gj ∈ V2,β2 (ST ), l1 , l2 > 2m, 0 < (l2 − β2 ) − (l1 − β1 ) 6 2 δ. Suppose further that the assumption (H) is fulfilled. Then there exists a neighborhood V of the origin of Rn such that in VT the solution u has representation

u(x, t) =

Λµ κµ,k −1 N X X X

µ=1 k=1

cµ,k,s (t)rλµ (t)

s=0

s X 1 (µ) (ln r)σ ϕk,s (ω, t) + w(x, t), σ! σ=0

(4.98)

l2 ,h where w ∈ V2,β (KT ), cµ,k,s (t) ∈ W2h ((0, T )). 2 l1 ,h 0,h Proof. We have u ∈ V2,β (GT ) ⊂ V2,β (GT ). Now we can apply Lemma 4.2 to assert 1 1 −l1

,h that u ∈ Vβl12−l (GT ). Thus, without loss of generality, we assume that l1 = l2 = l. 1 +l2 Rewrite (4.96), (4.97) in the form

Lu = f + (L − L)u ≡ f 1

Bj u = gj + (Bj − Bj )u ≡

in GT , gj1

on

ST , j = 1, . . . , m.

The Initial-Boundary Value Problems for Parabolic Equations...

235

Since the operators L, Bj , j = 1, . . . , m are δ-admissible, we have l−2m,h l−2m,h (L − L)u ∈ V2,β (GT ) ⊂ V2,β (GT ), 1 −δ 2 l−µj − 1 ,h

l−µj − 12 ,h

(Bj − Bj )u ∈ V2,β1 −δ 2 (ST ) ⊂ V2,β2 l−µj − 21 ,h

l−2m,h Thus f 1 ∈ V2,β (GT ), gj1 ∈ V2,β2 2 follows from Lemma 4.5.

(ST ).

(ST ), j = 1, . . . , m. Now formula (4.98)

Lemma 4.7. Let f =r

λ0 (t)−2m

s X 1 (ln r)σ fs−σ , σ!

σ=0

s X 1 λ0 (t)−µj gj = r (ln r)σ gj,s−σ , j = 1, . . . , m, σ! σ=0

l−µj − 1 ,h

2 where fσ ∈ W2l−2m,h (ΩT ), gj,σ ∈ W2 (∂ΩT ), σ = 0, . . . , s, j = 1, . . . , m. Suppose that if λ0 (t) is an eigenvalue of U (λ, t) for some t, then it is eigenvalue of U (λ, t) for all t ∈ [0, T ] with the geometric multiplicity and the partial multiplicities not depending on t ∈ [0, T ]. Then there exists a solution u of the problem (4.81), (4.82) which has the form

u=r

λ0 (t)

s+κ X 1 (ln r)σ us+κ−σ σ! σ=0

(4.99)

where uσ ∈ W2l,h (ΩT ), σ = 0, . . . , s + κ. Here κ = 0 if λ0 (t) is not an eigenvalue of U (λ, t), otherwise κ is the maximal partial multiplicity of λ0 (t). Proof. By Lemma 4.1, the inverse of U (λ, t) admits the representation U −1 (λ, t) =

+∞ X

k=−κ


k Pk (t) λ − λ0 (t) ,

where Pk (t) for k = −κ, . . . , −1 are defined in Lemma 4.1, and Pk (t) =

1 ∂kP (λ0 (t), t) n! ∂λk

for k = 0, 1, . . . . It is obvious that Pk (t), k = −κ, −κ + 1, . . . , are continuous operators from Y into X depending smoothly on t on [0, T ]. From the equality U (λ, t)U −1 (λ, t) =

+∞  κ+k  X 1 X U (q) (λ0 (t), t)Pk−q (t) (λ − λ0 (t))k = I q! q=0

k=−κ

it follows that κ+k X q=0

1 (q) U (λ0 (t), t)Pk−q (t) = δk,0 , k = −κ, −κ + 1, . . . , q!

(4.100)

236

Nguyen Manh Hung and Nguyen Thanh Anh

where δk,l is Kronecker symbol. Let u be the function given in (4.99). Then we have U (r∂r , t)u = rλ0 (t) U (λ0 (t) + r∂r , t)

s+κ X 1 (ln r)σ us+κ−σ σ!

σ=0

= rλ0 (t)

2m X

= rλ0 (t)

s+κ X

q=0

s+κ

X 1 1 (q) U (λ0 (t), t)(r∂r )q (ln r)σ us+κ−σ q! σ! σ=0

1 (ln r)σ σ! σ=0

s+κ−σ X q=0

1 (q) U (λ0 (t), t)us+κ−σ−q q!

Setting vσ = (fσ , g1,σ , . . . , gm,σ ), σ = 0, . . . , s, and min(k,s)

uk =

X

P−κ+k−p (t)vp , k = 1, . . . , s + κ,

p=0

we get uk ∈ W2l,h (ΩT ), k = 0, . . . , s + κ. Using the equality (4.100) we have U (r∂r , t)u =r

λ0 (t)

min(s+κ−σ−q,s) s+κ−σ s+κ X 1 X X 1 σ (q) (ln r) U (λ0 (t), t) Ps−σ−q−p (t)vp σ! q! q=0

σ=0

= rλ0 (t)

s+κ X 1 (ln r)σ σ!

σ=0

= rλ0 (t)

= rλ0 (t)

X

X

p=0

q=0

 s+κ−σ−p X

s s+κ−p X X

1 (ln r)σ σ!

s s+κ−p X X

1 (ln r)σ δs−σ−p,0 vp σ!

p=0 σ=0

= rλ0 (t)

p=0

min(s+κ−σ,s) s+κ−σ−p

p=0 σ=0 s X p=0

q=0

1 (q) U (λ0 (t), t)Ps−σ−p−q (t)vp q!

 1 (q) U (λ0 (t), t)Ps−σ−p−q (t) vp q!

s

X 1 1 (ln r)s−p vp = rλ0 (t) (ln r)σ vs−σ . (s − p)! σ! p=0

Rewrite this equality in the following form L (ω, t, ∂ω , r∂r )u = r2m f in GT , Bj (ω, t, ∂ω , r∂r )u = rµj gj on ST , j = 1, . . . , m. This implies u is a solution of the problem (4.81), (4.82), and the lemma is proved. Now let us give the main theorem of the present section. m (G), L (G)) be Theorem 4.8. Let h is a nonnegative integer and let u ∈ W21 ((0, T ); HB 2 the generalized solution of the problem (2.8)–(2.10). Let ǫ, β be real numbers such that 0 6 ǫ < 12 , 0 < β 6 m and ǫ + n2 , β + n2 are not integers in the case m > n2 . Suppose that

The Initial-Boundary Value Problems for Parabolic Equations...

237

the assumptions of Theorem 3.4 and the assumption (H) for l1 = l2 = 2m, β1 = m + ǫ, β2 = β are fulfilled. Suppose further that the operators L(x, t, ∂x ) and Bj (x, t, ∂x ) are (m + ǫ − β)- admissible near the origin 0. (i) For the case m < n2 , the solution u admits the decomposition u(x, t) =

Λµ κµ,k −1 N X X X

µ=1 k=1

cµ,k,s (t)rλµ (t)

s=0

s X 1 (µ) (ln r)σ ϕk,s (ω, t) + w(x, t), σ!

(4.101)

σ=0

2m,h where w ∈ V2,β (GT ), cµ,k,s ∈ W2h ((0, T )). (ii) For the case m > n2 , we assume further that if λk (t) = q is integer for some t ∈ [0, T ] and some k, 1 6 k 6 N, then λk (t) = q for all t ∈ [0, T ]. Then the solution u admits the decomposition

X

u(x, t) =

α

cα (t)x +

2m−ℓ X1 −1 k=2m−ℓ

|α|62m−ℓ−1 κµ,k −1

+

Λµ N X X X

µ=1 k=1

cµ,k,s (t)rλµ (t)

s=0

κk X 1 r (ln r)s ϕk,s (ω, t) s! s=0 k

s X 1 (µ) (ln r)σ ϕk,s (ω, t) + w(x, t), σ!

(4.102)

σ=0

2m,h where w ∈ V2,β (GT ), cα , cµ,k,s ∈ W2h ((0, T )), ℓ1 is the integer with ℓ1 − n2 < β < n ℓ1 + 1 − 2 , κk = 0 if k is not an eigenvalue of U (λ, t), otherwise κk is the maximal partial multiplicity of the eigenvalue k, and ϕk,s are functions from W22m,h (ΩT ). 2m,h Proof. According to Lemma 4.4, we have u ∈ W2,m+ǫ (GT ).

0,h (i) Suppose m < n2 . In this case it is known that u ∈ V2,−m (GT ) (see the proof of Lemma 4.4). Rewrite (2.8), (2.9) in the form

Lu = f1 := f − ut in

Bj u = 0

on

GT ,

(4.103)

ST , j = 1, . . . , m.

(4.104)

0,h 0,h (GT ). Hence, we can apply Lemma 4.6 for problem We have f1 ∈ V2,0 (GT ) ⊂ V2,β (4.103), (4.104) to obtain formula (4.128). (ii) Suppose m > n2 . Let ℓ = [m + n2 ] be the greatest integer not exceeding m + n2 . Then we have ℓ − n2 < m + ǫ < ℓ + 1 − n2 . It is similar as in the proof of Lemma 4.4, from 2m,h u ∈ W2,m+ǫ (GT ) there is the representation X cα (t)xα + v(x, t), (4.105) u(x, t) = |α|62m−ℓ−1

2m,h where v ∈ V2,m+ǫ (GT ), cα ∈ W2h ((0, T )), |α| 6 2m − ℓ − 1. Now rewrite (2.8), (2.9) in the form X Lv = f1 := f − ut − L cα (t)xα in GT ,

(4.106)

|α|62m−ℓ−1

Bj v = gj := −Bj

X

|α|62m−ℓ−1

cα (t)xα |ST

on

ST , j = 1, . . . , m.

(4.107)

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Nguyen Manh Hung and Nguyen Thanh Anh

0,h 0,h It is clear that (f1 )tk ∈ L2 (G This means f1 ∈ V2,0 (GT ) ⊂ V2,β (GT ). Write PT ) for k 6 h. α cα (t)x , j = 1, . . . , m, in the form the polynomials −Bj |α|62m−ℓ−1

X

pj,1 (x, t) + pj,2 (x, t) :=

pj,α (t)xα +

|α|62m−µj −ℓ−1

2m−ℓ−1 X

pj,α (t)xα ,

|α|=2m−µj −ℓ 2m−µ

where pj,α ∈ W h ((0, T )). From Lemma 4.3 it is known that pj,2 (., t) ∈ V2,m+ǫ j (G), 2m−µj − 21

and therefore, pj,2 (., t)|S ∈ V2,m+ǫ 2m−µj − 1 V2,m+ǫ 2 (S)

(S) for a.e. t ∈ [0, T ]. From this and that gj (., t) 2m−µj − 1

= Bj v(., t)|S ∈ it follows pj,1 (., t)|S ∈ V2,m+ǫ 2 (S). Consequently, pj,1 (., t)|S = 0 for a.e. t ∈ [0, T ] by [9, Le. 7.1.7]. Thus, gj = pj,2 |ST , j = 1, . . . , m. 2m−µj − 1

2m−µ

2 (S), if Since the monomials xα belong to V2,β j (G), and therefore, xα |S ∈ V2,β |α| > 2m − µj − ℓ1 (see Lemma 4.3), it is possible to write the functions gj , j = 1, . . . , m, in the form 2m−ℓ X1 −1 gj,k (ω, t)rk + gj′ (x, t), gj (x, t) = r−µj

k=2m−ℓ

2m−µj − 1 ,h

2 where gj,k ∈ W22m,h (ΓT ), gj′ ∈ V2,β (ST ). According to Lemma 4.7, there exists a functions κk 2m−ℓ X1 −1 X 1 rk (ln r)s ϕk,s (ω, t) u1 = s!

k=2m−ℓ

s=0

such that Lu1 = 0 Bj u1 = r−µj

2m−ℓ X1 −1

gj,k (ω, t)rk

in

GT ,

(4.108)

on

ST , j = 1, . . . , m.

(4.109)

k=2m−ℓ

Now we have L(v − u1 ) = f1

Bj (v − u1 ) =

gj′

in

GT ,

(4.110)

on

ST , j = 1, . . . , m.

(4.111)

Applying Lemma 4.6 we can conclude from (4.110), (4.111) that w := v − u1 ∈ 2m,h V2,ǫ (GT ). Consequently, we have the representation (4.102). The theorem is proved. If the boundary operators {Bj (x, t, ∂x )}m j=1 form a Dirichlet system, then the space ◦

m (G) coincides with the space W m (G), the closure of C ∞ (G) in W m (G). Here C ∞ (G) HB 2 0 2 0 is the set of all smooth functions with compact support in G. Moreover, it is known in this case that the trip m − n2 − 21 6 Re λ 6 m − n2 + 21 does not contains any eigenvalue of U (λ, t) (see [18, Th. 1]). Modify slightly the proof of Lemma 6.6.1 of [9] we can see ◦

m that W m 2 (G) is continuously imbedded into V2,0 (G). Thus, from the proof of the part (i) of Theorem 4.8, we get

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◦

Corollary 4.9. Let h is a nonnegative integer and let u ∈ W21 ((0, T ); W m 2 (G), L2 (G)) be the generalized solution of the problem (2.8)–(2.10) with the boundary operators {Bj (x, t, ∂x )}m j=1 being a Dirichlet system. Suppose that the assumptions of Theorem 3.4 and the assumption (H) for l1 = l2 = 2m, β1 = m, β2 = β (0 6 β 6 m) are fulfilled. Suppose further that the operators L(x, t, ∂x ) and Bj (x, t, ∂x ) are (m − β)− admissible near the origin 0. Then the solution u admits the decomposition u(x, t) =

Λµ κµ,k −1 N X X X

µ=1 k=1

cµ,k,s (t)r

λµ (t)

s=0

s X 1 (µ) (ln r)σ ϕk,s (ω, t) + w(x, t), σ! σ=0

(4.112)

2m,h where w ∈ V2,β (GT ), cµ,k,s ∈ W2h ((0, T )).

Example. We consider problem (2.8)-(2.10) in the case n = 2, m = 1 and (2.9) is Dirichlet condition, that means B1 u = u on ST . Assume that the domain G in a neighborhood U of the origin 0 coincides with the cone K = {x = (x1 , x2 ) ∈ R2 : r > 0, 0 < ω < ω0 }. Here r, ω are the polar coordinates of the point x = (x1 , x2 ) and 0 < ω0 < 2π, ω0 6= π. The problem (2.8)(2.10) is now restated as follows ut +Lu = f

in

GT ,

(4.113)

= 0,

on

ST , j = 1, . . . , m,

(4.114)

u|t=0 = φ

on

G,

(4.115)

u

where the operator L(x, t, ∂x ) has the form L = L(t, ∂x ) = −

2 X

ajk (x, t)∂xj ∂xk +

2 X

ak (x, t)∂xk + a0 (x, t).

k=1

j,k=1

We assume that the coefficients of the operator L(x, t, ∂x ) are real smooth functions on GT and ajk (x, t) = akj (x, t) for j, k = 1, 2. The leading part of L(x, t, ∂x ) at the point 0 is L = L(t, ∂x ) = −

2 X

ajk (0, t)∂xj ∂xk .

j,k=1

It follows from the elliptic property of the bilinear form associated to the operator L(x, t, ∂x ) that |ajk (0, t)| > ̺ for all t ∈ [0, T ], j, k = 1, 2, where is a positive constant independent of t. Thus, we can take advantage that a22 (0, t) ≡ 1 and the operator L is written in the form L(t, ∂x ) = −(∂x2 − a(t)∂x1 )(∂x2 − a(t)∂x1 ), where a(t) = (α(t) + iβ(t)), α(t), β(t) are real smooth functions defined on [0, T ], β(t) > ̺0 for all t ∈ [0, T ], ̺0 is a positive constant independent of t. Applying the formulas ∂x1 = r−1 (cos ω(r∂r ) − sin ω∂ω ), ∂x2 = r−1 (sin ω(r∂r ) + cos ω∂ω ),

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Nguyen Manh Hung and Nguyen Thanh Anh

we can write L(t, ∂x ) = r−2 L (t, ω, ∂ω , r∂r ), where L (t, ω, ∂ω , λ) =





 sin ω − a(t) cos ω λ + cos ω + a(t) sin ω ∂ω 

 sin ω − a(t) cos ω λ + cos ω + a(t) sin ω ∂ω .

(4.116)

The operator pencil U (λ, t) is now defined by   U (λ, t) = L (t, ω, ∂ω , λ), 1, 1 . Thus, a complex number λ is an eigenvalue of the operator pencil U (λ, t) if the following problem L (t, ω, ∂ω , λ)u = 0 in (0, α),

(4.117)

u|ω=0 = 0,

(4.118)

u|ω=α = 0

(4.119)

has a nontrivial solution. If λ = 0, w1 (ω, t) = 1, w2 (ω, t) = ω are two linearly independent solutions of the equation (4.117). We can verify directly that for each t ∈ [0, T ] there are no constants c1 (t), c2 (t) such that the function u = c1 (t)u1 + c2 (t)u2 satisfies the boundary conditions (4.118), (4.119). Hence, λ = 0 is not an eigenvalue of U (λ, t). For λ 6= 0, by calculating directly we can see that the functions v1 (ω, t) = eλϑ(ω,t) , v2 (ω, t) = eλϑ(ω,t) are solutions of the equation (4.117), where Z ω − sin ω + a(t) cos ω . ζ(θ, t)dθ, ζ(ω, t) = ϑ(ω, t) = cos ω + a(t) sin ω 0

(4.120)

(4.121)

The Wronskian of v1 (ω, t) and v2 (ω, t) at ω = 0 is 1 1 λa(t) λa(t) = λ(a(t) − a(t)) = −2λβ(t) 6= 0 for all t ∈ [0, T ].

Thus, for λ 6= 0, v1 (ω, t), v2 (ω, t) are two linearly independent solutions of the equation (4.117), and therefore, the general solution of its is u = c1 (t)v1 + c2 (t)v2 . Inserting this function into the boundary conditions (4.118), (4.119), we get the system ( c1 (t) + c2 (t) =0 (4.122) ,t) = 0. λϑ(ω ,t) λϑ(ω 0 0 c1 (t)e + c2 (t)e

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The coefficients determinant of this system is equal to D(λ, t) = eλϑ(ω0 ,t) − eλϑ(ω0 ,t) . A complex number λ is an eigenvalue of U (λ, t) if the system has nontrivial solutions c1 (t), c2 (t) or, in other words, λ is a solution of the following equation D(λ, t) = eλϑ(ω0 ,t) − eλϑ(ω0 ,t) = 0.

(4.123)

This equation is equivalent to e2iλ Im ϑ(ω0 ,t) = 1.

(4.124)

By calculating directly we can see Re ϑ(ω, t) = and

where

1 Zj2 (ω, t) + 1 |β(t)| + ln p ln 2 2 tan2 (ω) + 1 α (t) + β 2 (t)

 α(t) π   , ω ∈ (0, ], arctan Z(ω, t) − arctan   β(t) 2   π 3π α(t) + 2π, ω ∈ ( , ], Im ϑ(ω, t) = arctan Z(ω, t) − arctan  β(t) 2 2    α(t) 3π   + 4π, ω ∈ ( , 2π), arctan Z(ω, t) − arctan β(t) 2 Z(ω, t) =

α2 (t) + β 2 (t) α(t) tan ω + . β(t) β(t)

Here we set by convention tan

π π 3π π = +∞, arctan(+∞) = , tan = −∞, arctan(−∞) = − . 2 2 2 2

Note that Im ϑ(ω, t) > 0 for all ω ∈ (0, 2π) and all t ∈ [0, T ]. Therefore, the equation (4.124) has solutions λk (t) =

kπ , Im ϑ(ω0 , t)

k = ±1, ±2, . . . .

(4.125)

These are eigenvalues of U (λ, t). They are of course smooth functions on [0, T ]. For each t0 ∈ [0, T ] fixed, by lemmas 3.1.1 and 3.1.2 of [10], the algebraic multiplicity of an eigenvalue λ0 of U (λ, t0 ) coincides with the multiplicity of the zero λ = λ0 of the function D(λ, t0 ). Moreover, we have d D(λ, t)|λ=λk (t) = eλk (t)ϑ(ω0 ,t) ζ(ω0 , t) − eλk (t)ϑ(ω0 ,t) ζ(ω0 , t) dλ
= eλk (t)ϑ(ω0 ,t) ζ(ω0 , t) − ζ(ω0 , t) β(t) = −eλk (t)ϑ(ω0 ,t) 6= 0 (cos ω + a(t) sin ω)(cos ω + a(t) sin ω)

(4.126)

for all t ∈ [0, T ]. This means the zeros λk (t) of D(λ, t) have multiplicity one, and therefore, the eigenvalues λk (t)(k = ±1, ±2, . . .) of U (λ, t) are simple for all t ∈ [0, T ]. For each

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Nguyen Manh Hung and Nguyen Thanh Anh

eigenvalue λk (t), choosing c1 (t) = 1, c2 (t) = −1 in (4.122), we have the corresponding eigenvector is  1  λk (t)ϑ(ω,t) − eλk (t)ϑ(ω,t) e 2i k Re ϑ(ω,t)π k Im ϑ(ω, t)π , k = ±1, ±2, . . . . = e Im ϑ(ω0 ,t) sin Im ϑ(ω0 , t)

ϕk (ω, t) =

(4.127)

These are smooth functions on [0, ω0 ] × [0, T ]. Especially, if L(t, ∂x ) = −∆, then α(t) ≡ 0, β(t) ≡ 1, ϑ(ω, t) = ω, we have λk (t) = and ϕk (ω, t) = sin

kπω , ω0

kπ ω0 k = ±1, ±2, . . . .

Note that from the smoothness assumption on the coefficients of the operator L(x, t, ∂x ) it follows L(x, t, ∂x ) is 1− admissible. Now we can restate Corollary 4.9 for the problem (4.113)-(4.115) as follows ◦

Corollary 4.10. Let h is a nonnegative integer and let u ∈ W21 ((0, T ); W 12 (G), L2 (G)) 2h+1 (G), be the generalized solution of the problem (4.113)–(4.115). Suppose φ ∈ W2,loc

1 f ∈ W2h,h 2,loc (GT ) such that φk ∈ W2 (G), ftk ∈ L2 (GT ) for k = 0, . . . , h, and φk ∈ ◦

W 12 (G) for k = 1 . . . , h − 1 if h > 1. Let β be a real number, 0 6 β 6 1. Suppose that kπ Im ϑ(ω0 ,t) 6= 1 − β for all nonzero integer k. Denote by N the set of all positive integer k

such that

kπ Im ϑ(ω0 ,t)

u(x, t) =

< 1 − β for all t ∈ [0, T ]. Then the solution u admits the decomposition

X

kπ

ck (t)r Im ϑ(ω0 ,t) e

k Re ϑ(ω,t)π Im ϑ(ω0 ,t)

sin

k∈N 2,h where w ∈ V2,β (GT ), ck ∈ W2h ((0, T )).

Especially, if L(t, ∂x ) = −∆, and u(x, t) =

X

(1−β)ω0 π

0
k Im ϑ(ω, t)π + w(x, t), Im ϑ(ω0 , t)

is not an integer, then kπ

ck (t)r ω0 sin ω0 π

(4.128)

kπω + w(x, t), ω0

2,h where w ∈ V2,β (GT ), ck ∈ W2h ((0, T )), 0 < k < (1 − β) ωπ0 .

5.

Eigenvalues and Eigenvectors of the Operator Pencil U (λ, t)

In this section we use method of linearization ( [19]) to investigate the smoothness of the eigenvalues and the eigenvectors of U (λ, t) with respect to t. We will give a sufficient condition for assumption (H) to be fulfilled.

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Let M be a manifold in Rn and f (x, t) be a complex-valued function defined on MT = M × [0, T ]. We will say that f belongs to the class C ∞,a(MT ) if and only if f ∈ C a ([0, T ]; C l (M )) for all nonnegative integer l. We assume throughout this section that the coefficients of the operators L (ω, t, ∂ω , λ) and Bj (ω, t, ∂ω , λ), j = 1, . . . , m, are in C ∞,a(ΩT ) for all λ ∈ C. Now we deal with the operator pencil U (λ, t) introduced in section 4.. We can write U (λ, t) in the form U (λ, t) = A2m (t)λ2m + A2m−1 (t)λ2m−1 + · · · + A0 (t), where Ak (t), k = 2m, 2m − 1, . . . , 0 are differential operators in Ω of order 2m − k with coefficients belonging to C ∞,a(ΩT ), especially   X aα (0, t)ω α , 0, . . . , 0 . (5.129) A2m (t) = |α|=2m

Without loss of generality we can assume that the operator A0 (t) is invertible. Indeed, if λ0 is an eigenvalue of U (λ, t) for all t ∈ [0, T ] (for example, λ0 ∈ D with the set D defined in (4.54)), then 2m X ek (t)λk , U (λ0 + λ, t) = A k=0

e0 (t) = U (λ0 , t) is invertible for all t ∈ [0, T ]. Setting where A V (λ, t) = A−1 0 (t)U (λ, t),

Dk (t) = A−1 0 (t)Ak (t),

k = 1, . . . , 2m,

we have the pencils of continuous operators V (λ, t), Dk (t), k = 1, . . . , 2m, from X into itself, and V (λ, t) = D2m (t)λ2m + D2m−1 (t)λ2m−1 + · · · + D1 (t)λ + I,

(5.130)

where I is the identical operator in X. The eigenvalues and the eigenvectors of V (λ, t) are defined analogously as of U (λ, t). We can verify directly (or see [19, Le. 12.1]) that   V (λ, t)   I   I − λA (t) = C (t)E (λ, t)  (5.131)  F (λ, t), . .  .  I

where     −D1 · · · −D2m−1 −D2m I −D1 · · · −D2m−1  I    I     A = , C =   , .. ..     . . I I

244

Nguyen Manh Hung and Nguyen Thanh Anh   P2m P2m k−1 k−2 · · · D I 2m−1 + D2m λ k=1 Dk λ k=2 Dk λ   I   E = , . .   . I   I  −λI I   F = , . . . .   . . −λI I

(in operator matrices all the elements not indicated are assumed to be zero, and the argument t has been omitted for the sake of brevity) and I is the identical operator in X2m . Verifying directly we see that C (t), E (λ, t), F (λ, t) are invertible elements of B(X2m ) with

C

E −1

−1

   I D1 · · · D2m−1 I   −λI I    =  , F −1 =  ..    . I

I .. .

 ..

. −λI I

  , 

  P P2m k−1 − k−2 · · · −(D I − 2m 2m−1 + D2m λ) k=1 Dk λ k=2 Dk λ   I   = . ..   . I

It follows from the assumption on the analyticity of coefficients of differential operators L (ω, t, ∂ω , λ) and Bj (ω, t, ∂ω , λ), j = 1, . . . , m, that V (λ, t) is of the class C a([0, T ]; B(X)) and A (t) is of the class C a([0, T ]; B(X2m )). It is obvious that U (λ, t) and V (λ, t) have the same eigenvalues with the same multiplicities and the same corresponding eigenvectors. It follows from (5.131) that the spectra except the zero of the pencil V (λ, t) and the operator A (t) coincide for all t ∈ [0, T ]. We now show that for each t ∈ [0, T ] all eigenvalues of V (λ, t) and A (t) are nonzero. It is obvious for these of V (λ, t). Suppose ϕ = (ϕ(1) , . . . , ϕ(2m) ) ∈ X2m , ϕ 6= 0 such that A (t)ϕ = 0 for some t ∈ [0, T ]. Then ϕ(1) = · · · = ϕ(2m−1) = 0 and D2m (t)ϕ(2m) = 0. This implies A2m (t)ϕ(2m) = 0, but this do not occur since ker A2m (t) = {0} which follows from (5.129). Now we can apply [19, Le. 12.5, 12.8] to conclude that the complex number λ0 is an eigenvalue of the pencil V (λ, t) (for some t ∈ [0, T ]) if and only if σ0 = (λ0 )−1 is an eigenvalue of the operator A (t) with the same multiplicities. Hence for each t ∈ [0, T ] the spectrum of the operator A (t) is a bounded set consisting nonzero eigenvalues with finite multiplicities. Theorem 5.1. Let γ1 , γ2 be real numbers, γ1 < γ2 , such that the lines Re λ = γj , j = 1, 2, do not contain any eigenvalue of U (λ, t) and all eigenvalues of this pencil in the strip D1 := {λ ∈ C : γ1 < Re λ < γ2 }

(5.132)

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245

are simple for all t ∈ [0, T ]. Then there are complex-valued functions λk (t) and X-valued functions ϕk (t), k = 1, . . . , N, which are analytic on [0, T ] such that, for each t ∈ [0, T ], {λ1 (t), . . . , λN (t)} is the set of all eigenvalues of U (λ, t) in D1 and ϕk (t) are eigenvectors corresponding to the eigenvalues λk (t), k = 1, . . . , N , respectively. Proof. Since the set D defined in (4.54) does not contains eigenvalues of U (λ, t) for all t ∈ [0, T ], the eigenvalues of this pencil in the strip D actually are located in the bounded domain D2 = (C \ D) ∩ D1 . Moreover, the boundary ∂D2 of D2 contains no eigenvalues of U (λ, t) for all t ∈ [0, T ]. Let M be a positive number such that kA (t)k < M for all t ∈ [0, T ]. Put D0 = {σ ∈ C : (σ)−1 ∈ D2 , |σ| < M }. Then D0 is a connected bounded domain in C and for each t ∈ [0, T ] the spectrum of the operator A (t) consists a finite set of its simple eigenvalues and does not intersect with the boundary ∂D0 . Now let t0 ∈ [0, T ] and σ0 ∈ D0 be a simple eigenvalue of the operator A (t0 ). Then according to the results on analytic perturbation of linear operators (see [20, Th. XII.8]), there exists a complex-valued σ(t) defined and analytic on a subinterval containing t0 of [0, T ] such that σ(t) is a simple eigenvalue of A (t) for all t in such subinterval. We show now that σ(t) may be continued to be defined on [0, T ]. To see this, let I0 be the maximal interval of σ(t) considered and suppose that t1 is the right end of I0 and 0 < t1 < T . Since σ(t) does not go out of the domain D0 and the spectrum of A (t1 ) consists only a finite set of its eigenvalues, σ(t) must converge to an eigenvalue σ b0 ∈ D2 of A (t1 ) as t ↑ t1 (see [7, VII.3.5]). Thus, σ(t) must coincide with the analytic function σ b(t) representing eigenvalues of A (t) in a subinterval containing t1 , σ b(t1 ) = σ b0 . This implies that σ(t) admits an analytic continuation beyond t1 , contradicting the supposition that t1 is the right end of the maximal interval I0 of σ(t). Treating the other eigenvalues of A (t0 ) in D0 in the same way, we receive functions σ1 (t), . . . , σN (t) analytic on [0, T ] such that σk (t), k = 1, . . . , N, are simple eigenvalues of A (t) for all t ∈ [0, T ]. One can also choose X2m -valued functions ηk (t), k = 1, . . . , N, analytic on [0, T ] such that ηk (t) are eigenvectors corresponding to the eigenvalues σk (t) (see [20, Th. XII.8]). Set λk (t) = (δk (t))−1 , k = 1, . . . , N . Then these functions are analytic functions on [0, T ] and {λ1 (t), . . . , λN (t)} is the set of all eigenvalues of U (λ, t) in the strip D1 for each t ∈ [0, T ]. (1) (2m) Rewrite the function ηk (t) in the form of column vector (ηk (t), . . . , ηk (t)) (k = (1) 1, . . . , N ). Then X-valued function ϕk (t) = ηk (t) is analytic on [0, T ] and ϕk (t) is an eigenvector of U (λ, t) corresponding to eigenvalues λk (t) for each t ∈ [0, T ]. Remember that X = W2l (Ω), l is an arbitrary nonnegative integer. Thus, by Sobolev imbedding (1) theorem, we have ηk (t) ∈ C ∞,a(ΩT ). The proof is complete.

