Lie-B¨acklund-Darboux Transformationsfaculty.missouri.edu/~liyan/book3.pdf · 2019. 2. 27. · That brings the title of this book: Lie-B¨acklund-Darboux transformations which refer

Lie-Backlund-Darboux Transformations

Y. Charles Li

Artyom Yurov

Department of Mathematics, University of Missouri, Columbia,

MO 65211, USA

Theoretical Physics Department, Kaliningrad State University,

Kaliningrad, Russia

Contents

Preface xi

Chapter 1. Introduction 1

Chapter 2. A Brief Account on Backlund Transformations 32.1. A Warm-Up Approach 32.2. Chen’s Method 42.3. Clairin’s Method 52.4. Hirota’s Bilinear Operator Method 52.5. Wahlquist-Estabrook Procedure 6

Chapter 3. Nonlinear Schrodinger Equation 73.1. Physical Background 73.2. Lax Pair and Floquet Theory 83.3. Darboux Transformation and Homoclinic Orbit 93.4. Linear Instability 123.5. Quadratic Products of Eigenfunctions 133.6. Melnikov Vectors 133.7. Melnikov Integrals 16

Chapter 4. Sine-Gordon Equation 194.1. Background 194.2. Lax Pair 194.3. Darboux Transformation 204.4. Melnikov Vector 204.5. Heteroclinic Cycle 234.6. Melnikov Vector Along the Heteroclinic Cycle 26

Chapter 5. Heisenberg Ferromagnet Equation 295.1. Background 295.2. Lax Pair 295.3. Darboux Transformation 305.4. Figure Eight Connecting to the Domain Wall 315.5. Floquet Theory 335.6. Melnikov Vectors 355.7. Melnikov Vector Along the Figure Eight 375.8. A Melnikov Function for Landau-Lifshitz-Gilbert Equation 38

Chapter 6. Vector Nonlinear Schrodinger Equations 416.1. Physical Background 416.2. Lax Pair 41

vii

viii CONTENTS

6.3. Linearized Equations 426.4. Homoclinic Orbits and Figure Eight Structures 426.5. A Melnikov Vector 45

Chapter 7. Derivative Nonlinear Schrodinger Equations 477.1. Physical Background 477.2. Lax Pair 477.3. Darboux Transformations 487.4. Floquet Theory 497.5. Strange Tori 507.6. Whisker of the Strange T2 527.7. Whisker of the Circle 537.8. Diffusion 537.9. Diffusion Along the Strange T2 567.10. Diffusion Along the Whisker of the Circle 57

Chapter 8. Discrete Nonlinear Schrodinger Equation 598.1. Background 598.2. Hamiltonian Structure 598.3. Lax Pair and Floquet Theory 608.4. Examples of Floquet Spectra 618.5. Melnikov Vectors 628.6. Darboux Transformations 648.7. Homoclinic Orbits and Melnikov Vectors 66

Chapter 9. Davey-Stewartson II Equation 699.1. Background 699.2. Linear Stability 709.3. Lax Pairs and Darboux Transformation 709.4. Homoclinic Orbits 729.5. Melnikov Vectors 769.6. Extra Comments 80

Chapter 10. Acoustic Spectral Problem 8110.1. Physical Background 8110.2. Connection with Linear Schrodinger Operator 8210.3. Discrete Symmetries of the Acoustic Problem 8210.4. Crum Formulae and Dressing Chains for the Acoustic Problem 8310.5. Harry-Dym Equation 8610.6. Modified Harry-Dym Equation 8810.7. Moutard Transformations 89

Chapter 11. SUSY and Spectrum Reconstructions 9311.1. SUSY in Two Dimensions 9411.2. The Level Addition 9511.3. Potentials with Cylindrical Symmetry 9611.4. Extended Supersymmetry 97

Chapter 12. Darboux Transformation for Dirac Equation 9912.1. Dirac Equation 99

CONTENTS ix

12.2. Crum Laws 101

Chapter 13. Moutard Transformations for the 2D and 3D SchrodingerEquations 105

13.1. A 2D Moutard Transformation 10513.2. A 3D Moutard Transformation 106

Chapter 14. BLP Equation 10914.1. The Darboux Transformation of the BLP Equation 10914.2. Crum Law 11014.3. Exact Solutions 11314.4. Dressing From Burgers Equation 115

Chapter 15. Goursat Equation 11715.1. The Reduction Restriction 11815.2. Binary Darboux Transformation 12115.3. Moutard Transformation for 2D-MKdV Equation 122

Chapter 16. Links Among Integrable Systems 12516.1. Borisov-Zykov’s Method 12516.2. Higher Dimensional Systems 12616.3. Modified Nonlinear Schrodinger Equations 12816.4. NLS and Toda Lattice 129

Bibliography 131

Index 135

Preface

One of the mathematical miracles of the 20th century was the discovery ofa group of nonlinear wave equations being integrable. These integrable systemsare the infinite dimensional counterpart of the finite dimensional integrable Hamil-tonian systems of classical mechanics. Icons of integrable systems are the KdVequation, sine-Gordon equation, nonlinear Schrodinger equation etc.. The beautyof the integrable theory is reflected by the explicit formulas of nontrivial solutionsto the integrable systems. These explicit solutions bear the iconic names of soli-ton, multi-soliton, breather, quasi-periodic orbit, homoclinic orbit (focus of thisbook) etc.. There are several ways now available for obtaining these explicit so-lutions: Backlund transformation, Darboux transformation, and inverse scatteringtransform. The clear connection among these transforms is still an open questionalthough they are certainly closely related. These transformations can be regardedas the counterpart of the canonical transformation of the finite dimensional inte-grable Hamiltonian system. Backlund transformation originated from a quest forLie’s second type invariant transformation rather than his tangent transformation.That brings the title of this book: Lie-Backlund-Darboux transformations whichrefer to both Backlund transformations and Darboux transformations.

The most famous mathematical miracle of the 20th century was probably thediscovery of chaos. When the finite dimensional integrable Hamiltonian systemsare under perturbations, their regular solutions can turn into chaotic solutions. Forsuch near integrable systems, existence of chaos can sometimes be proved math-ematically rigorously. Following the same spirit, one may attempt to prove theexistence of chaos for near integrable nonlinear wave equations viewed as near in-tegrable Hamiltonian partial differential equations. This has been accomplished assummarized in the book [69]. The key ingredients in this theory of chaos in partialdifferential equations are the explicit formulas for the homoclinic orbit and Mel-nikov integral. The first author’s taste is to use Darboux transformation to obtainthe homoclinic orbit and Melnikov integral. This will be the focus of the first partof this book.

The second author’s taste is to use Darboux transformation in a diversity ofapplications especially in higher spatial dimensions. The range of applicationscrosses many different fields of physics. This will be the focus of the second part ofthis book. This book is a result of the second author’s several visits at Universityof Missouri as a Miller scholar.

The first author would like to thank his wife Sherry and his son Brandon, andthe second author would like to thank his wife Alla and his son Valerian, for lovingsupport during this work.

xi

CHAPTER 1

Introduction

The so-called Backlund transformation originated from studies by S. Lie [80][81] [82] and A. V. Backlund [11] [12] [13] [14] [15] on the Lie’s second questionon the existence of invariant multi-valued surface transformations [5]. Lie’s firstquestion was on the well-known Lie’s tangent transformations. The first example ofa Backlund transformation was studied on the Bianchi’s geometrical construction ofsurfaces of constant negative curvatures – pseudospheres [18]. The Gauss equationof a pseudosphere can be rewritten as the sine-Gordon equation. The Backlundtransformation for the sine-Gordon equation is an invariant transformation witha so-called Backlund parameter first introduced by Backlund. The Backlund pa-rameter is particularly important in Bianchi’s diagram of iterating the Backlundtransformation to generate a so-called nonlinear superposition law [18] [19]. Im-mediate further studies on Backlund transformations were conducted by J. Clairin[26] and E. Goursat [40].

Darboux transformation was first introduced by Gaston Darboux [29] for thenowadays well-known one-dimensional linear Schrodinger equation – a special formof the Sturm-Liouville equations [84]. Darboux found a covariant transformationfor the eigenfunction and the potential. The covariant transformation was builtupon a particular eigenfunction at a particular value of the spectral parameter.

At the beginning, it seemed that Backlund transformation and Darboux trans-formation are irrelevant. The first link of the two came about in 1967 when Gardner,Greene, Kruskal, and Miura related KdV equation to its Lax pair of which the spa-tial part is the one-dimensional linear Schrodinger equation [38]. Soon afterwards,the Backlund transformation for the KdV equation was found. This was the begin-ning of a renaissance of Backlund transformations and Darboux transformations.It turned out that the existence of a Lax pair for a nonlinear wave equation, thesolvability of the Cauchy problem for the nonlinear wave equation by the inversescattering transform [38], the existence of a Backlund transformation for the nonlin-ear wave equation, and the existence of a Darboux transformation for the nonlinearwave equation and its Lax pair are closely related (although clear relation is still notknown). Up to now, Backlund transformations and Darboux transformations formost of the nonlinear wave equations solvable by the inverse scattering transform,have been found [96] [84]. The potential lies at utilizing these transformations. Allthe earlier books [91] [5] [3] [96] [84] [27] [95] [41] focus on using Backlund orDarboux or inverse scattering transformation to construct multi-soliton solutions.Such multi-soliton solutions are defined on the whole spatial space with decayingboundary conditions. When the integrable system is posed with periodic boundaryconditions, the solutions are temporally quasi-periodic or periodic or homoclinic.The first part of this book will focus on homoclinic orbit. Chapter 3-9 contain manyvaluable formulas for homoclinic orbits and Melnikov integrals. Here the Darboux

1

2 1. INTRODUCTION

transformations are not only used to generate explicit formulas for the homoclinicorbits but also interlaced with the isospectral theory of the corresponding Lax pairsto generate Melnikov vectors crucial for building the Melnikov integrals. The inte-grable systems studied in Chapter 3-9 are the so-called canonical systems each ofwhich models a variety of different phenomena. The formulas can be directly usedby the readers to study their own near integrable systems. In Chapter 2, we brieflysummarize various methods for deriving Backlund transformations. These are all“experimental” or “trial-correction” methods. For a more detailed account on thesemethods, we refer the readers to the book [96]. There are not many methods forderiving Darboux transformations. The commonly used one is the dressing method[84]. Sometimes Chen’s method in Chapter 2 can be effective too. Again theseare all “trial-correction” methods. Chapter 10-16 contain applications of Darbouxtransformations in more specific physics problems, and various connections amongdifferent systems. Here no specific boundary condition is posed.

The future of Lie-Backlund-Darboux transformations is very bright. Besidesthe potential of their important applications and new transformations, it is pos-sible to broaden their notion and still end up with useful transformations. Thisbroadening process had begun long ago e.g. the group notion in [5], the jet bundlein [96], the Moutard transformation in this book. The broadened transformationseven reached the Euler equations of fluids [63] [79].

CHAPTER 2

A Brief Account on Backlund Transformations

Backlund transformation is an invariant transformation that transforms onesolution to another of a differential equation. It usually involves partial deriva-tives, and is easily solvable once the initial solution is given. Since J. Clairin[26], researchers have been trying to find an effective systematic way of obtain-ing a Backlund transformation. It was not successful. The commonly employedapproaches are Clairin’s method [26] [5] [96], Chen’s method [25] [91], Hirota’sbilinear operator approach [96], and Wahlquist-Estabrook procedure [103] [104][96]. These are all “trail and correction” approaches. The most common use ofBacklund transformations is to obtain multi-soliton solutions to integrable systems.For a detailed account, we refer the readers to [96]. Here we shall briefly go overthe above mentioned derivation methods.

2.1. A Warm-Up Approach

Take the sine-Gordon equation for example

(2.1) uxy = sinu ,

where u is a real-valued function of two variables x and y. Let us assume that aBacklund transformation transforms u into v where v also satisfies the same sine-Gordon equation

(2.2) vxy = sin v ,

and our goal is to find the Backlund transformation. Let w+ = 12 (u + v) and

w− = 12 (u− v) [59], then w+ and w− satisfy

(2.3) w+xy = sinw+ cosw− , w−

xy = cosw+ sinw− .

We assume that the Backlund transformation has the trial form

(2.4) w−x = f(w+) ,

then by the second equation in (2.3), one has

(2.5) f ′w+y = cosw+ sinw− .

By substituting (2.5) into the first equation of (2.3), one has

−w+x sinw−

f ′

[f ′′

f ′cosw+ + sinw+

]+ cosw−

[f

f ′cosw+ − sinw+

]= 0 ,

which can be satisfied if one demands

f ′′

f ′cosw+ + sinw+ = 0 , and

f

f ′cosw+ − sinw+ = 0 .

A solution of this over-determined system can be found

f = a sinw+ ,

3

4 2. A BRIEF ACCOUNT ON BACKLUND TRANSFORMATIONS

where a is an arbitrary constant called a Backlund parameter. (2.4) and (2.5) nowtake the form

(2.6) w−x = a sinw+ , w+

y =1

asinw− .

(2.6) implies (2.3) which is equivalent to (2.1)-(2.2). (2.6) is the Backlund transfor-mation of the sine-Gordon equation (2.1). The above approach of derivation wasdeveloped in [59]. After experimenting with different forms of assumptions like(2.4), one hopes to find a Backlund transformation for a given equation.

2.2. Chen’s Method

Chen’s method [25] is a quite efficient method. Take the KdV equation forexample,

ut + 6uux + uxxx = 0 ,

its Lax Pair is given by

Lψ = λψ , ∂tψ = Aψ ,

where

L = ∂2x + u , A = −4∂3

x − 3(u∂x + ∂xu) .

Let v = ψx/ψ, one gets

vx + v2 + u = λ ,(2.7)

−vt = 4vxxx + 12vvxx + 12v2vx + 12v2x

+6uxv + 6uvx + 3uxx .(2.8)

Eliminating u, one find that v satisfies the modified KdV equation

vt − 6v2vx + 6λvx + vxxx = 0 .

Relation (2.7) is the Miura transformation between KdV and modified KdV. If vsolves the modified KdV, so does −v, then one can find u solving KdV such that

−vx + v2 + u = λ ,(2.9)

vt = −4vxxx + 12vvxx − 12v2vx + 12v2x

−6uxv − 6uvx + 3uxx .(2.10)

Subtracting (2.9) from (2.7), one gets vx = 12 (u − u). Let wx = 1

2u and wx = 12 u,

then from (2.7) and (2.8), one gets the Backlund transformation for KdV

(w + w)x = λ− (w − w)2 ,

(w − w)t = 4(w − w)xxx + 12(w − w)(w − w)xx

+12(w − w)2(w − w)x + 12(w − w)2x

+12(w − w)wxx + 12wx(w − w)x + 6wxxx ,

where λ is the Backlund parameter.

2.4. HIROTA’S BILINEAR OPERATOR METHOD 5

2.3. Clairin’s Method

Clairin’s method is a direct trial method which involves tedious calculations.In general, let u = u(t, x) satisfy some equation, and let v = v(t, x) be anothervariable linked to u through a transformation of the form

ux = f(u, v, vx, vt) , ut = g(u, v, vx, vt) .

The compatibility conditionft = gx ,

hopefully can lead to an equation for v, of course, by virtue of the equation satisfiedby u. Consider the focusing cubic nonlinear Schrodinger equation

iqt + qxx + |q|2q = 0 ,

Lamb started with a transformation of the form [55]

qx = f(q, q, p, p, px, px) ,

qt = g(q, q, p, p, px, px, pt, pt) .

After some lengthy calculation, Lamb obtained the Backlund transformation [55],

qx = px −1

2iwξ + ikv ,

qt = pt +1

2ξwx − kζ +

1

4iv(|w|2 + |v|2) ,

where

ξ = ±i(b− 2|v|2)1/2 , w = q + p ,

v = q − p , ζ = −1

2iwξ + ikv ,

b and k are arbitrary real constants called Backlund parameters, and p and q satisfythe same equation.

2.4. Hirota’s Bilinear Operator Method

Hirota [45] [46] introduced certain bilinear operators to convert the nonlinearwave equations into bilinear equations for which Backlund transformations can beconstructed. The Hirota bilinear operators Dt and Dx act according to

Dmt D

nxf g = (∂t − ∂t)

m(∂x − ∂x)nf(t, x)g(t, x)|t=t,x=x .

Consider the KdV equation

ut + 6uux + uxxx = 0 ,

by setting u = 2(ln f)xx, one obtains

ut + 6uux + uxxx = ∂x[1

f2Dx(Dt +D3

x)f f ] .

Thus, if f solves the asociated bilinear equation

Dx(Dt +D3x)f f = 0 ,

then u solves the KdV equation. A Backlund transformation for the bilinear equa-tion can be obtained as [46],

(D2x − β)f f = 0 , (Dt + 3βDx +D3)f f = 0 ,

where β is the Backlund parameter.

6 2. A BRIEF ACCOUNT ON BACKLUND TRANSFORMATIONS

2.5. Wahlquist-Estabrook Procedure

Wahlquist-Estabrook Procedure [103] [104] offers a relatively more systematicapproach than that of Clairin [96]. Consider the equation (which is essentially thestationary 2D Euler equation) [58],

(2.11) ∆u = f(u) ,

where ∆ = ∂2x + ∂2

y , and f is an arbitrary function. Let M = R2 (with coordinates

x, y), N = R1 (with coordinate u), N ′ = R1 (with coordinate v), w denote thevolume form on M , w = dx ∧ dy, and θ denote the contact 1-form on the 1-jetJ1(M,N), θ = du − uxdx − uydy. The exterior differential system of 2-forms onJ1(M,N) associated with (2.11) is generated by

σ = dux ∧ dy − duy ∧ dx− f(u)w ,

η1 = du ∧ dy − uxdx ∧ dy ,η2 = du ∧ dx+ uydx ∧ dy .

We seek the following form of Backlund transformations for equation (2.11),

vz = ψz(u, ux, uy, v) , (z = x, y) .

The Wahlquist-Estabrook procedure [103] [104] [96] requires that

dψx ∧ dx+ dψy ∧ dy = f1η1 + f2η2 + gσ + ξ ∧ ζ ,where f1, f2, and g are arbitrary functions on J1(M,N) × J0(M,N ′), ζ = dv −ψxdx − ψydy, and ξ is a 1-form on J1(M,N) × J0(M,N ′). Solving this equation,one finds that if f satisfies [58] [98]

d2f

du2= λf ,

for arbitrary constant λ, then there is a Backlund transformation, and u and vsatisfy the same equation.