References [1] S. Agmon, The coerciveness problem for integro-differential forms. J. Analyse Math. 6 (1958), 183–223. [2] S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I. Comm. Pure Appl. Math. 12 (1959), 623–727.

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[3] M. S. Agranovich, Elliptic boundary problems. in: M. S. Agranovich. Yu. V. Egorov, M. A. Shubin (Eds.). Partial Differential Equations, IX, of Encyclopaedia Math. Sci. Springer, Berlin 79 (1997), pp. 1144. [4] M. Dauge,Elliptic boundary value problems on corner domains. Lecture Notes in Mathematics, Springer Verlag, Berlin, 1988. [5] L. C. Evans, Partial differential equations. Graduate Studies in Mathematics, Amer. Math. Soc., Providence, Rhode Island 19 (1998). [6] E. V. Frolova, An initial boundary-value problem with a noncoercive boundary condition in domains with edges, Zap. Nauchn. Semis. LOMI, 188 (1994), 206–223. [7] T. Kato, Perturbation theory for linear operators. Springer-Verlag, 1966. [8] V. A. Kondrat’ev, O. A. Oleinik, Boundary value problems for partial differential equations in nonsmooth domains, Usp. Math. Nauka 38 (1983), No. 3 (230), 3–76 (in Russian). [9] V. A. Kozlov, V. G. Maz’ya and J. Rossmann, Elliptic boundary problems in domains with point singularities. Mathematical Surveys and Monographs 52, Amer. Math. Soc., Providence, Rhode Island, 1997. [10] V. A. Kozlov, V. G. Maz’ya, J. Rossmann, Spectral problems associated with corner singularities of solutions to elliptic equations. Mathematical Surveys and Monographs 85, Amer. Math. Soc., Providence, Rhode Island, 2001. [11] N. M. Hung, Asymptotic behaviour of solutions of the first boundary value problem for strongly hyperbolic systems near a conical point at the boundary of the domain. Mat. Sb., 190 (1999), No. 7, 103–126. [12] N. M. Hung, P. T. Duong, On the smoothness of generalized solution for parabolic systems in domains with conic points on boundary. Ucrain. Math. J. 56 (2004), No. 6, 857–864. [13] N. M. Hung, N. T. Anh, Regularity of solutions of initial-boundary value problems for parabolic equations in domains with conical points. J. Differential Equations 245 (2008), 1801–1818. [14] W. P. Johnson, The Curious History of Fa`a di Bruno’s Formula. American Mathematical Monthly, 109 (2002), 217–234. [15] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type. Nauka, Moscow, 1967. [16] V. G. Maz’ya, B. A. Plamenevskii, Coefficients in the asymptotics of the solutions of an elliptic boundary value problem in a cone. J. Sov. Math. 9 (1978), 750–764. [17] V. G. Maz’ya, B. A. Plamenevskii, On the coefficients in the asymptotic of solutions of the elliptic boundary problem in domains with conical points. Amer. Math. Soc. Trans. (2) 123 (1984), 57–88.

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[18] V. A. Kozlov, V. G. Maz’ya, Spectral properties of the operator bundles generated by elliptic boundary value problems in a cone. Functional Anal. Appl., 22 (1988), 114– 121. [19] A. S. Markus, Introduction to the spectral theory of polynomial operator pencils. Translations of Mathematical Monographs, 71. American Mathematical Society, Providence, Rhode Island, 1988. [20] M. Reed, B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, London, 1978. [21] Solonnikov V. A., On the solvability of classical initial-boundary value problem for the heat equation in a dihedral angle. Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst., 127 (1983), 7–48.

In: Advances in Mathematics Research, Volume 10 ISBN 978-1-60876-265-1 c 2010 Nova Science Publishers, Inc. Editor: Albert R. Baswell, pp. 249-271

Chapter 5

T HE S ECOND B OUNDARY VALUE P ROBLEM FOR E QUATIONS OF V ISCOELASTIC D IFFUSION IN P OLYMERS Dmitry A. Vorotnikov ∗ Faculty of Mathematics, Voronezh State University, Universitetskaya Pl. 1, 394006, Voronezh, Russia

Abstract The classical approach to diffusion processes is based on Fick’s law that the flux is proportional to the concentration gradient. Various phenomena occurring during propagation of penetrating liquids in polymers show that this type of diffusion exhibits anomalous behavior and contradicts the just mentioned law. However, they can be explained in the framework of non-Fickian diffusion theories based on viscoelasticity of polymers (see e.g. D. A. Edwards and D. S. Cohen, AIChE J. 18 (1995), 2345–2355; T. P. Witelski, J. Polymer Science: Part B: Polymer Physics, 34 (1996), 141–150). Initial-boundary value problems for viscoelastic diffusion equations have been studied by several authors (see D. A. Vorotnikov, J. Differential Equations, 2009, Vol. 246, no. 3, 1038-1056). Most of the studies are devoted to the Dirichlet BVP (the concentration is given on the boundary of the domain). In this chapter we study the second BVP, i.e. when the normal component of the concentration flux is prescribed on the boundary, which is more realistic in many physical situations. We establish existence of weak solutions to this problem. We suggest some conditions on the coefficients and boundary data under which all the solutions tend to the homogeneous state as time goes to infinity.

Keywords: non-Fickian diffusion, polymer-penetrant systems, topological degree, weak solution, long-time behaviour. AMS Subject Classification: 35D05, 47H11, 76R50, 82D60. ∗

E-mail address: [email protected]

250

1.

Dmitry A. Vorotnikov

Introduction

The continuity equation for diffusion ∂u = −div J ∂t

(1.1)

states that variations of the concentration u(t, x) at any spatial point x and moment of time t can only be caused by inflow and outflow of a penetrant into and out of that area. Here J = J(t, x) is the concentration flux vector. The classical diffusion theory is based on Fick’s law (the flux is proportional to the concentration gradient with negative proportionality factor −D). The continuity equation and Fick’s law yield the classical diffusion equation ∂u = div (D(u)∇u), ∂t

(1.2)

∂u = D∆u ∂t

(1.3)

which becomes the heat equation

for constant diffusion coefficient D. The concentration behaviour for diffusion of penetrant liquids in polymers exhibits such phenomena as case II diffusion, sorption overshoot, literal skinning, trapping skinning and desorption overshoot, which contradict Fick’s law, see e.g. [5, 8, 9, 10, 20, 21, 23, 26]. There is a number of approaches which explain these non-Fickian properties of polymeric diffusion. They are usually based on taking into account the viscoelastic nature of polymers (cf. [15] and references therein) and on the possibility of glass-rubber phase transition (see e.g. [26] with some review). We are going to study the model which is due to Cohen et al. (see [4, 5, 11]; related models or particular cases of this one were suggested by other authors, see e.g. [7, 26]). It consists in combining the continuity equation with the following system J(t, x) = −D∇u − E∇σ + M u, (1.4) ∂σ (1.5) + βσ = µu + νu0 . ∂t Generally speaking, the coefficients β, D, E, M , µ and ν may depend u, or even on t, x and σ. Let us briefly discuss the meaning and typical behaviour of the coefficients. The scalar function β is the inverse of the relaxation time. A characteristic form of β is [5] u − u  1 1 RG (1.6) β(u) = (βR + βG ) + (βR − βG ) tanh 2 2 δ where βR , βG, δ, uRG are positive constants, βR > βG . The polymer-penetrant systems modeled with the help of (1.6) can be in two phases: glassy and rubbery. The glassy state corresponds to the areas of low concentration. Here the polymer network is severely entangled, and the relaxation time is high, so its inverse is low. Moreover, it is close to a certain value βG . In the high concentration areas the system is in the rubbery state: the network disentangles, so the relaxation time is small, and its inverse is close to βR > βG . The

The Second Boundary Value Problem for Equations...

251

glass-rubber phase transition occurs near a certain concentration uRG , and the value of δ determines the length of the transition segment. The coefficients D and E are non-negative scalars (more generally, they are positive-definite tensors) called the diffusion and stressdiffusion coefficients, respectively. As the concentration increases, the polymer network disentangles, so the diffusivity also increases. Thus, D should be an increasing function of concentration: in particular, D can depend on u in a way similar to (1.6) [10]. E is sometimes considered to be a constant, see [4] for some justification, but numerical simulations [6] have shown that, if E(0) 6= 0, then the concentration u may become negative, which is physically meaningless. Conversely, it can analytically be proved that, if E(0) = 0, then the concentration u remains non-negative provided it is non-negative at the initial moment of time [3]. In [24], we make related observations showing expediency of the condition E(1) = 0, which can maintain the concentration u of less than or equal to 100%. Thus, a modeling example is α1u(u − 1)2 , (1.7) E(u) = α2 + (u − 1)2 where α1 and α2 are positive constants, α2 is small. The functions µ and ν should be nonnegative and bounded [4], M is the convection velocity vector, assumptions on it will be given below. Let n(x) be the outward normal vector at the point x of the boundary ∂Ω of a domain Ω ⊂ Rn 1. Then system (1.1),(1.4),(1.5) may be completed with such boundary conditions as u(t, x) = φ(t, x), x ∈ ∂Ω (1.8) (the concentration on the boundary is prescribed) and −

n X

Ji (t, x)ni(x) = ϕ(t, x), x ∈ ∂Ω

(1.9)

i=1

(the influx 2 of the penetrant through the boundary is known). The initial-boundary value problems for system (1.1),(1.4),(1.5) possess maximal (not global in time) solutions for a more general boundary condition, which includes (1.8) and (1.9), see [3]. The global (in time) existence results are known for the Dirichlet condition (1.8). A theorem on global solvability is presented in [2] for f = µu, M ≡ 0 and D = E being a constant scalar. It is formulated for the one-dimensional case ( 0 < x < 1), but the technique used there seems to be applicable for x ∈ Ω, where Ω ⊂ Rn is a bounded domain with a smooth boundary. Another global existence result is given in [13]. They assume the stress-diffusion coefficient E to be a non-constant increasing function of concentration, E(0) = 0 (so E(1) = 0 is not allowed). However, it is required that the initial and boundary data for the concentration are bounded from below by a positive constant, so the solution is always strictly positive, and this approach does not permit to consider dry regions in a polymer. Paper [13] is mainly concerned with the one-dimensional case, but also they suggest a brief plan how to generalize the result for the multidimensional situation. Global existence of dissipative (ultra weak) solutions for constant scalar D and E and M ≡ 0 is 1 2

The most important particular cases are n = 2 (diffusion in polymer films) and n = 3. It can be negative.

252

Dmitry A. Vorotnikov

shown in [22] for Ω = Rn (again, the ideas used there seem to be suitable for Ω ⊂ Rn ). In [23], global (in time) weak solutions on a bounded domain Ω ⊂ Rn are constructed, under rather general assumptions on the coefficients. Further investigation of the weak solutions of the Dirichlet problem is carried out in [24]: it is proved that, for any sufficiently short time segment and any stress prescribed at the beginning of this segment, there exists a weak solution such that the concentrations at the beginning and at the end of the segment are the same, and, under an additional assumption on coefficients, existence of time-periodic weak solutions (without any restrictions of the period length) is shown. Paper [25] considers long-time behaviour issues for this problem: provided D and E are constant scalars and M ≡ 0, the solutions generate a dissipative semiflow, and there exist a minimal trajectory attractor and a global attractor. In this chapter, we construct (global in time) weak solutions for problem (1.1),(1.4),(1.5),(1.9) on a bounded domain Ω ⊂ Rn for given initial concentration and stress. The coefficients may depend on t, x, u and σ. In addition, we suggest some conditions on the coefficients and boundary data under which all the solutions tend to the homogeneous state u = const as time goes to infinity. The chapter is organized in the following way. In Section 2, we introduce the required notations. In Section 3, we give a weak formulation of the initial-boundary value problem and state the result on existence of weak solutions (Theorem 3.1), which is proved in Section 4. In Section 5, we touch the long-time behaviour.

2.

Notation

We use the standard notations Lp (Ω), Wpm (Ω), H m (Ω) = W2m (Ω) for Lebesgue and Sobolev spaces of functions defined on a bounded open set (domain) Ω ⊂ Rn , n ∈ N. The scalar product and the Euclidean norm in L2 (Ω)k = L2 (Ω, Rk ) are denoted by (u, v) and kuk, respectively (k is equal to 1 or n). In H m (Ω), m ∈ N, we use the scalar P α product (u, v)m = (D u, Dαv) and the corresponding Euclidean norm kukm . |α|≤m

The space of linear continuous functionals on H m (Ω) (the dual space) is denoted by −m −m (Ω) on an element from H m (Ω) is deHN (Ω). The value of a functional from HN noted by h·, ·i (the ”bra-ket” notation). Similarly, the dual space of Wpm (Ω) is denoted by −m Wq,N (Ω), 1p + 1q = 1, 1 < p < ∞, with the corresponding use of the ”bra-ket” notation. Sometimes we shall write simply Lp , H m for Lp (Ω)k , H m(Ω)k etc., k = 1, n. −1 Let us introduce some basic operators. The operator divN : Lq (Ω) → Wq,N (Ω) is determined by the formula Z 1 1 + = 1. (2.10) hdivN v, φi = − v(x)∇φ(x) dx, φ ∈ Wp1 (Ω), p q Ω −1 −2 The isomorphic operators A : H 1(Ω) → HN (Ω) and A2 : H 2(Ω) → HN (Ω) are given by the expressions

hAv, φi = (v, φ)1,

φ ∈ H 1(Ω),

(2.11)

hA2 v, φi = (v, φ)2,

φ ∈ H 2(Ω).

(2.12)

The Second Boundary Value Problem for Equations...

253

−1 −2 Note that L2 (Ω) ⊂ HN (Ω) ⊂ HN (Ω) with natural imbedding operators, and then Av = 1 v − divN ∇v, v ∈ H . Set XN = XN (Ω) = A−1 (H 1(Ω)). The scalar product and norm in XN are (u, v)X = −1 (Au, Av)1, kukX = kAuk1 . The duality between HN (Ω) and XN (Ω) is given by the formula −1 , v ∈ XN . (2.13) hu, vi1 = hu, Avi , u ∈ HN

Note that hu, vi1 = (u, v)1 for u ∈ H 1 , v ∈ XN . The elements of XN are solutions of the Neumann problem (2.14) v − ∆v = u ∈ H 1 (Ω), ∂v (x) = 0, x ∈ ∂Ω. ∂n

(2.15)

1 (Ω) XN (Ω) ⊂ H 2(Ω) ⊂ W2n/n−2

(2.16)

Thus, by Sobolev theorem (for sufficiently regular Ω). The symbols C(J; E), Cw (J; E), L2 (J; E) etc. denote the spaces of continuous, weakly continuous, quadratically integrable etc. functions on an interval J ⊂ R with values in a Banach space E. We recall that a function u : J → E is weakly continuous if for any linear continuous functional g on E the function g(u(·)) : J → R is continuous. If E is a function space (L2(Ω), H m(Ω) etc.), then we identify the elements of C(J; E), L2 (J; E) etc. with scalar functions defined on J × Ω according to the formula u(t)(x) = u(t, x), t ∈ J, x ∈ Ω. We shall also use the function spaces ( T is a positive number): −1 (Ω))} WN = WN (Ω, T ) = {τ ∈ L2 (0, T ; H 1(Ω)), τ 0 ∈ L2 (0, T ; HN

kτ kWN = kτ kL2 (0,T ;H 1(Ω)) + kτ 0 kL2 (0,T ;H −1(Ω)) ; N

−1 (Ω))} W1 = W1 (Ω, T ) = {τ ∈ L2 (0, T ; XN (Ω)), τ 0 ∈ L2 (0, T ; HN

kτ kW1 = kτ kL2(0,T ;XN (Ω)) + kτ 0kL2 (0,T ;H −1(Ω)); N

−2 W2 = W2 (Ω, T ) = {τ ∈ L2 (0, T ; H 2(Ω)), τ 0 ∈ L2 (0, T ; HN (Ω))}

kτ kW2 = kτ kL2 (0,T ;H 2(Ω)) + kτ 0kL2 (0,T ;H −2(Ω)) . N

Lemma III.1.2 from [19] implies continuous embeddings WN , W2 ⊂ C([0, T ]; L2(Ω)), W1 ⊂ C([0, T ]; H 1(Ω)) (see also [12]). We use the notation | · | for the absolute value of a number, for the Euclidean norm in n R , and in the following case. Denote by Rn×n the space of matrices of the order n × n with the norm |Q| =

max

ξ∈Rn,|ξ|=1

|Qξ|.

254

Dmitry A. Vorotnikov

Let Rn×n ⊂ Rn×n be the set of such matrices Q that + (Qξ, ξ)Rn ≥ d(Q)(ξ, ξ)Rn for some d(Q) ≥ 0 and all ξ ∈ Rn . The symbol C will stand for a generic positive constant that can take different values in different places.

3.

Weak Formulation of the Problem

We consider a polymer filling a sufficiently regular 3 bounded domain Ω ⊂ Rn , n ∈ N. We study the diffusion of a penetrant in this polymer which is described 4 by the following initial-boundary value problem 5: ∂u = div[D0(t, x, u, σ)∇u ∂t +E0 (t, x, u, σ)∇σ − M0 (t, x, u, σ)u], (t, x) ∈ [0, T ] × Ω, ∂σ ∂u + β0 (t, x, u, σ)σ = µ0 (u)u + ν0 (u) , (t, x) ∈ [0, T ] × Ω, ∂t ∂t n h X ∂u ∂σ + E0(t, x, u, σ)ij D0(t, x, u, σ)ij ∂xj ∂xj

(3.17) (3.18)

i,j=1

i −M0 (t, x, u, σ)iu ni (x) = ϕ(t, x), (t, x) ∈ [0, T ] × ∂Ω,

(3.19)

u(0, x) = u0 (x), σ(0, x) = σ0 (x), x ∈ Ω.

(3.20)

Here u = u(t, x) : [0, T ]×Ω → R is the unknown concentration of the penetrant (at the spatial point x at the moment of time t), σ = σ(t, x) : [0, T ]×Ω → R is the unknown stress, u0 = u0 (x), σ0 = σ0(x) : Ω → R are given initial data, ϕ : [0, T ] × ∂Ω → R is the influx of the liquid through the boundary, µ0 , ν0 : R → R, D0, E0 : Rn+3 = R × Rn × R × R → n+3 → R, M : Rn+3 → Rn are given functions, ν (·) ≥ 0. Rn×n 0 0 + , β0 : R Before giving a definition of a weak solution to this problem, it is convenient to make a change of variables. Denote

ς(t, x) = σ(t, x) −

u(t,x) Z

ν0 (y) dy,

0 3

Say, it is locally located on one side of its C 2 -smooth boundary. This problem makes sense for diffusion in polymers provided 0 ≤ u ≤ 1, i.e. the concentration is not less than 0% and does not exceed 100%. The assumptions on coefficients which guarantee this condition provided it is fulfilled at the initial moment of time are discussed in [24] for the Dirichlet problem, mainly based on the results of [3]; similar arguments are applicable in the Neumann case (1.9) with ϕ ≡ 0. However, this question is still not completely studied. So we consider here the general setting (3.1) – (3.4). 5 System (3.1), (3.2), (3.3) is obtained from (1.1),(1.4),(1.5),(1.9). For technical purposes, we assign subscript zero to the coefficients. 4

The Second Boundary Value Problem for Equations...

ς0 (x) = σ0 (x) −

255

uZ0 (x)

ν0 (y) dy,

0



D0 t, x, u, ς +

Zu



D(t, x, u, ς) = 

ν0 (y)dy  + ν0 (u)E0 t, x, u, ς +

0



0

E(t, x, u, ς) = E0 t, x, u, ς + 

Zu

β1 (t, x, u, ς) = −β0 t, x, u, ς +

γ (t, x, u, ς) = µ0 (u) −

ν0 (y)dy  ∈ Rn×n + , 

ν0 (y)dy  ,

Zu



ν0 (y)dy  ,

0



β0 t, x, u, ς +



0

f (t, x, u, ς) = −uM0 t, x, u, ς +



Zu

Zu

ν0 (y)dy  ,

0

Ru



ν0 (y)dy



0

Ru

ν0 (y) dy

0

. u Note that, if u vanishes, then, by continuity, we consider the last term to become −β0(t, x, 0, ς)ν0(0). Then we can rewrite (3.1) – (3.4) in the following form: ∂u = div[D(t, x, u, ς)∇u + E(t, x, u, ς)∇ς + f (t, x, u, ς)], ∂t ∂ς = β1 (t, x, u, ς)ς + γ(t, x, u, ς)u, ∂t n h X ∂u ∂ς + E(t, x, u, ς)ij D(t, x, u, ς)ij ∂xj ∂xj

(3.21) (3.22)

i,j=1

i +f (t, x, u, ς)i ni (x) = ϕ(t, x), (t, x) ∈ [0, T ] × ∂Ω,

(3.23)

u|t=0 = u0 , ς|t=0 = ς0 .

(3.24)

Now, before describing our assumptions on the coefficients, let us calculate the gradient of the right member of (3.6): ∇(β1(t, x, u, ς)ς) + ∇(γ(t, x, u, ς)u) = β1(t, x, u, ς)∇ς +

∂β1 ∂β1 ∂β1 (t, x, u, ς)ς + (t, x, u, ς)ς∇u + (t, x, u, ς)ς∇ς ∂x ∂u ∂ς

256

Dmitry A. Vorotnikov +γ(t, x, u, ς)∇u +

∂γ ∂γ ∂γ (t, x, u, ς)u + (t, x, u, ς)u∇u + (t, x, u, ς)u∇ς ∂x ∂u ∂ς

= β(t, x, u, ς)∇u + µ(t, x, u, ς)∇ς + g(t, x, u, ς),

(3.25)

where

∂γ ∂β1 (t, x, u, ς)ς + γ(t, x, u, ς) + (t, x, u, ς)u, (3.26) ∂u ∂u ∂β1 ∂γ µ(t, x, u, ς) = β1 (t, x, u, ς) + (t, x, u, ς)ς + (t, x, u, ς)u, (3.27) ∂ς ∂ς ∂γ ∂β1 (t, x, u, ς)ς + (t, x, u, ς)u. (3.28) g(t, x, u, ς) = ∂x ∂x We assume the following: i) D, E : Rn+3 → Rn×n ; f, g : Rn+3 → Rn ; µ, β, γ, β1 : Rn+3 → R. ii) Each of these eight functions (e.g. D(t, x, u, ς)) is measurable in (t, x) for fixed (u, ς). iii) Each of these functions is continuous in (u, ς) for fixed (t, x). iv) These functions satisfy the estimates β(t, x, u, ς) =

|D(t, x, u, ς)| ≤ KD ,

(3.29)

|E(t, x, u, ς)| ≤ KE ,

(3.30)

max(|β(t, x, u, ς)|, |γ(t, x, u, ς)|) ≤ Kβ ,

(3.31)

max(|µ(t, x, u, ς)|, |β1(t, x, u, ς)|) ≤ Kµ ,

(3.32)

|f (t, x, u, ς)| ≤ Kf (|u| + |ς|) + fe(t, x),

(3.33)

g(t, x) |g(t, x, u, ς)| ≤ Kg (|u| + |ς|) + e

(3.34)

with some constants KD , . . ., Kg and functions 6 fe, e g ∈ L2,loc (Rn+1 ). v) (D(t, x, u, ς)ξ, ξ)Rn ≥ d(ξ, ξ)Rn ,

(3.35)

where d > 0 is independent of (t, x, u, ς) ∈ Rn+3 and ξ ∈ Rn . vi) Relations (3.10) – (3.12) hold. It is easy to see that, if E0 and β0 are taken in the forms (1.7) and (1.6), then (3.14) and (3.15) are violated. It turns out that such deficiencies can be corrected without loss of generality of the model (see [24, Section 3] for a detailed discussion, cf. also [23, 25]). In brief, physically, the concentration u and the stress ς are uniformly bounded, therefore the coefficients of systems (3.1)–(3.2) (and, consequently, of (3.5)–(3.6)) can be experimentally determined only for bounded u and ς, whereas ”at infinity” we can choose them at discretion. Let us now rewrite (3.5) and (3.7) in a weak form. Assuming u and ς sufficiently regular, take the L2 (Ω)-scalar product of the members of (3.5) with a test function φ ∈ H 1(Ω), and integrate by parts in the right-hand side: (u0 , φ) = −(D(t, x, u, ς)∇u + E(t, x, u, ς)∇ς + f (t, x, u, ς), ∇φ) 6

Clearly, the behaviour of these functions outside (0, T ) × Ω does not matter.

The Second Boundary Value Problem for Equations... +

n Z h X

D(t, x, u, ς)ij

i,j=1∂Ω

257

i ∂u ∂ς + E(t, x, u, ς)ij + f (t, x, u, ς)i ni (x)φ(x) ds ∂xj ∂xj

= −(D(t, x, u, ς)∇u + E(t, x, u, ς)∇ς + f (t, x, u, ς), ∇φ) + Denote by ψ(t) the linear functional φ 7→

R

Z

ϕφ ds.

(3.36)

∂Ω

ϕ(t)φ ds. We assume that (for a.a. t) this

∂Ω −1 (Ω); clearly, this integral exists and continuously depends on φ ∈ H 1(Ω), so ψ(t) ∈ HN is true e.g. if ϕ(t) ∈ L2(∂Ω). Then we arrive at

∂u = divN [D(t, x, u, ς)∇u + E(t, x, u, ς)∇ς + f (t, x, u, ς)] + ψ, ∂t

(3.37)

−1 which should be understood as an equality of functionals from HN (Ω). Conversely, for each pair of sufficiently regular functions (u, ς), (3.21) implies (3.5) and (3.7).

Definition 3.1. A pair of functions (u, ς) from the class u ∈ WN (Ω, T ), ς ∈ H 1(0, T ; H 1(Ω))

(3.38)

is a weak solution to problem (3.5)-(3.8) if it satisfies (3.8), equality (3.21) holds in the −1 (Ω) a.e. on (0, T ), and (3.6) holds a.e. in (0, T ) × Ω. space HN Note that (3.8) makes sense due to the embeddings WN ⊂ C([0, T ]; L2(Ω)), H 1(0, T ; H 1(Ω)) ⊂ C([0, T ]; H 1(Ω)). −1 (Ω)), there exists Theorem 3.1. For every u0 ∈ L2 (Ω), ς0 ∈ H 1(Ω) and ψ ∈ L2(0, T ; HN a weak solution to problem (3.5)–(3.8) in class (3.22).

4.

Proof of the Existence Result

The proof of Theorem 3.1 is based on the study of the following auxiliary problem: ∂v + εA2 v = λdivN [D(t, x, v, τ )∇v + E(t, x, v, τ)∇τ + f (t, x, v, τ )] + λψ, (4.39) ∂t ∂τ + εA2 τ = λ[β1(t, x, v, τ )τ + γ(t, x, v, τ )v], ∂t v|t=0 = u0 , τ |t=0 = ς0 .

(4.40) (4.41) (4.42)

Here ε > 0, λ ∈ [0, 1] are parameters. We are going to derive some a priori estimates for the weak solutions of this problem. Then we shall show its solvability via topological degree arguments (the presence of the parameter λ is important at this stage). Finally, we shall put λ = 1 and pass to the limit as ε → 0.

258

Dmitry A. Vorotnikov

Definition 4.1. Given u0 ∈ L2 (Ω), ς0 ∈ H 1(Ω), a pair of functions (v, τ ) from the class v ∈ W2(Ω, T ), τ ∈ W1(Ω, T )

(4.43)

−2 (Ω) a.e. is a weak solution of problem (4.1)-(4.4) if equality (4.1) holds in the space HN −1 on (0, T ), (4.2) holds in the space HN (Ω) a.e. on (0, T ), (4.3) holds in L2 (Ω), and (4.4) holds in H 1(Ω).

The last two conditions make sense due to the embeddings W1 ⊂ C([0, T ]; H 1(Ω)), W2 ⊂ C([0, T ]; L2(Ω)). Lemma 4.1. Let (v, τ ) be a weak solution to problem (4.1)–(4.4). Then the following a priori estimate holds: εkvk2L2 (0,T ;H 2(Ω)) + εkτ k2L2 (0,T ;XN ) +kvk2L∞ (0,T ;L2(Ω)) + λkvk2L2(0,T ;H 1(Ω)) + kτ k2L∞ (0,T ;H 1(Ω)) ≤ C

(4.44)

where C is independent of λ and ε. −1 Proof. Take the ”bra-ket” of the terms of (4.2) (as elements of HN (Ω)) and Aτ (t) ∈ 1 H (Ω) at a.a. t ∈ [0, T ]:

0 

 τ , Aτ + εA2 τ, Aτ = λ (β1(t, x, v, τ )τ + γ(t, x, v, τ )v, Aτ ). (4.45)

Note that we can use parentheses instead of brackets in the right-hand side due to the equality hw1, w2i = (w1, w2), w1 ∈ L2 , w2 ∈ H 1. But



 1d τ 0, Aτ = τ 0 , τ 1 = kτ k21 2 dt (e.g. by [19, Lemma III.1.2]). Thus,

(4.46)

1d kτ k21 + ε(Aτ, Aτ )1 2 dt = λ (β1(t, x, v, τ )τ + γ(t, x, v, τ )v, τ ) + λ h∇[β1(t, x, v, τ )τ + γ(t, x, v, τ )v], ∇τ i = λ (β1 (t, x, v, τ )τ, τ) + λ (γ(t, x, v, τ )v, τ) +λ (β(t, x, v, τ )∇v + µ(t, x, v, τ )∇τ + g(t, x, v, τ ), ∇τ ).