CHAPTER 3

Nonlinear Schrodinger Equation

In the late 19th century, G. Darboux [29] introduced a type of transformations,now called the Darboux transformations, for Sturm-Liouville systems. The Dar-boux transformation is a covariant transformation which transforms the solutionand the coeffficient (potential) simultaneously.

For soliton systems, the corresponding Darboux transformation involves boththe evolution equation and its Lax pair. In this context, a Darboux transformationis another form of the Lie-Backlund-Darboux transformation. Like the Backlundtransformations, the derivation method for Darboux transformations is often of“experimental” nature. The popular ones include the “dressing method” [84] andthe Chen’s method [25]. As shown in the previous chapter, the main application of aBacklund transformation is at generating multisoliton solutions. Besides generatingmultisoliton solutions, a Darboux transformation has another novel application -generating homoclinic (heteroclinic) orbits. This new application was not heavilypublicized. Its importance is obvious. Homoclinic (heteroclinic) orbits are thefertile ground of chaos when the system is under perturbations [69] [66] [67] [68][76] [75] [61] [70] [71] [60] [77] [78], [62] [72]. These homoclinic (heteroclinic)orbits form a figure-eight structure also called a separatrix.

We take the focusing nonlinear Schrodinger equation (NLS) as our first exam-ple to show how to construct figure-eight structures [76] [70]. If one starts fromthe conservation laws of the NLS, it turns out that it is very difficult to get theseparatrices. On the contrary, starting from the Darboux transformation to bepresented, one can find the separatrices rather easily.

3.1. Physical Background

The term “nonlinear Schrodinger equation” of course comes from the well-known (linear) Schrodinger equation of quantum mechanics. Folklore says thatit was artificially written down at first and then discovered in many physical ap-plications. There are two types of nonlinear Schrodinger equations: focusing v.s.defocusing.

iqt = qxx ± 2|q|2qwhere q is a complex-valued function of the two real variables (t, x), the upper‘+’ sign corresponds to focusing, and the lower ‘-’ sign corresponds to defocus-ing. More general form may include multi-spatial dimensions with the second de-rivative replaced by Laplacian, and more general nonlinear terms. In the waterwave context, the focusing nonlinear Schrodinger equation models weakly nonlin-ear surface wave on deep water, while the defocusing nonlinear Schrodinger equa-tion models weakly nonlinear surface wave on shallow water. The focusing nonlinearSchrodinger equation has envelope soliton solutions, and homoclinic solutions under

7

8 3. NONLINEAR SCHRODINGER EQUATION

periodic boundary conditions; while the defocusing nonlinear Schrodinger equationhas neither. Shallow water solitons are modeled by KdV equation etc.. In thefiber optics context, the focusing nonlinear Schrodinger equation models the lightwave propagation through a nonlinear medium. An interesting note here is thatthe t-variable actually represents space, and the x-variable actually represents time.In the ferromagnet context, the focusing nonlinear Schrodinger equation is gaugeequivalent to the Heisenberg ferromagnet equation. There are many other contextswhere the nonlinear Schrodinger equation appears. In this sense, it is regardedas a canonical equation. The principal novelty of the more general form of thenonlinear Schrodinger equation is the finite-time blow-up solution. It is here forthe more general nonlinear Schrodinger equations where the finite-time blow-upsolutions are best understood. The nonlinear Schrodinger equation is also closelyrelated to Gross-Pitaevskii equation in Bose-Einstein condensate theory, complexGinzburg-Landau equation, and Davey-Stewartson equations.

3.2. Lax Pair and Floquet Theory

We consider the NLS

(3.1) iqt = qxx + 2|q|2q ,under periodic boundary condition q(x + 2π) = q(x). The NLS is an integrablesystem by virtue of the Lax pair [111],

ϕx = Uϕ ,(3.2)

ϕt = V ϕ ,(3.3)

where

U = iλσ3 + i

(0 q−r 0

),

V = 2 i λ2σ3 + iqrσ3 +

0 2iλq + qx

−2iλr + rx 0

,

where σ3 denotes the third Pauli matrix σ3 = diag(1,−1), r = −q, and λ is thespectral parameter. If q satisfies the NLS, then the compatibility of the over deter-mined system (3.2, 3.3) is guaranteed. Let M = M(x) be the fundamental matrixsolution to the ODE (3.2), M(0) is the 2 × 2 identity matrix. We introduce theso-called transfer matrix T = T (λ, ~q) where ~q = (q,−q), T = M(2π).

Lemma 3.1. Let Y (x) be any solution to the ODE (3.2), then

Y (2nπ) = T n Y (0) .

Proof: Since M(x) is the fundamental matrix,

Y (x) = M(x) Y (0) .

Thus,

Y (2π) = T Y (0) .

Assume that

Y (2lπ) = T l Y (0) .

Notice that Y (x+ 2lπ) also solves the ODE (3.2); then

Y (x + 2lπ) = M(x) Y (2lπ) ;

3.3. DARBOUX TRANSFORMATION AND HOMOCLINIC ORBIT 9

thus,

Y (2(l+ 1)π) = T Y (2lπ) = T l+1 Y (0) .

The lemma is proved. Q.E.D.

Definition 3.2. We define the Floquet discriminant ∆ as,

∆(λ, ~q) = trace T (λ, ~q) .We define the periodic and anti-periodic points λ(p) by the condition

|∆(λ(p), ~q)| = 2 .

We define the critical points λ(c) by the condition

∂∆(λ, ~q)

∂λ

∣∣∣∣λ=λ(c)

= 0 .

A multiple point, denoted λ(m), is a critical point for which

|∆(λ(m), ~q)| = 2.

The algebraic multiplicity of λ(m) is defined as the order of the zero of ∆(λ) ± 2.Ususally it is 2, but it can exceed 2; when it does equal 2, we call the multiple pointa double point, and denote it by λ(d). The geometric multiplicity of λ(m) is definedas the maximum number of linearly independent solutions to the ODE (3.2), andis either 1 or 2.

3.3. Darboux Transformation and Homoclinic Orbit

Let q(x, t) be a solution to the NLS (3.1) for which the linear system (3.2)has a complex double point ν of geometric multiplicity 2. We denote two linearlyindependent solutions of the Lax pair (3.2,3.3) at λ = ν by (φ+, φ−). Thus, ageneral solution of the linear systems at (q, ν) is given by

(3.4) φ(x, t) = c+φ+ + c−φ

− .

We use φ to define a Gauge matrix G by

(3.5) G = G(λ; ν;φ) = N

(λ− ν 0

0 λ− ν

)N−1 ,

where

(3.6) N =

(φ1 −φ2

φ2 φ1

).

Then we define Q and Ψ by

(3.7) Q(x, t) = q(x, t) + 2(ν − ν)φ1φ2

φ1φ1 + φ2φ2

and

(3.8) Ψ(x, t;λ) = G(λ; ν;φ) ψ(x, t;λ)

where ψ solves the Lax pair (3.2,3.3) at (q, ν). Formulas (3.7) and (3.8) are theBacklund-Darboux transformations for the potential and eigenfunctions, respec-tively. We have the following [76],


Theorem 3.3. Let q(x, t) be a solution to the NLS equation (3.1), for whichthe linear system (3.2) has a complex double point ν of geometric multiplicity 2,with eigenbasis (φ+, φ−) for the Lax pair (3.2,3.3), and define Q(x, t) and Ψ(x, t;λ)by (3.7) and (3.8). Then

(1) Q(x, t) is an solution of NLS, with spatial period 2π,(2) Q and q have the same Floquet spectrum,(3) Q(x, t) is homoclinic to q(x, t) in the sense that Q(x, t) −→ qθ±(x, t),

expontentially as exp(−σν |t|), as t −→ ±∞, where qθ± is a “torus trans-late” of q, σν is the nonvanishing growth rate associated to the complexdouble point ν, and explicit formulas exist for this growth rate and for thetranslation parameters θ±,

(4) Ψ(x, t;λ) solves the Lax pair (3.2,3.3) at (Q, λ).

This theorem is quite general, constructing homoclinic solutions from a wideclass of starting solutions q(x, t). It’s proof is one of direct verification [60].

We emphasize several qualitative features of these homoclinic orbits: (i) Q(x, t)is homoclinic to a torus which itself possesses rather complicated spatial and tem-poral structure, and is not just a fixed point. (ii) Nevertheless, the homoclinic orbittypically has still more complicated spatial structure than its “target torus”. (iii)When there are several complex double points, each with nonvanishing growth rate,one can iterate the Backlund-Darboux transformations to generate more compli-cated homoclinic orbits. (iv) The number of complex double points with nonvanish-ing growth rates counts the dimension of the unstable manifold of the critical torusin that two unstable directions are coordinatized by the complex ratio c+/c−. Un-der even symmetry only one real dimension satisfies the constraint of evenness, aswill be clearly illustrated in the following example. (v) These Backlund-Darbouxformulas provide global expressions for the stable and unstable manifolds of thecritical tori, which represent figure-eight structures.

Example: As a concrete example, we take q(x, t) to be the special solution

(3.9) qc = c exp−i[2c2t+ γ]

.

Solutions of the Lax pair (3.2,3.3) can be computed explicitly:

(3.10) φ(±)(x, t;λ) = e±iκ(x+2λt)

(ce−i(2c

2t+γ)/2

(±κ− λ)ei(2c2t+γ)/2

),

where

κ = κ(λ) =√c2 + λ2 .

With these solutions one can construct the fundamental matrix

(3.11) M(x;λ; qc) =

[cosκx+ iλκ sinκx i qc

κ sinκx

i qc

κ sinκx cosκx− iλκ sinκx

],

from which the Floquet discriminant can be computed:

(3.12) ∆(λ; qc) = 2 cos(2κπ) .

From ∆, spectral quantities can be computed:

(1) simple periodic points: λ± = ±i c ,(2) double points: κ(λ

(d)j ) = j/2 , j ∈ Z , j 6= 0 ,

(3) critical points: λ(c)j = λ

(d)j , j ∈ Z , j 6= 0 ,

3.3. DARBOUX TRANSFORMATION AND HOMOCLINIC ORBIT 11

(4) simple periodic points: λ(c)0 = 0 .

For this spectral data, there are 2N purely imaginary double points,

(3.13) (λ(d)j )2 = j2/4 − c2, j = 1, 2, · · · , N ;

where [N2/4 − c2

]< 0 <

[(N + 1)2/4 − c2

].

From this spectral data, the homoclinic orbits can be explicitly computed throughBacklund-Darboux transformation. Notice that to have temporal growth (and de-cay) in the eigenfunctions (3.10), one needs λ to be complex. Notice also thatthe Backlund-Darboux transformation is built with quadratic products in φ, thus

choosing ν = λ(d)j will guarantee periodicity of Q in x. When N = 1, the

Backlund-Darboux transformation at one purely imaginary double point λ(d)1 yields

Q = Q(x, t; c, γ; c+/c−) [76]:

Q =

[cos 2p− sin p sechτ cos(x+ ϑ) − i sin 2p tanh τ

]

[1 + sin p sechτ cos(x+ ϑ)

]−1

ce−i(2c2t+γ)(3.14)

→ e∓2ipce−i(2c2t+γ) as ρ→ ∓∞,

where c+/c− ≡ exp(ρ + iβ) and p is defined by 1/2 + i√c2 − 1/4 = c exp(ip),

τ ≡ σt− ρ, and ϑ = p− (β + π/2).Several points about this homoclinic orbit need to be made:

(1) The orbit depends only upon the ratio c+/c−, and not upon c+ and c−individually.

(2) Q is homoclinic to the plane wave orbit; however, a phase shift of −4poccurs when one compares the asymptotic behavior of the orbit as t →− ∞ with its behavior as t→ + ∞.

(3) For small p, the formula for Q becomes more transparent:

Q ≃[(cos 2p− i sin 2p tanh τ) − 2 sin p sech τ cos(x+ ϑ)

]ce−i(2c

2t+γ).

(4) An evenness constraint on Q in x can be enforced by restricting the phaseφ to be one of two values

φ = 0, π. (evenness)

In this manner, the even symmetry disconnects the level set. Each com-ponent constitutes one loop of the figure eight. While the target q is in-dependent of x, each of these loops has x dependence through the cos(x).One loop has exactly this dependence and can be interpreted as a spatialexcitation located near x = 0, while the second loop has the dependencecos(x − π), which we interpret as spatial structure located near x = π.In this example, the disconnected nature of the level set is clearly relatedto distinct spatial structures on the individual loops. See Figure 1 for anillustration.


Q

qc

Figure 1. An illustration of the figure-eight structure.

(5) Direct calculation shows that the transformation matrix M(1;λ(d)1 ;Q) is

similar to a Jordan form when t ∈ (−∞,∞),

M(1;λ(d)1 ;Q) ∼

(−1 10 −1

),

and when t→ ±∞, M(1;λ(d)1 ;Q) −→ −I (the negative of the 2x2 identity

matrix). Thus, when t is finite, the algebraic multiplicity (= 2) of λ = λ(d)1

with the potential Q is greater than the geometric multiplicity (= 1).

In this example the dimension of the loops need not be one, but is determinedby the number of purely imaginary double points which in turn is controlled by theamplitude c of the plane wave target and by the spatial period. (The dimensionof the loops increases linearly with the spatial period.) When there are severalcomplex double points, Backlund-Darboux transformations must be iterated toproduce complete representations. Thus, Backlund-Darboux transformations giveglobal representations of the figure-eight structures.

3.4. Linear Instability

The above figure-eight structure corresponds to the following linear instabilityof Benjamin-Feir type. Consider the uniform solution to the NLS (3.1),

qc = ceiθ(t) , θ(t) = −[2c2t+ γ] .

Let

q = [c+ q]eiθ(t) ,

and linearize equation (3.1) at qc, we have

iqt = qxx + 2c2[q + ¯q] .

Assume that q takes the form,

q =

[Aje

Ωjt +BjeΩjt

]cos kjx ,

where kj = 2jπ, (j = 0, 1, 2, · · ·), Aj and Bj are complex constants. Then,

Ω(±)j = ±kj

√4c2 − k2

j .

Thus, we have instabilities when c > 1/2.

3.6. MELNIKOV VECTORS 13

3.5. Quadratic Products of Eigenfunctions

Quadratic products of eigenfunctions play a crucial role in characterizing thehyperbolic structures of soliton equations. Its importance lies in the following as-pects: (i). Certain quadratic products of eigenfunctions solve the linearized solitonequation. (ii). Thus, they are the perfect candidates for building a basis to theinvariant linear subbundles. (iii). Also, they signify the instability of the solitonequation. (iv). Most importantly, quadratic products of eigenfunctions can serveas Melnikov vectors, e.g., for Davey-Stewartson equation [62].

Consider the linearized NLS equation at any solution q(t, x) written in thevector form:

i∂t(δq) = (δq)xx + 2[q2δq + 2|q|2δq] ,(3.15)

i∂t(δq) = −(δq)xx − 2[q2δq + 2|q|2δq] ,we have the following lemma [76].

Lemma 3.4. Let ϕ(j) = ϕ(j)(t, x;λ, q) (j = 1, 2) be any two eigenfunctionssolving the Lax pair (3.2,3.3) at an arbitrary λ. Then

δq

δq

,

ϕ(1)1 ϕ

(2)1

ϕ(1)2 ϕ

(2)2

, and S

ϕ(1)1 ϕ

(2)1

ϕ(1)2 ϕ

(2)2

−

, where S =

(0 11 0

)

solve the same equation (3.15); thus

Φ =

ϕ(1)1 ϕ

(2)1

ϕ(1)2 ϕ

(2)2

+ S

ϕ(1)1 ϕ

(2)1

ϕ(1)2 ϕ

(2)2

−

solves the equation (3.15) and satisfies the reality condition Φ2 = Φ1.

Proof: Direct calculation leads to the conclusion. Q.E.D.The periodicity condition Φ(x + 2π) = Φ(x) can be easily accomplished. For

example, we can take ϕ(j) (j = 1, 2) to be two linearly independent Bloch functionsϕ(j) = eσjxψ(j) (j = 1, 2), where σ2 = −σ1 and ψ(j) are periodic functions ψ(j)(x+2π) = ψ(j)(x). Often we choose λ to be a double point of geometric multiplicity 2,so that ϕ(j) are already periodic or antiperiodic functions.

3.6. Melnikov Vectors

Definition 3.5. Define the sequence of functionals Fj as follows,

(3.16) Fj(~q) = ∆(λcj(~q), ~q),

where λcj ’s are the critical points, ~q = (q,−q).

We have the lemma [76]:

Lemma 3.6. If λcj(~q) is a simple critical point of ∆(λ) [i.e., ∆′′(λcj) 6= 0], Fj isanalytic in a neighborhood of ~q, with first derivative given by

(3.17)δFjδq

=δ∆

δq

∣∣∣∣λ=λc

j

+∂∆

∂λ

∣∣∣∣λ=λc

j

δλcjδq

=δ∆

δq

∣∣∣∣λ=λc

j

,


where

(3.18)δ

δ~q∆(λ; ~q) = i

√∆2 − 4

W [ψ+, ψ−]

ψ+

2 (x;λ)ψ−2 (x;λ)

ψ+1 (x;λ)ψ−

1 (x;λ)

,

and the Bloch eigenfunctions ψ± have the property that

ψ±(x+ 2π;λ) = ρ±1ψ±(x;λ) ,(3.19)

for some ρ, also the Wronskian is given by

W [ψ+, ψ−] = ψ+1 ψ

−2 − ψ+

2 ψ−1 .

In addition, ∆′ is given by

d∆

dλ= −i

√∆2 − 4

W [ψ+, ψ−]

∫ 2π

0

[ψ+1 ψ

−2 + ψ+

2 ψ−1 ] dx .(3.20)

Proof: To prove this lemma, one calculates using variation of parameters:

δM(x;λ) = M(x)

∫ x

0

M−1(x′)δQ(x′)M(x′)dx′,

where

δQ ≡ i

0 δq

δq 0

.

Thus; one obtains the formula

δ∆(λ; ~q) = trace

[M(2π)

∫ 2π

0

M−1(x′) δQ(x′) M(x′)dx′],

which gives

δ∆(λ)

δq(x)= i trace

[M−1(x)

( 0 10 0

)M(x)M(2π)

],

(3.21)

δ∆(λ)

δq(x)= i trace

[M−1(x)

( 0 01 0

)M(x)M(2π)

].

Next, we use the Bloch eigenfunctions ψ± to form the matrix

N(x;λ) =

(ψ+

1 ψ−1

ψ+2 ψ−

2

).

Clearly,

N(x;λ) = M(x;λ)N(0;λ) ;

or equivalently,

(3.22) M(x;λ) = N(x;λ)[N(0;λ)]−1 .

Since ψ± are Bloch eigenfunctions, one also has

N(x+ 2π;λ) = N(x;λ)

(ρ 00 ρ−1

),


which implies

N(2π;λ) = M(2π;λ)N(0;λ) = N(0;λ)

(ρ 00 ρ−1

),

that is,

(3.23) M(2π;λ) = N(0;λ)

(ρ 00 ρ−1

)[N(0;λ)]−1 .

For any 2 × 2 matrix σ, equations (3.22) and (3.23) imply

trace [M(x)]−1 σ M(x) M(2π) = trace [N(x)]−1 σ N(x) diag(ρ, ρ−1),which, through an explicit evaluation of (3.21), proves formula (3.18). Formula(3.20) is established similarly. These formulas, together with the fact that λc(~q) is

differentiable because it is a simple zero of ∆′, provide the representation ofδFj

δ~q .

Q.E.D.

Remark 3.7. Formula (3.17) for theδFj

δ~q is actually valid even if ∆′′(λcj(~q); ~q) =

0. Consider a function ~q∗ at which

∆′′(λcj(~q∗); ~q∗) = 0,

and thus at which λcj(~q) fails to be analytic. For ~q near ~q∗, one has

∆′(λcj(~q); ~q) = 0;

δ

δ~q∆′(λcj(~q); ~q) = ∆′′ δ

δ~qλcj +

δ

δ~q∆′ = 0;

that is,δ

δ~qλcj = − 1

∆′′

δ

δ~q∆′ .

Thus,δ

δ~qFj =

δ

δ~q∆ + ∆′ δ

δ~qλcj =

δ

δ~q∆ − ∆′

∆′′

δ

δ~q∆′ |λ=λc

j(~q) .

Since ∆′

∆′′ → 0, as ~q → ~q∗, one still has formula (3.17) at ~q = ~q∗:

δ

δ~qFj =

δ

δ~q∆ |λ=λc

j(~q) .

The NLS (3.1) is a Hamiltonian system:

(3.24) iqt =δH

δq,

where

H =

∫ 2π

0

−|qx|2 + |q|4 dx .

Corollary 3.8. For any fixed λ ∈ C, ∆(λ, ~q) is a constant of motion of theNLS (3.1). In fact,

∆(λ, ~q), H(~q) = 0 , ∆(λ, ~q),∆(λ′, ~q) = 0 , ∀λ, λ′ ∈ C ,

where for any two functionals E and F , their Poisson bracket is defined as

E,F =

∫ 2π

0

[δE

δq

δF

δq− δE

δq

δF

δq

]dx .


Proof: The corollary follows from a direction calculation from the spatial part(3.2) of the Lax pair and the representation (3.18). Q.E.D.

For each fixed ~q, ∆ is an entire function of λ; therefore, can be determined byits values at a countable number of values of λ. The invariance of ∆ characterizesthe isospectral nature of the NLS equation.

Corollary 3.9. The functionals Fj are constants of motion of the NLS (3.1).Their gradients provide Melnikov vectors:

(3.25) grad Fj(~q) = i

√∆2 − 4

W [ψ+, ψ−]

ψ+

2 (x;λcj)ψ−2 (x;λcj)

ψ+1 (x;λcj)ψ

−1 (x;λcj)

.

The distribution of the critical points λcj are described by the following counting

lemma [76],

Lemma 3.10 (Counting Lemma for Critical Points). For q ∈ H1, set N =N(‖q‖1) ∈ Z+ by

N(‖q‖1) = 2

[‖q‖2

0 cosh

(‖q‖0

)+ 3‖q‖1 sinh

(‖q‖0

)],

where [x] = first integer greater than x. Consider

∆′(λ; ~q) =d

dλ∆(λ; ~q).

Then

(1) ∆′(λ; ~q) has exactly 2N + 1 zeros (counted according to multiplicity) inthe interior of the disc D = λ ∈ C : |λ| < (2N + 1)π2 ;

(2) ∀k ∈ Z, |k| > N,∆′(λ, ~q) has exactly one zero in each discλ ∈ C : |λ− kπ| < π

4 .(3) ∆′(λ; ~q) has no other zeros.(4) For |λ| > (2N + 1)π2 , the zeros of ∆′, λcj , |j| > N, are all real, simple,

and satisfy the asymptotics

λcj = jπ + o(1) as |j| → ∞.

3.7. Melnikov Integrals

When studying perturbed integrable systems, the figure-eight structures oftenlead to chaotic dynamics through homoclinic bifurcations. An extremely powerfultool for detecting homoclinic orbits is the so-called Melnikov integral method [85],which uses “Melnikov integrals” to provide estimates of the distance between thecenter-unstable manifold and the center-stable manifold of a normally hyperbolicinvariant manifold. The Melnikov integrals are often integrals in time of the in-ner products of certain Melnikov vectors with the perturbations in the perturbedintegrable systems. This implies that the Melnikov vectors play a key role in theMelnikov integral method. First, we consider the case of one unstable mode asso-ciated with a complex double point ν, for which the homoclinic orbit is given byBacklund-Darboux formula (3.7),

Q(x, t) ≡ q(x, t) + 2(ν − ν)φ1φ2

φ1φ1 + φ2φ2,

3.7. MELNIKOV INTEGRALS 17

where q lies in a normally hyperbolic invariant manifold and φ denotes a generalsolution to the Lax pair (3.2, 3.3) at (q, ν), φ = c+φ

+ + c−φ−, and φ± are Bloch

eigenfunctions. Next, we consider the perturbed NLS,

iqt = qxx + 2|q|2q + iǫf ,

where ǫ is the perturbation parameter and f may depend on q and q and theirspatial derivatives. The Melnikov integral can be defined using the constant ofmotion Fj , where λcj = ν [76]:

(3.26) Mj ≡∫ +∞

−∞

∫ 2π

0

δFjδq

f +δFjδq

fq=Q dxdt ,

where the integrand is evaluated along the unperturbed homoclinic orbit q = Q, and

the Melnikov vectorδFj

δ~q has been given in the last section, which can be expressed

rather explicitly using the Backlund-Darboux transformation [76] . We begin withthe expression (3.25),

δFjδ~q

= i

√∆2 − 4

W [Φ+,Φ−]

(Φ+

2 Φ−2

Φ+1 Φ−

1

)(3.27)

where Φ± are Bloch eigenfunctions at (Q, ν), which can be obtained from Backlund-Darboux formula (3.7):

Φ±(x, t; ν) ≡ G(ν; ν;φ) φ±(x, t; ν) ,

with the transformation matrix G given by

G = G(λ; ν;φ) = N

(λ− ν 0

0 λ− ν

)N−1,

N ≡[φ1 −φ2

φ2 φ1

].

These Backlund-Darboux formulas are rather easy to manipulate to obtain explicitinformation. For example, the transformation matrix G(λ, ν) has a simple limit asλ→ ν:

limλ→ν

G(λ, ν) =ν − ν

|φ|2(

φ2φ2 −φ1φ2

−φ2φ1 φ1φ1

)(3.28)

where |φ|2 is defined by

|~φ|2 ≡ φ1φ1 + φ2φ2.

With formula (3.28) one quickly calculates

Φ± = ±c∓ W [φ+, φ−]ν − ν

|φ|2(

φ2

−φ1

)

from which one sees that Φ+ and Φ− are linearly dependent at (Q, ν),

Φ+ = −c−c+

Φ−.

Remark 3.11. For Q on the figure-eight, the two Bloch eigenfunctions Φ± arelinearly dependent. Thus, the geometric multiplicity of ν is only one, even thoughits algebraic multiplicity is two or higher.


Using L’Hospital’s rule, one gets√

∆2 − 4

W [Φ+,Φ−]=

√∆(ν)∆′′(ν)

(ν − ν) W [φ+, φ−].(3.29)

With formulas (3.27, 3.28, 3.29), one obtains the explicit representation of thegrad Fj [76]:

δFjδ~q

= Cνc+c−W [ψ(+), ψ(−)]

|φ|4

φ22

−φ21

,(3.30)

where the constant Cν is given by

Cν ≡ i(ν − ν)√

∆(ν)∆′′(ν) .

With these ingredients, one obtains the following beautiful representation of theMelnikov function associated to the general complex double point ν [76]:

(3.31) Mj = Cν c+c−

∫ +∞

−∞

∫ 2π

0

W [φ+, φ−]

[(φ2

2)f(Q, Q) + (φ21)f(Q, Q)

|φ|4

]dxdt.

In the case of several complex double points, each associated with an instability,one can iterate the Backlund-Darboux tranformations and use those functionalsFj which are associated with each complex double point to obtain representations

Melnikov Vectors. In general, the relation betweenδFj

δ~q and double points can be

summarized in the following lemma [76],

Lemma 3.12. Except for the trivial case q = 0,

(a).δFjδq

= 0 ⇔ δFjδq

= 0 ⇔M(2π, λcj ; ~q∗) = ±I.

(b).δFjδ~q

|~q∗= 0 ⇒ ∆′(λcj(~q∗); ~q∗) = 0,⇒ |Fj(~q∗)| = 2,

⇒ λcj(~q∗) is a multiple point,

where I is the 2 × 2 identity matrix.

The Backlund-Darboux transformation theorem indicates that the figure-eightstructure is attached to a complex double point. The above lemma shows that atthe origin of the figure-eight, the gradient of Fj vanishes. Together they indicatethat the the gradient of Fj along the figure-eight is a perfect Melnikov vector.

Example: When 12 < c < 1 in (3.9), and choosing ϑ = π in (3.14), one can

get the Melnikov vector field along the homoclinic orbit (3.14),

δF1

δq= 2π sin2 p sechτ

[(− sin p+ i cosp tanh τ) cos x+ sechτ ]

[1 − sin p sech τ cosx]2c eiθ,(3.32)

δF1

δq=δF1

δq.

CHAPTER 4

Sine-Gordon Equation

4.1. Background

Sine-Gordon equation was the first equation for which a Backlund transforma-tion was found. The first application of the sine-Gordon equation was in differentialgeometry. It turned out that the Gauss equation of a pseudosphere can be rewrit-ten as the sine-Gordon equation [18]. It was in the pseudosphere study, the firstBacklund transformation was discovered and the first Bianchi diagram was devel-oped to build a nonlinear superposition principle for the solutions of the sine-Gordonequation [18] [15]. For a more detailed account, we refer the readers to [5]. Mod-ern applications of the sine-Gordon equation and its various generalizations havean extremely wide scope ranging from physics to biology. For instance, the fluxdynamics in a long Josephson junction is described by the sine-Gordon equation.The “sine-Gordon” was coined from the well-known Klein-Gordon equation. It isclosely connected to other integrable systems like elliptic sine-Gordon equation,sinh-Gordon equation, and elliptic sinh-Gordon equation.

4.2. Lax Pair

Here we consider the sine-Gordon equation

(4.1) utt = c2uxx + sinu , (c is a positive constant)

which is integrable through the Lax pair

ψx = Bψ ,(4.2)

ψt = Aψ ,(4.3)

where

B =1

c

i4 (cux + ut)

116λe

iu + λ

− 116λe

−iu − λ − i4 (cux + ut)

,

A =

i4 (cux + ut) − 1

16λeiu + λ

116λe

−iu − λ − i4 (cux + ut)

.

The Lax pair (4.2)-(4.3) possesses a symmetry.

Lemma 4.1. If ψ = (ψ1, ψ2)T solves the Lax pair (4.2)-(4.3) at (λ, u), then

(ψ2, ψ1)T solves the Lax pair (4.2)-(4.3) at (−λ, u).

For more details on this chapter, see [68] [73].

19

20 4. SINE-GORDON EQUATION

4.3. Darboux Transformation

There is a Darboux transformation for the Lax pair (4.2)-(4.3).

Theorem 4.2 (Darboux Transformation I). Let

U = u+ 2i ln

[iφ2

φ1

],

Ψ =

(−νφ2/φ1 λ−λ νφ1/φ2

)ψ ,

where φ = ψ|λ=ν for some ν, then Ψ solves the Lax pair (4.2)-(4.3) at (λ, U).

Often in order to guarantee the reality condition (i.e. U needs to be real-valued), one needs to iterate the Darboux transformation by virtue of Lemma 4.1.The result corresponds to the counterpart of the Darboux transformation for thecubic nonlinear Schrodinger equation [76] [68].

Theorem 4.3 (Darboux Transformation II). Let

U = u+ 2i ln

[ν|φ1|2 + ν|φ2|2ν|φ1|2 + ν|φ2|2

],

Ψ = Gψ ,

where

G =

(G1 G2

G3 G4

),

G1 = |ν|2 ν|φ1|2 + ν|φ2|2ν|φ1|2 + ν|φ2|2

− λ2 ,

G2 =λ(ν2 − ν2)φ1φ2

ν|φ1|2 + ν|φ2|2,

G3 =λ(ν2 − ν2)φ1φ2

ν|φ1|2 + ν|φ2|2,

G4 = |ν|2 ν|φ1|2 + ν|φ2|2ν|φ1|2 + ν|φ2|2

− λ2 ,

and φ = ψ|λ=ν for some ν, then Ψ solves the Lax pair (4.2)-(4.3) at (λ, U).

These Darboux transformations will be used to generate separatrices (figure-eight structures).

4.4. Melnikov Vector

Focusing upon the spatial part (4.2) of the Lax pair, one can develop a completeFloquet theory. Let M(x) be the fundamental matrix of (4.2), M(0) = I (2 × 2identity matrix), then the Floquet discriminant is given as

∆ = trace M(2π) .

The Floquet spectrum is given by

σ = λ ∈ C | − 2 ≤ ∆(λ) ≤ 2 .

4.4. MELNIKOV VECTOR 21

Periodic and anti-periodic points λ± (which correspond to periodic and anti-periodiceigenfunctions respectively) are defined by

∆(λ±) = ±2 .

A critical point λ(c) is defined by

d∆

dλ(λ(c)) = 0 .

A multiple point λ(m) is a periodic or anti-periodic point which is also a criticalpoint. The algebraic multiplicity of λ(m) is defined as the order of the zero of∆(λ) ± 2 at λ(m). When the order is 2, we call the multiple point a double point,and denote it by λ(d). The order can exceed 2. The geometric multiplicity of λ(m)

is defined as the dimension of the periodic or anti-periodic eigenspace at λ(m), andis either 1 or 2.

Counting lemmas for λ± and λ(c) can be established as in [90] [76], which lead

to the existence of the sequences λ±j and λ(c)j and their approximate locations.

Nevertheless, counting lemmas are not necessary here. For any λ ∈ C, ∆(λ) isinvariant under the sine-Gordon flow. This is the so-called isospectral theory.

Definition 4.4. An important sequence of invariants Fj of the sine-Gordonequation is defined by

Fj(u, ut) = ∆(λ(c)j (u, ut), u, ut) .

Lemma 4.5. If λ(c)j is a simple critical point of ∆, then

∂Fj∂w

=∂∆

∂w

∣∣∣∣λ=λ

(c)j

, w = u, ut .

Proof. We know that

∂Fj∂w

=∂∆

∂w

∣∣∣∣λ=λ

(c)j

+∂∆

∂λ

∣∣∣∣λ=λ

(c)j

∂λ(c)j

∂w.

Since∂∆

∂λ

∣∣∣∣λ=λ

(c)j

= 0 ,

we have

∂2∆

∂λ2

∣∣∣∣λ=λ

(c)j

∂λ(c)j

∂w+

∂2∆

∂λ∂w

∣∣∣∣λ=λ

(c)j

= 0 .

Since λ(c)j is a simple critical point of ∆,

∂2∆

∂λ2

∣∣∣∣λ=λ

(c)j

6= 0 .

Thus

∂λ(c)j

∂w= −

[∂2∆

∂λ2

∣∣∣∣λ=λ

(c)j

]−1∂2∆

∂λ∂w

∣∣∣∣λ=λ

(c)j

.


Notice that ∆ is an entire function of λ and w = u, ut [76], then we know that∂λ

(c)j

∂w is bounded, and

∂Fj∂w

=∂∆

∂w

∣∣∣∣λ=λ

(c)j

.

Theorem 4.6. As a function of three variables, ∆ = ∆(λ, u, ut) has the partial

derivatives given by Bloch functions ψ± (i.e. ψ±(x) = e±Λxψ±(x), where ψ± areperiodic in x of period 2π, and Λ is a complex constant):

∂∆

∂u=

−i16λc

√∆2 − 4

W (ψ+, ψ−)

[4λc∂x(ψ

+1 ψ

−2 + ψ+

2 ψ−1 ) + e−iuψ+

1 ψ−1 − eiuψ+

2 ψ−2

],

∂∆

∂ut=

i

4c

√∆2 − 4

W (ψ+, ψ−)

[ψ+

1 ψ−2 + ψ+

2 ψ−1

],

∂∆

∂λ=

−1

c

√∆2 − 4

W (ψ+, ψ−)

∫ 2π

0

[(1

16λ2eiu − 1

)ψ+

2 ψ−2 +

(1

16λ2e−iu − 1

)ψ+

1 ψ−1

]dx ,

where W (ψ+, ψ−) = ψ+1 ψ

−2 − ψ+

2 ψ−1 is the Wronskian.

Proof. Recall that M is the fundamental matrix solution of (4.2), we havethe equation for the differential of M

∂xdM = BdM + dBM , dM(0) = 0 .

Using the method of variation of parameters, we let

dM = MQ , Q(0) = 0 .

Thus

Q(x) =

∫ x

0

M−1dBMdx ,

and

dM(x) = M(x)

∫ x

0

M−1dBMdx .

Finally

d∆ = trace dM(2π)

= trace

M(2π)

∫ 2π

0

M−1dBMdx

.(4.4)

Let

N = (ψ+ψ−)

where ψ± are two linearly independent Bloch functions (For the case that there isonly one linearly independent Bloch function, L’Hospital’s rule has to be used, fordetails, see [76]), such that

ψ± = e±Λxψ± ,

where ψ± are periodic in x of period 2π and Λ is a complex constant (The existenceof such functions is the result of the well known Floquet theorem). Then

N(x) = M(x)N(0) , M(x) = N(x)[N(0)]−1 .