(4.47)

Denote v¯(t) = e−kt v(t), τ¯(t) = e−kt τ (t), where k > 0 will be defined below. Then 1 d kt 2 ke τ¯k1 + e2kt ε(A¯ τ , A¯ τ )1 2 dt     = λ β1 (t, x, ektv¯(t), ektτ¯(t))¯ τ ekt , τ¯(t)ekt + λ γ(t, x, ektv¯(t), ektτ¯(t))¯ v ekt , τ¯(t)ekt  vekt + µ(t, x, ektv¯(t), ektτ¯(t))∇¯ τ ekt +λ β(t, x, ektv¯(t), ektτ¯(t))∇¯

The Second Boundary Value Problem for Equations...  +g(t, x, ektv¯(t), ektτ¯(t)), ∇¯ τ (t)ekt .

259 (4.48)

Denote now v(t), τ¯(t)) = β(t, x, ektv¯(t), ektτ¯(t)), βk (t, x, ¯ µk (t, x, ¯ v(t), τ¯(t)) = µ(t, x, ekt v¯(t), ektτ¯(t)), v(t), τ¯(t)) = e−kt g(t, x, ektv¯(t), ektτ¯(t)), gk (t, x, ¯ β1k (t, x, ¯ v(t), τ¯(t)) = β1 (t, x, ektv¯(t), ektτ¯(t)), v(t), τ¯(t)) = γ(t, x, ektv¯(t), ektτ¯(t)). γk (t, x, ¯ Thus, 1d k¯ τ k21 + kk¯ τ k21 + ε(¯ τ , τ¯)X 2 dt     = λ β1k (t, x, v¯(t), τ¯(t))¯ τ (t), τ¯(t) + λ γk (t, x, ¯ v(t), τ¯(t))¯ v(t), τ¯(t)  v(t), τ¯(t))∇¯ v + µk (t, x, v¯(t), τ¯(t))∇¯ τ +λ βk (t, x, ¯  v(t), τ¯(t)), ∇¯ τ (t) . +gk (t, x, ¯

(4.49)

Integration from 0 to t ∈ [0, T ] yields 1 k¯ τ (t)k21 + k 2

Zt

k¯ τ (s)k21 ds + ε

0

1 = kς0 k21 + λ 2 

Zt

k¯ τ (s)k2X ds

0

Zt 

   ¯ (s) + γk (s, x, ¯ v(s), τ¯(s))∇¯ v(s), ∇τ v(s), τ¯(s))¯ v(s), τ¯(s) βk (s, x, ¯

0

   ¯ (s) + β1k (s, x, ¯ + µk (s, x, ¯ v(s), τ¯(s))∇¯ τ (s), ∇τ v(s), τ¯(s))¯ τ (s), τ¯(s)   v(s), τ¯(s)), ∇¯ τ (s) ds. (4.50) + gk (s, x, ¯ 1 2 b , (3.15) Applying the Cauchy-Buniakowski inequality, Cauchy’s inequality ab ≤ ca2 + 4c and (3.16) we obtain

1 k¯ τ (t)k21 + k 2

Zt

k¯ τ (s)k21 ds

0

λKβ2 1 2 ≤ kς0 k1 + 2 4

Zt 0

+

λ 4

Zt 0

k¯ v (s)k21 ds + λ

+ε

Zt

k¯ τ (s)k2X ds

0

Zt

k¯ τ (s)k21 ds + λKµ

0

kgk (s, ·, ¯ v(s, ·), ¯ τ (s, ·))k2 ds + λ

Zt

k¯ τ (s)k21 ds

0

Zt 0

k¯ τ (s)k21 ds.

(4.51)

260

Dmitry A. Vorotnikov

Note that Zt

2

kgk (s, ·, ¯ v(s, ·), ¯ τ (s, ·))k ds ≤

0

Zt 0

≤

3Kg2

Zt

2

k¯ v(s, ·)k ds +

Zt

3Kg2

0

≤

3Kg2

kKg [|¯ v(s, ·)| + |¯ τ (s, ·)|] + e g(s, ·)k2 ds

2

k¯ τ (s, ·)k ds + 3

0

Zt

k¯ v(s)k21 ds

+

3Kg2

0

Zt

ke g(s, ·)k2 ds

0

Zt

k¯ τ (s)k21 ds + 3ke gk2L2 ((0,T )×Ω).

0

Hence, 1 3 k¯ τ (t)k21 + (k − 2 − Kµ − Kg2 ) 2 4

Zt

k¯ τ (s)k21 ds + ε

0 2

Kβ 3 2 1 + Kg ) ≤ kς0 k21 + λ( 2 4 4

Zt

Zt

k¯ τ (s)k2X ds

0

3 gk2L2 ((0,T )×Ω). k¯ v (s)k21 ds + ke 4

(4.52)

0 3 2 2 Kg .

Take k ≥ 4 + 2Kµ + In particular, (4.14) implies Zt

C k¯ τ (s)k21 ds ≤ (1 + λ k

0

Zt

k¯ v(s)k21 ds).

(4.53)

0

−2 (Ω)) and v(t) ∈ H 2(Ω) at a.a. Now, take the “bra-ket” of (4.1) (as elements of HN t ∈ [0, T ]:

0  v , v + hεA2 v, vi

= λ hdivN [D(t, x, v, τ )∇v + E(t, x, v, τ )∇τ + f (t, x, v, τ )] + ψ, vi .

(4.54)

Again, by [19, Lemma III.1.2],  1 d v 0, v = kvk2. 2 dt



(4.55)

Thus, 1d kvk2 + ε(v, v)2 2 dt = −λ(D(t, x, v, τ )∇v + E(t, x, v, τ )∇τ + f (t, x, v, τ ), ∇v) + λ hψ, vi . Then

1 d kt 2 v, v¯)2 ke v¯k + e2kt ε(¯ 2 dt

 = −λ D(t, x, ektv¯(t), ektτ¯(t))∇¯ vekt + E(t, x, ektv¯(t), ektτ¯(t))∇¯ τ ekt

(4.56)

The Second Boundary Value Problem for Equations...  D E +f (t, x, ektv¯(t), ektτ¯(t)), ∇¯ v(t)ekt + λ ψ(t), ¯v(t)ekt .

261 (4.57)

Denote now Dk (t, x, ¯ v(t), τ¯(t)) = D(t, x, ektv¯(t), ektτ¯(t)), v(t), τ¯(t)) = E(t, x, ektv¯(t), ektτ¯(t)), Ek (t, x, ¯ fk (t, x, v¯(t), τ¯(t)) = e−kt f (t, x, ektv¯(t), ektτ¯(t)). Thus, 1d vk2 + ε(¯ v , v¯)2 k¯ vk2 + kk¯ 2 dt

 = −λ Dk (t, x, ¯ v(t), τ¯(t))∇¯ v + Ek (t, x, ¯ v(t), τ¯(t))∇¯ τ  +fk (t, x, ¯ v(t), τ¯(t)), ∇¯ v(t) + e−kt λ hψ(t), ¯ v(t)i . Therefore 1 k¯ v(t)k2 + k 2

Zt

Zt

k¯ v(s)k2 ds + ε

0

1 = ku0 k2 − λ 2

(4.58)

k¯ v(s)k22 ds

0

Zt 

v(s), τ¯(s))∇¯ v(s) + Ek (s, x, ¯ v(s), τ¯(s))∇¯ τ (s) Dk (s, x, ¯

0

 +fk (s, x, ¯ v(s), τ¯(s)), ∇¯ v(s) − e−ks hψ(s), ¯ v(s)i ds.

(4.59)

Using Cauchy’s inequality, (3.14) and (3.19), we get 1 k¯ v(t)k2 + k 2

Zt

2

k¯ v(s)k ds + ε

0

Zt

k¯ v (s)k22 ds

+ λd

0

2 λKE 1 ≤ ku0k2 + 2 d

Zt

Zt

(∇¯ v(s), ∇¯ v(s)) ds

0

λd k∇¯ τ (s)k2 ds + 4

0

λ + d

Zt

Zt

k∇¯ v(s)k2 ds

0

λd kfk (s, ·, ¯v(s, ·), ¯ τ (s, ·))k ds + 4 2

0

Zt

k∇¯ v(s)k2 ds.

0

λ + d

Zt 0

kψ(s)k2−1 ds

λd + 4

Zt

k¯ v(s)k21 ds.

0

As for gk above, we have Zt 0

kfk (s, ·, ¯ v(s, ·), ¯ τ (s, ·))k2 ds

(4.60)

262

Dmitry A. Vorotnikov ≤

3Kf2

Zt

2

k¯ v(s)k ds +

3Kf2

Zt

0

k¯ τ (s)k21 ds + 3kfek2L2 ((0,T )×Ω).

0

Hence, from (4.22) and (4.15), 3Kf2 d 1 k¯ v (t)k2 + (k − − ) 2 d 4

Zt

2

k¯ v(s)k ds + ε

0

K2 3 1 ≤ ku0 k2 + ( E + Kf2) 2 d d

Zt

Zt

k¯ v(s)k22 ds

λd + 4

0

Zt

k∇¯ v (s)k2 ds

0

3 1 k¯ τ (s)k21 ds + kfek2L2 ((0,T )×Ω) + kψk2L (0,T ;H −1(Ω)) . 2 N d d

0

C0 (1 + λ ≤ k

Zt

k¯ v (s)k21 ds) + C

0

C0 (1 + λ = k

Zt

k¯ v(s)k2 ds + λ

0

Take k ≥

3Kf2 d

+

8C0 d

+

Zt

d 4

+

Zt

k∇¯ v (s)k2 ds) + C.

(4.61)

0 C0 k

+ 1. Then (4.23) yields

λd k¯ v (s)k2 ds + 8

0

Zt

k∇¯ v(s)k2 ds ≤ C

0

(now C may depend on k), so

λ

Zt

k¯ v (s)k21 ds ≤ C.

0

Thus, the right-hand members of inequalities (4.14) and (4.23) are bounded, and we arrive at τ k2L2 (0,T ;XN )+ εk¯ v k2L2 (0,T ;H 2(Ω)) + εk¯ vk2L2 (0,T ;H 1(Ω)) + k¯ τ k2L∞ (0,T ;H 1(Ω)) ≤ C. k¯ v k2L∞ (0,T ;L2(Ω)) + λk¯

(4.62)

Since ekt ≤ ekT for t ∈ [0, T ], this implies (4.6). Lemma 4.2. Let (v, τ ) be a weak solution to problem (4.1)–(4.4). Then there is the following bound of the time derivatives: kv 0kL2 (0,T ;H −2(Ω)) + kτ 0kL2 (0,T ;H −1(Ω)) ≤ C(1 + N

where C is independent of λ and ε.

N

√

ε)

(4.63)

The Second Boundary Value Problem for Equations...

263

−1 −2 Proof. Really, since HN (Ω) ⊂ HN (Ω) continuously, (4.1) and (4.6) imply

kv0 kL2 (0,T ;H −2 (Ω)) ≤ εkA2 vkL2 (0,T ;H −2 (Ω)) + N

N

λkdivN [D(t, x, v, τ )∇v + E(t, x, v, τ )∇τ + f(t, x, v, τ )]kL2(0,T ;H −1 (Ω)) + λkψkL2 (0,T ;H −1 (Ω)) N N √ √ ≤ ε εkvkL2 (0,T ;H 2 (Ω))+ λkD(t, x, v, τ )∇v + E(t, x, v, τ )∇τ + f(t, x, v, τ )kL2(0,T ;L2 (Ω)) + kψkL2 (0,T ;H −1 (Ω)) N √ ≤ C ε + KD λkvkL2 (0,T ;H 1 (Ω)) + KE kτ kL2 (0,T ;H 1 (Ω)) + kf(t, x, v, τ )kL2(0,T ;L2 (Ω)) + C √ √ ≤ C ε + KD λkvkL2 (0,T ;H 1 (Ω)) + KE kτ kL2(0,T ;H 1 (Ω)) e L ((0,T )×Ω) + C +Kf kvkL2 (0,T ;L2 (Ω)) + Kf kτ kL2(0,T ;L2 (Ω)) + kfk 2 √ √ ≤ C ε + C[ λkvkL2 (0,T ;H 1 (Ω)) + kτ kL∞(0,T ;H 1 (Ω)) √ +kvkL∞ (0,T ;L2 (Ω)) + kτ kL∞ (0,T ;H 1 (Ω)) + 1] ≤ C(1 + ε). −1 Similarly, since L2 (Ω) ⊂ HN (Ω) continuously, (4.2) and (4.6) yield

kτ 0kL2 (0,T ;H −1(Ω)) ≤ εkA2 τ kL2 (0,T ;H −1(Ω)) + N

N

λkβ1(t, x, v, τ )τ + γ(t, x, v, τ )vkL2(0,T ;L2(Ω)) √ √ ≤ ε εkτ kL2 (0,T ;XN )+ Kµ kτ kL2(0,T ;L2(Ω)) + Kβ kvkL2(0,T ;L2(Ω)) ≤ C(1 +

√ ε).

Lemma 4.3. Given u0 ∈ L2 (Ω), ς0 ∈ H 1(Ω), there exists a weak solution to problem (4.1)–(4.4) in class (4.5). Proof. Let us introduce auxiliary operators by the following formulas: −2 Q1 : W2 × W1 → L2(0, T ; HN (Ω)),

Q1 (v, τ ) = divN [D(·, ·, v, τ )∇v], −2 (Ω)), Q2 : W2 × W1 → L2(0, T ; HN

Q2 (v, τ ) = divN [E(·, ·, v, τ )∇τ ], −2 Q3 : W2 × W1 → L2(0, T ; HN (Ω)),

Q3 (v, τ ) = divN [f (·, ·, v, τ )] + ψ, −1 (Ω)), Q4 : W2 × W1 → L2(0, T ; HN

Q4(v, τ ) = γ(·, ·, v, τ )v, −1 Q5 : W2 × W1 → L2(0, T ; HN (Ω)),

Q5(v, τ ) = β1(·, ·, v, τ )τ,

264

Dmitry A. Vorotnikov

−2 −1 Q : W2 × W1 → L2 (0, T ; HN (Ω)) × L2 (0, T ; HN (Ω)) × L2 (Ω) × H 1(Ω),

Q(v, τ ) = (−Q1 (v, τ ) − Q2 (v, τ ) − Q3 (v, τ ), −Q4(v, τ ) − Q5(v, τ ), 0, 0), −1 (Ω)) × H 1(Ω), A˜1 : W1 → L2 (0, T ; HN

A˜1(u) = (u0 + εA2 u, u|t=0), −2 A˜2 : W2 → L2(0, T ; HN (Ω)) × L2 (Ω),

A˜2(u) = (u0 + εA2 u, u|t=0), −2 −1 (Ω)) × L2(0, T ; HN (Ω)) × L2 (Ω) × H 1 (Ω), A˜ : W2 × W1 → L2 (0, T ; HN

˜ τ ) = (v 0 + εA2 v, τ 0 + εA2 τ, v|t=0, τ |t=0). A(v, Then the weak statement of problem (4.1) - (4.4) is equivalent to the operator equation ˜ τ ) + λQ(v, τ ) = (0, 0, u0, ς0). A(v,

(4.64)

Let us briefly explain the idea of the proof. We are going to show that the operator A˜ is invertible. This yields the solvability of equation (4.26) for λ = 0. On the other hand, Q turns out to be a compact operator. Then we can rewrite (4.26) in a form suitable for application of the Leray-Schauder degree theory, which will imply the existence of solutions for all λ ∈ [0, 1]. We recall that a non-linear operator K : X1 → X2 (X1 and X2 are Banach spaces) is called compact if it is continuous and the image of any bounded set in X1 is relatively compact in X2. In particular, if X1 is reflexive, and, for any sequence xm → x∗ which converges in X1 in the weak sense, one has K(xm) → K(x∗ ) strongly in X2 , then K is compact (since any bounded subset of X1 is relatively compact in the weak topology). For some q > 2, the embeddings W1 ⊂ Lq (0, T ; W q1(Ω)), W2 ⊂ Lq (0, T ; W q1(Ω)) are compact. Really, we have W1 ⊂ C([0, T ]; H 1(Ω)), W2 ⊂ C([0, T ]; L2(Ω)) continuously. Note that (by the Rellich-Kondrashov theorem) H 2 ⊂ L2 compactly. Furthermore, H 1 ⊂ −1 HN compactly, so the adjoint embedding XN ⊂ H 1 is also compact. Then, by [17, Corollary 6], W1 ⊂ Lp (0, T ; H 1(Ω)), W2 ⊂ Lp (0, T ; L2(Ω)) compactly for every p < ∞. 2n 2n and p0 = n−2 , cf. (2.7). Then p21 = 12 + p10 . For u ∈ XN , we have Let p1 = n−1 kuk2W 1 ≤ C(kuk2Lp + k∇uk2Lp ). p1

1

1

The second term is k|∇u|2kLp1 /2 ≤ k∇ukL2 k∇ukLp0 ≤ Ckuk1 kukX . The first term can be estimated similarly. If q1 > 2 is such that q21 = 12 + 1p with some p large enough, then, by [17, Lemma 11], W1 ⊂ Lq1 (0, T ; Wp11 ) compactly. Now, for u ∈ H 2, we have kuk4Wp1 ≤ C(kuk4Lp1 + k∇uk4Lp1 ), 1

The Second Boundary Value Problem for Equations...

265

and the second term is k|∇u|2k2Lp

1 /2

≤ k∇uk2L2 k∇uk2Lp0

≤ Ckuk21 kuk22 ≤ Ckukkuk32. The last inequality follows from [1, Theorem 4.17]. The first term can be estimated in the same way. If q2 > 2 is such that q42 = 32 + 1p with some p large enough, then, by [17, Lemma 11], W2 ⊂ Lq2 (0, T ; Wp11 ) compactly. Let us show that the operators Q1, . . . , Q5 are compact. Let vm → v∗ weakly in W2 , τm → τ∗ weakly in W1. Then vm → v∗ , τm → τ∗ strongly in Lq (0, T ; Wq1(Ω)) and in Lq (0, T ; Lq(Ω)), and ∇vm → ∇v∗ , ∇τm → ∇τ∗ strongly in Lq (0, T ; Lq(Ω)n). By Krasnoselskii’s theorem [14, 18] on continuity of Nemytskii operators we have D(·, ·, vm, τm) → D(·, ·, v∗, τ∗), E(·, ·, vm, τm) → E(·, ·, v∗, τ∗), n×n

strongly in Lp ((0, T ) × Ω)

,

β1(·, ·, vm, τm) → β1(·, ·, v∗, τ∗), γ(·, ·, vm, τm) → γ(·, ·, v∗, τ∗), strongly in Lp ((0, T ) × Ω) for all p < ∞, and f (·, ·, vm, τm) → f (·, ·, v∗, τ∗), strongly in L2 ((0, T ) × Ω)n . Clearly, if a sequence of functions ym converges in Lq ((0, T ) × Ω), and another sequence zm converges in Lp((0, T ) × Ω), 1p + 1q = 12 , then their pointwise products ym zm tend to the product of their limits in L2 ((0, T ) × Ω). Hence, D(·, ·, vm, τm)∇vm → D(·, ·, v∗, τ∗)∇v∗ in L2 (0, T ; L2(Ω)n ). Therefore Q1(vm , τm ) → Q1 (v∗, τ∗) in L2(0, T ; H −1(Ω)) (and all the more in L2 (0, T ; H −2(Ω))). Similarly, Qi (vm , τm) → Qi (v∗ , τ∗), i = 2, . . . , 5, in L2(0, T ; H −1(Ω)). Hence, the operator Q is also compact. Note that hA2 u, ui = kuk22, for u ∈ H 2(Ω), and 



2 A u, u 1 = A2u, Au = (Au, Au)1 = kuk2X for u ∈ X. Therefore the operators A˜1 and A˜2 are invertible (e.g. by Theorem 1.1 from [12], Chapter VI, or Lemma 3.1.3 from [27]). Hence, A˜ is also (continuously) invertible.

266

Dmitry A. Vorotnikov Rewrite equation (4.26) as (u, τ ) + λA˜−1 Q(u, τ ) = A˜−1 (0, 0, u0, ς0 ).

(4.65)

A priori bounds from Lemmas 4.1 and 4.2 imply that equation (4.27) has no solutions on the boundary of a sufficiently large ball B in W2 × W1 , independent of λ. Without loss of generality a0 = A˜−1 (0, 0, u0, ς0 ) belongs to this ball. Then we can consider the Leray ˜−1 Q (I is the identity map) on the ball Schauder degree (see e.g. [16]) of the map I + λA B with respect to the point a0 , ˜−1 Q, B, a0). degLS (I + λA By the homotopic invariance property of the degree we have degLS (I + λA˜−1 Q, B, a0) = degLS (I, B, a0) = 1 6= 0. Thus, equation (4.27) (and, therefore, problem (4.1) - (4.4)) has a solution in the ball B for every λ. Proof of Theorem 3.1. Take a decreasing sequence of positive numbers εm → 0. By Lemma 4.3, there is a pair (vm , τm) which is a weak solution to problem (4.1)-(4.4) with λ = 1, ε = εm . Due to a priori estimate (4.6), without loss of generality (passing to a subsequence if necessary) one may assume that there exist limits u = lim vm , which is ∗-weak in L∞ (0, T ; L2(Ω)) and weak in L2 (0, T ; H 1(Ω)); m→∞

ς = lim τm , which is ∗-weak in L∞ (0, T ; H 1(Ω)) and weak in L2 (0, T ; H 1(Ω)). m→∞

0 → u0 Moreover, due to Lemma 4.2, without loss of generality one may assume that vm −2 0 → ς 0 weakly in L (0, T ; H −1). Then, by [17, Corollary 4], weakly in L2 (0, T ; HN ), τm 2 N −1 ). Therefore u and ς satisfy (3.8). vm → u, τm → ς strongly in C([0, T ]; HN Furthermore, by [17, Corollary 4], vm → u, τm → ς strongly in L2 (0, T ; L2). By Krasnoselskii’s theorem [14, 18] we have again

D(·, ·, vm, τm ) → D(·, ·, u, ς), E(·, ·, vm, τm ) → E(·, ·, u, ς), strongly in Lp ((0, T ) × Ω)n×n , β1 (·, ·, vm, τm ) → β1 (·, ·, u, ς), γ(·, ·, vm, τm ) → γ(·, ·, u, ς), strongly in Lp ((0, T ) × Ω) for all p < ∞, and f (·, ·, vm, τm ) → f (·, ·, u, ς), strongly in L2 ((0, T ) × Ω)n . Observe that if a sequence of functions ym converges weakly in L2((0, T ) × Ω), and another sequence zm converges strongly in Lp ((0, T ) × Ω), then their pointwise products ym zm converge weakly to the product of their limits in Lq ((0, T ) × Ω), 1p + 12 = 1q .

The Second Boundary Value Problem for Equations...

267

Therefore, D(·, ·, vm, τm)∇vm → D(·, ·, u, ς)∇u, E(·, ·, vm, τm)∇τm → E(·, ·, u, ς)∇ς, n

weakly in Lq (0, T ; Lq(Ω) ), γ(·, ·, vm, τm)vm → β(·, ·, u, ς)u, β1(·, ·, vm, τm)τm → β1(·, ·, u, ς)ς weakly in Lq (0, T ; Lq(Ω)) for 1 ≤ q < 2. Therefore the right-hand members of (4.1) −1 (Ω)). converge to the corresponding right-hand members of (3.21) weakly in Lq (0, T ; Wq,N √ √ Due to Lemma 4.1, εm kvm kL2 (0,T ;H 2(Ω)) = εm εm kvm kL2 (0,T ;H 2(Ω)) → 0. Hence, εm kA2 vm kL2 (0,T ;H −2(Ω)) → 0. Similarly, εm kτm kL2 (0,T ;XN ) tends to zero, so N

εm kA2 τm kL2 (0,T ;H −1) → 0. N

2n . Then, by Sobolev theorem, W.l.o.g. we may assume, in addition, that q ≥ n+2 −2 2 1 −1 H (Ω) ⊂ Wq/q−1 (Ω), so Wq (Ω) ⊂ HN (Ω). Passing to the limit as m → ∞ in (4.1) and (4.2) with λ = 1, ε = εm , v = vm , τ = τm in the space of distributions on (0, T ) with −2 −1 values in HN (Ω) (for (4.2) it is possible for HN as well), we conclude that the pair (u, ς) is a solution to (3.5)–(3.8). It remains to observe that the right-hand side (and, hence, the left-hand side) of (3.21) −1 belongs to L2(0, T ; HN ), and, due to (3.9), the ones of (3.6) belong to L2(0, T ; H 1).

5.

Long-Time Behaviour

Theorem 3.1 implies that solutions to (3.5)–(3.8) can be continued, step by step, onto the whole positive semi-axis: −1 Corollary 5.1. Given u0 ∈ L2(Ω), ς0 ∈ H 1(Ω) and ψ ∈ L2,loc(0, ∞; HN (Ω)), there is a pair \ −1 1 1 (0, ∞; HN (Ω)), ς ∈ Hloc (0, ∞; H 1(Ω)) (5.66) u ∈ L2,loc (0, ∞; H 1(Ω)) Hloc −1 (Ω) a.e. on (0, ∞), and (3.6) holds which satisfies (3.8), whereas (3.21) holds true in HN a.e. in (0, ∞) × Ω.

Below we keep assuming conditions i)-vi) of Section 3, but we replace (3.17)–(3.19) with stronger requirements, namely vii) |f (t, x, v, τ )| ≤ fe(t, x), |g(t, x, v, τ )| ≤ e g(t, x) (5.67) with some known functions fe, e g ∈ L2 ((0, ∞) × Ω), and viii) there are7 positive numbers Γ and Γ0 such that    β(·) n n ξ, η ≥ Γ0 (|ξ|2 + |η|2) (D(·)ξ, ξ)R − (µ(·)η, η)R + E(·)Γ − Γ Rn 7

See [24, Section 5] for a discussion whether this assumption is realistic.

(5.68)

268

Dmitry A. Vorotnikov

for any ξ, η ∈ Rn . Consider any global weak solution (u, ς) existing by Corollary 5.1. Denote Ψ = A−1 ψ. Then Ψ ∈ L2,loc (0, ∞; H 1(Ω)). Assume, in addition, that Ψ ∈ L2 (0, ∞; H 1(Ω)) ∩ L1 (0, ∞; L2(Ω)). Lemma 5.1. The following estimate is valid: kukL∞ (0,∞;L2(Ω)) + k∇ukL2 (0,∞;L2 (Ω)) +k∇ςkL∞ (0,∞;L2 (Ω)) + k∇ςkL2 (0,∞;L2(Ω)) ≤ C.

(5.69)

Proof. The condition (5.3) can be rewritten as (D(·)Γ2ξ, ξ)Rn − (µ(·)η, η)Rn + E(·)Γ2ξ, η




Rn

− (β(·)ξ, η)Rn

≥ Γ0 (Γ2 |ξ|2 + |η|2)

(5.70)

for any ξ, η ∈ Rn (just substitute Γξ for ξ in (5.3)). −1 (Ω) and the terms of (3.6) (as elements of Take the ”bra-ket” of −divN ∇ς(t) ∈ HN 1 H (Ω)) at a.a. t ∈ [0, T ]:
∇ς 0 , ∇ς = (∇(β1(t, x, u, ς)ς + γ(t, x, u, ς)u), ∇ς). (5.71) Thus, 1d k∇ςk2 = (β(t, x, u, ς)∇u + µ(t, x, u, ς)∇ς + g(t, x, u, ς), ∇ς). 2 dt

(5.72)

−1 (Ω)) and u(t) ∈ H 1(Ω) at a.a. t ∈ Take the ”bra-ket” of (3.21) (as elements of HN [0, T ], arriving at (cf. the proof of Lemma 4.1)

1d kuk2 = −(D(t, x, u, ς)∇u + E(t, x, u, ς)∇ς + f (t, x, u, ς), ∇u) + hψ, ui . (5.73) 2 dt Multiply it by Γ2 and add this with (5.7): 1d Γ2 d kuk2 + k∇ςk2 2 dt 2 dt = −(D(t, x, u, ς)Γ2∇u + E(t, x, u, ς)Γ2∇ς, ∇u)   + β(t, x, u, ς)∇u + µ(t, x, u, ς)∇ς, ∇ς   + g(t, x, u, ς), ∇ς − Γ2 (f (t, x, u, ς), ∇u) + Γ2 hψ, ui.

(5.74)

Using (5.5), we conclude that 1d Γ2 d kuk2 + k∇ςk2 + Γ0 (Γ2k∇uk2 + k∇ςk2) 2 dt 2 dt   ≤ g(t, x, u, ς), ∇ς − Γ2 (f (t, x, u, ς), ∇u) + Γ2 (Ψ, u)1.

(5.75)

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269

Integrating along the interval (0, t), t > 0, we get 1 Γ2 ku(t)k2 + k∇ς(t)k2 + Γ0 Γ2 2 2

Zt

2

k∇u(s)k ds + Γ0

0

1 Γ2 ≤ ku0k2 + k∇ς0 k2 + 2 2

Zt

k∇ς(s)k2 ds

0

Zt 

 g(s, x, u(s), ς(s)), ∇ς(s) ds

0

+

Zt

Γ2 (Ψ(s), u(s)) ds

0

+

Zt

h

 i Γ2 (∇Ψ(s), ∇u(s)) − Γ2 f (s, x, u(s), ς(s)), ∇u(s) ds.