4.5. HETEROCLINIC CYCLE 23

Notice that

N(2π) = N(0)E , where E =

(eΛ2π 00 e−Λ2π

).

ThenM(2π) = N(0)E[N(0)]−1 .

Thus∆ = trace M(2π) = trace E = eΛ2π + e−Λ2π ,

and

e±Λ2π =1

2[∆ ±

√∆2 − 4] .

In terms of N , d∆ as given in (4.4) takes the form

d∆ = trace

N(0)E[N(0)]−1

∫ 2π

0

N(0)[N(x)]−1dB(x)N(x)[N(0)]−1dx

= trace

E

∫ 2π

0

[N(x)]−1dB(x)N(x)dx

,

from which one obtains the partial derivatives of ∆ as stated in the theorem.

It turns out that the partial derivatives of Fj provide the perfect Melnikovvectors rather than those of the Hamiltonian or other invariants [76], in the sensethat Fj is the invariant whose level sets are the separatrices.

4.5. Heteroclinic Cycle

In this section, we will focus upon an example starting from u = 0. Linearizingthe sine-Gordon equation at u = 0 leads to

utt = c2uxx + u .

Let u =∑∞k=0 uk(t) cos kx, then

d2

dt2uk = (1 − c2k2)uk .

Let uk ∼ eΩkt, then

Ωk = ±√

1 − c2k2 , k = 0, 1, 2, · · · .Since 1/2 < c < 1, there are only two unstable modes k = 0 and 1. The cor-responding nonlinear unstable foliation can be represented through the Darbouxtransformations given in Theorems 4.2 and 4.3. When u = 0, the Bloch functionsof the Lax pair (4.2)-(4.3) are

ψ± = e±i(κx+ωt)(

1±i

),

where κ = 1c (λ+ 1

16λ ) and ω = λ− 116λ . Thus,

λ =1

2

[κc±

√(κc)2 − 1

4

].

The Floquet discriminant is given by

(4.5) ∆ = 2 cos(2πκ) .

The spectral data are depicted in Figure 1. First we choose ν0 = λ(d)0 = i

4 and use


l

0 1/4

lNFÉG

lOFÉG

Figure 1. Floquet spectrum of the Lax pair at u = 0, • doublepoint, critical point.

Theorem 4.2 to generate the foliation corresponding to k = 0. The correspondingBloch functions are

φ± = exp

∓1

2t

(1±i

).

The wise choice for φ is

φ =

√c+

c−φ+ +

√c−

c+φ− ,

where c± are arbitrary complex constants. Let

(4.6)c+

c−= eρ+iθ , τ = t− ρ ,

then

(4.7) φ = 2

(cosh τ

2 cos θ2 − i sinh τ2 sin θ

2

− cosh τ2 sin θ

2 − i sinh τ2 cos θ2

).

By Theorem 4.2, we have

U = 2i ln

[iφ2

φ1

],

where

iφ2

φ1=

tanh τ − i sin θ sech τ

1 + cos θ sech τ.

For U to be real-valued, θ = ±π/2; and we have

(4.8) U± = ±2ϑ ,

where

ϑ = arccos(tanh τ) , ϑ ∈ [0, π] .

4.5. HETEROCLINIC CYCLE 25

In order to obtain the full foliation for the two unstable modes k = 0 and 1, one

needs to apply the Darboux transformation II in Theorem 4.3 at ν1 = λ(d)1 =

14 [c+ i

√1 − c2] and u = U±. At (λ

(d)1 , u = 0), the Bloch functions are

(4.9) ϕ± = exp

±i1

2x∓ σ

2t

(1±i

), σ =

√1 − c2 ,

Let

ϕ =

√c+c−ϕ+ +

√c−c+ϕ− ,

where c± are arbitrary complex constants. Let

(4.10)c+c−

= eρ+iθ , τ = σt− ρ , ξ = x+ θ ;

then

ϕ = 2

(cosh τ

2 cos ξ2 − i sinh τ2 sin ξ

2

− cosh τ2 sin ξ

2 − i sinh τ2 cos ξ2

).

Applying the Darboux transformation in Theorem 4.2 to ϕ, one gets

(4.11) Φ =

(−ν0φ2/φ1 ν1−ν1 ν0φ1/φ2

)ϕ =

(iν0e

∓iϑϕ1 + ν1ϕ2

−ν1ϕ1 + iν0e±iϑϕ2

),

which solves the Lax pair at (ν1, U±). Applying the Darboux transformation II inTheorem 4.3 at (ν1, U±) with Φ, we have

U = ±2ϑ− 4ϑ

where ϑ is given in (4.8), and ϑ ∈ (−π/2, π/2),

ϑ = arctan

[σ(tanh τ sech τ sin ξ ∓ sech τ tanh τ )

1 − σ(tanh τ tanh τ ± sech τ sech τ sin ξ)

].

For ϑ to be even in x, we need to choose θ = π/2 or −π/2, and obtain

U±,δ = ±2ϑ− 4ϑ , δ = + or − ;

where

ϑ = arctan

[σ(δ tanh τ sech τ cosx∓ sech τ tanh τ )

1 − σ(tanh τ tanh τ ± δ sech τ sech τ cosx)

].

The heteroclinic cycle is given by

U1 = U+,δ = 2 arccos(tanh τ)

−4 arctan

[σ(± tanh τ sech τ cosx− sech τ tanh τ)

1 − σ(tanh τ tanh τ ± sech τ sech τ cosx)

],(4.12)

U2 = 2π + U−,δ = 2π − 2 arccos(tanh τ)

−4 arctan

[σ(± tanh τ sech τ cosx+ sech τ tanh τ)

1 − σ(tanh τ tanh τ ∓ sech τ sech τ cosx)

],(4.13)

where the ranges of arccos and arctan are [0, π] and (−π/2, π/2), and (τ, σ, τ ) aredefined in (4.6), (4.9), and (4.10).


4.6. Melnikov Vector Along the Heteroclinic Cycle

Since the sine-Gordon equation is invariant under the transformations u →u + 2π and u → −u, we have the following symmetries of the Lax pair (4.2) and(4.3).

Lemma 4.7. If ψ = (ψ1, ψ2)T solves the Lax pair (4.2) and (4.3) at (λ, u), then

it also solves the Lax pair (4.2) and (4.3) at (λ, u+ 2π), and (ψ2,−ψ1)T solves the

Lax pair (4.2) and (4.3) at (λ,−u).

Next we calculate the Melnikov vectors which will be given by∂Fj

∂ut(j = 0, 1)

where λ(c)j = νj . By Lemma 4.7, the Melnikov vectors along U1 and U2 are the

same as along U+,δ and U−,δ. Expressions of the Melnikov vectors are given inTheorem 4.6. According to Theorem 4.2, let

φ± =

(−ν0φ2/φ1 ν0−ν0 ν0φ1/φ2

)φ±

= ∓ν0 exp

∓ρ

2∓ i

θ

2

W (φ+, φ−)

(1/φ1

1/φ2

),

where θ = π/2 or −π/2. φ± solves the Lax pair at (ν0, U+) or (ν0, U−) dependingupon the choice of θ. According to Theorem 4.3, let

(4.14) Φ± =

(G1 G2

G3 G4

)φ± ,

where

G1 = |ν1|2ν1|Φ1|2 + ν1|Φ2|2ν1|Φ1|2 + ν1|Φ2|2

− ν20 ,

G2 =ν0(ν

21 − ν1

2)Φ1Φ2

ν1|Φ1|2 + ν1|Φ2|2,

G3 =ν0(ν

21 − ν1

2)Φ1Φ2

ν1|Φ1|2 + ν1|Φ2|2,

G4 = |ν1|2ν1|Φ1|2 + ν1|Φ2|2ν1|Φ1|2 + ν1|Φ2|2

− ν20 .

Φ± solves the Lax pair at (ν0, U±,δ) where δ = + or − depending on the choice of

θ = π/2 or −π/2.Similarly, according to Theorem 4.2, let

Φ± =

(−ν0φ2/φ1 ν1−ν1 ν0φ1/φ2

)ϕ± ,

then

Φ =

√c+c−

Φ+ +

√c−c+

Φ− .

Φ± solves the Lax pair at (ν1, U+) or (ν1, U−) depending on the choice of θ = π/2or −π/2. According to Theorem 4.3, let

Φ± =

(G1 G2

G3 G4

)Φ± ,

4.6. MELNIKOV VECTOR ALONG THE HETEROCLINIC CYCLE 27

where

G1 =ν1(ν1

2 − ν21)|Φ2|2

ν1|Φ1|2 + ν1|Φ2|2,

G2 =ν1(ν

21 − ν1

2)Φ1Φ2

ν1|Φ1|2 + ν1|Φ2|2,

G3 =ν1(ν

21 − ν1

2)Φ1Φ2

ν1|Φ1|2 + ν1|Φ2|2,

G4 =ν1(ν1

2 − ν21)|Φ1|2

ν1|Φ1|2 + ν1|Φ2|2.

Then

(4.15) Φ± = ±ν1(ν21 − ν1

2) exp

∓ ρ

2∓ i

θ

2

W (Φ+,Φ−)

−Φ2

ν1|Φ1|2+ν1|Φ2|2

Φ1

ν1|Φ1|2+ν1|Φ2|2

.

Φ± solves the Lax pair at (ν1, U±,δ) where δ = + or − depending on the choice of

θ = π/2 or −π/2. Recall the process of obtaining (4.14) and (4.15), let ψ± be apair of Bloch functions solving the Lax pair at (λ, u = 0), then by Theorems 4.2and 4.3,

ψ± ν0−→ ψ± ν1−→ ψ± .

Notice that in Theorem 4.3,

det G = (λ2 − ν2)(λ2 − ν2) .

Thus

W (ψ+, ψ−) = (λ2 − ν21)(λ2 − ν1

2)W (ψ+, ψ−)

= (λ2 − ν21)(λ2 − ν1

2)(λ2 − ν20 )W (ψ+, ψ−) .

By the isospectral property, U±,δ, U± and u = 0 have the same Floquet discriminant∆ = ∆(λ) as given in (4.5), ∆(ν0) = 2, ∆(ν1) = −2. In the neighborhood of ν0 orν1, to the leading order,

∆2 − 4 = (∆ + 2)(∆ − 2) = ∆(ν)∆′′(ν)(λ − ν)2 , ν = ν0, ν1;

where

∆′′(ν) = − 4π2

c2ν2(ν − ν)2∆(ν) .

Thus √∆2 − 4 = i

2π

cν(ν − ν)∆(ν)(λ − ν) .

By L’Hospital’s rule, as λ→ ν0

(4.16)i

4c

√∆2 − 4

W (ψ+, ψ−)→ π(ν0 − ν0)

2c2ν20(ν2

0 − ν21)(ν2

0 − ν12)W (φ+, φ−)

,

and as λ→ ν1

(4.17)i

4c

√∆2 − 4

W (ψ+, ψ−)→ π

2c2ν21 (ν1 + ν1)(ν2

1 − ν20)W (ϕ+, ϕ−)

.


Finally, from Theorem 4.6 and relations (4.14), (4.15), (4.16), and (4.17), we have

∂F0

∂ut=

π(ν0 − ν0)W (φ+, φ−)

c2(ν20 − ν2

1 )(ν20 − ν1

2)(G1/φ1 + G2/φ2)(G3/φ1 + G4/φ2)

=64π

c4(G1/φ1 + G2/φ2)(G3/φ1 + G4/φ2) ,(4.18)

∂F1

∂ut=

π

c2(ν1 − ν1)(ν

21 − ν1

2)(ν21 − ν2

0)W (ϕ+, ϕ−)

Φ1Φ2

(ν1|Φ1|2 + ν1|Φ2|2)(ν1|Φ1|2 + ν1|Φ2|2)

=π(1 − c2)

4ci(ν2

1 − ν20 )

Φ1Φ2

(ν1|Φ1|2 + ν1|Φ2|2)(ν1|Φ1|2 + ν1|Φ2|2),(4.19)

where φ is given in (4.7) with θ = ±π/2, G is given in (4.14), and Φ is given in(4.11).

CHAPTER 5

Heisenberg Ferromagnet Equation

5.1. Background

The Heisenberg model is a model in statistical physics to model ferromagnetism.The continuum limit of the Heisenberg model leads to the Heisenberg ferromagnetequation. It extends to the more realistic ferromagnet equations like the Landau-Lifschitz equation and Landau-Lifschitz-Gilbert equation when other effects of theferromagnet are taken into account.

The Heisenberg ferromagnet equation is another important integrable system[34]. Although there exists a correspondence between the Heisenberg ferromagnetequation and the integrable nonlinear Schrodinger equation, this correspondencedoes not render the two problems equivalent. We start with the Heisenberg ferro-magnet equation in the form

(5.1) ∂tm = −m×mxx ,

subject to the periodic boundary condition

(5.2) m(t, x+ 2π) = m(t, x) ,

where m = (m1,M2,m3)T and |m|(t, x) = 1. Introduce the matrix

(5.3) Γ = mjσj =

(m3 m−

m+ −m3

),

where σj are the Pauli matrices

(5.4) σ1 =

(0 11 0

), σ2 =

(0 −ii 0

), σ3 =

(1 00 −1

),

and

(5.5) m+ = m1 + im2 , m− = m1 − im2 ,

i.e. m+ = m−. Using the matrix Γ introduced in (5.3), the Heisenberg equation(5.1) has the form

(5.6) i∂tΓ = −1

2[Γ,Γxx] ,

where [Γ,Γxx] = ΓΓxx − ΓxxΓ.

5.2. Lax Pair

The Heisenberg equation (5.1) is an integrable system with the following Laxpair,

∂xψ = iλΓψ ,(5.7)

∂tψ = −λ2

(4iλΓ + [Γ,Γx])ψ ,(5.8)

29

30 5. HEISENBERG FERROMAGNET EQUATION

where ψ = (ψ1, ψ2)T is complex-valued, λ is a complex parameter, Γ is the matrix

defined in (5.3), and [Γ,Γx] = ΓΓx − ΓxΓ. For more details on this chapter, see[56].

5.3. Darboux Transformation

A Darboux transformation for (5.7)-(5.8) can be obtained.

Theorem 5.1. Let φ = (φ1, φ2)T be a solution to the Lax pair (5.7)-(5.8) at

(Γ, ν). Define the matrix

G = N

((ν − λ)/ν 0

0 (ν − λ)/ν

)N−1 ,

where

N =

(φ1 −φ2

φ2 φ1

).

Then if ψ solves the Lax pair (5.7)-(5.8) at (Γ, λ),

(5.9) ψ = Gψ

solves the Lax pair (5.7)-(5.8) at (Γ, λ), where Γ is given by

(5.10) Γ = N

(e−iθ 00 eiθ

)N−1ΓN

(eiθ 00 e−iθ

)N−1 ,

where eiθ = ν/|ν|.

The transformation (5.9)-(5.10) is the Darboux transformation. This theoremcan be proved either through the connection between the Heisenberg equation andthe NLS equation (with a well-known Darboux transformation) [23], or through adirect calculation.

Notice also that Γ2 = I. Let(

Φ1 −Φ2

Φ2 Φ1

)= N

(e−iθ 00 eiθ

)N−1

=1

|φ1|2 + |φ2|2(e−iθ|φ1|2 + eiθ|φ2|2 −2i sin θ φ1φ2

−2i sin θ φ1φ2 eiθ|φ1|2 + e−iθ|φ2|2).(5.11)

Then

(5.12) Γ =

(m3 m1 − im2

m1 + im2 −m3

),

where

m+ = m1 + im2 = Φ12(m1 + im2) − Φ2

2(m1 − im2) + 2Φ1Φ2m3 ,

m3 =(|Φ1|2 − |Φ2|2

)m3 − Φ1Φ2(m1 + im2) − Φ1Φ2(m1 − im2) .

One can generate the figure eight connecting to the domain wall, as the nonlinearfoliation of the modulational instability, via the above Darboux transfomation.

5.4. FIGURE EIGHT CONNECTING TO THE DOMAIN WALL 31

5.4. Figure Eight Connecting to the Domain Wall

Let Γ be the domain wall

Γ =

(0 e−i2x

ei2x 0

),

i.e. m1 = cos 2x, m2 = sin 2x, and m3 = 0. Solving the Lax pair (5.7)-(5.8), onegets two Bloch eigenfunctions

(5.13) ψ = eΩt(

2λ exp i2 (k − 2)x(k − 2) exp i2 (k + 2)x

), Ω = −iλk , k = ±2

√1 + λ2 .

To apply the Darboux transformation (5.10), we start with the two Bloch functionswith k = ±1,

φ+ =

( √3e−ix

ieix

)exp

√3

2t+ i

1

2x

,

(5.14)

φ− =

(−ie−ix√

3eix

)exp

−√

3

2t− i

1

2x

.

The wise choice for φ used in (5.10) is:

(5.15) φ =

√c+

c−φ+ +

√c−

c+φ− =

( (√3eτ+iχ − ie−τ−iχ

)e−ix(

ieτ+iχ +√

3e−τ−iχ)eix

),

where c+/c− = expσ + iγ, τ = 12 (√

3t + σ), and χ = 12 (x + γ). Then from the

Darboux transformation (5.10), one gets

m1 + im2 = −ei2x

1 − 2 sech2τ cos 2χ

(2 −√

3 sech2τ sin 2χ)2

[sech2τ cos 2χ

+i(√

3 − 2 sech2τ sin 2χ)]

,(5.16)

m3 =2 sech2τ tanh 2τ cos 2χ

(2 −√

3 sech2τ sin 2χ)2.(5.17)

As t→ ±∞,

m1 → − cos 2x , m2 → − sin 2x , m3 → 0 .

The expressions (5.16)-(5.17) represent the two dimensional figure eight separatrixconnecting to the domain wall (m+ = −ei2x, m3 = 0), parametrized by σ andγ. See Figure 1 for an illustration. Choosing γ = 0, π, one gets the figure eightcurve section of Figure 1. The spatial-temporal profiles corresponding to the twolobes of the figure eight curve are shown in Figure 2. In fact, the two profilescorresponding the two lobes are spatial translates of each other by π. Inside oneof the lobe, the spatial-temporal profile is shown in Figure 3(a). Outside the figureeight curve, the spatial-temporal profile is shown in Figure 3(b). Here the insideand outside spatial-temporal profiles are calculated by using the integrable finitedifference discretization [50] of the Heisenberg equation (5.1),

(5.18)d

dtm(j) = − 2

h2m(j) ×

(m(j + 1)

1 +m(j) ·m(j + 1)+

m(j − 1)

1 +m(j − 1) ·m(j)

),


where m(j) = m(t, jh), j = 1, · · · , N , Nh = 2π, and h is the spatial mesh size. Forthe computation of Figure 3, we choose N = 128.