(5.76)

0

Applying the Cauchy-Buniakowski inequality, Cauchy’s inequality and (5.2), we observe that Zt   ∇Ψ(s) + f (s, x, u(s), ς(s)), ∇u(s) ds 0

≤ [k∇ΨkL2((0,∞)×Ω) + kfekL2 ((0,∞)×Ω) ]k∇ukL2((0,t)×Ω) 1 Γ0 [k∇ΨkL2(0,∞;L2 (Ω)) + kfekL2 ((0,∞)×Ω)]2 + k∇uk2L2 (0,t;L2(Ω)) . 2Γ0 2 Similarly, ≤



(5.77)

Zt 

 Γ0 1 k∇ςk2L2(0,t;L2(Ω)) . ke gk2L2 ((0,∞)×Ω) + g(s, x, u(s), ς(s)), ∇ς(s) ds ≤ 2Γ0 2

0

(5.78) And, obviously,

Zt

Γ2 (Ψ(s), u(s)) ds ≤ Γ2 kΨkL1 (0,∞;L2 (Ω))kukL∞ (0,t;L2(Ω)) .

(5.79)

0

Inequalities (5.11)–(5.14) yield Γ2 1 Γ0 Γ2 ku(t)k2 + k∇ς(t)k2 + 2 2 2

Zt

Γ0 k∇u(s)k ds + 2 2

0

Zt

k∇ς(s)k2 ds

0

≤ C + CkukL∞ (0,t;L2(Ω)) ,

(5.80)

where C is independent of t, thus we have the same inequality between the essential supremums of both members on (0, t), in particular, Γ2 kuk2L∞ (0,t;L2(Ω)) ≤ C + CkukL∞ (0,t;L2(Ω)), 2

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Dmitry A. Vorotnikov

so kukL∞ (0,t;L2(Ω)) ≤ C, and (5.4) follows from (5.15). Estimate (5.4) means, in particular, that in a certain sense the concentration u(t) tends to a constant as t → ∞.

References [1] R. A. Adams, Sobolev spaces. Acad. Press, New York-San Francisco-London, 1975. [2] H. Amann, Global existence for a class of highly degenerate parabolic systems. Japan J. Indust. Appl. Math. 8 (1991), 143–151. [3] H. Amann, Highly degenerate quasilinear parabolic systems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 18 (1991), 135–166. [4] D. S. Cohen and A. B. White, Jr., Sharp fronts due to diffusion and viscoelastic relaxation in polymers. SIAM J. Appl. Math. 51 (1991), No. 2, 472–483. [5] D. S. Cohen, A. B. White, Jr., and T.P. Witelski, Shock formation in a multidimensional viscoelastic diffusive system. SIAM J. Appl. Math. 55 (1995), No. 2, 348–368. [6] R. W. Cox, A Model for Stress-Driven Diffusion in Polymers. Ph.D. thesis, California Institute of Technology, (1988). [7] C. J. Durning, Differential sorption in viscoelastic fluids. J. Polymer Sci., Polymer Phys. Ed. 23 (1985), 1831–1855. [8] D. A. Edwards, A mathematical model for trapping skinning in polymers. Studies in Applied Mathematics 99 (1997), 49–80. [9] D. A. Edwards, A spatially nonlocal model for polymer desorption. Journal of Engineering Mathematics 53 (2005), 221–238. [10] D. A. Edwards and R. A. Cairncross, Desorption overshoot in polymer-penetrant systems: Asymptotic and computational results. SIAM J. Appl. Math. 63 (2002), 98–115. [11] D. A. Edwards and D. S. Cohen, A mathematical model for a dissolving polymer. AIChE J. 18 (1995), 2345–2355. [12] H. Gajewski, K. Groeger, K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Akademie-Verlag, Berlin, 1974. [13] Hu, Bei and Zhang, Jianhua. Global existence for a class of non-Fickian polymerpenetrant systems. J. Partial Diff. Eqs. 9 (1996), 193–208. [14] M. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations. Gostehizdat, 1956 (Russian); Engl. transl.: Macmillan, 1964.

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[15] Sang-Wha Lee, Relaxation Characteristics of Poly(vinylidene fluoride) and Ethylenechlorotrifluoroethylene in the Transient Uptake of Aromatic Solvents. Korean J. Chem. Eng. 21(6) (2004), 1119–1125. [16] N. G. Lloyd, Degree Theory. Cambridge University Press, 1978. [17] J. Simon. Compact sets in the space Lp (0, T ; B). Ann. Mat. Pura Appl. 146 (1987), 65–96. [18] I. V. Skrypnik, Methods for analysis of nonlinear elliptic boundary value problems. Translations of Mathematical Monographs, Amer. Math. Soc. 139 (1994). [19] R. Temam, Navier-Stokes equations. Theory and numerical analysis. Studies in Mathematics and its Applications 2. North-Holland Publishing Co., Amsterdam-New YorkOxford, 1977. [20] N. Thomas and A. H. Windle, Transport of methanol in poly-(methyl-methocry-late). Polymer 19 (1978), 255–265. [21] N. Thomas and A. H. Windle, A theory of Case II diffusion. Polymer 23 (1982), 529– 542. [22] D. A. Vorotnikov, Dissipative solutions for equations of viscoelastic diffusion in polymers. J. Math. Anal. Appl. 339 (2008), 876–888. [23] D. A. Vorotnikov, Weak solvability for equations of viscoelastic diffusion in polymers with variable coefficients. J. Differential Equations 246 (2009), No. 3, 1038–1056. [24] D. A. Vorotnikov. On iterating concentration and periodic regimes at the anomalous diffusion in polymers. Mat. Sbornik (submitted). [25] D. A. Vorotnikov, Anomalous diffusion in polymers: long-time behaviour (submitted). [26] T. P. Witelski, Traveling wave solutions for case II diffusion in polymers. Journal of Polymer Science: Part B: Polymer Physics 34 (1996), 141–150. [27] V. G. Zvyagin, D. A. Vorotnikov, Topological approximation methods for evolutionary problems of nonlinear hydrodynamics. de Gruyter Series in Nonlinear Analysis and Applications, 12. Walter de Gruyter & Co., Berlin, 2008.

In: Advances in Mathematics Research, Volume 10 ISBN 978-1-60876-265-1 c 2010 Nova Science Publishers, Inc. Editor: Albert R. Baswell, pp. 273-302

Chapter 6

Z IGZAG AND A RMCHAIR N ANOTUBES IN E XTERNAL F IELDS Evgeny Korotyaev1,∗ and Anton Kutsenko2,† 1 School of Mathematics, Cardiff Univ., Senghennydd Road, Cardiff, CF24 4AG, UK 2 Department of Mathematics of Sankt-Petersburg State University, Russia

Abstract We consider the Schr¨odinger operator on the zigzag and armchair nanotubes (tightbinding models) in an uniform magnetic field B and in an external periodic electric potential. The magnetic and electric fields are parallel to the axis of the nanotube. We show that this operator is unitarily equivalent to the finite orthogonal sum of Jacobi operators. We describe all spectral bands and all eigenvalues (with infinite multiplicity, i.e., flat bands). Moreover, we determine the asymptotics of the spectral bands both for small and large potentials. We describe the spectrum as a function of |B|. For exπ π πk ample, if |B| → 16 3 ( 2 − N + πs) tan 2N , k = 1, 2, . . . , N, s ∈ Z, then some spectral band for zigzag nanotube shrinks into a flat band and the corresponding asymptotics are determined.

1.

Introduction

After their discovery [5], carbon nanotubes remain in both theoretical and applied research [21]. Structure of nanotubes are formed by rolling up a graphene sheet into a cylinder. Such nanomodels were introduced by Pauling [17] in 1936 to simulate aromatic molecules. They were described in more detail by Ruedenberg and Scherr [20] in 1953. Various physical properties of carbon nanotubes can be found in [21]. There are mathematical results about Schr¨odinger operators on carbon nanotubes (zigzag, armchair and chiral) (see [2], [11], [12], [6], [13], [18]). All these papers consider the ∗ †

E-mail addresses: [email protected] E-mail addresses: [email protected], [email protected]

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Evgeny Korotyaev and Anton Kutsenko

so called continuous models. But in the physical literature the most commonly used model is the tight-binding model, see [21], [4]. For applications of our models see ref. in [1], [21], [4]. In this paper we concentrate on carbon nanotubes which arise from graphene: zigzag and armchair nanotubes (see physical propereties in [21]). We will study and compare spectral properties of Shr¨odinger operator on zigzag and armchair nanotubes. We will show that these operators have different spectral properties. For example: 1) The Shr¨odinger operator Hzi on the zigzag nanotube is unitarily equivalent to the direct sum of scalar Jacobi matrices (see Theorem 2.1). But the Shr¨odinger operator on armchair nanotube Har is unitarily equivalent to the direct sum of Jacobi matrices with 2 × 2 matrix valued coefficients (see Theorem 6.1). Then the spectral analysis of Har is more difficult. 2) For some amplitude of the constant magnetic field the spectrum of Hzi has absolutely continuous part and eigenvalues (flat bands, see Theorem 2.2). But the spectrum of Har is purely absolutely continuous for any amplitude of the magnetic field. 3) The spectral bands of operators Hzi and Har are different. But in some cases the spectra of these operators has the same part (see Theorem 6.2). 4) In the simple case, when the magnetic field is absent and external electric potential has minimal period 2 the spectrum of Hzi and Har are coincide. Remark that the multiplicity of some spectral zones is different (see Sect 4 and Sect. 6.2). 5) The structure of spectral zones of Har and Hzi for large electric potentials is similar, since the spectrum is a union of small clusters, but asymptotics of this clusters are different (see Theorem 2.6 and Theorem 6.5). Moreover, we have similar situation for small potentials. In the proof of our theorems we determine various asymptotics for periodic Jacobi operators with specific coefficients, see (2.5). Note that there exist a lot of papers devoted to asymptotics and estimates both for periodic Jacobi operators and Schr¨odinger operators, see e.g. [8], [14], [24], [22], [23].

2.

Zigzag Nanotube

In this Section we consider the Schr¨odinger operator H b on the zigzag nanotube Γ ⊂ R3 (1D models tight-binding model of zigzag single-wall nanotubes, see [21], [16]) in a uniform magnetic field B = |B|e0 , e0 = (0, 0, 1) ∈ R3 and in an external electric potential. Our model nanotube Γ is a graph (see Fig. 2. and 2) embedded in R3 oriented in the zdirection e0 with unit edge length. Γ is a set of vertices (atoms) rω connecting by bonds (edges) Γn,k,j and 3n e0 , rn,1,k = rn,0,k + e0 , ω = (n, j, k) ∈ Z, 2 √  πk 3 πk  = Z/(N Z), κk = R cos , sin ,0 , R = π . N N 4 sin 2N (2.1)

Γ = ∪ω∈Zrω , rn,0,k = κn+2k + Z = Z × {0, 1} × ZN , ZN

Zigzag and Armchair Nanotubes in External Fields (2, 1, 0)

(2, 1, 1)

275

(2, 1, 2)

(2, 0, 0) (2, 0, 1) (2, 0, 2) b1 b2 (1, 1, 0) (1, 1, 1) (1, 1, 2) b3 (1, 0, 0) (1, 0, 1) (1, 0, 2) b1 b2 (0, 1, 0) (0, 1, 1) (0, 1, 2) (0, 0, 0)

(0, 0, 1)

(0, 0, 2)

Figure 1. A piece of zigzag nanotube.

Our carbon model nanotube is the honeycomb lattice of a graphene sheet rolled into a cylinder. This nanotube Γ has N hexagons around the cylinder embedded in R3 . Here n ∈ Z labels the position in the axial direction of the tube, j = 0, 1 is a label for the two types of vertices (atoms) (see Fig. 2.), and k ∈ ZN labels the position around the cylinder. The points r0,1,k , k ∈ ZN are vertices of the regular N-gon P0 and r1,0,k are the vertices of the regular N-gon P1 . P1 arises from P0 by combination of the rotation around the axis π and of the translation by 21 e0 . Repeating this procedure of the cylinder C by the angle N we obtain Γ. Introduce the Hilbert space ℓ2 (Γ) of functions f = (fω )ω∈Z on Γ equipped with the P norm kf k2ℓ2 (Γ) = ω∈Z |fω |2 . The tight-binding Hamiltonian H b on the nanotube Γ has

the form H b = H0b + V on ℓ2 (Γ), where H0b is the Hamiltonian of the nanotube in the magnetic field and is given by (H0b f )n,0,k = eib2 fn−1,1,k + eib1 fn−1,1,k−1 + eib3 fn,1,k , (H0b f )n,1,k = e−ib1 fn+1,0,k+1 + e−ib2 fn+1,0,k + e−ib3 fn,0,k , ω = (n, j, k) ∈ Z × {0, 1} × ZN ,

f = (fω )ω∈Z, 3|B| π b3 = 0, b1 = −b2 = b = cot , (2.2) 16 2N

(the last line in (2.2) was obtained in [12]) and the operator V corresponding to the external electric potential is given by v = (vn )n∈Z ∈ ℓ∞ . (2.3) Such potentials can be realized using optical methods, by gating, or by an acoustic field (see [16]). For example, if an external potential is given by A0 cos(ξ0 z + β0 ) for some constant A0 , ξ0 , β0 , then we obtain (V f )ω = Vω fω ,

where

Vn−1,1,k = v2n ,

   1 v2n = A cos 2πξ n − +β , 3

Vn,0,k = v2n+1 ,

v2n+1 = A cos(2πξn + β),

n ∈ Z,

(2.4)

276

Evgeny Korotyaev and Anton Kutsenko

Figure 2. Nanotube in the magnetic field.

for some constant A, ξ, β. If ξ is rational, then the sequence vn , n ∈ Z is periodic. If ξ is irrational, then the sequence vn , n ∈ Z is almost periodic. Below we use notation Nj for the set {1, . . . , j}, j > 1.

Theorem 2.1. i) Let v = (vn )n∈Z ∈ ℓ∞ . Then each operator H b , b ∈ R is unitarily b b 2 equivalent to the operator ⊕N 1 Jk , where Jk is a Jacobi operator, acting on ℓ (Z) and given by (Jkb y)n = an−1 yn−1 + an yn+1 + vn yn , a2n ≡ ak,2n = 2|ck |,

a2n+1 ≡ ak,2n+1

y = (yn )n∈Z ∈ ℓ2 ,  πk  , n ∈ Z, (2.5) = 1, ck = cos b + N

b+ π

b , J −b = J b ±b and Jk N = Jk+1 N −k for all (k, b) ∈ ZN × R. Moreover, the operators H k π and H b+ N are unitarily equivalent for all b ∈ R. ii) Let, in addition, ck = cos(b + πk N ) = 0 for some (k, b) ∈ ZN × R. Then   1 v2n−1 ± v2n ± b b + j −2 , (n, j) ∈ Z × N2 . σ(Jk ) = σpp (Jk ) = zn,j = vn +(−1) |vn +1| 2 , vn = 2 (2.6)

Remark. 1) The matrix of the operator Jkb is given by  ... ... ... ... ... ... ...  . . . 2|ck | v1 1 0 0 ...   ... 0 1 v 2|c | 0 . .. 2 k  b  0 0 2|ck | v3 1 ... Jk =  . . .  ... 0 0 0 1 v . .. 4   ... 0 0 0 0 2|ck | . . . ......................................



    .    

(2.7)

2) If |ck | = 12 , then Jkb is the Schr¨odinger operator with an = 1 for all n ∈ Z. In particular, if b = 0, N3 ∈ N, then J 0N is the Schr¨odinger operator. 3

Zigzag and Armchair Nanotubes in External Fields

277

3) In the continuous models similar results were obtained in [11], [12]. 4) Exner [3] obtained a duality between Schr¨odinger operators on graphs and certain Jacobi matrices, which depend on energy. In our case the Jacobi matrices do not depend on energy. 1. Periodic electric potentials v. Introduce the class ℓper of real s-periodic sequences s b v = (vn )n∈Z ∈ ℓ∞ and vn+s = vn , for all n ∈ Z. If v ∈ ℓper p∗ , p∗ > 1, then Jk is 2p-periodic matrix where ( p∗ if p∗ is even p= 2 . (2.8) p∗ if p∗ is odd If ck 6= 0 for some (k, b) ∈ ZN × R, then the spectrum of Jkb has the form b σ(Jkb ) = σac (Jkb ) = ∪2p 1 σk,n ,

b,+ b,− b σk,n = [zk,n−1 , zk,n ],

n ∈ N2p ,

b,+ b,− b,+ b,− b,+ b,− zk,0 < zk,1 6 zk,1 < zk,2 6 zk,2 < · · · < zk,2p ,

(2.9)

b,± see [15], where zk,n are 4p-periodic eigenvalues for the equation an−1 yn−1 + an yn+1 + b , σb b vn yn = zyn , y = (yn )n∈Z . The intervals σk,n k,n+1 are separated by a gap γk,n = b,− b,+ b | > 0. If a gap γ b b (zk,n , zk,n ) of length |γk,n k,n is degenerate, i.e., |γk,n | = 0, then the b , σb corresponding segments σk,n k,n+1 merge. If ck = 0 for some (k, b) ∈ ZN × R, then (2.6) gives σ(Jkb ) = σpp (Jkb ), where   q v2n−1 ± v2n −2 ± b + j , (n, j) ∈ Np × N2 , σpp (Jk ) = zn,j = vn + (−1) vn + 1, vn = 2 (2.10) and each eigenvalue of Jkb is a flat band, i.e. it has infinite multiplicity. In Theorem 2.2 b we show that the spectral band σk,n shrinks to the flat band {λn } as ck → 0 and the corresponding asymptotics are determined. R⊕ dt Each operator Jkb is unitarily equivalent to the operator [0,2π) K(eit , a) 2π , a = 2|ck |, where 2p×2p matrix K(τ, a) ≡ K(τ, a, v) is a Jacobi operator, acting on C2p and given by   0 1 0 . . . τa  1 0 a ... 0    0 0  , B = diag(vn )2p , 0 a 0 . . . 0 K(τ, a) = K (τ, a) + B, K (τ, a) =  1    ..................  τa 0 . . . 1 0 (2.11) where τ ∈ S1 = {τ ∈ C : |τ | = 1}. Let µ1 (τ, a) 6 µ2 (τ, a) 6 µ3 (τ, a) 6 · · · 6 µ2p (τ, a) be eigenvalues of K(τ, a), τ ∈ S1 , here µn (·, a) is analytic function in τ ∈ S1 . Note that b for all (k, n) ∈ Z × N . If c 6= 0, then each µ (·, a), n ∈ N is not a µn (S1 , a) = σk,n 2p n 2p N k b constant and |σk,n | > 0. If ck = 0 for some k ∈ ZN , then each µn (·, 0) = const = λn , n ∈ b = {λ } is a flat band. N2p and σk,n n

2. The case B = 0. Consider the Schr¨odinger operator H 0 at B = 0. By Theorem 2.1, the 0 0 b operator H 0 is unitarily equivalent to the operator ⊕N 1 Jk , where Jk is a Jacobi operator Jk πk N 0 0 at b = 0 and here a2n = 2| cos N |, a2n+1 = 1. Note that if k 6= 2 , then σ(Jk ) = σac (Jk ) and if k = N2 , then σ(Jk0 ) = σpp (Jk0 ).

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Evgeny Korotyaev and Anton Kutsenko

3. Example of simple periodic potentials v. Consider the potential v = v2k+1 = −v2k ∈ R, k ∈ Z. In Section 4 we will show that  b,+ b,−  b,− b,+
b b , γk,1 = zk,1 , zk,1 , σ(Jkb ) = zk,0 , zk,0 \ γk,1 p p b,∓ b,± zk,0 = ± v 2 + (2|ck | + 1)2 , zk,1 = ± v 2 + (2|ck | − 1)2 , k ∈ ZN ,

b is the gap in the spectrum of J b . This gives where γk,1 k ( ∅ b b b b σ(Jk ) = σac (Jk ) ∪ σpp (Jk ), σpp (Jk ) =  √ ± 1 + v2

if ck 6= 0 , if ck = 0

and then we deduce that the spectrum of H b is given by

σ(H b ) = σac (H b ) ∪ σpp (H b ), ( ∅ if ck 6= 0, any k ∈ ZN σpp (H b ) =  √ , 2 ± 1+v if ck = 0, some k ∈ ZN  
σac (H b ) = z0b,+ , z0b,− \ γ(H b ), γ(H b ) = z1b,− , z1b,+ ,

(2.12)

(2.13)

where γ(H b ) is the √ gap in the spectrum of H b . Note that if ck = 0 for some k ∈ ZN π then σpp (H b ) = {± 1 + v 2 } ⊂ γ(H b ). Theorem 2.1.i yields σ(H b+ N ) = σ(H b ) for all π b ∈ R. Then we need to consider only the case b ∈ [0, N ) and in this case we get  b,+ π z0,0 if b 6 2N . z0b,+ = (2.14) π z b,+ if b > N −1,0 2N Moreover, in particular case B = 0 we obtain N ∈ N, b = 0. (2.15) 3 Now we return to the general case of periodic potentials. First theorem is devoted to the asymptotics of small spectral bands that degenerate to the flat band. γ(H 0 ) = (−|v|, |v|),

if

π πk Theorem 2.2. Let v ∈ ℓper p∗ , p∗ > 1 and ck → 0 as b → b0 = 2 − N for some k ∈ ZN and b,+ b,− let λ1 6 λ2 6 · · · 6 λ2p be eigenvalues of K(1, 0, v). Then the endpoints zk,s−1 , zk,s ,s∈   b,+ b,− b = z N2p of the spectral bands σk,s k,s−1 , zk,s are analytic functions in b ∈ {|b − b0 | < ε} for some ε > 0 and satisfy b,+ b,− zk,s−1 = λs + O(c2k ), zk,s = λs + O(c2k ) as

ck → 0.

(2.16)

Let in addition λs−1 < λs < λs+1 for some s ∈ N2p , where λ0 = −∞, λ2p+1 = +∞. Then b,− zk,s = λs −

X 2 |2ck |p + Ck,n (2ck )2n +O(cp+1 k ), Λs 262n6p

|p

Λs =

2p Y

n=1, n6=s

4|2ck + O(cp+1 k ) Λs as ck → 0 for some constants Ck,n , which depend only on v. b,− b,+ b |σk,s | = zk,s − zk,s−1 =

|λs −λn |, (2.17) (2.18)

Zigzag and Armchair Nanotubes in External Fields

279

b ,n ∈ N Remark. By (2.16), each spectral band σk,n 2p shrinks to the flat band {λn } as ck → 0.

We consider the nanotube in weak electric fields. Our operator has the form H b (t) = + tV , where a coupling constant t → 0. In this case the corresponding Jacobi operator depend on t and is given by

H0b

(Jkb (t)y)n = an−1 yn−1 + an yn+1 + tvn yn , y = (yn )n∈Z ∈ ℓ2 , n ∈ Z,

(2.19)

b (t) = We study how the spectral bands σk,n

where a2n = 2|ck |, a2n+1 = 1.

b,− b,+ [zk,n−1 (t), zk,n (t)], n ∈ N2p of the operator Jkb (t) depend on the couple constant t → 0. per For v ∈ ℓp∗ we define two vectors v 0 = (v2n )p1 , v 1 = (v2n−1 )p1 ∈ Rp and

u ˆn = hu, en i, u ∈ Cp , en =

1 2j p i πn ˆp+n = u ˆp−n , n ∈ Np . (τn )j=1 ∈ Cp , τn = e p , u 2p (2.20)

Here en , n ∈ Np is a basis in Cp and hu, en i is the scalar product in Cp . Define ℓper 0,p∗ = per P2p {v ∈ ℓp∗ : 1 vn = 0} and the sets ( N2p−1 if 2|ck | = 1 . Nk,p = N2p−1 \ {p} if 2|ck | = 6 1 0 Theorem 2.3. Let ck 6= 0 for some (k, b) ∈ ZN × R. Let v ∈ ℓper 0,p∗ and let v = b (t) = (v2n )p , v 1 = (v2n−1 )p1 ∈ Rp . Then the asymptotic of the spectral bands σk,n  b,+ 1  b,− zk,n−1 (t), zk,n (t) , n ∈ N2p of the operator Jkb (t) hold true b,± ± zk,n (t) = zn,k (0) ± tψk,n (v) + O(t2 ), n ∈ Nk,p , ( |ˆ vn0 + e2i arg(2|ck |+τn ) vˆn1 |, n 6= p , ψk,n (v) = |ˆ vp0 − vˆp1 |, 2|ck | = 1, n = p

(2.21)

b,+ b,+ b,− b,− zk,0 (t) = zk,0 (0) + O(t2 ), zk,2p (t) = zk,2p (0) + O(t2 ),

and if 2|ck | 6= 1

⇒

b,± b,± zk,p (t) = zk,p (0) + O(t2 ),

(2.22)

b,± b,± zk,n (0) = |2|ck | + τn | sign(n − p), n ∈ N2p−1 \ {p}, zk,p (0) = ±|2|ck | − 1|, (2.23)

as t ↓ 0. Moreover, if p∗ is odd, then for all n ∈ Nk,p the following identities hold true vˆn0 = τnp+1 vˆn1 ,

ψk,n (v) = |ˆ vn0 |ρk,n ,

( |(−1)n τn +e2i arg(2|ck |+τn ) |, n 6= p ρk,n = 0, if 2|ck | = 1 and n = p

   ρk,n 6= 0, if |ck | 6=     , ρk,n 6= 0, if |ck | =      ρk,n = 0, if |ck | =

1 , 2 1 , even n . 2 1 , odd n 2 (2.24)

280

Evgeny Korotyaev and Anton Kutsenko To describe some examples of external fields which create the open gaps we define the

set X p∗ =

(

v ∈ ℓper 0,p∗

))

( vˆn0 +ˆ vn1 6= 0, vˆn0 vˆn1 = 0, all n ∈ Np−1 , vˆp0 6= 0, p∗ ∈ 2N : vˆn0 6= 0, all n ∈ Np−1 , p∗ is odd

.

(2.25)

Proposition 2.4. i) The set Xp∗ 6= ∅ for any p∗ > 2. ii) If v ∈ Xp∗ , p∗ ∈ 2N, then b,± ± zk,n (t) = zk,n (0)±tξn +O(t2 ),

ξn = |ˆ vn1 +ˆ vn0 | > 0 as t ↓ 0,

all n ∈ Nk,p . (2.26)

iii) If v ∈ Xp∗ is sufficiently small and p∗ is odd, then b 6= 0. If 2|ck | 6= 1, then each ψk,n (v) 6= 0, n ∈ N2p−1 \ {p} and γk,n ( 6= 0 all even n ∈ N2p−1 If 2|ck | = 1, then each ψk,n (v) = 0 all odd n ∈ N2p−1 and γk,n 6= 0 for any even n ∈ N2p−1 .

b,+ b,− Remark. (2.26) gives the asymptotics of the gap length zk,n (t) − zk,n (t) = t2|ˆ vnj | + O(t2 ) as t → 0 where j = 0 or j = 1. Note that the first term does not depend on k ∈ ZN . If p∗ b,− b,+ is even, then for large class of potentials v ∈ Xp all gaps (zk,n (t), zk,n (t)) are open.

We formulate the theorem, motivated by the physical paper of Novikov [16]. Theorem 2.5. Let v ∈ ℓper p∗ and let t > 0, b ∈ R be sufficiently small. i) Let b = 0. If N ∈ 2N and p are coprime numbers, then σpp (H 0 (t)) ⊂ N −1 ∩n=1 σ(Jk0 (t)). ii) If p > 2N , then the spectrum of H b (t) on the set σ(H b (t)) ∩ ([−ρ, −r] ∪ [r, ρ]) has multiplicity 2 and satisfies σ(H

b

2p−1 [ b b (t))∩[r, ρ] = σ(JN (t))∩[r, ρ] = [r, ρ]\ (t), γN,n p 2p−1−N

iπ

3+|2+e p | r = |2+e |, ρ = , 2 iπ N

p

b (t)) ∩ [−ρ, −r] = [−ρ, −r] \ σ(H b (t)) ∩ [−ρ, −r] = σ(JN

N [

b (t). γN,n

(2.27)

1

Moreover, if v ∈ Xp∗ , then each |γN,n (t)| > 0, n ∈ N2p−1 . iii) If N 6∈ 3N, then σ(H b (t)) ∩ [−r, r] = ∅ for some r > 0. iv) If N ∈ 3N and p > 2N , then the spectrum of H b (t) on the set σ(H b (t)) ∩ [−r, r] has multiplicity 2 and satisfies 1 ) p(1+ N

b

b

σ(H (t))∩[−r, r] = σ(J N (t))∩[−r, r] = [−r, r]\ 3

|γ N ,n (t)| > 0 if 3

(

[

1 ) p(1− N

p∗ ∈ 2N, p∗ is odd,

iπ

γ bN ,n (t), r = |1−e N |, (2.28) 3

n ∈ N2p−1 even n ∈ N2p−1

,

v ∈ X p∗ .

(2.29)

Zigzag and Armchair Nanotubes in External Fields

281

b (t) in (2.27) and γ b Remark. 1) The gaps γN,n (t) in (2.28) are also the gaps in the specN ,n 3

trum of H b (t). Then we may choose the potentials v such that all these gaps are open (for wide set of potentials). 2) Due to iii) σ(H b ) has a gap contained the interval [−r, r] We consider the nanotube in strong electric fields. Our operator has the form H b (t) = 2p H0b + tV , where a coupling constant t → ∞. For each (vn )2p 1 ∈ R there exists a permutation α : N2p → N2p such that hn = vα(n) and h1 6 h2 6 · · · 6 h2p . Let vn 6= vj for n+1 ,n ∈ all n 6= j, n, j ∈ N2p . Defining disjoint intervals Cn = [th0n−1 , th0n ), h0n = hn +h 2 2p 0 0 b N2p , h0 = −∞, h2p+1 = ∞, we obtain the inclusion σ(H (t)) ⊂ ∪n=0 Cn = R. We shall call the set σ(H b (t)) ∩ Cn the n’th spectral bands cluster. Our goal is to study the asymptotic distribution of eigenvalues in the n’th cluster as t → ∞. πk Theorem 2.6. Let v ∈ ℓper p∗ , vn 6= vj for all n 6= j, n, j ∈ N2p and let ck = cos(b+ N ) 6= 0 for some (k, b) ∈ NN × R. Let vα(n) < vα(j) for all n < j and some permutation b (t) = α : N2p → N2p . If n e = α−1 (n) for some n ∈ N2p , then the spectral bands σk,e n b,+ b,− [zk,e n−1 (t), zk,e n (t)] satisfy b,+ zk,e n−1 (t) = tvn −

Cn + O(t−1 ) , t

b,+ b,− zk,e n (t) − zk,e n−1 (t) =

Cn =

1 + O(t−1 ) , En t2p−1

a2k,n−1 vn−1 − vn

En =

+

a2k,n vn+1 − vn

Y 1 |vn − vj |, p 2|2ck |

,

(2.30)

(2.31)

j6=n

as t → ∞. Moreover, b

σ(H (t)) ∩ Cne (t) = b (t) σk,n

N [

b σk,e n (t)

k=1

∩

σkb ′ ,n (t)

  δ δ 2 ⊂ vn t − , vn t + , δ = max , (2.32) n t t |vn − vn+1 |

= ∅,

if

( k 6= k ′ , |ck | 6= |ck′ |,

b∈ / b∈

π 2N N π 2N N

,

(2.33)

b (t) has multiplicity 2 if c 6= 0 and σ b (t) is a flat where the spectrum of H b (t) on σk,e k n k,e n band if ck = 0.

Remark. 1) Theorems 2.3, 1.4 describe the case t → 0 and Theorem 2.6 describe the case t → ∞. These two cases are quite different, see Fig. 3 and Fig 4. −3

0

3

Figure 3. Open small spectral gaps for the potential tV as t → 0. 2) The spectral bands cluster σ(J b (t)) ∩ Cne (t) is a union of N non overlapping bands b (t), k ∈ N , see (2.32). Recall that if |c | = |c ′ |, then J b (t) = J b (t). σk,e N k k k k′ n

282

Evgeny Korotyaev and Anton Kutsenko

Figure 4. Spectral clasters for the potential tV as t → ∞ for the case N = 4.

We present the plan of our paper. In Sect. 2 we prove Theorem 2.1 and 2.2. In the proof Theorem 2.1 we use arguments from [11], [12]. In the proof Theorem 2.2 we use arguments from [8]. In Sect. 3 we consider the simple examples for the case p = 1, in fact, we study unperturbed Hamiltonians. In Sect. 4 we prove Theorem 2.3 -2.6. In Sect. 6 we apply some of these methods to analyze the spectral properties of Shr¨odinger operator on armchair nanotubes.

3.

Proof of Theorems 2.1 and 2.2

Proof of Theorem 2.1. i) Define an operator J b : (ℓ2 )N → (ℓ2 )N acting on a vector-valued function ψ = (ψn )n∈Z ∈ (ℓ2 )N , ψ2n+1 = (fn,0,k )k∈ZN , ψ2n = (fn−1,1,k )k∈ZN ∈ CN , by (J b ψ)2n = ((H b f )n,0,k )k∈ZN ,

(J b ψ)2n+1 = ((H b f )n,1,k )k∈ZN .

(3.1)

Define a matrix-valued operators Pn : CN → CN by P2n+1 h = (Vn,0,k hk )k∈ZN , P2n h = (Vn,1,k hk )k∈ZN , h = (hk )k∈ZN .

(3.2)

N Define the operator S in CN by Su = (uN , u1 , . . . , uN −1 )⊤ , u = (un )N 1 ∈ C . Using ∗ −1 ib −ib ∗ (3.1), (2.2), (2.3),(3.2) and S = S , A = e IN + e S we obtain

(J b ψ)2n+1 = (eib S+e−ib )ψ2n +ψ2n+2 +P2n+1 ψ2n+1 = A∗ ψ2n +ψ2n+2 +P2n+1 ψ2n+1 , (J b ψ)2n = ψ2n−1 + (eib + e−ib S∗ )ψ2n+1 + P2n ψ2n = ψ2n−1 + Aψ2n+1 + P2n ψ2n . Finally we rewrite the operator J b : (ℓ2 )N → (ℓ2 )N in the form of the operator Jacobi by (J b ψ)n = A∗n−1 ψn−1 + An ψn+1 + Pn ψn , n ∈ Z, and then



    b J =    

A2n = A = eib IN + e−ib S∗ , A2n+1 = IN , (3.3)

............................... . . . A∗ P1 IN 0 0 ... . . . 0 IN P2 A 0 . . . ... 0 0 A∗ P3 IN . . . ... 0 0 0 IN P4 . . . ... 0 0 0 0 A∗ . . . ...............................



    .    

(3.4)

The matrix-valued function Pn is 2p-periodic. Then the operator J b is a 2p-periodic Jacobi operator with N × N matrix -valued coefficients. Note that such operators were considered in [9].

Zigzag and Armchair Nanotubes in External Fields 283 PN k The unitary operator S has the form S = ek = sk eek and eek = 1 s Pk , where Se 1 −k , s−2k , . . . , s−kN +k )⊤ is an eigenvector (recall that s = ei 2π N ); Pk u = 1 (1, s

N2

N is a projector. Define the operators e eek (u, eek ), u = (un )N Sψ = (Sψn )n∈Z and 1 ∈ C b b be e e Pk ψ = (Pk ψn )n∈Z . The operators S and J commute, then J = ⊕N 1 (J Pk ). Using e k is unitarily equivalent to the operator J b given by (3.3), (3.4) we deduce that J b P k

(Jkb y)n = e a∗k,n−1 yn−1 + e ak,n yn+1 + vn yn , y = (yn )n∈Z ∈ ℓ2 ,  πk 2π πk  , e ak,2n = eib + e−ib s−k = 2e−i N ck , s = ei N , ck = cos b + N

e ak,2n+1 = 1,

(3.5)

and using Lemma 3.1 we obtain (2.5). ii) If ck = 0, then the Jacobi operator Jkb has the form 

    b Jk =     

........................... . . . 0 v1 1 0 0 . . . . . . 0 1 v2 0 0 . . . . . . 0 0 0 v3 1 . . . . . . 0 0 0 1 v4 . . . ... 0 0 0 0 0 ... ...........................



     = ⊕n∈Z Jn ,    

The eigenvalues of Jn are given by zn,j = vn+ + (−1)j (n, j) ∈ Z × N2 , which yields (2.6).

Jn =



v2n−1 1 1 v2n

q 2 vn− + 1, vn± =



. (3.6)

v2n−1 ±v2n 2

for

Recall results from [15] about our 2p-periodic Jacobi operator J(a) : ℓ2 → ℓ2 given by (J(a)y)n = an−1 yn−1 +an yn+1 +vn yn , a2n = a > 0, a2n+1 = 1, n ∈ Z, y = (yn )n∈Z . (3.7) + b). Introduce fundamental solutions Note that Jkb = J(a), where a = 2|ck |, ck = cos( πk n ϕ = (ϕn (z, a))n∈Z and ϑ = (ϑn (z, a))n∈Z for the equation an−1 yn−1 + an yn+1 + vn yn = zyn , (z, n) ∈ C × Z,

a2n+1 = 1, a2n = a,

(3.8)

with initial conditions ϕ0 ≡ ϑ1 ≡ 0, ϕ1 ≡ ϑ0 ≡ 1. The function ∆ = 12 (ϕ2p+1 + ϑ2p ) is called the Lyapunov function for the operator J(a). The functions ∆, ϕn and ϑn , n > 1 are polynomials of (z, a, v) ∈ C2p+2 . It is well known that σ(J(a)) = σac (J(a)), where σac (J(a)) = {z ∈ R : ∆(z, a) ∈ [−1, 1]} = ∪2p 1 σn (a),

+ σn (a) = [zn−1 (a), zn− (a)], (3.9)

− , where zn± = zn± (a). Note that ∆(zn± , a) = (−1)p−n and z0+ < z1− 6 z1+ < · · · 6 z2p for all n = 0, . . . , p. Below we will sometimes write σ(a, v), J(a, v), . . . , instead of σ(a), J(a), . . . , when several potentials are being dealt with. Recall that the 2p × 2p matrix

284

Evgeny Korotyaev and Anton Kutsenko

K(τ, a) is given by

0

K(τ, a) = K (τ, a) + B,



  K (τ, a) =    0

0 1 0 . . . τa 1 0 a ... 0 0 a 0 ... 0 .................. τa 0 . . . 1 0



   , B = diag(vn )2p , 1  

(3.10) where τ ∈ S1 = {τ ∈ C : |τ | = 1}. Fix a, φ ∈ [0, 2π], then eigenvalues of K(eiφ , a) are all zeros of the polynomial ∆(z, a) − cos φ. Then the fundamental solutions ϕk,n , ϑk,n , the b for the operator J b satisfy (see also (2.9)) Lyapunov function and the spectral bands σk,n k b,± ϕk,n = ϕn (z, a), ϑk,n = ϑn (z, a), ∆k = ∆(z, a) zk,n = zn± (a), (3.11)   b,+ b,−  b b , σ(Jkb ) = σac (Jkb ) = z ∈ R : ∆k (z) ∈ [−1, 1] = ∪2p σk,n = zk,n−1 , zk,n 1 σk,n , (3.12) Proof of Theorem  2.2. Let a = 2|ck | → 0. We consider the matrix K(τ, a) as a → 0, τ ∈ S1 = τ ∈ C : |τ | = 1 . If a = 0, then we get K(τ, 0) = ⊕p1 Jn , where Jn is given by (3.6). Let λ1 6 λ2 6 . . . .λ2p be the of  +eigenvalues  K(τ, 0). The endpoints + − − zn−1 (a), zn (a) of the spectral bands σn (a) = zn−1 (a), zn (a) of the operator J(a) are the eigenvalues of K(±1, a, v). By the perturbation theory [19], they are analytic function from a and if a → 0, then the spectral bands converge to the set {λ1 , λ2 , . . . .λ2p }. The number of spectral bands converging to λn coincides with the multiplicity of λn as a → 0. In particular, if some λn , n ∈ N2p is simple, then σn (a) → {λn }. Recall that the monodromy matrix M2p for the operator J(a) is given by   ϑ2p ϕ2p M2p (z) = = Tp . . . T2 T1 , ϑ2p+1 ϕ2p+1      1 0 −a z − v2n−1 a 0 1 Tn = = , (3.13) −a z − v2n v2n − z φn /a a −1 z − v2n+1

where φn = (z − v2n )(z − v2n−1 ) − 1. Let    0 φn v2n − z , E= Xn = ETn E1 = 1 z − v2n−1 −1    2 1 a 0 a = A1 , A1 = A = (EE1 )−1 = 1 0 a 0 a

a 0 0 1

 

, E1 =



0 a1 1 0



,

.

Then M2p = E −1 Xp AXp−1 A . . . AX1 E1−1 , which yields the Lyapunov function ∆ given by 2∆ = Tr M2p

p 1 1 X 2n = Tr Xp AXp−1 A . . . AX1 A = p Tr Xp A1 Xp−1 A1 . . . AX1 A1 = p a Φn (z), a a n=0

and ∆(z, a) =

Φ0 (z) + a2 Φ(z, a2 ) , 2ap

Φ0 =

2p Y

n=1

(z − λn ),

Φ(z, t) =

p X

tn−1 Φn (z),

n=1

(3.14)

Zigzag and Armchair Nanotubes in External Fields

285

for some polynomials Φn . By the perturbation theory (see [19]), the endpoints z+ , z− of + the spectral band σs (a) = [zs−1 (a), zs− (a)] = [z+ , z− ] are analytic functions in some disk {a ∈ C : |a| < ε}, ε > 0 and satisfy the equation ∆(z± , a) = ∓(−1)s , which has the form Φ0 (z± ) + a2 Φ(z± , a2 ) = ∓(−1)s 2ap .

(3.15)

Moreover, they satisfy z± (a) = λs + O(a2 ) as a → 0 at p > 2 (see the case p = 1 in Sect. 3). Let λs be a simple eigenvalue for some s ∈ N2p . The differentiation of (3.15) yields
′ z± (a)Ω + ∂a a2 Φ(z± , a2 ) = ∓p(−1)s 2ap−1 ,


Ω(z, a) = ∂z Φ0 (z) + a2 Φ(z, a2 ) . (3.16)

The differentiation of (3.15) r ∈ [1, p] times yields (r−1)

(r)

z± Ω(z± ) + Gr z±


, . . . , z± , a = ∓

p! (−1)s 2ap−r , (p − r)!

(3.17)

for some polynomial Gr . Then at a = 0 this gives (r−1)

(r)

z± (0)(−1)s Λs + Gr z±


(0), . . . , z± (0), 0 = ∓

p! (−1)s 2ap−r |a=0 . (p − r)!

(3.18)

(2r+1)

Thus we obtain z± (0) = 0 for all 2r + 1 < p, since the polynomial Φ = Φ(z, a2 ). (r) (r) Moreover, using z− (0) = z+ (0) we obtain z− (0) = z+ (0) for all r < p. Consider the case r = p. Identity (3.17) implies (p−1)

(p)

z± (0)(−1)s Λs + Gp z+


(0), . . . , z+ (0), 0 = ∓p!2(−1)s ,

(3.19)

(p)

which yields z± (0) = Λp!s (Cp ∓ 2) for some constant Cp ∈ R. Using this and σs (a) = + [zs−1 (a), zs− (a)] = [z+ , z− ] and (3.9), (3.11) we obtain (2.17), (2.18). Lemma 3.1. Let a Jacobi operator J : ℓ2 → ℓ2 is given by (Jy)n = a∗n−1 yn−1 + an yn+1 + vn yn ,

y = (yn )n∈Z ∈ ℓ2 , an+p = an ∈ C, vn ∈ R, (3.20)

n ∈ Z, for some p > 1. Then Ψ∗ JΨ = J + ,

(J + y)n = |an−1 |yn−1 + |an |yn+1 + vn yn ,

(3.21)

where the unitary diagonal operator Ψ is given by Ψy = (un yn )n∈Z , un =

n Y

εj , n > 0, un =

1

Proof. Direct calculations give (3.21).

n Y 1

a  n εj , n < 0, εn = |an | 1

if an 6= 0

. if an = 0 (3.22)

286

4.

Evgeny Korotyaev and Anton Kutsenko

Example for the Case p = 1

In this section we consider the Jacobi operator Jkb , k ∈ ZN given by 

    b Jk =      a = 2|ck |,

........................... ... a v 1 0 0 ... . . . 0 1 −v a 0 . . . ... 0 0 a v 1 ... . . . 0 0 0 1 −v . . . ... 0 0 0 0 a ... ...........................



    ,    

v = v2n+1 = −v2n ∈ R, n ∈ Z,

i.e., the case p = 1. The monodromy matrix M2 satisfies (see (3.13)) !    0 1 ϑ 2 ϕ2 1 0 M2 (z) = = 1 z+v ϑ 3 ϕ3 −a z − v − a a   −a z−v = z2 − v2 − 1  . −z − v a Let ∆0 =

Tr M2 2

∆k =

=

z 2 −v 2 −5 4

(4.1)

(4.2)

be the Lyapunov function for the case a = 1. This yields

z 2 − v 2 − 4c2k − 1 ∆0 + s2k Tr M2 = = , 2 4|ck | |ck |

 πk  . ck = cos b + N

(4.3)

b,± satisfy the equation ∆k (z) = 1 and anti-periodic eigenvalues The periodic eigenvalues zk,0 b,± zk,1 satisfy the equation ∆k (z) = −1 and they are given by

p b,∓ zk,0 = ± v 2 + (2|ck | + 1)2 ,

The spectrum of Jkb has the form

p b,± zk,1 = ± v 2 + (2|ck | − 1)2 .

 b,+ b,−   b,+ b,−   b,+ b,−  σ(Jkb ) = zk,0 , zk,1 ∪ zk,1 , zk,0 = zk,0 , zk,1 \ γk,1 ,

(4.4)

(4.5)

b = (z b,− , z b,+ ) is a gap. Note that where γk,1 k,1 k,1

b,− b,+
b 6 ∅, γk,1 = zk,1 , zk,1 =

if

1 |ck | 6= . 2

(4.6)

Let ck → 0. Then (2.17), (2.18) yield b,− b,+ |sb1 | = zk,1 − zk,0 =− b,− zk,1 = −w +

4|ck | + O(c2k ), w

2|ck | + O(c2k ), w

b,+ zk,0 = −w −

w=

p 1 + v2,

2|ck | + O(c2k ). w

(4.7)

Zigzag and Armchair Nanotubes in External Fields

287

1. The operator H 0 , no magnetic field, b = 0. In this case using (4.4), (4.5), we obtain  n N 2N o   , = ∅ if k ∈ p 0,+ 0,+ 0,± 0,− 0,+ 3 3 zk,0 < z0,0 , z0,0 = ± v 2 + 9, γk,1 = (zk,1 , zk,1 ) n N 2N o ,  6= ∅ if k ∈ / , 3 3 (4.8) and then  N  ∅ ∈ /N if 0 0 0 0 2 σ(H ) = σac (H ) ∪ σpp (H ), σpp (H ) =  √ , (4.9) N   ± 1 + v2 if ∈N 2  N  ∅ ∈N if   0,+ 0,− 3 , , z0,0 \ γ(H 0 ), γ(H 0 ) = σac (H 0 ) = z0,0
N   z 0,− , z 0,+ 6= ∅ ∈ /N if m,0 m,0 3 (4.10) for some m ∈ ZN , where roughly speaking m ∼ N3 . π

2. Magnetic field, b 6= 0. Using i) of Theorem 2.1 we obtain σ(H b+ N ) = σ(H b ), π Then we need to consider only the case b ∈ (0, N ). Using 2.1 we obtain

b ∈ R.

σ(H b ) = σac (H b ) ∪ σpp (H b ),

( ∅ if ck 6= 0, all k ∈ ZN √ , σpp (H b ) = 2 6 0, some k ∈ ZN {± 1 + v } if ck =  
σac (H b ) = z0b,+ , z0b,− \ γ(H b ), γ(H b ) = z1b,− , z1b,+ ,

where γ(H b ) is the gap in the spectrum of H b and  b,+ z0,0 if z0b,+ = z b,+ if N −1,0

π 2N , π b> 2N

(4.11) (4.12)

b6

(4.13)

and


γ(H b ) = z1b,− , z1b,+ ,

p z1b,± = ± v 2 + (2|ck | − 1)2 , for some k ∈ ZN ,

(4.14)

where roughly speaking 2|ck | ∼ 1.

5.

Proof of Theorems 2.3–2.6.

Proof of Theorem 2.3. In order to determine the asymptotics (2.21) we need the following fact from the perturbation theory [19]: Let A(t) = A0 + tA1 , t ∈ R, where A0 = A∗0 , A1 = A∗1 are operators in C2p . Let µ be an eigenvalue of A0 of multiplicity 2 and let h± be the corresponding orthonormalized eigenvectors. Then there are 2 functions µ± (t) analytic in a neighborhood of 0, which are all the eigenvalues. Moreover, µ± (t) = µ+µ′± (0)t +O(t2 )

288

Evgeny Korotyaev and Anton Kutsenko

as t → 0, where µ′± (0) are the eigenvalues of P ∗ A1 P and P = (h− , h+ ) is the 2p × 2 matrix. b,± We determine the asymptotics (2.21) of zk,n (t) for k ∈ Np , n 6= 0, p, 2p, the proof of other cases is similar. We apply the perturbation theory to the operator K(±1, a, tv) = b,± K 0 (τ, a) + tB as t → 0, where K is given by (3.10) and a = 2|ck |. Recall that zk,n (t) are 0 eigenvalues of K(±1, a, tv), (see (3.7)-(3.11)). The operators K (±1, a) has eigenvalues b,+ b,− zk,n (0) = zk,n (0) = λ± n (a) (with multiplicity 2) and the corresponding eigenvectors ± Zk,n = Zn± (a),

n ∈ Z2p−1 ,

(5.1)

± ′ see Corollary 7.2 and (3.11), Then by this fact, the derivatives (zk,n ) (0) are eigenvalues of + − ∗ the 2 × 2-matrix Pk,n BPk,n , where Pk,n = (Zk,n , Zk,n ) is the p × 2-matrix. Define the vectors 2j 2i arg(2|ck |+τn ) Fn = (2p)−1 (fj )2p , f2j = τn2j , τn = e 1 , f2j+1 = τn e

iπn p

, j ∈ Np . (5.2) = vep−n , n ∈ Np−1 . Using (5.1), Corollary 7.2, (7.2)

Let ven = hv, Fn i, n ∈ Np and vep+n we obtain     0 ven Tr B hb, Fn i ∗ = Pk,n BPk,n = , hFn , bi Tr B ven 0

where B = diag(vj )2p 1 .

± ′ The eigenvalues of the last matrix have the form ±|e vn |, which yields (zk,n ) (0) = ±|e vn |. 2j p 1 2p (τn )j=1 , (v2n−1 )pn=1 , vˆnj =

Recall that the orthogonal basis in Cp is given by en = i πn p

and the vectors v 0 = (v2n )pn=1 and v 1 = Then (5.2) gives ven = vˆn0 + e2i arg(a+τn ) vˆn1 and

n ∈ Np , where

τn = e hv j , en i, n ∈ Np , j = 0, 1. we obtain (2.21). Let S(u1 , . . . , up ) = (up , u1 , . . . , up−1 ) be a shift operator. If p∗ is odd, then p = p∗ p+1 and v 1 = S 2 v 0 and using (2.20), we obtain vˆn1 = hv 1 , en i = hS

p+1 2

v 0 , en i = hv 0 , S −

p+1 2

en i = hv 0 , τnp+1 en i = τn−p−1 vˆn0 = τnp−1 vˆn0 ,

since τnp = τn−p . Then we get vˆn0 + e2i arg(2|ck |+τn ) vˆn1 = vˆn0 (1 + τnp−1 e2i arg(2|ck |+τn ) ). p−1 2i arg(2|ck |+τn ) Simple calculations gives: if 2|ck | 6= 1 and n ∈ N( 6= 0, k,p , then 1 + τn e 6= 0 n is even . and if 2|ck | = 1, then 1 + τnp−1 e2i arg(2|ck |+τn ) = 0, n is odd Proof of Proposition 2.4. i) Consider the case p∗ is even. Denote z = (z n )p1 for z = (zn )p1 ∈ Cp . Using (2.20), we obtain ep−n = en , n ∈ Np−1 and ep = (2p)−1 (1, . . . , 1)⊤ ∈ Rp . P 1 p If v 1 = p−1 n=1 αn en + αp ep , αn = αp−n 6= 0, n ∈ Np−1 , 0 6= αp ∈ R, then v ∈ R p 1 p 0 1 and vˆn = αn 6= 0, n ∈ Np , since {en }1 is orthogonal basis in C . Consider v = −ˆ vp ep , then v 0 ∈ Rp , since ep ∈ Rp and vˆp1 P = αp ∈ R. Also vˆp0 = −αp 6= 0. Then the vector v = (v10 , v11 , . . . , vp0 , vp1 ) ∈ Xp∗ , since pn=1 (vn1 + vn0 ) = vˆp0 + vˆp1 = 0. Then Xp∗ 6= ∅. The proof of the case of odd p∗ is similar. The statements ii) and iii) follows from Theorem 2.3, (2.24).

Zigzag and Armchair Nanotubes in External Fields

289

Proof of Theorem 2.5. i) Using (5.4), we obtain that (1 − δ, 1 + δ) ∪ (−1 − δ, −1 + δ) ⊂ σ(Jk0 (0)) for any k ∈ NN −1 \ { N2 } and for some δ > 0. If k = N2 then we obtain σ(Jk0 (t)) = σpp (H b (t)). Moreover, we have that σpp (H b (t)) ∈ ((1 − δ, 1 + δ) ∪ (−1 − δ, −1 + δ)) for small t, then in order to prove i) we have to show that there are no gaps in b,± small neighborhood of {±1}, i.e. we need to show that zk,n 6∈ {±1}, i.e. |2|ck | + τn | 6= 1, n ∈ Np or

or

  2 cos kπ + cos nπ + i sin nπ = 1 + 4 cos kπ cos nπ + cos kπ 6= 1 N p p N p N cos

kπ nπ + cos 6= 0, p N

(5.3)

N since cos kπ N 6= 0 for k 6= 2 . The identity (5.3) holds true, since p and N are coprime. ii) Consider the case σ(H b (t)) ∩ [−ρ, −r] the proof of other cases is similar. Theorem 2.3 gives     σ(Jk0 (0)) = − 2|ck | − 1, −|2|ck | − 1| ∪ |2|ck | − 1|, 2|ck | + 1 , k ∈ ZN , (5.4)

0 (0)) = [−3, −1] ∪ [1, 3] and which yields σ(JN 0 [−ρ − δ, −r + δ] ⊂ JN (0),

[−ρ − δ, −r + δ] ∩ Jn0 (0) = ∅, k ∈ NN −1

for some small δ > 0 (see (2.5) and before (2.27)). Then the spectrum in σ(J b (t)) ∩ [−ρ, −r] has multiplicity 2 for all sufficiently small t and b. Also, using (2.23), we obtain 0 (0) ∈ [−ρ, −r], 1 6 n 6 p and z 0 (0) 6∈ [−ρ, −r] for n > p , which yield(2.27). zN,n N,n N N The inequality |γN,n (t)| > 0 follows from Proposition 2.4. iii) follows from (5.4), since σ(Jk0 (0)) ∩ [−r, r] = ∅ for any k and sufficiently small r > 0. The proof of iv) is similar to the proof of ii). Proof of Theorem 2.6. Recall that (Jkb (t)y)n = an−1 yn−1 + an yn+1 + tvn yn , y = (yn )n∈Z ∈ ℓ2 , n ∈ Z, where a2n = 2|ck |, a2n+1 = 1. Using (3.10) we obtain Kk (τ, a, tv) = t(B + εK 0 (τ, a)),

B = diag(vj )2p 1 as ε =

1 → 0, t

a = 2|ck |.

Then the perturbation theory [19] for B + εK 0 (τ, a) gives λn (t) = t(vn + εun,n + αn ε2 + O(ε3 )),

αn = −

X un,j uj,n j6=n

vj − vn

,

uj,n = (e0j , K 0 (τ )e0n ),

2p where Be0j = vj e0j and the vector e0j = (δj,n )2p n=1 ∈ C . The definition of uj,n yields

un−1,n = un,n−1 = an−1 , un+1,n = un,n+1 = an , and un,j = 0 if |j − n| 6= 1. b,± These imply (2.30), since zk,n (t) are eigenvalues of Kk (±1, a, tv).

290

Evgeny Korotyaev and Anton Kutsenko

We show (2.31) for the case v1 < · · · < v2p , the proof of other cases is similar. Using the identity 2∆k (z, t) ≡ 2∆(z, a, tv) = w−1 det(zI2p − K(i, a, tv)) (see reasoning Q p 2p between (3.7) and (3.12)), where w = 2p 1 an = |2ck | ε , we obtain 2∆k (z, t) = w−1 det(zεI2p −B+εK 0 (i, a)) =

F0 (λ) + εF (λ, ε) , |2ck |p ε2p

λ=

z = zε, (5.5) t

Q where F0 (λ) = det(λI2p −B) = 2p j=1 (λ−vj ) and F is some polynomial of two variables λ, ε. b,+ b,− Let λ+ (ε) = zk,n−1 (t)/t, λ− (ε) = zk,n (t)/t for some n ∈ N2p . These λ± (ε) are the solutions of the equation F (λ± , ε) = ±1, where F (λ, ε) = ∆k (z, t) and (2.30) yields λ± (ε) = vn + O(ε2 ) as ε → 0. By the perturbation theory [19], the functions λ± (ε) are analytic in some disk {|ε| < r}, r > 0. Now we repeat the arguments from the proof of Theorem 2.2 after (3.15). Differentiating (5.5) 2p times we obtain (λ+ )(j) (0) = (λ− )(j) (0), j < 2p, (λ+ )(2p) (0) − (λ− )(2p) (0) =

(2p)! , En

i.e., |λ+ (ε) − λ− (ε)| =

ε2p + O(ε2p+1 ) as ε → 0, En

En =

Y 1 (vn − vj ), 2|2ck |p j6=n

b,+ b,− which yields (2.31), since λ+ (ε) = zk,n−1 (t)/t, λ− (ε) = zk,n (t)/t. Using (2.30), we obtain (2.32). If |ck | 6= |ck′ |, then (2.31) implies  b,+    b,− b,− b σk,n (t) ∩ σkb ′ ,n (t) = zk,n−1 (t), zk,n (t) ∩ zkb,+ ′ ,n−1 (t), zk ′ ,n (t) = ∅

for sufficiently large t. This yields (2.33) for the second case. If k 6= k ′ for k, k ′ ∈ NN and π b∈ / 2N N, then |ck | 6= |ck′ | and we obtain (2.33) for the first case. Using (2.33), we obtain σ(Jkb (t)) ∩ σ(Jkb′ (t)) = ∅, k 6= k ′ . Then σ(Jkb (t)) has multib (t) has multiplicity 2. plicity 2 and σk,n

6.

Armchair Nanotube

We consider the Schr¨odinger operator H b (v) with a real periodic potential v on the armchair nanotube Γ ⊂ R3 in a uniform magnetic field B = B(0, 0, 1) ∈ R3 , B ∈ R. Our model armchair nanotube Γ is a graph (see Fig. 6.) embedded in R3 oriented in the z-direction e0 . Γ is a set of vertices (atoms) rω connecting by bonds (edges) and Γ = ∪ω∈Zrω , ω = (n, j, k) ∈ Z = Z × {0, 1} × ZN ,

ZN = Z/(N Z),

(6.1)

where N is a number of vertices in any ring of nanotube. The detail information about 3D coordinates of rω and about constants bj see in Appendix. Introduce the P Hilbert space ℓ2 (Γ) of functions f = (fω )ω∈Z on Γ equipped with the norm kf k2ℓ2 (Γ) = ω∈Z |fω |2 . The tight-binding Hamiltonian H b (where b = (b1 , b2 , b3 ))

Zigzag and Armchair Nanotubes in External Fields (3,1,0)

291

(3,1,1) (3,0,1)

(3,0,2)

(2,1,0)

(2,1,1) (2,0,1)

(2,0,2)

(1,1,0)

(1,1,1) (1,0,1) (0,1,0)

b1

b2

(0,0,1) (−1,1,0)

b1

b3

b2

Figure 5. A piece of armchair nanotube.

on the nanotube Γ has the form H b = H0b + V on ℓ2 (Γ), where H0b is the Hamiltonian of the nanotube in the magnetic field and is given by (H0b f )n,0,k = eib2 fn+1,1,k + eib1 fn−1,1,k−1 + eib3 fn,1,k , (H0b f )n,1,k = e−ib1 fn+1,0,k+1 + e−ib2 fn−1,0,k + e−ib3 fn,0,k ,

f = (fω )ω∈Z,

ω = (n, j, k) ∈ Z × {0, 1} × ZN

(6.2)

and the operator V corresponding to the external electric potential is given by (V f )ω = Vω fω ,

where Vn,0,k = v2n ,

Vn,1,k = v2n+1 ,

k ∈ ZN ,

v = (vn )n∈Z ∈ ℓ∞ . (6.3)

1. The operator H b is an orthogonal sum of Jacobi operators. Theorem 6.1. Let v = (vn )n∈Z ∈ ℓ∞ . Then the operator H b is unitarily equivalent to the b b 2 2 operator ⊕N 1 Jk , where Jk is a Jacobi operator, acting on ℓ (Z) ⊕ ℓ (Z) and given by (Jkb y)n = ayn−1 + a∗ yn+1 + dn yn , y = (yn )n∈Z ∈ ℓ2 ⊕ ℓ2 ,     ib3 2πi 0 eib1 sk v e 2n , s = e N , dn ≡ a ≡ ak = , e−ib2 0 e−ib3 v2n+1

n ∈ Z.