γ

σ

Figure 1. The separatrix connecting to the domain wall m+ =−ei2x,m3 = 0.

02

46

0

10

20

30

−1

0

1

xt

m1

(a) γ = 0

02

46

0

10

20

30

−1

0

1

xt

m1

(b) γ = π

Figure 2. The spatial-temporal profiles corresponding to the twolobes of the figure eight curve.

By a translation x→ x+ θ, one can generate a circle of domain walls:

m+ = −ei2(x+θ) , m3 = 0 ,

where θ is the phase parameter. The three dimensional figure eight separatrixconnecting to the circle of domain walls, parametrized by σ, γ and θ; is illustratedin Figure 4.

In general, the unimodal equilibrium manifold can be sought as follows: Let

mj = cj cos 2x+ sj sin 2x , j = 1, 2, 3,

then the uni-length condition |m|(x) = 1 leads to

|c| = 1 , |s| = 1 , c · s = 0 ,

where c and s are the two vectors with components cj and sj . Thus the unimodalequilibrium manifold is three dimensional and can be represented as in Figure 5.

5.5. FLOQUET THEORY 33

02

46

10

20

30

40

−1

0

1

xt

m1

(a) inside

01

23

45

6

10

15

20

25

30

35

40

−1

0

1

xt

m1

(b) outside

Figure 3. The spatial-temporal profiles corresponding to the in-side and outside of the figure eight curve.

γσ

θ

Figure 4. The separatrix connecting to the circle of domain wallsm+ = ei2(x+θ),m3 = 0.

sc

Figure 5. A representation of the 3 dimensional unimodal equi-librium manifold.

5.5. Floquet Theory

Focusing on the spatial part (5.7) of the Lax pair (5.7)-(5.8), let Y (x) be thefundamental matrix solution of (5.7), Y (0) = I (2 × 2 identity matrix), then the


Floquet discriminant is defined by

∆ = trace Y (2π) .


σ = λ ∈ C | − 2 ≤ ∆(λ) ≤ 2 .Periodic and anti-periodic points λ± (which correspond to periodic and anti-periodiceigenfunctions respectively) are defined by

∆(λ±) = ±2 .


d∆

dλ(λ(c)) = 0 .

A multiple point λ(n) is a periodic or anti-periodic point which is also a criticalpoint. The algebraic multiplicity of λ(n) is defined as the order of the zero of∆(λ) ± 2 at λ(n). When the order is 2, we call the multiple point a double point,and denote it by λ(d). The order can exceed 2. The geometric multiplicity of λ(n)

is defined as the dimension of the periodic or anti-periodic eigenspace at λ(n), andis either 1 or 2.



For any λ ∈ C, ∆(λ) is a constant of motion of the Heisenberg equation (5.1).

Example 5.2. For the domain wall m1 = cos 2x, m2 = sin 2x, and m3 = 0; thetwo Bloch eigenfunctions are given in (5.13). The Floquet discriminant is given by

∆ = 2 cos[2π√

1 + λ2].

The periodic points are given by

λ = ±√n2

4− 1 , n ∈ Z , n is even .

The anti-periodic points are given by

λ = ±√n2

4− 1 , n ∈ Z , n is odd .

The choice of φ+ and φ− correspond to n = ±1 and λ = ν = i√

3/2 with k = ±1.

∆′ = −4πλ√

1 + λ2sin[2π√

1 + λ2].

∆′′ = −4π(1 + λ2)−3/2 sin[2π√

1 + λ2]− 8π2 λ2

1 + λ2cos[2π√

1 + λ2].

When n = 0, i.e.√

1 + λ2 = 0, by L’Hospital’s rule

∆′ → −8π2λ , λ = ±i .That is, λ = ±i are periodic points, not critical points. When n = ±1, we have twoimaginary double points

λ = ±i√

3/2 .

When n = ±2, λ = 0 is a multiple point of order 4. The rest periodic and anti-periodic points are all real double points. Figure 6 is an illustration of these spectralpoints.


λ

Figure 6. The periodic and anti-periodic points corresponding tothe potential of domain wall m+ = ei2x,m3 = 0. The open circlesare double points, the solid circle at the origin is a multiple pointof order 4, and the two bars intersect the imaginary axis at twoperiodic points which are not critical points.


Starting from the Floquet theory, one can build Melnikov vectors.

Definition 5.3. An important sequence of invariants Fj of the Heisenbergequation is defined by

Fj(m) = ∆(λ(c)j (m),m) .


∂Fj∂m

=∂∆

∂m

∣∣∣∣λ=λ

(c)j

.

Proof. We know that

∂Fj∂m

=∂∆

∂m

∣∣∣∣λ=λ

(c)j

+∂∆

∂λ

∣∣∣∣λ=λ

(c)j

∂λ(c)j

∂m.

Since∂∆

∂λ

∣∣∣∣λ=λ

(c)j

= 0 ,

we have

∂2∆

∂λ2

∣∣∣∣λ=λ

(c)j

∂λ(c)j

∂m+

∂2∆

∂λ∂m

∣∣∣∣λ=λ

(c)j

= 0 .


∂2∆

∂λ2

∣∣∣∣λ=λ

(c)j

6= 0 .

Thus

∂λ(c)j

∂m= −

[∂2∆

∂λ2

∣∣∣∣λ=λ

(c)j

]−1∂2∆

∂λ∂m

∣∣∣∣λ=λ

(c)j

.


Notice that ∆ is an entire function of λ and m [76], then we know that∂λ

(c)j

∂m isbounded, and

∂Fj∂m

=∂∆

∂m

∣∣∣∣λ=λ

(c)j

.

Theorem 5.5. As a function of two variables, ∆ = ∆(λ,m) has the partial


∂∆

∂m+= −iλ

√∆2 − 4

W (ψ+, ψ−)ψ+

1 ψ−1 ,

∂∆

∂m−= iλ

√∆2 − 4

W (ψ+, ψ−)ψ+

2 ψ−2 ,

∂∆

∂m3= iλ

√∆2 − 4

W (ψ+, ψ−)

(ψ+

1 ψ−2 + ψ+

2 ψ−1

),

∂∆

∂λ= i

√∆2 − 4

W (ψ+, ψ−)

∫ 2π

0

[m3

(ψ+

1 ψ−2 + ψ+

2 ψ−1

)−m+ψ

+1 ψ

−1 +m−ψ

+2 ψ

−2

]dx ,


−2 − ψ+


Proof. Recall that Y is the fundamental matrix solution of (5.7), we have theequation for the differential of Y

∂xdY = iλΓdY + i(dλΓ + λdΓ)Y , dY (0) = 0 .


dY = Y Q , Q(0) = 0 .

Thus

Q(x) = i

∫ x

0

Y −1(dλΓ + λdΓ)Y dx ,

and

dY (x) = iY

∫ x

0

Y −1(dλΓ + λdΓ)Y dx .

Finally

d∆ = trace dY (2π)

= i trace

Y (2π)

∫ 2π

0

Y −1(dλΓ + λdΓ)Y dx

.(5.19)

Let

Z = (ψ+ψ−)


ψ± = e±Λxψ± ,


Z(x) = Y (x)Z(0) , Y (x) = Z(x)[Z(0)]−1 .

5.7. MELNIKOV VECTOR ALONG THE FIGURE EIGHT 37

Notice that

Z(2π) = Z(0)E , where E =


).

Then

Y (2π) = Z(0)E[Z(0)]−1 .

Thus

∆ = trace Y (2π) = trace E = eΛ2π + e−Λ2π ,

and

e±Λ2π =1

2[∆ ±

√∆2 − 4] .

In terms of Z, d∆ as given in (5.19) takes the form

d∆ = i trace

Z(0)E[Z(0)]−1

∫ 2π

0

Z(0)[Z(x)]−1(dλΓ + λdΓ)Z(x)[Z(0)]−1dx

= i trace

E

∫ 2π

0

[Z(x)]−1(dλΓ + λdΓ)Z(x)dx

,


It turns out that the partial derivatives of Fj provide the perfect Melnikovvectors rather than those of the Hamiltonian or other invariants [76], in the sensethat Fj is the invariant whose level sets are the separatrices.

5.7. Melnikov Vector Along the Figure Eight

We continue the calculation in subsection 5.4 to obtain an explicit expressionof the Melnikov vector along the figure eight connecting to the domain wall. Applythe Darboux transformation (5.9) to φ± (5.14) at λ = ν, we obtain

φ± = ± ν − ν

ν

exp∓ 12σ ∓ i 12γW (φ+, φ−)

|φ1|2 + |φ2|2

φ2

−φ1

.

In the formula (5.9), for general λ,

detG =(ν − λ)(ν − λ)

|ν|2 , W (ψ+, ψ−) = detG W (ψ+, ψ−) .

In a neighborhood of λ = ν,

∆2 − 4 = ∆(ν)∆′′(ν)(λ − ν)2 + higher order terms in (λ− ν) .

As λ→ ν, by L’Hospital’s rule√

∆2 − 4

W (ψ+, ψ−)→

√∆(ν)∆′′(ν)

ν−ν|ν|2 W (φ+, φ−)

.

Notice, by the calculation in Example 5.2, that

ν = i

√3

2, ∆(ν) = −2 , ∆′′(ν) = −24π2 ,


then by Theorem 5.5,

∂∆

∂m+

∣∣∣∣m=m

= 12√

3πi

(|φ1|2 + |φ2|2)2φ2

2,

∂∆

∂m−

∣∣∣∣m=m

= 12√

3π−i

(|φ1|2 + |φ2|2)2φ1

2,

∂∆

∂m3

∣∣∣∣m=m

= 12√

3π2i

(|φ1|2 + |φ2|2)2φ1φ2 ,

where m is given in (5.16)-(5.17). With the explicit expression (5.15) of φ, weobtain the explicit expressions of the Melnikov vector,

∂∆

∂m+

∣∣∣∣m=m

=3√

3π

2

i sech2τ

(2 −√

3 sech2τ sin 2χ)2

[(1 − 2 tanh2τ) cos 2χ

+i(2 − tanh 2τ) sin 2χ− i√

3 sech2τ

]e−i2x ,(5.20)

∂∆

∂m−

∣∣∣∣m=m

=3√

3π

2

−i sech2τ

(2 −√

3 sech2τ sin 2χ)2

[(1 + 2 tanh2τ) cos 2χ

−i(2 + tanh 2τ) sin 2χ+ i√

3 sech2τ

]ei2x ,(5.21)

∂∆

∂m3

∣∣∣∣m=m

=3√

3π

2

2i sech2τ

(2 −√

3 sech2τ sin 2χ)2

[2 sech2τ −

√3 sin 2χ

−i√

3 tanh 2τ cos 2χ

],(5.22)

where again

m± = m1 ± im2 , τ =

√3

2t+

σ

2, χ =

1

2(x+ γ) ,

and σ and γ are two real parameters.

5.8. A Melnikov Function for Landau-Lifshitz-Gilbert Equation

Discovered by Slonczewski [99] and Berger [17], electrical current can applya large torque to a ferromagnet. If electrical current can be used to achieve mag-netization reversal, such a nanotechnique will dramatically increase the magneticmemory capacity and speed. This subject has just recently emerged as one of themost important subjects in science and technology [106, 51]. Taking into accountof the torque induced by electrical current, the governing equation is given by theLandau-Lifshitz-Gilbert (LLG) equation which can be written in the dimensionlessform as

(5.23) ∂tm = −m×H − αm× (m×H) + βm× (m× ex) ,

subject to the periodic boundary condition (5.2), where |m|(t, x) = 1, the effectivemagnetic field H has several terms

H = Hexch +Hext +Hdem +Hani

= ∂2xm+ aex − bm3ez + cm1ex ,(5.24)

whereHexch = ∂2xm is the exchange field, Hext = aex is the external field, Hdem =

−bm3ez is the demagnetization field, and Hani = cm1ex is the anisotropy field. For

5.8. A MELNIKOV FUNCTION FOR LANDAU-LIFSHITZ-GILBERT EQUATION 39

the materials of the experimental interest, the dimensionless parameters are in theranges

a ∈ [−0.05, 0.05] , b ∈ [0, 0.005] , c ∈ [0.05, 0.1] ,

α = [0.008, 0.02] , β ∈ [−0.08, 0.08] .(5.25)

The Landau-Lifshitz-Gilbert (LLG) equation (5.23) can be rewritten in the form,

(5.26) ∂tm = −m×mxx + f

where f is the perturbation

f = −am× ex + bm3(m× ez) − cm1(m× ex)

−αm× (m×H) + βm× (m× ex) ,

andH = mxx + aex − bm3ez + cm1ex .

The Melnikov function for the LLG (5.23) is given as

M =

∫ ∞

−∞

∫ 2π

0

[∂∆

∂m+(f1 + if2) +

∂∆

∂m−(f1 − if2) +

∂∆

∂m3f3

]∣∣∣∣m=m

dxdt ,

where m is given in (5.16)-(5.17), and ∂∆∂w (w = m+,m−,m3) are given in (5.20)-

(5.22). The Melnikov function depends on several external and internal parametersM = M(a, b, c, α, β, γ) where γ is an internal parameter.

CHAPTER 6

Vector Nonlinear Schrodinger Equations


In birefringent optical fibers, optical signals have two polarization directions.The propagation of two optical pulse envelopes is described by the vector nonlinearSchrodinger equation [86] [87]

ipt + pxx +1

2[|p|2 + κ|q|2]p = 0,

iqt + qxx +1

2[κ|p|2 + |q|2]q = 0,

where p and q are complex-valued functions of (t, x), κ ∈ [0, 1]. When κ = 0, 1; itreduces to an integrable system. Otherwise, it is non-integrable. Optical fibers nowhave important industrial applications in fiber communication systems ( see [42]for an introductory reference), and all-optical switching devices [48]. This Chapterwill focus on the κ = 1 case, for more details, see [64]. In general, the followingn-vector nonlinear Schrodinger equation is integrable [2]

i~qt + ~qxx + |~q|2~q = 0,

where ~q = (q1, · · · , qn).In the parametric wave context [89], Faraday surface waves in a liquid container

are generated by oscillatory vertical shaking of the container. The envelope equationof the weakly nonlinear Faraday waves is the vector Ginzburg-Landau equationwhich is closely connected to the vector nonlinear Schrodinger equation.

6.2. Lax Pair

Consider the integrable vector nonlinear Schrodinger equations (VNLS),

ipt + pxx +1

2[(|p|2 + |q|2) − ω2]p = 0 ,(6.1)

iqt + qxx +1

2[(|p|2 + |q|2) − ω2]q = 0 ,(6.2)

where ω ∈ (1, 2). Periodic boundary condition is imposed

p(t, x+ 2π) = p(t, x) , q(t, x+ 2π) = q(t, x) .

The Lax pair is given as [83],

(6.3) ψx = Uψ , ψt = V ψ ,

where

U = λA0 +A1 , V = λ2A0 + λA1 +A2 ,

41

42 6. VECTOR NONLINEAR SCHRODINGER EQUATIONS

A0 =

− 23 i 0 0

0 13 i 0

0 0 13 i

, A1 =

0 12p

12q

− 12 p 0 0

− 12 q 0 0

,

A2 = − i

4

−(|p|2 + |q|2) + 4ω2 −2px −2qx−2px |p|2 + 2ω2 qp−2qx qp |q|2 + 2ω2

.

Notice that here the spatial part of the Lax pair is a 3 × 3 problem for which theFloquet theory is more complicated than the 2×2 problems before. In particular,itis difficult to build Melnikov vectors via Floquet theory in this case.

6.3. Linearized Equations

We start with the solution,

(6.4) p = aeiδ1 , q = beiδ2 ,

where

δ1 =1

2(a2 + b2 − ω2)t+ γ1 , δ2 =

1

2(a2 + b2 − ω2)t+ γ2 .

Linearizing (6.1,6.2) at this solution, by setting

p = eiδ1 [a+ p] , q = eiδ2 [b+ q] ,

one gets

ipt + pxx +1

2a[a(p+ ¯p) + b(q + ¯q)] = 0 ,

iqt + qxx +1

2b[a(p+ ¯p) + b(q + ¯q)] = 0 .

Setting

p = f+eikx+Ωt + f−e

−ikx+Ωt , q = g+eikx+Ωt + g−e

−ikx+Ωt , (k ∈ Z)

one gets the dispersion relations

(6.5) Ω1,2 = ±k√a2 + b2 − k2 , Ω3,4 = ±ik2 .

Thus, if we choose 1 <√a2 + b2 < 2, then there is only one unstable mode cosx.

This is the reason that we restricts ω as in (6.1,6.2). When k = 1,

(6.6) Ω1,2 = ±σ , σ =√a2 + b2 − 1 .

The corresponding eigenfunctions are

p = ae±iϕe±σt cosx , q = be±iϕe±σt cosx , ϕ = arctanσ .

6.4. Homoclinic Orbits and Figure Eight Structures

The Backlund-Darboux transformation given in [107] [35] will be utilized toconstruct homoclinic orbits asymptotic to the periodic orbits (6.4) up to phaseshifts.

6.4. HOMOCLINIC ORBITS AND FIGURE EIGHT STRUCTURES 43

Theorem 6.1 ([107] [35]). Let p and q be a solution of the vector nonlinearSchrodinger equations (6.1,6.2), and let ψ = (ψ1, ψ2, ψ3)

T be an eigenfunction ofthe Lax pair (6.3) at (λ, p, q). If one defines

p = p+ 2i(λ− λ)ψ1ψ2

|ψ1|2 + |ψ2|2 + |ψ3|2,(6.7)

q = q + 2i(λ− λ)ψ1ψ3

|ψ1|2 + |ψ2|2 + |ψ3|2,(6.8)

then p and q also solve the vector nonlinear Schrodinger equations (6.1,6.2).