(6.4)

Each Jkb has absolutely continuous spectrum. Proof. We give compressed Proof because this one is similar to the Proof of Theorem 2.1. Define the operator J b : (ℓ2 )2N → (ℓ2 )2N acting on a vector-valued function ψ = 2N , by (ψn )n∈Z ∈ (ℓ2 )2N , ψn = (fn,0,k , fn,1,k )⊤ k∈ZN ∈ C (J b ψ)n = (H b f )n,0,k , (H b f )n,1,k


⊤

k∈ZN

.

(6.5)

292

Evgeny Korotyaev and Anton Kutsenko

Figure 6. 3D model of armchair nanotube.

N Define the operator S in CN by Su = (uN , u1 , . . . , uN −1 )⊤ , u = (un )N 1 ∈ C . Using (6.5), (6.2), (6.3) and S∗ = S−1 we obtain

(J b ψ)n = Aψn−1 + A∗ ψn+1 + Cn ψn , where A=



0 e−ib2 IN

eib1 S 0



,

The unitary operator S has the form S = Se ek = sk eek , eek =

Cn =

PN 1

1 N

1 2



v2n IN e−ib3 IN

eib3 IN v2n+1 IN

(6.6) 

.

(6.7)

sk Pk , where

(1, s−k , s−2k , . . . , s−kN +k )⊤

2π

N is an eigenvector (recall that s = ei N ); Pk u = eek (u, eek ), u = (un )N 1 ∈ C is a projector. e k ψ = (Pk ψn )n∈Z . The operators e Define the operators e Sψ = (Sψn )n∈Z and P S and J b be be commute, then J b = ⊕N 1 (J Pk ). Using (6.6), (6.7) we deduce that J Pk is unitarily b equivalent to the operator Jk .

Below we use notation a ≡ a(b, v) and dn = dn (b, v).

2. The spectrum of unperturbed operator H 0 . We consider the case when all v2n+1 = −v2n = ve and b = 0, i.e. all Jk0 are 1-periodic Jacobi matrices. For this case we denote e a = a(0, v), de = dn (0, v). The monodromy matrix for Jk0 is Mk (z) =



0 I2 2 e −(e a) e a(z − d)



 0 0 1 0   0 0 0 1 . = k k  −sk 0 −s (z − ve)s  0 −sk z + ve −1 

Zigzag and Armchair Nanotubes in External Fields

293

The determinant is Dk (z, τ ) = det(Mk (z) − τ I4 ) = τ 4 + τ 3 (sk + 1) + τ 2 sk (3 + ve2 − z 2 ) + τ sk (sk + 1) + s2k   2k 4 3 2 2 2 =s τe + τe 2ck + τe (3 + ve − z ) + τe2ck + 1   2k 2 −1 2 −1 2 2 = s τe (e τ + τe ) + 2ck (e τ + τe ) + 1 + ve − z = k

= s2k τe2 (e τ + τe−1 − ∆− τ + τe−1 − ∆+ e = s− 2 τ and k (z))(e k (z)), where τ q πk πk ∆± (z) = ± z 2 − ve2 − s2k − ck , where ck = cos , sk = sin . (6.8) k N N

The spectrum of Jk0 is

 σ(Jk0 ) = z ∈ R : Dk (z, τ ) = 0 for some τ ∈ S1 =  1 2 2 1 = z ∈ R : −2 6 ∆± k (z) 6 2 = (−σk ) ∪ (−σk ) ∪ (σk ) ∪ (σk ), where hq i hq i p p 1 2 2 2 2 σk = ve + sk , 5 + ve + 4ck , σk = ve2 + s2k , 5 + ve2 − 4ck .

(6.9)

The spectrum of H 0 is 0

σ(H ) =

N [

k=1

i h p p v |, |e v |). σ(Jk0 ) = − 9 + ve2 , 9 + ve2 \ (−|e

(6.10)

In particular case, if ve = 0, then σ(H 0 ) = [−3, 3].

3. Small 2p-periodic real potentials. We consider the case b = 0. Firstly let J ≡ J(q) : ℓ2 (Z) → ℓ2 (Z) is a p-periodic Shr¨odinger operator, i.e (Jf )n = fn−1 + fn+1 + qn fn , f = (fn )n∈Z ∈ ℓ2 (Z), where q = (qn )pn=1 ∈ ℓ∞ R (Z) and qn+p = qn for all n ∈ Z. It is well known (see [8]), that the spectrum of this operator is absolutely continuous and has a form σ(J) = σac (J) = ∪p1 σn ,

+ σn = [zn−1 , zn− ],

n ∈ Np ,

z0+ < z1− 6 z1+ < z2− 6 z2+ < · · · < zp− .

(6.11) (6.12)

We denote zn± (q) ≡ zn± . Also we introduce spectral gaps γn ≡ γn (q) as γn = (zn− , zn+ ),

n ∈ Np−1 .

(6.13)

πn , n ∈ Np−1 , −z0+ (0) = zp− (0) = 2. p

(6.14)

If q = 0 then zn± (0) = −2 cos

For sufficiently small q we have (see [8]) zn± (q) = −2 cos

πn + qˆ0 ± |ˆ qn | + O(kqk2 ), q → 0, n ∈ Np−1 , p

(6.15)

294

Evgeny Korotyaev and Anton Kutsenko z0+ (q) = −2 + qˆ0 + O(kqk2 ), zp− (q) = 2 + qˆ0 + O(kqk2 ), q → 0,

(6.16)

iπn

p . where we denote qˆn = (q, eˆn ), eˆn = p−1 (τn2j )p−1 j=0 , τn = e p Introduce the set Ξp ⊂ R by nX o Ξp = αn (ˆ en + eˆp−n ), all αn 6= 0 .

(6.17)

n6 p2

0 (v) (zigzag) and H 0 (v) (armchair). Now we compare the spectrum of Hzi ar

Theorem 6.2. i) Let v2n = v2n+1 , vn+2p = vn for all n ∈ Z. Let v ev = (v2n )p1 and J ≡ J(v ev ), then 0 (σ(J) + 1) ∪ (σ(J) − 1) ⊂ σ(Har ). (6.18) ii) Let N ∈ 3Z and vn+p = vn for all n ∈ Z. Let v = (vn )p1 and J ≡ J(v), then 0 σ(J) ⊂ σ(Hzi ).

(6.19)

Proof. i) In our case (see (6.4)) we have     −1 0 0 1 N C ∗, =C a (0, v) = 0 1 1 0 dn (0, v) =



v2n 1 1 v2n



=C



v2n − 1 0 0 v2n + 1



C∗

0 (see (6.4)) unitarily equivalent to for some unitary matrix C (CC ∗ = I2 ). Then JN ev ev (J(v ) − I) ⊕ (J(v ) + I), where I is identity operator on ℓ2 (Z). The statement ii) was proved in Theorem 2.3 (see also Remark 2) on page 5).

For example we describe the spectrum of H 0 (v) (armchair) near z = 0 and near z = ±3 for small potentials v (recall that σ(H 0 (0)) = [−3, 3]). ev = (v )p . Let Theorem 6.3. Let v2n = v2n+1 n for all n ∈ Z and denote v 2n 1  , vn+2p = v 1 1 π p > 2N > 4 and r± = 2 cos 3 ∓ 2N ∓ 6p − 1. Then for sufficiently small v we have 0

σ(H ) ∩ [r− , r+ ] =



[r− , r+ ] \

[

p |n− p3 |6 2N

  (γn + 1) ∪ [r− , r+ ] \

[

p |n− 2p |6 2N 3

 (γn − 1) , (6.20)

where first set and second set in the union has multiplicity 2. Also let   π 1 1 re− = 1 + 2 cos , re+ = 1 + 2 cos . + 2N 6p 6p

Then for sufficiently small v we have

σ(H 0 ) ∩ [−e r+ , −e r− ] = [−e r+ , −e r− ] \

[

p 16n6 2N

(γn − 1),

(6.21)

Zigzag and Armchair Nanotubes in External Fields [ (γn + 1), σ(H 0 ) ∩ [e r− , re+ ] = [e r− , re+ ] \

295 (6.22)

p 6n6p−1 p− 2N

where set on the right side has multiplicity 2. Moreover if v ev ∈ Ξp then all |γn | = 6 0 in (6.20)–(6.22). Proof. We consider only the statement (6.21), the proof of other statements is similar. We have (see (6.14)) 1

−3 > −e r+ > z1− (0) − 1 > z[+p ] (0) − 1 > −e r− > z[+p

]+1 2N

2N

(0) − 1 > (5 + 4c1 ) 2 > −1.

This inequalities shows (see (6.18)) that for sufficiently small v we have 0 σ(H 0 ) ∩ [−e r+ , −e r− ] = σ(JN ) ∩ [−e r+ , −e r− ] = (σ(J) − 1) ∩ [−e r+ , −e r− ],

(6.23)

since [−e r+ , −e r− ] ∩ Jk0 = ∅, k ∈ NN −1 (see (6.4), (6.9)). Using identities (6.23) and (6.11)-(6.15) we obtain (6.21). Let v be sufficiently small. We denote by Gar , Gzi is a maximal possible number 0 ) ∩ [−3, −3 + α] and of the open gaps on the edge of spectrum, i.e. in the set σ(Har 0 σ(Hzi ) ∩ [−3, −3 + α] respectively, where α is a some sufficiently small value. Now we estimate Gar , Gzi for sufficiently large period 2p. Corollary 6.4. For sufficiently large p we have   p α Gar = arccos 1 − + o(p), π 2   p 6α − α2 Gzi = arccos 1 − + o(p) π 4 as p → ∞. 4. Large 4p-periodic real potentials. Now we consider Shr¨odinger operator H on armchair nanotube with large periodic potentials. We show that in this case the structure of the spectrum is the same in the essential as for zigzag nanotube (see Theorem 2.6), but the Proofs are different. Theorem 6.5. i) Let v = (vn )+∞ −∞ be a 4p-periodic (p > 2) real potential such that vi 6= vj , 1 6 i 6= j 6 4p. Let σ(t) = σ(H b (tv)) and σk (t) = σ(Jk (tv)). Then σ(t) =

N [

σk (t), σk (t) =

4p [

σk,j (t),

(6.24)

j=1

k=1

where intervals σk,j (t) satisfy |σk,j | =

t2p−1

Q

4 n∈(Qi \j) (vj − vn )

+ O(t−2p ),

ej | = O(t−3 ), t → ∞. max |λ − λ

λ∈σk,j

(6.25) ej are defined in (6.46) and j ∈ Qi are defined in (6.33) (j determines i = 1, 2 Here λ uniquely). 1 and b is sufficiently small, then all intervals σ (t) Moreover, if v11 < v21 < · · · < v4p k,j are disjoint for sufficiently large t.

296

Evgeny Korotyaev and Anton Kutsenko

Proof. Recall that a ≡ ak = dn ≡





v2n e−ib3

0 e−ib2 eib3 v2n+1

 2πi eib1 sk , s=eN , 0  , n ∈ Z, aa∗ = I4 .

(6.26) (6.27)

Also we use notation dn ≡ dn (v), where v = (v1 , . . . , v4p ). The monodromy matrix for operator Jk (v) is   0 I2 Mk ≡ Mk (z) ≡ Mk (z, v) = M2p . . . M1 , Mn = . (6.28) −a2k ak (z − dn ) It is well known that  σ(Jkb ) = z : det(Mk (z) − τ ) = 0 for some τ ∈ S1 .

Using (6.28) we obtain     P1 P2 0 0 + , Mk = 0 a2p (z − d2p ) . . . a1 (z − d1 ) P3 P4

(6.29)

(6.30)

where Pj ≡ Pj (z − d2p , . . . , z − d1 ) is a 2 × 2 matrix polynomial and deg Pj < 2p for all j = 1, . . . , 4. Also, using (6.27) and periodicity of v, we deduce that a2p (z − d2p ) . . . a1 (z − d1 )     Q (z − vn ) 0 Q Q 1 2 p n∈Q 1 Q + , = (det ak ) 0 Q3 Q4 n∈Q2 (z − vn )

(6.31) (6.32)

where Qj ≡ Qj (z − v4p , . . . , z − v1 ) are polynomials and deg Qj < 2p, sets Qj are p−1 Q1 = ∪p−1 j=0 {4j + 1, 4j + 2}, Q2 = N4p \ Q1 = ∪j=0 {4j + 3, 4j + 4}.

(6.33)

Let Dk (z, τ ) ≡ Dk (z, τ, v) = det(Mk − τ I2 ). Using (6.30)-(6.33) we get Y Y (z − vn ) + R1 ) (z − vn ) + Dk (z, τ ) = τ 4 + det apk τ 3 ( n∈Q1

4p Y 2p 2 + det ak τ ( (z n=1

where polynomials

n∈Q2

e1 + R e2 , − vn ) + R2 ) + τ R

R1 ≡ R1 (z − v4p , . . . , z − v1 ), deg R1 < 2p,

R2 ≡ R2 (z − v4p , . . . , z − v1 ), deg R2 < 4p,

e1 ≡ R e1 (z − v4p , . . . , z − v1 ), R e2 ≡ R2 (z − v4p , . . . , z − v1 ). R

(6.34)

(6.35) (6.36)

are not depended on τ . Let τ ∈ S1 , z ∈ R, then it is well known, that the polynomial e k (z) ≡ D e k (z, τ ) ≡ D e k (z, τ, v) = (det a−2p )τ −2 Dk (z, τ ) D k

Zigzag and Armchair Nanotubes in External Fields =

4p Y

n=1

(z − vn ) + O(z 4p−1 ), z → ∞.

297 (6.37)

is real, since it has only real zeroes, because the spectrum of Jk is real. Let τ ∈ S1 , z ∈ R, e k (z, τ ) ≡ D e k (z, τ ), a∗ = a−1 we deduce that then using (6.37), (6.34) and D k k  Y  Y 3p e1 = det a e2 = det a4p . (z − v ) + R (z − v ) + R (6.38) , R n n 1 k k n∈Q2

n∈Q1

Substituting (6.38) into (6.34) and using (6.37) we deduce that e k (z) = D

4p Y

n=1

 Y  Y (z − vn ) + R2 + 2 Re(τ det a−p ) (z − v ) + (z − v ) (6.39) n n k n∈Q1

n∈Q2

−2p 2 +2 Re(τ det a−p k R1 ) + 2 Re(τ det ak ).

e k (z, τ, tv). Now we denote a = 1t , λ = zt and Fk (λ) ≡ Fk (λ, a) ≡ Fk (λ, τ, a) = t−4p D Then, using (6.39), (6.35), we deduce that Fk =

4p Y

n=1

(λ − vn ) + aG1 (λ, a) + a2p 2 Re(τ det a−p k )

 Y  Y × (λ − vn ) + a2p+1 G2 (λ, a), (λ − vn ) +

(6.40)

n∈Q2

n∈Q1

where G1 , G2 are polynomials and G1 is not depended on τ . Let λj (a) ≡ λj (a, τ ) be zeroes of Fk (λ) such that λj (0) = vj1 , these are analytic functions. Using similar arguments as in ”zigzag case”, we deduce that derivatives (λj )(r) (0) are not depended on τ for all j ∈ N4p , r ∈ N2p−1 and first derivative which depends on τ is −2 Re(τ det a−p k ) + R3 , where j ∈ Qi for some i = 1, 2 (λj )(2p) (0) = Q n∈(Qi \j) (vj − vn )

(6.41)

and R3 doesn’t depend on τ . These yield |λj (a, S1 )| = Q

4a2p + O(a2p+1 ), a → 0, (v − v ) j n n∈(Qi \j)

(6.42)

where j ∈ Qi for some i = 1, 2. Let zj (t) ≡ zj (t, τ ), j ∈ N4p be zeroes of Dk (z, τ, tv), then zj = tλj and |σk,j (t)| = |zj (t, S1 )| =

t2p−1

Q

4 n∈(Qi \j) (vj

− vn )

+ O(t−2p ), t → ∞,

(6.43)

4p×4p matrices L (τ ) ≡ where the spectrum σ(Jk (tv)) = ∪4p k 1 σk,j (t). Introduce the C Lk (τ, t) and Bk (τ ) ≡ Bk (τ, t) by   d a∗k 0 . . . aτk  ak d a∗ . . . 0  k    Lk = Bk + diag(tv) =  (6.44)  0 ak d . . . 0  + diag(tv),  .....................  τ a∗k 0 . . . ak d

298

Evgeny Korotyaev and Anton Kutsenko

where d=



0 e−ib3

eib3 0



.

(6.45)

ej (t) ≡ λ ej (t, τ ) be eigenvalues of Lk , it is well known, that σk,j (t) = λj (t, S1 ). Then Let λ perturbation theory gives us ej = vj t + (Bk ej , ej ) − 1 λ t

X

n∈N4p \j

(vn − vj )|(Bk ej , en )|2 + · · · =

(6.46)

 Vj,−1 + Vj,1 + Vj,3   vj t −   t      − sek (Vj,−2 Vj,−1 Vj,1 + Vj,1 Vj,2 Vj,3 ) + O(t−3 ) j ∈ 2N t2 , t → ∞, = V + V + V j,−1 j,1  1 t − j,−3  v  j  t    sek (Vj,−3 Vj,−2 Vj,−1 + Vj,−1 Vj,1 Vj,2 )   − + O(t−3 ) j ∈ 2N + 1 t2 where Vj,k = (vj+k − vj )−1 , sek = 2 Re(sk ei(b1 +b2 −2b3 ) ).

(6.47)

The theorem is proved.

7.

Appendix

Below we consider the unperturbed Jacobi operator J 0 (a) = J(a, 0) given by (see (3.7)) (J 0 (a)y)n = an−1 yn−1 + an yn+1 , a2n = a > 0, a2n+1 = 1, n ∈ Z, y = (yn )n∈Z . (7.1) Lemma 7.1. The eigenvalues zns and the eigenvectors esn , (n, s) ∈ Np × N2 of the matrix K 0 (eiφ , a), φ ∈ R (given by (3.10)) have the forms: , then if εn = a + eirn 6= 0, rn = φ+2πn p εn . |εn | (7.2) If εn = 0, then the eigenvalue zn1 = zn2 = 0 has the multiplicity two and the corresponding orthogonal eigenvectors are given by 1

2p s s ijrn , en,2j+1 = eijrn zns = (−1)s |εn |, esn = (2p)− 2 (esn,j )2p j=1 ∈ C , en,2j = (−1) e

e1n = (1, 1, −1, −1, 1, 1, . . . )⊤ , e2n = (1, −1, −1, 1, 1, −1, . . . )⊤ ∈ C2p . Proof. We need the simple fact. Let K 0 (τ )e = ze for some z, τ and the eigenvector −1 f , f e = (fn )2p 2p 2p+1 = τ f1 . Then 1 . Introduce two numbers f0 = τ M2 (z)(fn−1 , fn )⊤ = (fn+1 , fn+2 )⊤ , M2 (z)p (f0 , f1 )⊤ = (f2p , f2p+1 )⊤ = τ (f0 , f1 )⊤ , and (f0 , f1 )⊤ is the eigenvector of the monodromy matrix M2 given by (4.) at v = 0.

Zigzag and Armchair Nanotubes in External Fields

299

Conversely, let M2 (z1 )(f0 , f1 )⊤ = τ (f0 , f1 )⊤ for some τ, z1 . We introduce the vectors (fn+1 , fn+2 )⊤ = M2 (z1 )(fn−1 , fn )⊤ , n ∈ N2p−2 . Then K 0 (τ, a)e1 = z1 e1 ,

where

e1 = (fn )2p 1 .

(7.3)

Recall that (see (4.3)) the Lyapunov function ∆2 (corresponding to M2 ) is given by ∆ = 1 1 2 2 2 Tr M2 (z) = 2a (z − a − 1). Using these arguments we will determine the eigenvalues and the eigenvectors of the matrix K 0 (τ, a). Firstly, let zs = zs (r) be solutions of the equation ∆(z) = cos r for fixed r ∈ R. Then (zs )2 = a2 +2a cos r+1 = |ε|2 , ε = a+eir , which yields z1 = −|ε|, z2 = |ε|. We will determine the eigenvectors of the monodromy matrix M2 (zs ), s = 1, 2 for the eigenvalue τ = eir , since ∆(zs ) = cos r. Firstly, if ε 6= 0, then we obtain     −a − eir zs −ε (−1)s |ε| ir , M2 (zs ) − e I2 = = (−1)s+1 |ε| ε −zs a + 2 cos r − eir and the corresponding eigenvectors are given by  s    (−1)s η1 s = η = , s = 1, 2. ε η2s |ε| Define the vectors es = (esn )2p 1 by  s   s   s  e2j e0 e0 j s = eijr η s (r). = M2 (zs ) = η (r), es2j+1 es1 es1

(7.4)

(7.5)

Then using (7.3) we deduce that K 0 (eipr , a)es = zs es , where identities (7.4), (7.5) give the components of es by s s ijr s ijr ε es = (esj )2p , j=1 , e2j = (−1) e , e2j−1 = e |ε| which yields (7.2), since solutions of the equation eipr = eiφ has the form rn = φp + 2πn p , n ∈ Np . Secondly, if ε = a + eir = 0, then we deduce that a = 1, eir = −1, zs = 0, s = 1, 2 and the matrix M2 (zs ) − eir I2 = 0. The corresponding eigenvectors have the forms η 1 = (−1, 1)⊤ , η 2 = (1, 1)⊤ and using arguments as above, we obtain the proof of the case ε = 0. Corollary 7.2. The spectrum of the operator J 0 (a) given by (7.1) has the form + − + 0 0 − ± ± ± σ(J 0 (a)) = ∪2p n=1 σn , σn = [λn−1 , λn ], λn ≡ λn (a) = zn (a, 0), λ2p = −λ0 = a + 1, i πn p

± λ± n = νn |a + e

|, νn± = (±1)δn,p sign(n − p), n ∈ N2p−1 , sign(0) = 1,

± 0 0 where λ± 2n (and λ2n+1 ) are all eigenvalue of the matrix K (1, a) (and K (−1, a)) given by 0 0 (3.10). Corresponding eigenvectors of K (1, a) (and K (−1, a)) are given by

Zn± ≡ Zn± (a) =

1 (2p)

1 2

± 2p ± ± (fj,n )j=1 , f2j,n = νn± τn±j , f2j+1,n = τn±j e±i arg(a+τn ) ,

300

Evgeny Korotyaev and Anton Kutsenko τn = e

iπn p

, j ∈ Np , a + τn 6= 0,

(7.6)

and 1

1

Zn+ = (2p)− 2 (1, 1, −1, −1, 1, 1, . . . )⊤ , Zn− = (2p)− 2 (1, −1, −1, 1, 1, −1, . . . )⊤ , a+τn = 0, + − + and λ− n (a) = λn (a), n ∈ N2p−1 \ {p} has multiplicity two. Also λp (a) < λp (a), a 6= 1 − + + − and λp (1) = λp (1). The vectors Zn and Zn , n ∈ N2p−1 are orthogonal.

Proof follows from Lemma 7.1.P In particular, we have the following identity (2p)hZn+ , Zn− i = (1 + e2i arg(a+τn ) ) pj=1 τn2j = 0.

3D coordinates of rω and bj in the case of armchair nanotube. We rewrite similar formulas from [2] adapted for our case rn,j,k = (R cos αn,j,k , R sin αn,j,k , nh), n ∈ Z, j ∈ {0, 1}, k ∈ ZN , where

(7.7)

2π(k − n) 2π(k − n) + α0,j , α2n+1,j,k = + α1,j , N N π e α0,1 = 2π , α1,0 = βe − α α0,0 = 2β, e, α1,1 = , N N q π + 54 cos N 1 1 e , sin β = , R = , sin α e= π 2R R sin N q p jR)2 − 1, e j = 1, 2, h = 2 + R1 R2 − 2R2 , Rej = (e

α2n,j,k =

and the magnetic constants are

b1 = b2 =

B(R2 − R1 ) BR2 , b3 = − . 4 4

(7.8)

Acknowledgments The research was partially supported by EPSRC grant EP/D054621. The various parts of this paper were written at Mathematical Institute of the Tsukuba Univ., Japan and Institut f¨ur Mathematik, Humboldt Universit¨at zu Berlin. The first author E.K. is grateful to the Institutes for the hospitality.

References [1] J. E. Avron, A. Raveh, B. Zur, Adiabatic quantum transport in multiply connected systems. Rev. Modern Phys. 60 (1988), No. 4, 873–915. [2] A. Badanin, E. Korotyaev, Magnetic Schr¨odinger operators on armchair nanotubes. http://arxiv.org/abs/0804.0171 [3] P. Exner, A duality between Schr¨odinger operators on graphs and certain Jacobi matrices. Ann. Inst. H. Poincar´e Phys. Theor. 66 (1997), No. 4, 359–371.

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[4] P. Harris, Carbon Nanotubes and Related Structures. Cambridge Univ. Press., Cambridge, 1999. [5] Iijima S. Helical microtubules of graphitic carbon. Nature. 354 (1991), 56–58. [6] E. Korotyaev, Effective masses for zigzag nanotubes in magnetic fields. Lett. Math. Phys. 83 (2008), No 1, 83–95. [7] E. Korotyaev, I. Krasovsky, Spectral estimates for periodic Jacobi matrices. Commun. Math. Phys. 234 (2003), 517–532. [8] E. Korotyaev, A. Kutsenko, Inverse problem for the discrete 1D Schr¨odinger operator with small periodic potentials. Commun. Math. Phys. 261 (2006), 673–692. [9] E. Korotyaev, A. Kutsenko, Lyapunov functions for periodic matrix-valued Jacobi operators, will be published in American Mathematical Society Translations. [10] E. Korotyaev, A. Kutsenko, Armchair nanotubes in magnetic field, tight-binding models. (in preparation). [11] E. Korotyaev, I. Lobanov, Schr¨odinger operators on zigzag periodic graphs. Ann. Henri Poincar´e 8 (2007), 1151–1176. [12] E. Korotyaev, I. Lobanov, Zigzag periodic nanotube in magnetic field. (preprint 2006). [13] P. Kuchment, O. Post, On the spectra of carbon nano-structures Commun. Math. Phys. 275 (2007), 805–826. [14] Y. Last, On the measure of gaps and spectra for discrete 1D Schrodinger operators. Comm. Math. Phys. 149 (1992), No. 2, 347–360. [15] van P. Moerbeke, The spectrum of Jacobi matrices. Invent. Math. 37 (1976), No. 1, 45–81. [16] D. S. Novikov, Electron properties of carbon nanotubes in a periodic potential. Physical Rev. B 72 (2005), 235428–1–22. [17] L. Pauling, The diamagnetic anisotropy of aromatic molecules. Journal of Chemical Physics, 4 (1936), 673–677. [18] K. Pankrashkin, Spectra of Schr¨odinger operators on equilateral quantum graphs. Lett. Math. Phys. 77 (2006), 139–154. [19] M. Reed, B. Simon, Methods of modern mathematical physics. IV. Analysis of operators. Academic Press, New York-London, 1978. [20] K. Ruedenberg and C. W. Scherr, Free-electron network model for conjugated systems. I. Theory. The Journal of Chemical Physics 21 (1953), 1565–1581. [21] R. Saito, G. Dresselhaus, M. Dresselhaus, Physical properties of carbon nanotubes. Imperial College Press, 1998.

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[22] B. Simon, Orthogonal polynomials on the unit circle. Part 1. Classical theory. American Mathematical Society Colloquium Publications, 54, Part 1. American Mathematical Society, Providence, RI, 2005. [23] B. Simon, Orthogonal polynomials on the unit circle. Part 2. Spectral theory. American Mathematical Society Colloquium Publications, 54, Part 2. American Mathematical Society, Providence, RI, 2005. [24] P. van Mouche, Spectral asymptotics of periodic discrete Schrodinger operators. I. Asymptotic Anal. 11 (1995), No. 3, 263–287.

In: Advances in Mathematics Research, Volume 10 ISBN 978-1-60876-265-1 c 2010 Nova Science Publishers, Inc. Editor: Albert R. Baswell, pp. 303-314

Chapter 7

G LOBAL W ELL -P OSEDNESS AND L IMIT B EHAVIOR FOR S OME N ONLINEAR E VOLUTION E QUATION Lijia Han∗ Institute of Applied Physics and Computational Mathematics, PO Box 8009, Beijing 100088, China

Abstract In this chapter, we will talk about global well-posedness and limit behavior for the Cauchy problem of some nonlinear evolution equations with derivative nonlinear term. We will introduce some results and show some method for solving this kind of problems. In the first section, we introduce generalized finite-depth-fluid equation  uk+1  = 0, u(0) = u0 , ∂t u − G(∂x2 u) + ∂x k+1   1 where G = −iF −1 coth(2πδξ) − 2πδξ F , k > 4 is an integer. We will show that its solution will converge to those of the Benjamin-Ono and KdV equations as the depth parameter δ → ∞ and δ → 0, respectively. I will introduce some basic idea for solving global well-posedness and limit behavior. Meanwhile, I will emphasize Kato smoothing effect method with Besov space and Littlewood-Paley decomposition here. In the second section, we introduce the Cauchy problem for the generalized Benjamin-Ono-Burgers equation  uk+1  ∂t u − H(uxx ) − νuxx + ∂x = 0, u(0) = u0 , k+1 where H denotes Hilbert transform, k > 6 is an integer. We will show its solution will converge to that of generalized Benjamin-Ono equation. We will emphasize some new estimates here. In the third section, we introduce the Cauchy problem for the Korteweg-de VriesBurgers equation ut + uxxx + ǫ|∂x |2 u + (u2 )x = 0,

u(0) = u0 ,

where ǫ > 0. We will introduce the global well-posedness and inviscid limit for Korteweg-de Vries equation obtained in [10]. We will emphasize a l1 variant Bougaintype space method here. ∗

E-mail address: [email protected]

304

1.

Lijia Han

Generalized Finite-Depth-Fluid Equation

In this section we introduce the Cauchy problem for the generalized Finite-depth-fluid (FDF) equation: ∂t u − G(∂x2 u) + ∂x

 uk+1 

= 0, u(0, x) = u0 (x), or k+1  uk+1  3 = 0, u(0, x) = u0 (x), G(∂x2 u) + ∂x ∂t u − 2πδ k+1

(1.1) (1.2)

where u(t, x) is a complex (or real) valued function of (t, x) ∈ R1+1 , k > 4 is an integer,  G(f ) = −iF −1 coth(2πδξ) −

1  F f, 2πδξ

(1.3)

δ is a positive real number which characterizes the depth of the fluid layer. Eq. (1.1) was first derived by Joseph [15, 31, 38] to describe the propagation of internal waves in the stratified fluid of finite depth. From the physical point of view, if the depth δ tends to zero and infinity, then Eqs. (1.2)1 and (1.1) reduce to the generalized Korteweg-de Vrise (KdV) equation ∂t v + ∂x3 v + ∂x

 v k+1  k+1

=0

(1.4)

and the generalized Benjamin-Ono (BO) equation ∂t v − H(∂x2 v) + ∂x

 v k+1  k+1

= 0,

(1.5)

respectively, where H := −iF −1 sign(ξ)F denotes the Hilbert transform. Using the energy method, Abdelouhab, Bona, Felland and Saut [1] studied Eq. (1.1) in the case k = 1 and they showed the global well-posedness in H s with s > 3/2, and the limit behavior as δ → ∞ and δ → 0 of the solutions of Eqs. (1.1) and (1.2) in C k (0, T ; H s−2k ) (s > 3/2) and C(0, T ; H s ) (s > 2), respectively. Using Kato smoothing effect estimates with Besov space and Littlewood-Paley decomposition method, weTshow in [11] the global sk s˜k well-posedness for Eq. (1.1) and (1.2)in critical Besov space B˙ 2,∞ B˙ 2,1 , with sk = 1/2− 2/k, s˜k = 1/2 − 1/k and u0 is sufficiently small. Meanwhile, we show the limitTbehavior sk sk +3 for the solutions of Eqs. (1.1) and (1.2) as δ → ∞ and δ → 0 in C(0, T ; B˙ 2,1 B˙ 2,1 ) s˜k −1/2 s˜k +1 ˙ ˙ and C(0, T ; B2,1 ∩ B2,1 ), for any T > 0, respectively. For 2 6 k 6 3, there remain some open problem for the global well-posedness and limit behavior for generalized FDF equaiton. Studying global well-posedness of evolution equation with derivative nonlinear term, the largest difficulty is how to absorb derivative in the nonlinear term. Here we apply Kato smoothing effect estimate and Littlewood-Paley decomposition to overcome this difficulty. We were impulsed by Molinet and Ribaud [22, 23], which solve the global well-posedness of generalized KdV and BO equations. They obtained that for k > 4, the generalized KdV 1

Under the scaling u(t, x) →

q k

3 3 u 2πδ t, 2πδ


x , Eq. (1.1) reduces to (1.2).

Global Well-Posedness and Limit Behavior. . .

305

1/2−2/k 1/2−1/k and BO equations are globally wellposed if the initial data in B˙ 2,∞ and B˙ 2,1 are sufficiently small, respectively. Recall that sk := 1/2 − 2/k (˜ sk := 1/2 − 1/k) corresponds to the critical regularity indices for the generalized KdV (BO) equation, which means that the generalized KdV (BO) equation is illposed in H s if s < sk (s < s˜k ); cf. [2], [23]. Now we take Eq.(1.1) for example to give a brief explanation to our method. Step1: Writing into integral equation We will prove our results by applying classical contraction mapping method. we can rewrite (1.1) into an integral equation: Z t 1 u = W (t)u0 − W (t − s)∂x (uk+1 (s))ds, (1.6) k+1 0   1 W (t) = F −1 eitϕ(ξ) F , ϕ(ξ) = coth(2πδξ) − ξ2. (1.7) 2πδξ

Step2: Uniform linear estimates In order to solve (1.6), we R tneed some linear estimates for the semi-group W (t) and the integral operator L : f → 0 W (t − s)∂x f (s)ds. Using Kenig, Ponce and Vega’s results in [18], we showed that W (t) satisfies the following sharp Kato smoothing effect and the maximal inequality: C kW (t)u0 kL∞ 2 6 CkP>M/2πδ u0 k ˙ −1/2 + √ kP6M/2πδ u0 k ˙ −1 , H H x Lt δ kW (t)u0 kLkx L∞ 6 Cku0 kH˙ 1/2−1/k , k > 4 t

(1.8) (1.9)

where M > 0 is a fixed constant and C is independent of δ > 0. (1.8) indicates that the Kato’s smooth effect (see [16]) is similar to the KdV equation in the low frequency and the BO equation in the high frequency, respectively. The maximal inequality (1.9) is the same one as both the KdV and the BO equations, cf. [11, 18]. On the basis of (1.8) and (1.9), together with the dual estimate method R t and the oscillatory integral techniques, we can get that the integral operator L : f → 0 W (t − s)∂x f (s)ds, has the same estimates as the KdV equation in the low frequency and as the BO equation in the high frequency, respectively, which can be found in [11]. Among them, the estimate Z t W (t − s)(∂x f )dskL∞ j > j0 (1.10) k∆j 2 6 Ck∆j f kL1 L2 , x Lt x t 0

M can absorb 1-order derivative in the nonlinear term, where j0 ∼ ln 2πδ . As our final purpose is to obtain limit behavior results, we will emphasize that, all the linear estimates must be independent of parameter δ. It is the necessary condition for solving limit behavior, meanwhile, it is also the largest difficulty in solving limit behavior problem. Step3: Constructing resolution space After establishing the linear estimates above, we need to choose a proper resolution space to treat the nonlinear term. We will Construct the resolution space based on the linear estimates. From (1.8), it is evident that the estimates for W (t) become worse than those of the KdV and BO equations, it is not expected that the resolution space for the KdV equation

306

Lijia Han

(or the BO equation) can be directly applied for the FDF equation. So we choose a more complicated resolution space E as following: M Let j0 ∼ ln 2πδ (we can assume, without loss of generality that j0 < 0), and for i = 1, 2, j( p1 + q3 − 21 )

Ai (u) = sup 2jsk · 2

i

i

A˜i (u) =

X

j( p1 + q2 − 21 )

j0 <j60

Bi (u) =

X j>0

k∆j ukLpxi Lqi ,

2jsk · 2

i

i

i

k∆j ukLpxi Lqi , t

j( p1 + q2 − 21 )

2j˜sk · 2

(1.11)

t

j6j0

i

k∆j ukLpxi Lqi , t

(1.12) (1.13)

Qi (u) = Ai (u) + A˜i (u) + Bi (u); ∞ X 2j˜sk k∆j ukL∞ N (u) = 2, t Lx

(1.14)

2j/2 2j˜sk k∆j ukL∞ 2, x Lt

(1.16)

(1.15)

j=−∞

T (u) = M (u) =

X

j>0 ∞ X

j=−∞

k∆j ukLkx L∞ . t

(1.17)

Define E = {u : kukE := Q1 (u) + Q2 (u) + N (u) + T (u) + M (u) 6 Cρ}, where sk =

1 2

− k2 , s˜k =

1 2

(1.18)

− k1 .

(p1 , q1 ) = (6k/5 , 3k) and (p2 , q2 ) = (12 , 3).

(1.19)

E contains both the KdV-type space Ai (u), Bi (u) and the BO-type space N (u), T (u), M (u) ( [11, 22, 23]). In this way the solutions can be controlled in both the low frequency and high frequency part. A˜i (u) appears to avoid the dependence of u on parameter δ. The key idea is to divide frequency space into different part. For high frequency part, we use BO-type estimates, while for low frequency part, we use KdV-type estimates. The detailed techniques of estimating the nonlinear term is complex, we use the following para product decomposition ∆j uk+1 = ∆j

∞  X

r=−∞

∆r+1 u

k X ℓ=0

 (Sr+1 u)ℓ (Sr u)k−ℓ ,

I will not introduce it here, please look at [11]. In this way, we can obtain global well-posedness result and uniform bound for the solution of FDF equation kukE 6 Cku0 kB˙ sk

2,∞

T ˙ s˜k . B2,1

(1.20)

Global Well-Posedness and Limit Behavior. . .

307

Step4: Limit behavior For the proof of limit behavior of Eq.(1.1), we resort to some ideas as in [1, 35, 36] and rewrite (1.1) as  uk+1  ∂t u − H(∂x2 u) + ∂x + H(∂x2 u) − G(∂x2 u) = 0, (1.21) k+1 and so, w = u − v satisfies the following equation: ∂t w − H(∂x2 w) +

1 ∂x (uk+1 − v k+1 ) + H(∂x2 u) − G(∂x2 u) = 0. k+1

(1.22)

Then we can regard (1.22) as a generalized BO equation and treat H(∂x2 u)−G(∂x2 u) as a part of the nonlinearity. According to the equivalent integral equation, the regularity estimates of u seems necessary in order to control H(∂x2 u) − G(∂x2 u). This is why we assume that s˜k −1/2 s˜k +1 initial data in B˙ 2,1 ∩ B˙ 2,1 . In this way, follow similar process as the proof of global well-posedness of generalized BO equation, we can finish the proof of limit behavior, as desired.

Follow the basic process introduced above, we can solve a series of problem on limit behavior or inviscid limit. The difficulty is how to obtain uniform linear estimates and constructing proper resolution space in step2 and step3. Next I will use the method above to solve inviscid limit between generalized Benjamin-Ono-Burgers equation and generalized Benjamin-Ono equation.

2.

Generalized Benjamin-Ono-Burgers Equation

In this section, we introduce Cauchy problem for the generalized Benjamin-Ono-Burgers (BOB) equation:  k+1  u ∂t u − H(uxx ) − νuxx + ∂x = 0, u(0, x) = u0 (x). (2.1) k+1 Where u(t, x) is a complex (or real) valued function of (t, x) ∈ R+ × R, R+ = [0, ∞). k > 6 is an integer, ν > 0 is a parameter. H is the Hilbert transform. Benjamin-Ono-Burgers equation was originally derived as a model in the study of internal waves in deep, stratified fluids [3], [27]. If the dissipation effects cannot be neglected, then the term −νuxx must be added [7], and the so-called Benjamin-Ono-Burgers equation ∂t u − H(uxx ) − νuxx + uux = 0

(2.2)

is obtained. Recently, Masanori Otani in [29], obtained global well-posedness for equation (2.2) in space H s , where s > −1/2. A. S. Fokas and L. Luo [8] also studied generalized BOB equation. However, their results were obtained by fixing the parameter ν. Terence Tao in [33] conjecture it feasible for the solution of (2.2) converges to Benjamin-Ono equation: ∂t u − H(uxx ) + uux = 0

(2.3)

308

Lijia Han

when ν → 0. In our recent work [12], we have obtained the following theorems: s , where s = 1 − 1 Theorem 2.1. Let k > 6, assume u0 ∈ B˙ 2,1 k and there exists a small ρ > 0 such that ku0 kB˙ s 6 ρ. Then equation (2.1) has a unique global solution u ∈ 2,1 L∞ (R+ , B˙ s ) ∩ X, and 2,1

kukL∞ (R+ ,B˙ s

2,1 )

6 Cρ

(2.4)

the constant C is independent of ν, X is defined in (2.10). 1

3− Theorem 2.2. Let u0 ∈ B˙ 2,1 k , k > 6, and there exists a small ρ > 0 such that ku0 kB˙ s 6 2,1 ρ. Assume that uν is the solution of (2.1) and v is the solution of generalized Benjamin-Ono equation (2.3). Then for any T > 0, we have

kuν − vkY . νT → 0,

ν → 0.

(2.5)

where kukY =

X j∈Z

j

2j˜sk k∆j ukL∞

L2 t∈(0,T ] x

+ 2 2 2j˜sk k∆j ukL∞ 2 x L

t∈(0,T ]

+ k∆j ukLkx L∞

t∈(0,T ]



.

(2.6) The basic idea is similar to that of generalized FDF equation introduced above. Here we only introduce new difficulty for solving this problem. Denote 2

VB (t) = F −1 eitξ|ξ| e−νtξ F,

L f (t, x) =

Z

t 0

Bν (t − τ )f (τ, x)dτ.

(2.7)

First, as Kenig, Ponce and Vega’s Lemmas in [18] are invalid for the semigroup of BOB equation VB (t), we will use techniques in harmonic analysis to construct the following new Kato smoothing effect estimates and maximal estimates for the semigroup of BOB equation (see [13] for detail proof): kVB (t)∆j φ(x)kL∞ 2 x L

6 C2−j/2 k∆j φ(x)kL2 ,

(2.8)

kVB (t)∆j φ(x)kLkx L∞

6 C2(1/2−1/k)j k∆j φ(x)kL2 .

(2.9)

t∈(0,∞)

t∈(0,∞)

We were emphasize that C is independent of ν. We were also troubled by the difference 2 between dispersive semigroup F −1 eitξ|ξ| F and the fixed semigroup F −1 eitξ|ξ| e−νtξ F when obtaining similar linear estimate as (1.10). Second, the semigroup of BOB equation VB (t) dosen’t have conjugate symmetry property, this means VB (t) 6= VB (t). So we can not use standard T T ∗ argument to construct some good Lpx1 Lqt 1 → Lpx2 Lqt 2 estimates for integral operators L , which effect us in choosing the resolution space. It is

Global Well-Posedness and Limit Behavior. . .

309

the reason that we add Strichartz type norm L6t,x (R+ × R) to N (u) in our resolution space X: N (u) = T (u) = M (u) =

∞ X

j=−∞ ∞ X

j=−∞ ∞ X j=−∞

1

1

3

1

2( 2 − k )j k∆j ukL∞ , 6 + 2 t Lx ∩Lt,x (R ×R) 2( 2 − k )j k∆j ukL∞ 2 x L

t∈[0,∞)

k∆j ukLkx L∞

t∈[0,∞)

,

.

X := {u ∈ S ′ : kukG := N (u) + T (u) + M (u) 6 ∞}

(2.10)

The key idea to estimate nonlinear term, is to divide frequency space into different part. For high frequency part, we use smooth effect estimates, while for low frequency part, we use Strichartz-type estimates. It was known that the global maximal estimates (1.9) and (2.9) were still unknown for the case 2 6 k 6 3, even for the semigroup of Kdv or BO equation (local case is known, see [19, 20]), so it is difficult to use this method to obtain global well-posedness results in the cases k < 4 (the condition 4 6 k 6 6 in Theorem 2.1 is added because of the nonlinear estimates). Recently, we also successfully to use this method in high dimension space to solve derivative Ginzburg-Landau equation, see our recent work [13]. Finally I will introduce a l1 – variant Bougain-type space method in the following section, which can solve the inviscid limit problem when k < 4.

3.

Korteweg-de Vries-Burgers Equation

In this section, we introduce the Cauchy problem for the Korteweg-de Vries-Burgers equation ut + uxxx + ǫ|∂x |2 u + (u2 )x = 0,

u(0) = u0 ,

(3.1)

where ǫ > 0, u(t, x) is a real valued function of (t, x) ∈ R+ ×R. Eq. (3.1) was first derived by E. Ott [28] as a model for the propagation of weakly nonlinear dispersive long waves in some physical contexts when dissipative effects occur. When fixing the parameter ǫ, there are some recent results obtained by using Bougain space with dissipation for Eq. (3.1), cf. [24, 25, 37]. Impused by their results, using a l1 – variant Bougain-type space method, Z. Guo , L. Peng, B. Wang [10] obtained the global well-posedness and inviscid limit for Korteweg-de Vries equation ut + uxxx + (u2 )x = 0, when ǫ → 0.

u(0) = u0 ,

(3.2)

310

Lijia Han I will introduce the method used in [10] here. For Eq.(3.2), the standard Bougain space was first used by Bougain [4], and was defined by Kenig, Ponce and Vega [34] as

X s,b

kukX s,b = khτ − ξ 3 ib hξis u ˆkL2 (R2 ) .

(3.3)

Where h·i = (1 + | · |2 )1/2 . We can look upon s as an index for variable x, and b as an index for variable t. Standard Bougain space method was used successfully to solve many interesting problems by J. Colliander, M. Kill, G. Staffilani, H. Takaoka and T. Tao [34] and so on. But in order to study the uniform global well-posedness of (3.1) and the limit behavior, this method is not enough. We can also look upon the l1 Besov-type space as dyadic Bougain-type space. It was first used by A. D. Ionescu and C. E. Kenig, for solving the global well-posedness of the Benjamin-Ono equation in low-regularity spaces [14]. Z. Guo , L. Peng, B. Wang [10] absorbed the idea of [14] and defined the dyadic X s,b -type norm spaces Xk = Xk (R2 ) as following: n Xk = f ∈ L2 (R2 ) : f (ξ, τ ) is supported in Ik × R and kf kX k =

∞ X j=0

o 2j/2 kηj (τ − ξ 3 ) · f kL2 ,

(3.4)

where k ∈ Z+ , Ik = {ξ : |ξ| ∈ [2k−1 , 2k+1 ]}, k > 1, I0 = {ξ : |ξ| 6 2}. For −3/4 < s 6 0, they define the following spaces: o n X 22sk kηk (ξ)F (u)k2Xk < ∞ , F s = u ∈ S ′ (R2 ) : kuk2F s = k∈Z

o n X 22sk k(i + τ − ξ 3 )−1 ηk (ξ)F (u)k2Xk < ∞ . N s = u ∈ S ′ (R2 ) : kuk2N s = k∈Z

They obtained the following uniform linear estimates: kψ(t)Wǫ (t)φkF s 6 CkφkH s , Z t Wǫ (t − τ )(u(s))dskF σ 6 CkukN σ , kψ(t) 0

2

3

where Wǫ (t) = F −1 e−ǫ|ξ| t eiξ t F , ǫ > 0. The main resolution space B is defined by B = {u ∈ F s :

kuk 6 2cr}.

From the basic idea introduced in Section1, with some nonlinear estimates which are so complex, the uniform local well-posedness results were obtained. I do not introduce the nonlinear estimates here, which can be found in [10, 26]. But how to extend the local solution to global solution? One standard method is using conservation law, when u0 ∈ L2 . But when u0 ∈ H s , −3/4 < s < 0, there is no such conservation law. We can follow the idea in [6], which we call I−method to obtain global well-posedness. We do not introduce the detailed process here, cf. [10].

Global Well-Posedness and Limit Behavior. . .

311

As the uniform bound for the solution were obtained, from the basic idea for limit behavior introduced in Section1, then repeat the process of proving the global well-possedness results, one can obtained the inviscid limit for equation (3.1), as desired. Compared with the method in Section1 and Section2, Bougain-type space method is not so useful when the nonlinear term is ∂x (uk u), k > 4 or ∂x (|u|k u).

A.

Appendix

In this section, I will introduce the notations used in this chapter, then show brief introduction to the definition of Littlewood-Paley decomposition and homogeneous Besov space, cf [32]. Let S (R) be the Schwartz space. Let P ̺ : R → [0, 1] be a Schwartz function supported by the set {ξ : 2−1 6P|ξ| 6 2} and j∈Z ̺(2−j ξ) = 1, ξ 6= 0. Define ̺j = ̺(2−j ξ) and define ̺0 := 1 − j>1 ̺j and observe that ̺0 ∈ S (R), ̺0 is supported in the ball {ξ| |ξ| 6 2} and ̺0 = 1 for |ξ| 6 1. We denote by ∆j and Sj the convolution operators whose symbols are respectively given by ̺(2−j ξ) and ̺0 (2−j ξ). ∆j := F −1 ̺(2−j ξ)Ff,

Sj := F −1 ̺0 (2−j ξ)Ff.

It is easy to see ∞ X k=0

△k = I.

We call {△k }∞ 0 the Littlewood-Paley decomposition operatous. Also we define the oper˜ ˜ j ∆j = ∆j . We define the projection ator ∆j := ∆j−1 + ∆j + ∆j+1 which satisfies ∆ operators P6M and P>M by: P6M f := F −1 ̺0 (ξ/M )F , P>M f = I − P6M f. For any s ∈ R, 1 6 q 6 ∞, the homogeneous Sobolev space H˙ s (R) is defined by s (R) denotes the completion of S (R) (−∆)−s/2 L2 . the homogeneous Besov space B˙ 2,q with respect to the semi-norm,

kf k ˙ s = {2js k∆j f kL2 } q . B2,q

l (Z)

sk T ˙ s˜k sk s˜k It is easy to see that u0 ∈ B˙ 2,∞ B2,1 is equivalent to P61 u0 ∈ B˙ 2,∞ and P>1 u0 ∈ B˙ 2,1 , where P>a = F −1 χ(|ξ|>a) F , P6a = I − P>a , χ is the cut-off function.

References [1] L. Abdelouhab, J. L. Bona, M. Felland, J. C. Saut, Nonlocal models for nonlinear dispersive waves. Physica D., 40 (1989), 360–392.

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[2] B. Binir, C. E. Kenig, G. Ponce, N. Svanstedt, and L. Vega, On the ill-posedness of the IVP for the generalized Korteweg de Vries and nonlinear Schr¨odinger equation. ,J. London Math. Soc. 53 (1996), 551–559. [3] T. B. Benjamin, Internal waves of permanent form in fluids of great depth. J. Fluid. Mech., 29 (1967), 559–592. [4] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I, II. Geom. Funct. Anal., 3 (1993), 107– 156, 209–262. [5] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Global well–posedness and scattering for the energy–critical nonlinear Schr¨odinger equation in R3 . Ann. of Math., 166 (2007), 1–100. [6] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, The sharp global well– posedness for KdV and modified KdV on R and T. J. Amer. Math. Soc., 16 (2003), 705–749. [7] P. M. Edxin and B. Roberts, The Benjamin-Ono-Burgers equation. An application in solar physics, Wave Motion., 8 (1986), 151–158. [8] A. S. Fokas and L. Luo, Global solutions and their asymptotic behavior for BenjaminOno-Burgers type equations. Differential and Integral Equations., 13(1-3) (2000), 115–124. [9] B. Guo, S. Tan, Long time behavior for the equation of finite-depth fluids. Commum. Math. Phys., 163 (1994), 1–15. [10] Z. Guo, B. Wang, Global well posedness and inviscid limit for the Korteweg-de VriesBurgers equation, J. Differential Equation., 10(2009), 3864–3901. [11] L. J. Han, B. Wang, Global wellposedness and limit behavior for the generalized finitedepth-fluid equation with small critical data. J. Differential Equation., 245 (2008), 2103–2144. [12] L. Han, L. Zhao, Global wellposedness and limit behavior for the generalized Benjamin-Ono Burgers equation. (Preprint) [13] L. Han, B. Wang, B. Guo, inviscid limit for the derivative Ginzburg–Landau equation with small data in higher spacial dimensions. (Preprint). [14] A. D. Ionescu and C. E. Kenig, Global well-posedness of the Benjamin-Ono equation in low-regularity spaces. J. Amer. Math. Soc., 20 (2007), 753–789. [15] R. I. Joseph, Solitary waves in a finite depth fluid. J. Phys. A., 10 (1977), L225–L227. [16] T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Advances in Mathematics Supplementary Studies. Studies in Applied Math., 8 (1983), 93–128.

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[17] D. J. Korteweg, G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos. Mag., 539 (1895), 422–443. [18] C. E. Kenig, G. Ponce, L. Vega, Oscillatory integrals and regularity of dispersive equations. Indiana Univ. Math. J., 40 (1991), 32–69. [19] C. E. Kenig, G. Ponce, L. Vega, Well-posedness of the initial value problem for the Korteweg de Vries equation. J. Amer. Math. Soc., 4 (1991), 323–347. [20] C. E. Kenig, G. Ponce, L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via contraction principle. Comm. Pure Appl. Math., 46 (1993), 527–620. [21] T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation. Advances in Mathematics Supplementary Studies, Studies in Applied Math., 8 (1983), 93–128. [22] L. Molinet, F. Ribaud, On the Cauchy problem for the generalized Korteweg de Vries equation. Comm. Part. Diff. Equat., 28 (2003), 2065–2091. [23] L. Molinet, F. Ribaud, Well-posedness results for the generalized Benjamin-Ono equation with small initial data. J. Math. Pures Appl. 83 (2004), 277–311. [24] L. Molinet, F. Ribaud, On the low regularity of the Korteweg-de Vries-Burgers equation. Internet. Math. Res. Notices., 37, (2002). [25] L. Molinet, F. Ribaud, The Cauchy problem for dissipative Korteweg-de VriesBurgers equations in Sobolev spaces of negative order. Indiana Univ. Math. J., 50 (2001), No. 4, 1745–1776. [26] L. Molinet, F. Ribaud, The Cauchy problem for dissipative Korteweg de Vries equations in Sobolev sapces of negative order, Indiana Univ. Math. J., 50(2001), 1745– 1776. [27] H. Ono, Algebraic solitary waves in stratified fluids. J. Phys. Soc. Japan., 39 (1975), 1082–1091. [28] E. Ott, N. Sudan, Damping of solitary waves. J. Phys. Fluids., 13 (1970), No. 6, 1432– 1434. [29] Masanori Otani, Well-posedness of the genereralized Benjamin-Ono-Burgers equations in sobolev spaces of negative order. Osaka J. Math., 43 (2006), 935–965. [30] G. Ponce, On the global well-posedness of the Benjamin-Ono equation. Differential Integral Equations., 4 (1991), 527–542. [31] J. Satsuma, M. J. Ablowitz, Y. Kodama, On an internal Wave equation describing a stratified fluid weth finite depth. Phys. Lett., 73 (1979), 283–286. [32] H. Triebel, Theory of Function Spaces. Birkh¨auser–Verlag, 1983.

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[33] T. Tao, Global Well-Posedness of the Benjamin-Ono equation in H 1 (R). J. Hyperbolic Diff. Eq., 1 (2004), 27–49. [34] T. Tao, Multilinear weighted convolution of L2 functions, and applications to nonlinear dispersive equations. Amer. J. Math., 123 (2001), 839–908. [35] B. X. Wang, The limit behavior of solutions for the Cauchy problem of the complex Ginzburg–Landau equation. Commun. Pure Appl. Math., 55 (2002), 481–508. [36] B. X. Wang, Y. D. Wang, The inviscid limit for the derivative Ginzburg-Landau equations. J. Math. Pures Appl., 83 (2004), 477–502. [37] S. Vento, Global well-posedness for dissipative Korteweg-de vries equations. arXiv: 0706.1730v1. [38] Y. Kodama, J. Satsuma, and M. J. Ablowitz, Nonlinear intermediate long-wave equation: analysis and method of solution. Phys. Rev. Lett., 46 (1981), 687–690.

In: Advances in Mathematics Research, Volume 10 ISBN 978-1-60876-265-1 c 2010 Nova Science Publishers, Inc. Editor: Albert R. Baswell, pp. 315-321

Chapter 8

S TRUCTURE OF C ERTAIN P ERIODIC N EAR R INGS Asma Ali∗ Department of Mathematics, Aligarh Muslim University Aligarh-202002 (India)

Abstract Let R be a left near ring and let m = m(x, y) ≥ 1, n = n(x, y) ≥ 1 be positive integers depending on x and y in R. Further, let p(xy) be an element of R which is finite sum of powers (xy)k ; k ≥ 2 and additive inverses of such powers. In the present paper we establish some decomposition theorems for a near ring R satisfying any one of the conditions, (i) xy = (xy)n p(xy), (ii) xy = (yx)n p(xy), (iii) xy = (yx)n p(yx), (iv) xy = xm y n p(xy), (v) xy = y m xn p(xy), (vi) xy = xm p(xy)xn and (vii) xy = y m p(xy)y n .

AMS Subject Classifications (1991): 16W25, 16U80. Key Words and Phrases: Distributively generated (d-g)-near ring, D-near ring, zerosymmetric near ring and zero-commutative near ring.

1.

Preliminaries

Throughout the paper R will denote a left near ring with multiplicative center Z(R). We shall denote by N the set of nilpotent elements and by P the set of potent elements that is, the set {x ∈ R | xn = x, for some positive integer n(x) > 1}. If P = R, we shall call R a J-ring. By the well known x = xn(x) theorem of Jacobson [6], J-rings are necessarily commutative. The set of commutators is denoted by C. An element x of R is said to be distributive if (y + z)x = yx + zx, for all y, z ∈ R. If every element of R is distributive, then R is said to be a distributive near ring. A near ring is said to be distributively generated (d−g) if it contains a multiplicative subsemigroup of distributive elements which generates the additive group (R, +). A near ring R is called zero-commutative if for all x, y ∈ R, ∗

E-mail address: [email protected]

316

Asma Ali

xy = 0, implies that yx = 0, and if for all x ∈ R, 0x = 0, R is called zero-symmetric (we may recall that the left distributivity of R yields x0 = 0). A near ring R is called periodic if for every x ∈ R there exist distinct positive integers m = m(x), n = n(x) such that xm = xn . A sufficient condition for a ring R to be periodic is Chacron’s criterion : For each x ∈ R there exists an integer m = m(x) ≥ 1 and a polynomial f (X) ∈ Z[X], the ring of polynomials in X such that xm = xm+1 f (x) ([4]). For each element x of a near ring R, the subnear ring generated by x will be denoted by < x >. Note that if x is a distributive element, then the element t(x) ∈< x > ( or of < 1, x > if R has unity 1) may be assumed to be a finite sum of powers of x. An ideal of a near ring R is a normal subgroup I of R+ such that (i) RI ⊆ I and (ii) (x + i)y − xy ∈ I for all x, y ∈ R and i ∈ I. A near ring R is called a D-near ring if every nonzero homomorphic image T of R satisfies the following conditions: (i) T has a nonzero right distributive element. (ii) (T, +) is abelian implies that (T, +, .) is a ring. It is evident by definition that all distributive near rings are examples of D-near rings. However, the example 2.5 # 6 of [5] illustrates that the class of D- near rings is larger than the class of d − g near rings.

2.