We solve the Lax pair (6.3) at (λ, p, q) with p and q given by the periodic orbits(6.4), we get

(6.9) ψ± =

ψ±

1

ψ±2

ψ±3

=

(iλ∓ i)ei(δ1+δ2)

aeiδ2

beiδ1

e±σ1t+iσ2te±iκ±x ,

where

κ± = −1

6λ± 1

2

√a2 + b2 + λ2 ,

√a2 + b2 + λ2 = 1 ,(6.10)

σ1 =iλ

2, σ2 = − 1

12[7(a2 + b2) + 2] .(6.11)

Remark 6.2. From (6.11), we see that in order to have temporal growth, theimaginary part of λ can not be zero. Also in order to have a nontrivial Backlund-Darboux transformation (Theorem 6.1), the imaginary part of λ can not be zero.From (6.10), if a2+b2 > 1, then λ is purely imaginary. If we also require a2+b2 < 4,

i.e., in the one unstable mode regime (6.5), then√a2 + b2 + λ2 ≥ 2 will lead to real

λ and no temporal growth.

Remark 6.3. Notice that the imaginary part of κ± is not zero, thus ψ± arenot periodic or antiperiodic functions in x. In fact, they grow or decay in x. Thisfact shows that complex double points are not necessary in constructing homoclinicorbits through Backlund-Darboux transformations (see also [62]).

Let

(6.12) ψ = c+ψ+ + c−ψ− ,

where c+ and c− are two arbitrary complex constants. We introduce the notations

c+/c− = eρ+iϑ , λ = −iσ , σ =√a2 + b2 − 1 ,(6.13)

σ + i =√a2 + b2ei(

π2 −ϑ0) , ϑ0 = arctanσ , σ1 = σ/2 ,(6.14)

τ =σ

2t+

ρ

2, y =

1

2x+

ϑ

2,(6.15)

44 6. VECTOR NONLINEAR SCHRODINGER EQUATIONS

where ρ and ϑ are called the Backlund parameters. If one chooses λ = iσ, one willget the same result. We can rewrite ψ in the following form,

ψ1 =

[cosh τ cos(y − π

2+ ϑ0) + i sinh τ sin(y − π

2+ ϑ0)

]

2√c+c−

√a2 + b2ei(δ1+δ2)eiσ2te−

16σx ,

ψ2 =

[cosh τ cos y + i sinh τ sin y

]2√c+c−aeiδ2eiσ2te−

16σx ,

ψ3 =

[cosh τ cos y + i sinh τ sin y

]2√c+c−beiδ1eiσ2te−

16σx ,

where the factor 2√c+c−eiσ2te−

16σx will be canceled away. By (6.7,6.8), one gets

(6.16) p = aeiδ1h , q = beiδ2h ,

where

h =

[cos(2ϑ0) + i sin(2ϑ0) tanh(2τ) − sinϑ0 sech(2τ) cos(2y + ϑ0 −

π

2)

]

[1 + sinϑ0 sech(2τ) cos(2y + ϑ0 −

π

2)

]−1

,(6.17)

where δ1 and δ2 are given in (6.4), other notations are given in (6.13-6.15). Forthe moment, the even-in-x condition has not been inforced. For fixed a and b, thephases γ1 and γ2 (6.4), and the Backlund parameters ρ and ϑ parametrize a fourdimensional submanifold with a figure eight structure. If a = b and γ1 = γ2, thenthe homoclinic orbit reduces to that for the scalar nonlinear Schrodinger equation[70].

As t→ ±∞,

(6.18) h→ e±i2ϑ0 .

The even-in-x condition can be achieved by restricting the Backlund parameter ϑto special values. That is, if one requires that

(6.19) ϑ+ ϑ0 −π

2= 0 , π ,

then h is even in x,

h =

[cos(2ϑ0) + i sin(2ϑ0) tanh(2τ) ∓ sinϑ0 sech(2τ) cosx

]

[1 ± sinϑ0 sech(2τ) cosx

]−1

,(6.20)

where the upper sign corresponds to ϑ+ ϑ0 − π2 = 0. For fixed a and b, the phases

γ1 and γ2 (6.4), and the Backlund parameter ρ parametrize a three dimensionalsubmanifold with a figure eight structure. If one further chooses γ1 = γ2 (6.4),then (6.4) represents a periodic orbit. In such case, the phase γ1 = γ2 and theBacklund parameter ρ parametrize a two dimensional submanifold with a figureeight structure.

6.5. A MELNIKOV VECTOR 45

6.5. A Melnikov Vector

The Hamiltonian for the integrable vector nonlinear Schrodinger equations(6.1,6.2) is given as,

(6.21) H =

∫ 2π

0

|px|2 + |qx|2 −1

4[(|p|2 + |q|2)2 − 2ω2(|p|2 + |q|2)]dx .

In fact, (6.1,6.2) can be rewritten in the Hamiltonian form,

ipt =δH

δp, ipt = −δH

δp, iqt =

δH

δq, iqt = −δH

δq.

The two L2 norms

(6.22) E1 =

∫ 2π

0

|p|2dx , E2 =

∫ 2π

0

|q|2dx ,

are also invariant functionals. We can build the Melnikov vector by the gradient ofthe invariant functional G obtained through a combination out of H , E1, and E2,

G = H +1

2

[1

4π(E1 + E2) − ω2

](E1 + E2)

=

∫ 2π

0

|px|2 + |qx|2 −1

4(|p|2 + |q|2)2dx+

1

8π(E1 + E2)

2 .(6.23)

As mentioned before, because of the fact that the spatial part of the Lax pair (6.3)is a 3 × 3 system, it is difficult to build the Melnikov vector out of the Floquettheory. The above Melnikov vector offers an alternative.

CHAPTER 7

Derivative Nonlinear Schrodinger Equations


In plasma physics, the propagation of Alfven waves is described by the deriva-tive nonlinear Schrodinger equation. It also describes the propagation of ultra-shortoptical pulses in optical fibers, the propagation of electromagnet waves in ferromag-netism, and many other physical phenomena [100]. Several forms of the derivativenonlinear Schrodinger equations are integrable, e.g. [33]

iqt = qxx + iα(|q|2q)x + 2|q|2q, αis a real parameter,

and [100] [74]

iqt = qxx − i(|q|2q)x.In this Chapter, we shall focus on the second one. For more details on this chapter,see [74].

7.2. Lax Pair

Here we study the derivative nonlinear Schrodinger equation (DNLS) in theform,

(7.1) iqt = qxx − i(|q|2q

)x,

where q is a complex-valued function of two variables t and x. Periodic boundarycondition is imposed:

q(t, x+ 2π) = q(t, x) .

DNLS is an integrable system, and its Lax pair is given as

∂xψ = Uψ ,(7.2)

∂tψ = V ψ ,(7.3)

where

U =

(−iλ2 λqλq iλ2

),

V =

(i2λ4 + iλ2|q|2 −2λ3q − λ(iqx + |q|2q)−2λ3q + λ(iqx − |q|2q) −i2λ4 − iλ2|q|2

).

The Lax pair possesses some symmetries as stated in the following lemma.

Lemma 7.1. If ψ = (ψ1, ψ2)T solves the Lax pair (7.2)-(7.3) at (λ, q), then

(ψ2, ψ1)T solves the Lax pair at (λ, q) and (ψ1,−ψ2)

T solves the Lax pair at (−λ, q).

47

48 7. DERIVATIVE NONLINEAR SCHRODINGER EQUATIONS

7.3. Darboux Transformations

There are several forms of the Darboux transformation for the DNLS. Here weadopt the form obtained in [100].

Theorem 7.2 (Darboux Transformation I). Let q be a solution of the DNLS(7.1), let φ = (φ1, φ2)

T be a solution to the Lax pair (7.2)-(7.3) at λ = ν for someν. Define the matrix

G =

(λφ2/φ1 −ν−ν λφ1/φ2

).

Let

Q =φ2

2

φ21

q − 2iνφ2

φ1, Ψ = Gψ ;

then Q is also a solution of the DNLS, and Ψ solves the Lax pair at (λ,Q); providedthat

(7.4) Q =φ2

1

φ22

q + 2iνφ1

φ2.

The Darboux transformation in this theorem does not guarantee reality condi-tion (7.4). In some cases, for instance ν is real and |φ1| = |φ2|, reality condition(7.4) can be achieved. It is such a case that leads us to the strange tori to bediscussed later. When ν is complex, an iteration of the Darboux transformation inTheorem 7.2 leads to a new form of the Darboux transformation which guaranteesreality condition.

Theorem 7.3 (Darboux Transformation II). Let q be a solution of the DNLS(7.1), let φ = (φ1, φ2)

T be a solution to the Lax pair (7.2)-(7.3) at λ = ν for someν (ν complex). Define the matrix

G =

(G11 G12

G21 G22

),

where

G11 = λ2 ν|φ1|2 − ν|φ2|2ν|φ2|2 − ν|φ1|2

+ |ν|2 ,

G12 =λ(ν2 − ν2)φ1φ2

ν|φ2|2 − ν|φ1|2,

G21 =λ(ν2 − ν2)φ2φ1

ν|φ1|2 − ν|φ2|2,

G22 = λ2 ν|φ2|2 − ν|φ1|2ν|φ1|2 − ν|φ2|2

+ |ν|2 .

Let

Q =ν|φ1|2 − ν|φ2|2ν|φ2|2 − ν|φ1|2

[ν|φ1|2 − ν|φ2|2ν|φ2|2 − ν|φ1|2

q +2i(ν2 − ν2)φ1φ2

ν|φ2|2 − ν|φ1|2],

and

Ψ = Gψ .

Then Q is also a solution of the DNLS, and Ψ solves the Lax pair at (λ,Q).

7.4. FLOQUET THEORY 49

Proof. Since φ solves the Lax pair at λ = ν, by Lemma 7.1, (φ2, φ1)T solves

the Lax pair at λ = ν. Iterating the Darboux transformation in Theorem 7.2 at ν,we have

Φ =

(Φ1

Φ2

)=

(νφ2/φ1 −ν−ν νφ1/φ2

)(φ2

φ1

)

=

((ν|φ2|2 − ν|φ1|2)/φ1

(ν|φ1|2 − ν|φ2|2)/φ2

),

G =

(λΦ2/Φ1 −ν−ν λΦ1/Φ2

).

ThenΨ = GΨ = Gψ ,

where

G = GG =

(G11 G12

G21 G22

),

with Gmn (m,n = 1, 2) are given as in the Theorem 7.3 (without˜). Similarly

Q =Φ2

2

Φ21

Q− 2iνΦ2

Φ1,

which leads to the expression given as in the Theorem 7.3 (without˜). The corre-sponding transform of (7.4) is

Φ21

Φ22

[φ2

1

φ22

q + 2iνφ1

φ2

]+ 2iν

Φ1

Φ2,

which can be verified directly to be just ¯Q.

7.4. Floquet Theory

Focusing upon the spatial part (7.2) of the Lax pair, one can develop a completeFloquet theory. Let M(x) be the fundamental matrix of (7.2), M(0) = I (2 × 2identity matrix), then the Floquet discriminant is given as

∆ = trace M(2π) .


σ = λ ∈ C | − 2 ≤ ∆(λ) ≤ 2 .Periodic and anti-periodic points λ± (which correspond to periodic and anti-periodiceigenfunctions respectively) are defined by

∆(λ±) = ±2 .


d∆

dλ(λ(c)) = 0 .

A multiple point λ(m) is a periodic or anti-periodic point which is also a criticalpoint. The algebraic multiplicity of λ(m) is defined as the order of the zero of∆(λ) ± 2 at λ(m). When the order is 2, we call the multiple point a double point,and denote it by λ(d). The order can exceed 2. The geometric multiplicity of λ(m)

is defined as the dimension of the periodic or anti-periodic eigenspace at λ(m), andis either 1 or 2.




For any λ ∈ C, ∆(λ) is invariant under the DNLS flow.

7.5. Strange Tori

For the monochromatic wave solution

(7.5) q = aeiϑ , ϑ = kx+ ωt+ θ , ω = k(k − a2) ,

where k ∈ Z, a > 0 and θ ∈ [0, 2π]; the eigenfunctions of the Lax pair (7.2)-(7.3)can be calculated,

(7.6) ψ± =

(λaeiϑ/2

i(λ2 + k2 + ξ±)e−iϑ/2

)expiξ±x+ Ω±t ,

where

ξ± = ±√

(λ2 +k

2)2 − λ2a2 , Ω± = iξ±(k − a2 − 2λ2) .

The Floquet discriminant is given as

∆ = 2 cos

[2π(ξ+ +

k

2)

]= 2 cos

[2π

√(λ2 +

k

2)2 − λ2a2 + kπ

].

From now on, I choose k = 2 and (a2 − 2)2 < 3, i.e.√

2 −√

3 < a <

√2 +

√3 ,

so that there are only two linearly unstable modes at the monochromatic wave. Inthis case,

∆ = 2 cos[2π√

(λ2 + 1)2 − λ2a2],

and

∆′ = 4πλ2(λ2 + 1) − a2

√(λ2 + 1)2 − λ2a2

sin[2π√

(λ2 + 1)2 − λ2a2].

Periodic and antiperiodic points are given by

(7.7) λ2 =1

2(a2 − 2) ± 1

2

√(a2 − 2)2 + (n2 − 4)

which correspond to ξ+ = n2 (n ≥ 0, n ∈ Z). Except for n = 0, all the rest periodic

or antiperiodic points are also double points. See Figure 1 for an illustration.Let ξ+ = n/2, n ≥ 2, n ∈ Z. Then pick λ = νn > 0 where

ν2n =

1

2(a2 − 2) +

1

2

√(a2 − 2)2 + (n2 − 4) .

Let

φ = c+φ+ + c−φ−

=

νnaeiϑ/2

[c+ei

n2 x+Ω+t + c−e−i

n2 x+Ω−t

]

ie−iϑ/2[c+(ν2

n + 1 + n2 )ei

n2 x+Ω+t + c−(ν2

n + 1 − n2 )e−i

n2 x+Ω−t

]

,

7.5. STRANGE TORI 51

λ2

•

•

•• ••

Figure 1. Periodic and antiperiodic spectra.

where Ω± = ±in2 (2− a2 − 2ν2n). To apply the Darboux transformation in Theorem

7.2, we need to have |φ1| = |φ2|. This leads to the condition

c+/c− =

√ν2n + 1 − n

2

ν2n + 1 + n

2

eiγ

where γ is a phase parameter, γ ∈ [0, 2π]. In this case,

(7.8) φ2/φ1 = ie−iϑ√ν2n + 1 + n

2 einx+2Ω+t+iγ +

√ν2n + 1 − n

2√ν2n + 1 − n

2 einx+2Ω+t+iγ +

√ν2n + 1 + n

2

,

and

(7.9) Q =φ2

φ1

[φ2

φ1q − 2iνn

],

which in general represents a 2-torus T2. The two frequencies are

ω = 2(2 − a2) , ωn = −i2Ω+ = n[2(2 − a2) −

√(a2 − 2)2 + (n2 − 4)

].

The strangeness of the T2 comes from the fact that the Floquet discriminants forq and Q are identical

(7.10) ∆(λ, q) = ∆(λ,Q) .

That is, the T2 represented by Q and the circle represented by q lie on the samelevel set determined by ∆(λ) for all λ ∈ C. The T2 and the circle are obviouslydisconnected pieces. For Q, the geometric multiplicity at λ = νn is 1; while it is 2for q. The algebraic multiplicity at λ = νn for both Q and q is 2. Such a multiplicityfeature is also true for a torus and its whisker. This was well understood before[76].


To see the relation (7.10), one notices that φ2/φ1 in (7.8) is periodic in x, thusthe matrix G in Theorem 7.2 is periodic in x, and Q and q have the same Floquetdiscriminant.

Of course, Q can reduce to a periodic orbit or a fixed point in some cases. Forinstance, when n = 2, ωn/ω is rational, thus Q is a periodic orbit. When n = 2

and a =√

2, Q is a fixed point

Q = −√

2ei2x+i(2γ−θ) = qei(2γ−2θ+π) .

When (a2 −2)2 = ℓ2

j2−ℓ2 (n2−4) for some integers j and ℓ (such that (a2−2)2 < 3),

ωn/ω is rational, and Q is a periodic orbit.To see the geometric multiplicity at λ = νn for Q, we start with φ±. Under the

Darboux transformation in Theorem 7.2, they are transformed into

(7.11) Φ± = Gφ± = ±νnc∓W (φ+, φ−)

(1/φ1

−1/φ2

),

where W (φ+, φ−) is the Wronskian of φ+ and φ−.

7.6. Whisker of the Strange T2

Let ξ+ = 1/2 in (7.6) which corresponds to n = 1 in (7.7) and denote the λ byν1:

ν21 =

1

2(a2 − 2) +

i

2

√3 − (a2 − 2)2 .

Then the eigenfunctions (7.6) have the form

(7.12) ϕ± =

(ν1ae

iϑ/2

i(ν21 + 1 ± 1

2 )e−iϑ/2

)e±i

12x+Ω±t

where

Ω± = ±1

2

√3 − (a2 − 2)2 ± i(2 − a2) .

Let

(7.13) ϕ = c+ϕ+ + c−ϕ

− , c+/c− = eρ+iα .

Under the Darboux transformation leading to (7.9), ϕ is transformed into

ϕ =

(ν1

φ2

φ1ϕ1 − νnϕ2

−νnϕ1 + ν1φ1

φ2ϕ2

),

where ϕ solves the Lax piar (7.2)-(7.3) at (ν1, Q). Introducing the new notations

F = φ2/φ1 , F = ϕ2/ϕ1 , F = ϕ2/ϕ1 ,

we have that F is given in (7.8), F has the expression

(7.14) F =i

ν1ae−iϑ

(ν21 + 3

2 ) expix+ 2Ω+t+ ρ+ iα + ν21 + 1

2

expix+ 2Ω+t+ ρ+ iα + 1,

and F is given by

(7.15) F =−νn + ν1F /F

ν1F − νnF.

7.8. DIFFUSION 53

Applying the Darboux transformation in Theorem 7.3 at (ν1, Q), one obtains that

(7.16) Q =ν1 − ν1|F |2

(ν1|F |2 − ν1)2

[(ν1 − ν1|F |2)Q+ 2i(ν2

1 − ν12)

¯F].

7.7. Whisker of the Circle

Directly using ϕ (7.13) in the Darboux transformation in Theorem 7.3, onegenerates the whisker of the circle represented by q (7.5),

(7.17) Q =ν1 − ν1|F |2

(ν1|F |2 − ν1)2

[(ν1 − ν1|F |2)q + 2i(ν2

1 − ν12) ¯F

].