Introduction

The property xn = x has been among the favourites of many ring theorists over the last few decades since Jacobson[6] first studied the commutativity of rings satisfying this condition in order to generalize the classical Wedderburn theorem [10]. Later, this property has been weakened by Searcoid and MacHale [9] who proved that a ring satisfying the condition (xy)n(x,y) = xy must be commutative. Ligh and Luh [7] established that, a ring R satisfying the above condition is a direct sum of a J-ring and a zero ring. Later, Bell and Ligh [3] obtained structure of rings satisfying the related properties like xy = (xy)2 p(xy) and xy = (yx)2 p(xy), for p(XY ) ∈ ZZ[X, Y ], the ring of polynomials in two noncommuting indeterminates X and Y. It is natural to question whether the analogous hypotheses yield the direct sum decomposition in the case of near rings. An example due to Clay (cf. [5, Example 2.5 # 29]) shows that it is not possible to obtain such a decomposition. Example 2.1. Consider the non-abelian additive group (R, +), isomorphic to the symmetric group S3 and define addition and multiplication in R as follows :

Structure of Certain Periodic Near Rings

+ 0 a b c u v

0 0 a b c u v

a a o u v b c

b b v 0 u c a

c c u v 0 a b

u u c a b v 0

v v b c a 0 u

. 0 a b c u v

0 0 0 0 0 0 0

a 0 a a a 0 0

b 0 a a a 0 0

317

c 0 a a a 0 0

u 0 0 0 0 0 0

v 0 0 0 0 0 0

It is easy to see that (R, +, ·) is a commutative (distributive) near ring satisfying xy = (xy)2 = xy 2 x, for all x, y ∈ R. However P = {0, a} is not an ideal of R. Hence, following [3], we define a weaker notion of orthogonal sum. A near ring R is an orthogonal sum of subnear rings A and B denoted by R = A⊎B if AB = BA = {0} and each element of R has a unique representation in the form a + b with a ∈ A and b ∈ B. In this paper we shall obtain an orthogonal sum decomposition for a near ring R satisfying either of the following conditions, where p(xy) denotes an element of R which is finite sum of powers (xy)k ; k ≥ 2 and additive inverses of such powers : (P1 ) For every pair of elements x, y ∈ R, there exists a positive integer n = n(x, y) ≥ 1 such that xy = (yx)n p(yx). (P2 ) For every pair of elements x, y ∈ R, there exists a positive integer n = n(x, y) ≥ 1 such that xy = (yx)n p(xy). (P3 ) For every pair of elements x, y ∈ R, there exist positive integers m = m(x, y) ≥ 1 and n = n(x, y) ≥ 1 such that xy = y m xn p(xy). (P4 ) For every pair of elements x, y ∈ R, there exist positive integers m = m(x, y) ≥ 1 and n = n(x, y) ≥ 1 such that xy = y m p(xy)y n . (P5 ) For every pair of elements x, y ∈ R, there exists a positive integer n = n(x, y) ≥ 1 such that xy = (xy)n p(xy). (P6 ) For every pair of elements x, y ∈ R, there exist positive integers m = m(x, y) ≥ 1 and n = n(x, y) ≥ 1 such that xy = xm y n p(xy). (P7 ) For every pair of elements x, y ∈ R, there exist positive integers m = m(x, y) ≥ 1 and n = n(x, y) ≥ 1 such that xy = xm p(xy)xn .

318

3.

Asma Ali

Main Results

Theorem 3.1. Let R be a D-near ring satisfying condition (P1 ). Then R is periodic and R = P ⊎ N , where P is a subring, N is a subnear ring with trivial multiplication. Theorem 3.2. Let R be a zero-symmetric D-near ring satisfying either of the conditions (P2 ) − (P4 ). Then R is periodic and R = P ⊎ N , where P is a subring, N is a subnear ring with trivial multiplication. Theorem 3.3. Let R be a D-near ring satisfying condition (P7 ). If idempotent elements of R are multiplicatively central, then R is periodic. Moreover, R = P ⊎ N , where P is a subring and N is a subnear ring with trivial multiplication. Theorem 3.4. Let R be a zero-symmetric D-near ring satisfying either of the conditions (P5 ) and (P6 ). If idempotent elements of R are central, then R is periodic. Moreover, R = P ⊎ N , where P is a subring and N is a subnear ring with trivial multiplication. Before starting the proof of the above theorems we state the following lemmas which are essentially proved in [1], [2], [3] and [8] respectively. Lemma 3.1. Let R be a zero-symmetric near ring satisfying the following properties : (i) For each x in R, there exists a positive integer n(x) > 1 such that xn(x) = x. (ii) Every non trivial homomorphic image of R contains a nonzero central idempotent. Then (R, +) is commutative. Lemma 3.2. Let R be a near ring with unity 1. < x >= x < 1, x > .

Then for every x ∈ R,

Lemma 3.3. If R is a zero-commutative periodic near ring , then R = P + N . Lemma 3.4. Let R be a near ring in which idempotents are multiplicatively central. If e and f are any idempotents, there exists an idempotent g such that ge = e and gf = f . Lemma 3.5. If R is a zero-commutative near ring, then N is an ideal of R. The following Lemma was obtained for d.g near ring by Bell [2, Theorem 12]. Now we shall extend the result for D-near ring. Lemma 3.6. Let R be a D-near ring satisfying xn = xn p(x), where n = n(x) a positive integer and p(x) ∈< x >. If N ⊆ Z(R), then R/N is a periodic and commutative ring.

Structure of Certain Periodic Near Rings

319

Proof. Since N ⊆ Z(R), N is an ideal by Lemma 3.5. Consider the near ring R/N . Since R/N can be written as a subdirect product of near rings without zero divisors, we may assume R/N has no nonzero divisors of zero. Let d be a nonzero distributive element of R/N . Note that in view of the property that xn = xn p(x), d = dt(d) for some t(d) ∈< d >. It follows that e = t(d) is a nonzero idempotent in R/N . Since e(er − r) = 0, for every r ∈ R/N , e is the left identity in R/N . Considering arbitrary x, y ∈ R/N and using the fact that e commutes with d and d = de, we have, 0 = (x + y)de − (xde − yde) = (x + y)e − (xe − ye)d. Again since R/N has no nonzero divisors of zero, e is a distributive element of R/N . Hence e is the multiplicative identity in R/N . If x is an arbitrary nonzero element of R/N , then x = t(x), where t(x) ∈< x >. Using Lemma 3.2, we can write t(x) = xt′ (x) for some t′ (x) ∈< 1, x >. Now x = x2 t′ (x) yields that xt′ (x) = 1 i.e., R/N is a division near ring. Thus R/N is additively commutative and is a ring. HenceR/N is periodic by Chacron’s criterion [4] and a commutative ring by [2, Theorem 2].

xn

The proof of the above lemma runs on the same lines if we replace the property = xn p(x) by xn = xn p(x)xn .

Lemma 3.7. Let R be a near ring satisfying either of the conditions (P1 ) and (P7 ). Then RN = N R = {0}. Proof. Clearly R satisfying (P1 ) is zero-commutative. Hence N is an ideal by Lemma 3.5. Replacing y by x in (P1 ), we find that x2 = x2m p(x2 ) ; for m = m(x) ≥ 1.

(3.1)

If u ∈ N , then making repeated use of (3.1), we have u2 = 0. Now for any u ∈ N by condition (P1 ), we have ux = (xu)l p(xu) = 0, for l = l(u, x) ≥ 1. Zero-commutativity yields that xu = 0, for u ∈ N and x ∈ R. Hence RN = N R = {0}. The proof runs on the parallel lines for condition (P7 ). Proceeding on the same lines we can prove the following lemma : Lemma 3.8. Let R be a zero-symmetric near ring satisfying either of the conditions (P2 ) − (P6 ). Then RN = N R = {0}. Lemma 3.9. Let R be a D-near ring satisfying condition (P1 ). Then idempotent elements of R are central. Proof. Let e be an idempotent and x ∈ R. Then by condition (P1 ) there exists an integer t = t(x, e) ≥ 1 such that xe = (ex)t h(ex). Multiplying by e on the left we get exe = xe. Application of Lemma 3.6 and Lemma 3.7 yields that C ⊆ N ⊆ Z(R) and we have e(xe − ex) = 0, for all x ∈ R. Hence ex = xe, for all x ∈ R.

320

Asma Ali In the similar manner we can prove the following :

Lemma 3.10. Let R be a zero-symmetric D-near ring satisfying either of the conditions (P2 ) − (P4 ). Then idempotent elements of R are central. Proof of Theorem 3.1. In view of Lemma 3.6 we see that for each x ∈ R, there exist distinct positive integers m = m(x), n = n(x) such that xm − xn ∈ N . Hence using Lemma 3.7, we have xm+1 = xn+1 , for each x ∈ R and R is periodic. Next we show that P is a subring. Let a, b ∈ P and choose integers p = p(a) > 1 and q = q(b) > 1 such that ap = a and bq = b. Let r = (p − 1)q − (p − 2) = (q − 1)p − (q − 2). Then it is clear that ar = a and br = b. Note that e = ar−1 , f = br−1 are idempotents with ea = a and f b = b. Since C ⊆ N ⊆ Z(R), we can write a2 b = aba for all a, b in P using Lemma 3.7. Obviously ab = ar br = (ab)r , hence ab ∈ P for all a, b in P . Moreover since R/N has the xl = x property we have an integer j > 1 such that (a − b)j = a − b + u ; u ∈ N.

(3.2)

Using Lemma 3.4 we can choose an idempotent g for which ge = e and gf = f . Therefore ga = a and gb = b. Now multiplying (3.2) by g we have (a − b)j = a − b i.e. a − b ∈ P . Also by Lemma 3.1 (P, +) is abelian. Hence P is a subring. Trivially P ∩ N = {0}. Let a + u = b + v where a, b ∈ P and u, v ∈ N . Then a − b = v − u ∈ P ∩ N = {0}, which yields a = b and v = u. Hence R = P ⊎ N . The proof of Theorem 3.2 - 3.4 can be obtained in the similar manner.

References [1] H. E. Bell, Near rings in which each element is a power of itself, Bull. Austral. Math. Soc. 2 (1970), 363-368. [2] H. E. Bell, On commutativity of periodic rings and near rings, Acta Math. Acad. Sci. Hungar. 36 (1980), 293-302. [3] H. E. Bell and S. Ligh , Some decomposition theorems for periodic rings and near rings, Math. J. Okayama Univ. 31 (1989), 93-99. [4] M. Chacron, On a theorem of Herstein, Canad. J. Math. 21 (1969), 1348-1353. [5] J. R. Clay , The near rings on groups of low order, Math Z. 104 (1968), 364-371. [6] N. Jacobson, Structure theory of algebraic algebras of bounded degree, Ann. of Math. 46 (1945),695-707. [7] S. Ligh and J. Luh, Direct sum of J-rings and zero rings, Amer. Math. Monthly 96 (1989), 40-41.

Structure of Certain Periodic Near Rings

321

[8] Y. F. Reddy and C. V. L. N. Murty, Semi-symmetric ideals in near rings, Indian J. Pure Appl. Math. 16 (1985), 17-21. [9] M. O. Searcoid and D. MacHale, Two elementary generalizations of Boolean rings, Amer. Math. Monthly 93 (1986), 121-122. [10] J.H.M. Wedderburn, A theorem on finite algebras, Trans. Math. Soc. 6 (1905), 349352.

INDEX Brownian motion, 33

A C abstraction, 47 action potential, viii, 38, 53, 59 activity rate, 60, 135 actuators, 29, 30, 32 adaptation, 38, 49, 99 adaptations, 38, 39 amplitude, vii, viii, 1, 2, 3, 16, 17, 18, 19, 20, 21, 22, 24, 26, 29, 30, 37, 39, 42, 46, 48, 274 anisotropy, 301 applied research, 273 assignment, 48 assimilation, 2, 25 assumptions, 69, 77, 83, 100, 143, 163, 164, 186, 206, 211, 226, 237, 239, 251, 252, 254, 255 asymptotics, ix, 205, 206, 222, 246, 273, 274, 277, 278, 280, 287, 288, 302 atoms, 274, 275, 290 Australia, 29 authors, ix, 45, 206, 249, 250 axons, 35, 36, 42, 44, 58

B background, 75 Banach spaces, 207, 264 behavior, ix, 13, 36, 39, 45, 144, 201, 249, 303, 304, 305, 307, 310, 311, 312, 314 Beijing, 203, 303 biological control, 29 Boltzmann distribution, 47 bonds, 274, 290 bones, 29 boundary value problem, 122, 205, 220, 221, 226, 246, 247, 271 bounds, 266 brain, 41, 42, 52, 55, 98, 201 brain activity, 201 brain structure, 42 brainstem, 41, 42

calcium, 54, 55 carbon, 273, 274, 275, 301 carbon nanotubes, 273, 274, 301 Cauchy problem, ix, x, 186, 303, 304, 307, 312, 313, 314 cell, 36, 42, 43, 44, 54, 58, 59 cell body, 58, 59 central nervous system, 52 cerebellum, 29, 30, 31, 35, 36, 37, 38, 41, 42, 43, 44, 45, 46, 49, 51, 52 cerebral cortex, 44 cerebrum, 44 channels, 59 China, 303 circulation, 25 classes, 12, 35, 60, 66, 75, 86, 136, 140, 145 closure, 7, 207, 238 clusters, 274 cognition, 39 collateral, 36 combustion, 198 compatibility, 100, 106, 130, 215, 216 compensation, 38, 49 complement, 3, 11, 220 complex numbers, 81, 93, 95, 96, 106, 134, 147, 149 complexity, 29 compliance, 41 components, 3, 4, 12, 14, 20, 31, 55, 299 composition, 10 concentration, ix, 59, 249, 250, 251, 252, 254, 256, 270, 271 concreteness, 73 conditioning, 42, 46 conduction, 200, 202 configuration, 33, 34, 47, 48 conjecture, 307 connectivity, 198 conservation, vii, 1, 2, 9, 11, 16, 17, 22, 27, 38, 39, 49, 55, 58, 65, 310

324

Index

continuity, 250, 255, 265 control, 30, 31, 34, 35, 36, 37, 39, 40, 42, 43, 45, 46, 49, 50, 52, 307 convergence, 82 conversion, 32 conversion rate, 32 cortex, 35, 36, 38, 42, 43, 44, 45, 55, 56, 60, 202 coupling, viii, 30, 33, 39, 53, 55, 56, 59, 60, 65, 66, 73, 79, 83, 99, 105, 135, 143, 144, 145, 163, 166, 167, 168, 170, 171, 173, 174, 175, 182, 185, 186, 187, 203, 279, 281 covering, 8 critical value, 22 cytoplasm, 54, 58

D decay, 18, 24, 69, 71, 75, 77, 80, 99, 197 decomposition, ix, x, 14, 234, 237, 239, 242, 303, 304, 306, 311, 315, 316, 317, 320 defence, 29 definition, 11, 15, 47, 80, 177, 179, 207, 208, 221, 254, 289, 311, 316 degenerate, 270, 277, 278 dendrites, 42, 43, 58 depression, 43, 46, 99 derivatives, 4, 7, 9, 32, 47, 65, 103, 107, 114, 152, 207, 232, 262, 288, 297 desorption, 250, 270 differential approach, 205 differential equations, viii, 53, 54, 55, 57, 58, 65, 70, 72, 73, 74, 75, 79, 80, 85, 87, 97, 99, 100, 111, 118, 119, 120, 121, 122, 125, 126, 127, 129, 135, 156, 157, 161, 188, 189, 192, 193, 199, 203, 246 differentiation, 285 diffusion, ix, 30, 33, 34, 55, 58, 63, 70, 71, 74, 189, 198, 200, 249, 250, 251, 254, 270, 271 diffusion process, ix, 30, 249 diffusivity, 251 dimensionality, 36 discretization, 38 distribution, 13, 25, 47, 48, 54, 281 distribution function, 48 division, 36, 45, 319 duality, 253, 277, 300 dynamical systems, 39, 47, 100, 202

entropy, 38, 39, 46, 47, 48 environment, 38, 47 environmental control, 38 epilepsy, 56 equality, 17, 24, 165, 166, 167, 210, 213, 214, 217, 230, 235, 236, 257, 258 equilibrium, 42, 54 estimating, 9, 306 evolution, ix, 3, 12, 13, 14, 20, 23, 25, 199, 303, 304, 312 excitation, 30, 32, 35, 39, 40, 41, 42, 54, 60, 66, 67, 68, 69, 75, 79, 86, 99, 143, 144, 161, 165, 166, 167, 175, 176, 178, 179, 180, 200, 202 excitatory synapses, 42, 43 execution, 35, 51 extensor, 40

F family, 12, 31, 60, 61, 62, 63, 64, 65 feedback, 35, 38, 39, 41, 42, 43, 49, 99, 188, 198 feedback inhibition, 43 fibers, 35, 36, 40, 41, 42, 43, 44 fine tuning, 30 fires, 43, 58, 59 fish, 42 flexor, 40 fluctuations, 33, 34 fluid, vii, 1, 2, 4, 9, 13, 22, 24, 25, 27, 304, 312, 313 free energy, 46, 47 freedom, 29 functional analysis, viii, 53, 57, 70, 135 fuzzy sets, 34

G ganglion, 44 gene, 321 generalization, 39, 48 generation, 30, 200 genes, 58 genetics, 198 graph, 85, 136, 139, 274, 290 graphene sheet, 273, 275 groups, 4, 21, 55, 320 growth, vii, 1, 2, 3, 16, 17, 18, 21, 22, 24, 28 growth rate, vii, 2, 3, 16, 18, 22, 24, 28

E earth, 25 Education, 205 electric field, ix, 273, 279, 281 electromagnetic, 201 encoding, 35, 45 energy, vii, 1, 3, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 27, 32, 33, 38, 47, 48, 59, 277, 304

H Hamiltonian, 27, 33, 34, 37, 38, 47, 48, 51, 275, 290, 291 heat, 47, 50, 247, 250 highways, 44 Hilbert space, 2, 5, 25, 275, 290 hospitality, 300

Index House, 203 human brain, 42, 58 hybrid, 32 hyperbolic systems, 246 hypothesis, 38, 217, 218 hysteresis, 39

L

I ideal, vii, 1, 2, 4, 9, 13, 22, 25, 26, 60, 316, 317, 318, 319 ideals, 321 identity, 19, 72, 91, 97, 266, 289, 290, 294, 300, 319 image, 30, 264, 316, 318 imitation, 52 in vitro, 202 inclusion, 281 India, 315 indices, 48, 305 induction, 139, 140, 217, 218, 224, 226, 230, 233 inequality, 8, 17, 18, 137, 144, 170, 213, 214, 218, 224, 231, 233, 259, 261, 265, 269, 289, 305 inertia, 33, 34 infinite, ix, 26, 27, 36, 273, 277 information processing, 36, 45, 49 inhibition, 30, 35, 39, 40, 41, 43, 66, 67, 68, 79, 86, 100, 144, 161, 166, 167, 175, 176, 179, 180, 201 initiation, 59 innovation, 31 instability, vii, viii, 1, 2, 3, 11, 12, 13, 14, 15, 16, 17, 18, 22, 23, 24, 25, 26, 27, 28, 39, 73, 187, 198 integration, 30, 35, 42, 58, 119, 121, 122, 125, 126, 128, 157, 214 interaction, 22, 41, 55 interactions, 4, 36, 45, 46, 55, 59, 75, 99, 199, 201, 203 interneuron, 44 interneurons, 35, 41, 46 interval, 22, 77, 79, 85, 86, 137, 138, 142, 245, 253, 269, 281 invariants, 51 ion channels, 59, 99 ions, 59 ipsilateral, 44, 46 iteration, 31, 89

J Japan, 270, 300, 313 joints, 29, 30, 32, 33, 34, 46 Jordan, 220, 221, 222, 228 justification, 99, 251

K knees, 76

325

laws, 9, 22, 27, 29, 55, 65 learning, 30, 31, 35, 36, 37, 38, 42, 46, 49, 52 likelihood, 38, 48 limitation, 98 line, 82, 109, 136, 144, 145, 229, 275 linear function, 75, 257 linearity, 52 liquids, ix, 249, 250 Lyapunov function, 283, 284, 286, 299, 301

M magnetic field, ix, 273, 274, 275, 276, 287, 290, 291, 301 manifolds, 37 manipulation, 36 mapping, 305 mastery, 25 mathematics, vii matrix, 72, 88, 89, 91, 92, 133, 192, 193, 274, 276, 277, 282, 283, 284, 286, 288, 292, 294, 296, 298, 299 measures, 166 mechanical properties, 41, 52 media, 54, 199 membranes, 59 messages, 59 methanol, 271 Mexico, 24 mice, 42 mitochondria, 58 model, viii, 29, 33, 35, 36, 39, 45, 46, 49, 50, 51, 53, 55, 58, 59, 65, 75, 83, 99, 100, 116, 135, 188, 189, 197, 198, 199, 200, 201, 203, 250, 256, 270, 274, 275, 290, 292, 301, 307 model system, 59, 75, 189 modeling, 37, 55, 60, 188, 189, 251 models, ix, 55, 56, 60, 135, 201, 202, 250, 273, 274, 277, 301, 311 modules, 45 molecules, 47, 55, 273, 301 momentum, 9, 11, 47 monodromy, 284, 286, 292, 296, 298, 299 motion, vii, 1, 4, 9, 13, 14, 22, 25, 26, 29, 30, 31, 33, 200 motion control, 29, 30 motivation, 81 motor behavior, 39 motor control, 35, 36, 37, 38, 45 motor neurons, 41 motor system, 39, 49 motor task, 38 movement, 30, 36, 38, 39, 42, 44, 45, 51, 52 multidimensional, 251, 270 multiplication, 316, 318

326

Index

muscle mass, 40 muscles, 29, 30, 40, 41, 42, 45, 49 musculoskeletal system, 50

N nanotube, ix, 273, 274, 275, 279, 281, 290, 291, 292, 295, 300, 301 neocortex, 199, 200, 202 nerve, 54, 55, 56, 60, 72, 75, 188, 198, 200, 202 nervous system, 55, 56 network, 36, 52, 55, 56, 65, 75, 98, 165, 199, 202, 250, 251, 301 neural network, 29, 31, 36, 51, 199, 200 neural networks, 36, 200 neurobiology, 58 neurons, 31, 35, 36, 39, 45, 55, 58, 59, 60, 66, 74, 75, 98, 99, 165, 179, 202, 203 neuroscience, 55, 56, 75, 118, 188, 189, 190 noise, 31 nonlinear optics, 55 nonlinear parabolic systems, 202 nonlinear systems, viii, 53, 54, 192 normalization constant, 175 nuclei, 36, 40, 42, 43, 44 nucleus, 44, 58, 59 numerical analysis, 271

O observations, 251 operator, viii, ix, 2, 4, 6, 7, 8, 14, 19, 27, 53, 55, 70, 72, 77, 78, 79, 80, 82, 88, 94, 95, 96, 108, 111, 130, 134, 147, 184, 185, 188, 206, 208, 209, 211, 218, 219, 221, 229, 239, 240, 242, 243, 244, 245, 247, 252, 264, 265, 273, 274, 275, 276, 277, 279, 281, 282, 283, 284, 285, 286, 287, 288, 290, 291, 292, 293, 294, 295, 296, 298, 299, 301, 305 Operators, 27, 247 orbit, 16, 85, 86, 94, 97, 117, 120, 121, 122, 124, 125, 126, 127, 128, 129, 134, 156 order, 22, 30, 39, 58, 72, 80, 82, 94, 103, 106, 107, 109, 111, 114, 134, 205, 206, 207, 208, 218, 219, 221, 232, 234, 243, 253, 287, 289, 305, 307, 310, 313, 316, 320 ordinary differential equations, 57, 85, 87, 100 organ, 41 organelles, 58 organism, 38 orthogonality, vii, 1, 3, 19, 24

P

parameter, ix, 12, 38, 39, 47, 59, 72, 76, 77, 85, 86, 94, 95, 99, 100, 103, 147, 154, 167, 177, 179, 197, 206, 257, 303, 305, 306, 307 parameters, 2, 32, 34, 39, 46, 55, 70, 82, 83, 86, 87, 90, 99, 107, 116, 118, 135, 161, 166, 167, 168, 188, 193, 257 partial differential equations, 54, 57, 60, 73, 74, 111, 245 particles, 47 partition, viii, 29, 38, 46, 47, 48 passive, 59 patella, 29 pathogenesis, 201 pathways, 31, 35, 37, 39, 40, 41, 42, 44 periodicity, 296 permit, 251 phase transitions, 39, 49, 198 physics, 54, 55, 201, 301, 312 physiology, 30, 201 piano, 42 pitch, 29 plasma, 55 plasticity, 46, 49 Poincaré, vii polymer, 250, 251, 254, 270 polymer films, 251 polymers, ix, 249, 250, 254, 270, 271 pons, 44 population, 40, 55, 198 portraits, 102, 103, 117, 124 positive feedback, 35, 41 posture, 42 potassium, 59 power, 320 pressure, 46 primacy, 39 probability, 18, 24, 47, 48, 59, 166 probability density function, 48, 166 probability distribution, 47, 48 probability theory, 47 production, 35, 59 program, 18, 24 propagation, ix, 56, 59, 75, 188, 198, 200, 202, 249, 304 proportionality, 250 protein synthesis, 59 pulse, 54, 70, 72, 77, 83, 84, 85, 86, 87, 96, 98, 99, 117, 118, 119, 120, 121, 135, 156, 157, 158, 189, 190, 198, 203

Q quantum field theory, 48

R

pairing, 209 radius, 16

Index range, 13, 198 real numbers, 122, 220, 236, 244 real time, 48 reason, 42, 55, 309 reasoning, 290 recall, 105, 136, 148, 163, 253, 264, 283, 292, 294, 316 receptive field, 46 receptors, 35, 39, 40, 41, 59 reconstruction, 25 recovery, 75, 99 reflex action, 41 reflexes, 30, 35, 39, 41, 45, 50 region, 41, 45, 66, 81, 94 regulation, 52 rehabilitation, 32 reinforcement, 31 reinforcement learning, 31 relationship, 41, 150, 190 relaxation, 250, 270 resistance, 59 resolution, 305, 306, 307, 308, 309, 310 responsiveness, 40 rings, 316, 317, 319, 320, 321 robotics, 32, 36 rolling, 273 rotations, 33 Royal Society, 198, 199, 201 rubbery state, 250 Russia, 249, 273

S scaling, 304 scattering, 312, 313 search, 36, 45 self-organization, 39 semigroup, 308 senses, 70 sensitivity, 2, 40 shape, 54, 86 shaping, 55 shares, 95 signals, 35, 36, 45, 58 signs, 20 simulation, 31 Singapore, 50, 51 skeletal muscle, 35, 49, 50 skeleton, 29, 30, 32 skin, 46 smoothing, ix, 303, 304, 305, 308 smoothness, 206, 242, 246 sodium, 58, 59, 76, 190, 204 soft matter, 201 soleus, 49 sorption, 250, 270 space, vii, ix, x, 1, 3, 12, 13, 14, 16, 23, 30, 33, 42, 47, 48, 49, 70, 72, 85, 88, 93, 108, 113, 118, 119,

327

120, 125, 126, 127, 130, 134, 147, 197, 200, 206, 207, 208, 209, 238, 252, 253, 257, 258, 267, 271, 303, 304, 305, 306, 307, 308, 309, 310, 311 space-time, 46, 52 spectrum, vii, viii, ix, 2, 53, 70, 71, 80, 81, 82, 95, 109, 111, 147, 184, 188, 220, 244, 245, 273, 274, 277, 278, 280, 281, 286, 287, 289, 291, 292, 293, 294, 295, 297, 299, 301 speed, viii, 5, 39, 53, 54, 57, 60, 69, 70, 77, 83, 84, 85, 86, 97, 98, 101, 102, 104, 105, 106, 107, 120, 121, 122, 125, 127, 128, 129, 135, 136, 140, 143, 148, 150, 153, 154, 156, 157, 161, 163, 164, 165, 167, 168, 169, 170, 173, 175, 178, 179, 189, 190, 197, 198 spinal cord, 42, 44, 45 spindle, 35, 40, 41 stability, viii, 2, 3, 5, 10, 11, 13, 14, 15, 18, 19, 22, 24, 25, 26, 27, 53, 54, 57, 69, 70, 71, 72, 73, 75, 77, 78, 81, 86, 96, 98, 106, 109, 111, 115, 123, 129, 135, 147, 155, 161, 185, 187, 188, 189, 190, 192, 197, 198, 199, 200, 201, 202, 203 stimulus, 31, 32, 35, 39, 41, 54 strength, 41, 59 stress, 251, 252, 254, 256 subgroups, 33 Sudan, 313 swelling, 42 symbols, 253, 311 symmetry, 4, 14, 26, 47, 97, 308 synapse, 41, 42, 44, 60 synaptic plasticity, 46 synergetics, 39

T task demands, 38 temperature, 46, 47 tendon, 35, 39, 40, 41 tension, 30, 32 terminals, 42 thalamus, 44, 55 thermodynamic equilibrium, 46 threat, 81, 109 three-dimensional space, 85 threshold, 59, 75, 77, 85, 86, 99, 100, 101, 112, 117, 118, 121, 122, 124, 125, 126, 135, 142, 143, 144, 145, 146, 161, 185, 189, 197 thresholds, 69, 100, 112, 117, 121, 122, 124, 127, 129 timing, 35, 42, 46, 51 tin, 37, 38 tissue, 60, 203 tonic, 51 topology, 51, 264 total energy, 46, 47 training, 35 trajectory, 252 transformation, 46, 55, 73, 74, 166, 229

328

Index

transformations, 45, 46, 66, 68 transition, 30, 37, 38, 39, 48, 54, 250, 251 transitions, 25, 39 translation, 26, 69, 103, 106, 111, 141, 184, 275 transmission, 198, 201 transport, 300 traveling waves, 54, 55, 56, 69, 70, 72, 73, 77, 100, 116, 135, 161, 188, 189, 199, 200, 202, 203 trial, 48 turtle, 56, 202

U uncertainty, 38, 39 uniform, ix, 171, 273, 274, 290, 306, 307, 310, 311 USSR, 25

variables, 15, 46, 47, 48, 161, 163, 254, 290 vector, 4, 33, 63, 69, 90, 93, 133, 195, 245, 250, 251, 288, 289 velocity, 4, 5, 18, 19, 20, 21, 22, 32, 251 Vietnam, 205 vinylidene fluoride, 271

W walking, 42 wave propagation, 25, 100 white matter, 44 winter, 28 withdrawal, 45 working memory, 201

Y V yin, x variability, 2, 22