As t→ +∞,

F → ie−iϑν21 + 3

2

ν1a.

As t→ −∞,

F → ie−iϑν21 + 1

2

ν1a.

As t→ +∞,

(7.18) Q→ qeiβ+ ,

where

eiβ+ =ν21

ν12

(ν12 + 3

2 )2

(ν21 + 3

2 )2−1 − (ν2

1 − ν12)

−1 + (ν21 − ν1

2).

As t→ −∞,

(7.19) Q→ qeiβ− ,

where

eiβ− =ν21

ν12

(ν12 + 1

2 )2

(ν21 + 1

2 )21 − (ν2

1 − ν12)

1 + (ν21 − ν1

2).

Under the Darboux transformation (Theorem 7.3), ϕ± are transformed into

(7.20) ϕ± = Gϕ± = ±c∓ν1(ν21 − ν1

2)W (ϕ+, ϕ−)

−ϕ2

ν1|ϕ2|2−ν1|ϕ1|2

ϕ1

ν1|ϕ1|2−ν1|ϕ2|2

.

7.8. Diffusion

Here diffusion refers to variations of invariants when DNLS is under pertur-bations. When the perturbation is Hamiltonian, such a diffusion is often calledan Arnold diffusion. Along homoclinics or heteroclinics, such a diffusion is oftenmeasured by a Melnikov integral.

First we define the invariants that we are interested in.

Definition 7.4. An important sequence of invariants Fj of the DNLS is definedby

Fj(q, q) = ∆(λ(c)j (q, q), q, q) .



∂Fj∂w

=∂∆

∂w

∣∣∣∣λ=λ

(c)j

, w = q, q .

Proof. We know that

∂Fj∂w

=∂∆

∂w

∣∣∣∣λ=λ

(c)j

+∂∆

∂λ

∣∣∣∣λ=λ

(c)j

∂λ(c)j

∂w.

Since∂∆

∂λ

∣∣∣∣λ=λ

(c)j

= 0 ,

we have

∂2∆

∂λ2

∣∣∣∣λ=λ

(c)j

∂λ(c)j

∂w+

∂2∆

∂λ∂w

∣∣∣∣λ=λ

(c)j

= 0 .


∂2∆

∂λ2

∣∣∣∣λ=λ

(c)j

6= 0 .

Thus

∂λ(c)j

∂w= −

[∂2∆

∂λ2

∣∣∣∣λ=λ

(c)j

]−1∂2∆

∂λ∂w

∣∣∣∣λ=λ

(c)j

.

Notice that ∆ is an entire function of λ and w = q, q [76], then we know that∂λ

(c)j

∂wis bounded, and

∂Fj∂w

=∂∆

∂w

∣∣∣∣λ=λ

(c)j

.

Theorem 7.6. As a function of three variables, ∆ = ∆(λ, q, q) has the partial


∂∆

∂q=

λ√

∆2 − 4

W (ψ+, ψ−)ψ+

2 ψ−2 ,(7.21)

∂∆

∂q= − λ

√∆2 − 4

W (ψ+, ψ−)ψ+

1 ψ−1 ,(7.22)

∂∆

∂λ=

√∆2 − 4

W (ψ+, ψ−)

∫ 2π

0

[qψ+

2 ψ−2 − qψ+

1 ψ−1

−i2λ(ψ+

1 ψ−2 + ψ+

2 ψ−1

) ]dx .


−2 − ψ+


7.8. DIFFUSION 55

Proof. Recall that M is the fundamental matrix solution of (7.2), we havethe equation for the differential of M

∂xdM = UdM + dUM , dM(0) = 0 .


dM = MR , R(0) = 0 .

Thus

R(x) =

∫ x

0

M−1dUMdx ,

and

dM(x) = M(x)

∫ x

0

M−1dUMdx .

Finally

d∆ = trace dM(2π)

= trace

M(2π)

∫ 2π

0

M−1dUMdx

.(7.23)

Let

N = (ψ+ψ−)


ψ± = e±Λxψ± ,


N(x) = M(x)N(0) , M(x) = N(x)[N(0)]−1 .

Notice that

N(2π) = N(0)E , where E =


).

Then

M(2π) = N(0)E[N(0)]−1 .

Thus

∆ = trace M(2π) = trace E = eΛ2π + e−Λ2π ,

and

e±Λ2π =1

2[∆ ±

√∆2 − 4] .

In terms of N , d∆ as given in (7.23) takes the form

d∆ = trace

N(0)E[N(0)]−1

∫ 2π

0

N(0)[N(x)]−1dU(x)N(x)[N(0)]−1dx

= trace

E

∫ 2π

0

[N(x)]−1dU(x)N(x)dx

,



Consider a perturbed derivative nonlinear Schrodinger equation,

(7.24) iqt = qxx − i(|q|2q

)x

+ iǫf ,

where ǫ is the perturbation parameter, and f may depend on q and its spatialderivatives, x and t. Then we have the diffusion formula,

dFjdt

=

∫ 2π

0

[∂Fj∂q

qt +∂Fj∂q

qt

]dx

= ǫ

∫ 2π

0

[∂Fj∂q

f +∂Fj∂q

f

]dx

= ǫλ

(c)j

√∆2 − 4

W (ψ+, ψ−)

∫ 2π

0

[ (ψ+

2 ψ−2

)f −

(ψ+

1 ψ−1

)f]dx .(7.25)

Here Fj Poisson commutes with the Hamiltonian of DNLS. For any solution q(t, x)

of (7.24), at any fixed time t,∂Fj

∂q evaluated at q(t, x) is given by (7.21); similarly

for∂Fj

∂q . For specific examples, the leading order term of (7.25) can be calculated.

7.9. Diffusion Along the Strange T2

Here we are interested in the diffusion of Fn with λ(c)n = νn (n ≥ 2). First we

need to evaluate the gradient of Fn as given in (7.21)-(7.22) along the strange T2

(7.9). In the neighborhood of νn (n ≥ 1), to the leading order

(7.26) ∆2 − 4 = (∆ + 2)(∆ − 2) = ∆(νn)∆′′(νn)(λ− νn)

2 ,

where

∆(νn) = 2(−1)n , ∆′′(νn) = (−1)n32π2ν2nn

−2[2(ν2n + 1) − a2]2 .

Under the Darboux transformation in Theorem 7.2,

W (Ψ+,Ψ−) = (λ2 − ν2n)W (ψ+, ψ−) ,

where as λ→ νn, ψ± → φ± and Ψ± → Φ± as given in (7.11). By L’Hospital’s rule,

limλ→νn

√∆2(λ) − 4

W (Ψ+,Ψ−)=

√∆2(νn) − 4

W (Φ+,Φ−)=

√∆(νn)∆′′(νn)

2νnW (φ+, φ−),

Thus

∂Fn∂q

∣∣∣∣q=Q

=

√∆(νn)∆′′(νn)

2W (φ+, φ−)Φ+

2 Φ−2 ,

∂Fn∂q

∣∣∣∣q=Q

= −√

∆(νn)∆′′(νn)

2W (φ+, φ−)Φ+

1 Φ−1 ,

where Q is given in (7.9) and Φ± is given in (7.11). Then, to the leading order in ǫ,

dFndt

= ǫ

√∆(νn)∆′′(νn)

2W (φ+, φ−)

∫ 2π

0

[(Φ+

2 Φ−2

)f(Q) −

(Φ+

1 Φ−1

)f(Q)

]dx .

For a finite T , the leading order diffusion of Fn is given by∫ T

0

dFndt

dt = ǫ

√∆(νn)∆′′(νn)

2W (φ+, φ−)

∫ T

0

∫ 2π

0

[(Φ+

2 Φ−2

)f(Q) −

(Φ+

1 Φ−1

)f(Q)

]dxdt .

7.10. DIFFUSION ALONG THE WHISKER OF THE CIRCLE 57

7.10. Diffusion Along the Whisker of the Circle

Notice that here under the Darboux transformation in Theorem 7.3,

det G = (λ2 − ν21 )(λ2 − ν1

2) .

Thus by the L’Hospital’s rule as above,√

∆2(ν1) − 4

W (ϕ+, ϕ−)=

√∆(ν1)∆′′(ν1)

2ν1(ν21 − ν1

2)W (ϕ+, ϕ−),

where ϕ± is given in (7.20). Thus

∂F1

∂q

∣∣∣∣q=Q

=

√∆(ν1)∆′′(ν1)

2(ν21 − ν1

2)W (ϕ+, ϕ−)ϕ+

2 ϕ−2 ,

∂F1

∂q

∣∣∣∣q=Q

= −√

∆(ν1)∆′′(ν1)

2(ν21 − ν1

2)W (ϕ+, ϕ−)ϕ+

1 ϕ−1 ,

where Q is given in (7.17). Then, to the leading order in ǫ,

dF1

dt= ǫ

√∆(ν1)∆′′(ν1)

2(ν21 − ν1

2)W (ϕ+, ϕ−)

∫ 2π

0

[(ϕ+

2 ϕ−2

)f(Q) −

(ϕ+

1 ϕ−1

)f(Q)

]dx .

From their expressions (7.20), as |t| → ∞, ϕ± → 0. Also notice the asymptotics of

Q (7.18)-(7.19). In this case, often the infinite time leading order diffusion of F1

can be given by the Melnikov integral∫ ∞

−∞

dF1

dtdt = ǫ

√∆(ν1)∆′′(ν1)

2(ν21 − ν1

2)W (ϕ+, ϕ−)

∫ ∞

−∞

∫ 2π

0

[ (ϕ+

2 ϕ−2

)f(Q)

−(ϕ+

1 ϕ−1

)f(Q)

]dxdt .

CHAPTER 8

Discrete Nonlinear Schrodinger Equation

8.1. Background

The discrete nonlinear Schrodinger equation models the dynamics of a lat-tice of anharmonic oscillators. Its continuous version is the well-known nonlinearSchrodinger equation. It is actually a finite difference discretization of the nonlinearSchrodinger equation. Take the integrable nonlinear Schrodinger equation as theexample

iqt = qxx + 2|q|2q.There are two natural discretizations, the diagonal one

iqn,t =1

h2[qn+1 − 2qn + qn−1] + 2|qn|2qn,

and the central one

iqn,t =1

h2[qn+1 − 2qn + qn−1] + |qn|2[qn+1 + qn−1];

where h is the mesh length. It turn out that the diagonal one is not integrable,while the central one is [1]. In fact the dynamics of the diagonal one is chaotic [43].This Chapter will focus on the central one. For more details on this Chapter, werefer the readers to [60] [77] [65].

8.2. Hamiltonian Structure

We consider the discrete nonlinear Schrodinger equation (DNLS)

(8.1) iqn =1

h2[qn+1 − 2qn + qn−1] + |qn|2(qn+1 + qn−1) − 2ω2qn ,

where i =√−1, qn’s are complex variables, n ∈ Z, and ω is a positive parameter;

under periodic boundary condition and even constraint,

qn+N = qn , q−n = qn .

The DNLS can be re-written in the Hamiltonian form

(8.2) iqn = ρn∂H/∂qn ,

where

H =1

h2

N−1∑

n=0

qn(qn+1 + qn−1) −

2

h2(1 + ω2h2) ln ρn

,

and ρn = 1+h2|qn|2. Notice that∑N−1

n=0 qn(qn+1 + qn−1) itself is also a constant

of motion. This invariant, together with H , implies that∑N−1

n=0 ln ρn is a constant

59

60 8. DISCRETE NONLINEAR SCHRODINGER EQUATION

of motion too. Therefore,

(8.3) D2 ≡N−1∏

n=0

ρn

is a constant of motion.

8.3. Lax Pair and Floquet Theory

The DNLS (8.1) has the discretized Lax pair [1]:

ϕn+1 = L(z)n ϕn,(8.4)

ϕn = B(z)n ϕn,(8.5)

where

L(z)n =

(z ihqn

ihqn 1/z

), B(z)

n =i

h2

(B11 B12

B21 B22

),

B11 = 1 − z2 + 2iλh− h2qnqn−1 + ω2h2,

B12 = −izhqn + (1/z)ihqn−1,

B21 = −izhqn−1 + (1/z)ihqn,

B22 = 1/z2 − 1 + 2iλh+ h2qnqn−1 − ω2h2,

and z = exp(iλh). Compatibility of the over determined system (8.4,8.5) gives the“Lax representation”

Ln = Bn+1Ln − LnBn

of the discrete DNLS (8.1). Focusing attention upon the discrete spatial flow (8.4),we let Y (1), Y (2) be the fundamental solutions of the ODE (8.4), i.e. solutions withthe initial conditions:

Y(1)0 =

(10

), Y

(2)0 =

(01

).

The Floquet discriminant is defined by

(8.6) ∆(z; ~q) ≡ trM(N ; z; ~q) ,

where M(n; z; ~q) = columnsY (1)n , Y

(2)n is the fundamental matrix of (8.4).

Remark 8.1. ∆(z; ~q) is a constant of motion for the integrable system (8.1) forany z ∈ C. Since ∆(z; ~q) is a meromorphic function in z of degree (+N,−N), theFloquet discriminant ∆(z; ~q) acts as a generating function for (M +1) functionallyindependent constants of motion, and is the key to the complete integrability ofthe system (8.1), where M = N/2 (N even), M = (N − 1)/2 (N odd).

The Floquet theory here is not standard as can be seen from the Wronskian relation:

WN (ψ+, ψ−) = D2 W0(ψ+, ψ−),

where D is defined in (8.3),

Wn(ψ+, ψ−) ≡ ψ(+,1)

n ψ(−,2)n − ψ(+,2)

n ψ(−,1)n ,

ψ+ and ψ− are any two solutions to the linear system (8.4). In fact,Wn+1(ψ+, ψ−) =

ρnWn(ψ+, ψ−). Due to this nonstandardness, modifications of the usual definitions

8.4. EXAMPLES OF FLOQUET SPECTRA 61

of spectral quantities [60] are required. The Floquet spectrum is defined as the clo-sure of the complex λ for which there exists a bounded eigenfunction to the ODE(8.4). In terms of the Floquet discriminant ∆, this is given by

σ(L) =

z ∈ C

∣∣∣∣ − 2D ≤ ∆(z; ~q) ≤ 2D

.

Periodic and antiperiodic points zs are defined by

∆(zs; ~q) = ±2D.

A critical point zc is defined by the condition

d∆

dz

∣∣∣∣(zc;~q)

= 0.

A multiple point zm is a critical point which is also a periodic or antiperiodic point.The algebraic multiplicity of zm is defined as the order of the zero of ∆(z)± 2D.Usually it is 2, but it can exceed 2; when it does equal 2, we call the multiple pointa double point, and denote it by zd. The geometric multiplicity of zm is definedas the dimension of the periodic (or antiperiodic) eigenspace of (8.4) at zm, and iseither 1 or 2.

The normalized Floquet discriminant ∆ is defined as

∆ = ∆/D .

Remark 8.2 (Continuum Limit). In the continuum limit (i.e. h → 0), theHamiltonian has a limit in the manner: H/h → Hc, where Hc is the Hamiltonian

for NLS PDE studied in Chapter 3, Hc = i∫ 1

0qxqx + 2ω2|q|2 − |q|4dx. The Lax

pair (8.4;8.5) also tends to the corresponding Lax pair for NLS PDE with spectralparameter λ (z = eiλh) [76]. If Q ≡ maxn|qn| is finite, then ρn → 1 as h → 0.

Therefore, D2 ≡ ∏N−1n=0 ρn → 1 as h → 0. The nonstandard Floquet theory for

the spatial part of the Lax pair (8.4) becomes the standard Floquet theory in thecontinuum limit.

8.4. Examples of Floquet Spectra

Consider the uniform solution: qn = qc, ∀n

qc(t) = a exp

− i[2(a2 − ω2)t− γ]

.

The corresponding Bloch functions of the Lax pair are given by:

ψ+n = (

√ρeiβ)neΩ+t

(1z −

√ρeiβ) exp−i[(a2 − ω2)t− γ/2]

−iha expi[(a2 − ω2)t− γ/2]

,

ψ−n = (

√ρe−iβ)neΩ−t

−iha exp−i[(a2 − ω2)t− γ/2]

(z −√ρe−iβ) expi[(a2 − ω2)t− γ/2]

,

where

z =√ρ cosβ +

√ρ cos2 β − 1 , ρ = 1 + h2a2 ,

Ω+ =i

h2

(1

z− z)

√ρeiβ + i2λh

,


Ω− =i

h2

(1

z− z)

√ρe−iβ + i2λh

.

The Floquet discriminant is given by:

∆ = 2D cos(Nβ) ,

where D = ρN/2. Thus the Floquet spectra are given by:

−1 ≤ cos(Nβ) ≤ 1 .

Periodic and antiperiodic points are given by:

z(s)m =

√ρ cos

m

Nπ +

√ρ cos2

m

Nπ − 1 ,

where z(s)m is a periodic point when m is even, and z

(s)m is an antiperiodic point

when m is odd. The following facts are obvious,

• If N is odd, all the periodic and antiperiodic points are on the real axisfor sufficiently large |qc|.

• If N is even, except the two points z = ±i, all other periodic and antiperi-odic points are on the real axis for sufficiently large |qc|.

Derivatives of the Floquet discriminant ∆ with respect to z are given by:

d∆/dz = 2ND sin(Nβ)[z√ρ sinβ]−1

√ρ cos2 β − 1 ,

d2∆/dz2 = −2ND[ρz2 sin3 β]−1

[N cos(Nβ) sin β (ρ cos2 β − 1)

+ (1 − ρ) cosβ sin(Nβ) +√ρ sin(Nβ) sin2 β

√ρ cos2 β − 1

].

The critical points are given by:

β =m

Nπ (β 6= 0, π), or cos2 β = 1/ρ .

There can be multiple points with algebraic multiplicity greater than 2. For ex-ample, when two symmetric double points on the circle collide at the intersectionpoints z = ±1, we have multiple points of algebraic multiplicity 4. In such case,ρ cos2 β − 1 = 0 and sin(Nβ) = 0; then d2∆/dz2 = 0, at z = ±1.


Definition 8.3. The sequence of invariants Fj is defined as:

(8.7) Fj(~q) = ∆(z(c)j (~q); ~q) .

These invariants Fj ’s are perfect candidate for building Melnikov functions.The Melnikov vectors are given by the gradients of these invariants.

Lemma 8.4. Let z(c)j (~q) be a simple critical point; then

(8.8)δFjδ~qn

(~q) =δ∆

δ~qn(z

(c)j (~q); ~q) .

(8.9)δ∆

δ~qn(z; ~q) =

ih(ζ − ζ−1)

2Wn+1

ψ(+,2)n+1 ψ

(−,2)n + ψ

(+,2)n ψ

(−,2)n+1

ψ(+,1)n+1 ψ

(−,1)n + ψ

(+,1)n ψ

(−,1)n+1

,


where ψ±n = (ψ

(±,1)n , ψ

(±,2)n )T are two Bloch functions of the Lax pair (8.4,8.5),

such that

ψ±n = Dn/Nζ±n/N ψ±

n ,

where ψ±n are periodic in n with period N , Wn = det (ψ+

n , ψ−n ).

Proof: By the definition of critical points,

∆′(z(c)j (~q); ~q) = 0 .

Differentiating this equation, we have

δz(c)j

δ~qn= − 1

∆′′

δ∆′

δ~qn.

Since z(c)j (~q) is a simple critical point, z

(c)j is a differentiable function. Thus

δFjδ~qn

=δ∆

δ~qn

∣∣∣∣z=z

(c)j

+∂∆

∂z

∣∣∣∣z=z

(c)j

δz(c)j

δ~qn

=δ∆

δ~qn

∣∣∣∣z=z

(c)j

.

This proves equation (8.8). Next we derive the formula (8.9). Let Mn be thefundamental matrix to the system (8.4), i.e. the matrix solution to (8.4) withinitial condition M0 being a 2 x 2 identity matrix. Variation of ~q leads to thevariational equation for the variation of Mn at fixed z,

δMn+1 =

z ihqn

ihqn 1/z

δMn +

0 ih δqn

ih δqn 0

Mn ,

δM0 = 0 .

Let δMn = MnAn, where An is a 2 x 2 matrix to be determined, we have

An+1 −An = M−1n+1δUnMn ,

(8.10)

A0 = 0 ,

where

δUn =

0 ih δqn

ih δqn 0

.

Solving the system (8.10), we have

δMn = Mn

[ n∑

j=1

M−1j δUj−1Mj−1

],

δM0 = 0 .

Then,

δ∆(z, ~q) = trace

MN [

N∑

j=1

M−1j δUj−1Mj−1]

.


Thus,

δ∆

δqn= ih trace

M−1n+1

(0 10 0

)MnMN

,(8.11)

δ∆

δrn= −ih trace

M−1n+1

(0 01 0

)MnMN

,(8.12)

where rn = −qn. Let ψ+ and ψ− be two Bloch functions for the discrete Lax pairs(8.4,8.5),

(8.13) ψ±n = Dn/Nζ±n/N ψ±

n ,

where ψ±n are periodic in n with period N . Let Bn be the 2 x 2 matrix with ψ+

n

and ψ−n as the column vectors,

(8.14) Bn = (ψ+n , ψ

−n ) .

Then

(8.15) Mn = BnB−10 , MN = B0

(Dζ 00 Dζ−1

)B−1

0 .

Substitute the representations (8.14) and (8.15) into (8.11,8.12), we have

δ∆

δ~qn= ∆

δD

δ~qn+iDh(ζ − ζ−1)

2Wn+1

ψ(+,2)n+1 ψ

(−,2)n + ψ

(+,2)n ψ

(−,2)n+1

ψ(+,1)n+1 ψ

(−,1)n + ψ

(+,1)n ψ

(−,1)n+1

,

where

Wn = det Bn , ψ±n = (ψ(±,1)

n , ψ(±,2)n )T .

Thus,

δ∆

δ~qn=ih(ζ − ζ−1)

2Wn+1

ψ(+,2)n+1 ψ

(−,2)n + ψ

(+,2)n ψ

(−,2)n+1

ψ(+,1)n+1 ψ

(−,1)n + ψ

(+,1)n ψ

(−,1)n+1

,

which is the formula (8.9). The lemma is proved.

8.6. Darboux Transformations

The hyperbolic structure and homoclinic orbits for (8.1) are constructed throughthe Darboux transformations, which were built in [60]. First, we present the Dar-boux transformations. Then, we show how to construct homoclinic orbits.

Fix a solution qn(t) of the system (8.1), for which the linear operator Ln hasa double point zd of geometric multiplicity 2, which is not on the unit circle. Wedenote two linearly independent solutions (Bloch functions) of the discrete Lax pair(8.4;8.5) at z = zd by (φ+

n , φ−n ). Thus, a general solution of the discrete Lax pair

(8.4;8.5) at (qn(t), zd) is given by

φn(t; zd, c+, c−) = c+φ+n + c−φ−n ,

where c+ and c− are complex parameters. We use φn to define a transformationmatrix Γn by

Γn =

(z + (1/z)an bn

cn −1/z + zdn

),

8.6. DARBOUX TRANSFORMATIONS 65

where,

an =zd

(zd)2∆n

[|φn2|2 + |zd|2|φn1|2

],

dn = − 1

zd∆n

[|φn2|2 + |zd|2|φn1|2

],

bn =|zd|4 − 1

(zd)2∆nφn1φn2,

cn =|zd|4 − 1

zdzd∆nφn1φn2,

∆n = − 1

zd

[|φn1|2 + |zd|2|φn2|2

].

From these formulae, we see that

an = −dn, bn = cn.

Then we define Qn and Ψn by

(8.16) Qn ≡ i

hbn+1 − an+1qn

and

(8.17) Ψn(t; z) ≡ Γn(z; zd;φn)ψn(t; z)

where ψn solves the discrete Lax pair (8.4;8.5) at (qn(t), z). Formulas (8.16) and(8.17) are the Backlund-Darboux transformations for the potential and eigenfunc-tions, respectively. We have the following theorem [60].

Theorem 8.5 (Darboux Transformations). Let qn(t) denote a solution of thesystem (8.1), for which the linear operator Ln has a double point zd of geomet-ric multiplicity 2, which is not on the unit circle and which is associated with aninstability. We denote two linearly independent solutions (Bloch functions) of thediscrete Lax pair (8.4;8.5) at (qn, z

d) by (φ+n , φ

−n ). We define Qn(t) and Ψn(t; z)

by (8.16) and (8.17). Then

(1) Qn(t) is also a solution of the system (8.1). (The eveness of Qn can beobtained by choosing the complex Backlund parameter c+/c− to lie on acertain curve, as shown in the example below.)

(2) Ψn(t; z) solves the discrete Lax pair (8.4;8.5) at (Qn(t), z).(3) ∆(z;Qn) = ∆(z; qn), for all z ∈ C.(4) Qn(t) is homoclinic to qn(t) in the sense that Qn(t) → eiθ± qn(t), expo-

nentially as exp(−σ|t|) as t → ±∞. Here θ± are the phase shifts, σ isa nonvanishing growth rate associated to the double point zd, and explicitformulas can be developed for this growth rate and for the phase shifts θ±.

This theorem is quite general, constructing homoclinic solutions from a wideclass of starting solutions qn(t). Its proof is by direct verification [60].


8.7. Homoclinic Orbits and Melnikov Vectors

Through the Darboux transformations,homoclinic orbits Qn can be generatedvia the formula (8.16)

Qn = ih−1bn+1 − an+1qn

=

[z(d)(|φ(1)

n+1|2 + |z(d)|2|φ(2)n+1|2)

]−1

×[ih−1(1 − |z(d)|4)φ(1)

n+1φ(2)n+1 + z(d)qn(|φ(2)

n+1|2 + |z(d)|2|φ(1)n+1|2)

].

The Melnikov vector field located on this homoclinic orbit is given by

(8.18)δ∆(z(d); ~Q)

δ ~Qn= K

Wn

EnAn+1

[z(d)]−2 φ(1)n φ

(1)n+1

[z(d)]−2 φ(2)n φ

(2)n+1

,

where

φn = (φ(1)n , φ(2)

n )T = c+ψ+n + c−ψ

−n ,

Wn =∣∣ ψ+

n ψ−n

∣∣ ,En = |φ(1)

n |2 + |z(d)|2|φ(2)n |2 ,

An = |φ(2)n |2 + |z(d)|2|φ(1)

n |2 ,

K = − ihc+c−2

|z(d)|4(|z(d)|4 − 1)[z(d)]−1

√∆(z(d); ~q)∆′′(z(d); ~q) ,

where

∆′′(z(d); ~q) =∂2∆(z(d); ~q)

∂z2.

Next we shall study a specific example. we start with the uniform solution of(8.1)

(8.19) qn = qc , ∀n; qc = a exp

− i[2(a2 − ω2)t− γ]

.

We choose the amplitude a in the range

N tanπ

N< a < N tan

2π

N, when N > 3 ,

3 tanπ

3< a <∞ , when N = 3 ;

so that there is only one set of quadruplets of double points which are not on

the unit circle, and denote one of them by z = z(d)1 = z

(c)1 which corresponds to

β = π/N . The homoclinic orbit Qn is given by

(8.20) Qn = qc(En+1)−1

[An+1 − 2 cosβ

√ρ cos2 β − 1Bn+1

],

and the Melnikov vectors evaluated on this homoclinic orbit are given by

(8.21)δF1

δ ~Qn= K

[EnAn+1

]−1

sech [2µt+ 2p]

(X

(1)n

−X(2)n

),

where

En = ha cosβ +√ρ cos2 β − 1 sech [2µt+ 2p] cos[(2n− 1)β + ϑ] ,

8.7. HOMOCLINIC ORBITS AND MELNIKOV VECTORS 67

An+1 = ha cosβ +√ρ cos2 β − 1 sech [2µt+ 2p] cos[(2n+ 3)β + ϑ] ,

Bn+1 = cosϕ+ i sinϕ tanh[2µt+ 2p] + sech [2µt+ 2p] cos[2(n+ 1)β + ϑ] ,

X(1)n =

[cosβ sech [2µt+ 2p] + cos[(2n+ 1)β + ϑ+ ϕ]

−i tanh[2µt+ 2p] sin[(2n+ 1)β + ϑ+ ϕ]

]ei2θ(t) ,

X(2)n =

[cosβ sech [2µt+ 2p] + cos[(2n+ 1)β + ϑ− ϕ]

−i tanh[2µt+ 2p] sin[(2n+ 1)β + ϑ− ϕ]

]e−i2θ(t) ,

K = −2Nh2a(1 − z4)[8ρ3/2z2]−1√ρ cos2 β − 1 ,

β = π/N , ρ = 1 + h2a2 , µ = 2h−2√ρ sinβ√ρ cos2 β − 1 ,

h = 1/N, c+/c− = ie2peiϑ , ϑ ∈ [0, 2π] , p ∈ (−∞,∞) ,

z =√ρ cosβ +

√ρ cos2 β − 1 , θ(t) = (a2 − ω2)t− γ/2 ,

√ρ cos2 β − 1 + i

√ρ sinβ = haeiϕ ,

where ϕ = sin−1[√ρ(ha)−1 sinβ], ϕ ∈ (0, π/2).

Next we study the “evenness” condition: Q−n = Qn. It turns out that thechoices ϑ = −β , −β + π in the formula of Qn lead to the evenness of Qn in n. Interms of figure eight structure of Qn, ϑ = −β corresponds to one ear of the figureeight, and ϑ = −β + π corresponds to the other ear. The even formula for Qn isgiven by,

(8.22) Qn = qc

[Γ/Λn − 1

],

where

Γ = 1 − cos 2ϕ− i sin 2ϕ tanh[2µt+ 2p] ,

Λn = 1 ± cosϕ[cos β]−1 sech[2µt+ 2p] cos[2nβ] ,

where (‘+’ corresponds to ϑ = −β). The Melnikov vectors evaluated on thesehomoclinic orbits are not necessarily even and are in fact not even in n. For thepurpose of calculating the Melnikov functions, only the even parts of the Melnikovvectors are needed, which are given by

(8.23)δF1

δ ~Qn

∣∣∣∣even

= K(e) sech[2µt+ 2p][Πn]−1

(X

(1,e)n

−X(2,e)n

),

where

K(e) = −2N(1 − z4)[8aρ3/2z2]−1√ρ cos2 β − 1 ,

Πn =

[cosβ ± cosϕ sech[2µt+ 2p] cos[2(n− 1)β]

]×

[cosβ ± cosϕ sech[2µt+ 2p] cos[2(n+ 1)β]

],


X(1,e)n =

[cosβ sech [2µt+ 2p] ± (cosϕ

−i sinϕ tanh[2µt+ 2p]) cos[2nβ]

]ei2θ(t) ,

X(2,e)n =

[cosβ sech [2µt+ 2p] ± (cosϕ

+i sinϕ tanh[2µt+ 2p]) cos[2nβ]

]e−i2θ(t) .

The homoclinic orbit (8.22) represents the figure eight structure. If we denoteby S the circle, we have the topological identification:

(figure 8) ⊗ S =⋃

p∈(−∞,∞), γ∈[0,2π]

Qn(p, γ, a, ω,±, N) .

CHAPTER 9

Davey-Stewartson II Equation

9.1. Background

Davey-Stewartson Equation is a canonical high dimensional integrable system.It has a variety of applications. The original form was derived by A. Davey and K.Stewartson [30] as a model of water wave. It is this form that we shall focus on inthis chapter. For more details on this chapter, see [62] [72].

Consider the Davey-Stewartson II equation (DSII)

(9.1)

iqt = q + [2(|q|2 − ω2) + uy]q ,∆u = −4∂y|q|2 ,

where q is a complex-valued function of the three variables (t, x, y), u is a real-valuedfunction of the three variables (t, x, y), ω is a parameter, and

= ∂xx − ∂yy, ∆ = ∂xx + ∂yy, i =√−1.

Periodic boundary condition is imposed,

q(t, x+ 2π/κ1, y) = q(t, x, y) = q(t, x, y + 2π/κ2),

u(t, x+ 2π/κ1, y) = u(t, x, y) = u(t, x, y + 2π/κ2),

where κ1 and κ2 are positive constants. Even constraint is also imposed,

q(t,−x, y) = q(t, x, y) = q(t, x,−y),

u(t,−x, y) = u(t, x, y) = u(t, x,−y).The following constraint guarantees the existence of only two unstable modes,

κ2 < κ1 < 2κ2, or κ1 < κ2 < 2κ1,

as shown in next section. The Davey-Stewartson equation is Gauge equivalent tothe Ishimori equation [49] in the same way as the nonlinear Schrodinger equationis Gauge equivalent to the Heisenberg ferromagnet equation. Davey-Stewartsonequation can be regarded as a higher dimensional generalization of the nonlinearSchrodinger equation studied in Chapter 3. In fact, it is a nontrivial generalizationin the sense that the spatial part of the Lax pair of the DSII is a system of two firstorder partial differential equations, for which there is no convenient Floquet theory.It turns out that Melnikov vectors can still be obtained through quadratic productsof Bloch eigenfunctions, instead of the gradient of the Floquet discriminant as inthe NLS case.

At the moment, there is no global well-posedness for DSII in Sobolev spaces.In fact, DSII has finite-time blow-up solutions in Hs(R2) (0 < s < 1) [94].

69

70 9. DAVEY-STEWARTSON II EQUATION

9.2. Linear Stability

Consider the spatially uniform solution

(9.2) qc = ηe−2i[η2−ω2]t+iγ ,

where η is the amplitude and γ is the initial phase. Making the change of variable

q = e−2i[η2−ω2]t+iγ(η + q) ,

and linearizing the DSII with regard to

q = ei(k1x+k2y)+Ωt, k1, k2 ∈ R,

we have

Ω = ± |k21 − k2

2 |√k21 + k2

2

√4η2 − (k2

1 + k22) .

Therefore, the unstable modes (k1, k2) ∈ R2 lie inside the circle k21 + k2

2 = 4η2 andaway from the lines k1 = k2 and k1 = −k2 on the plane.

9.3. Lax Pairs and Darboux Transformation

The DSII has the Lax pair

Lψ = λψ ,(9.3)

∂tψ = Aψ ,(9.4)

where ψ = (ψ1, ψ2), and

L =

D− q

q D+

,

A = i

[2

(−∂2

x q∂xq∂x ∂2

x

)+

(r1 (D+q)

−(D−q) r2

)],

(9.5) D+ = α∂y + ∂x , D− = α∂y − ∂x , α2 = −1 .

r1 and r2 have the expressions,

(9.6) r1 =1

2[−w + iv] , r2 =

1

2[w + iv] ,

where u and v are real-valued functions satisfying

[∂2x + ∂2

y ]w = 2[∂2x − ∂2

y ]|q|2 ,(9.7)

[∂2x + ∂2

y ]v = i4α∂x∂y|q|2 ,(9.8)

and w = 2(|q|2 − ω2) + uy. Notice that DSII (9.1) is invariant under the transfor-mation σ:

(9.9) σ (q, q, r1, r2;α) = (q, q,−r2,−r1;−α) .

Applying the transformation σ (9.9) to the Lax pair (9.3, 9.4), we have a congruentLax pair for which the compatibility condition gives the same DSII. The congruentLax pair is given as:

Lψ = λψ ,(9.10)

∂tψ = Aψ ,(9.11)

9.3. LAX PAIRS AND DARBOUX TRANSFORMATION 71

where ψ = (ψ1, ψ2), and

L =

−D+ q

q −D−

,

A = i

[2

(−∂2

x q∂xq∂x ∂2

x

)+

(−r2 −(D−q)

(D+q) −r1

)].

The compatibility condition of the Lax pair (9.3, 9.4),

∂tL = [A,L] ,

where [A,L] = AL−LA, and the compatibility condition of the congruent Lax pair(9.10, 9.11),

∂tL = [A, L]

give the same DSII (9.1). Let (q, u) be a solution to the DSII (9.1), and let λ0

be any value of λ. Let ψ = (ψ1, ψ2) be a solution to the Lax pair (9.3, 9.4) at(q, q, r1, r2;λ0). Define the matrix operator:

Γ =

[∧ + a bc ∧ + d

],

where ∧ = α∂y − λ, and a, b, c, d are functions defined as:

a =1

∆

[ψ2 ∧2 ψ2 + ψ1 ∧1 ψ1

],

b =1

∆

[ψ2 ∧1 ψ1 − ψ1 ∧2 ψ2

],

c =1

∆

[ψ1 ∧1 ψ2 − ψ2 ∧2 ψ1

],

d =1

∆

[ψ2 ∧1 ψ2 + ψ1 ∧2 ψ1

],

in which ∧1 = α∂y − λ0, ∧2 = α∂y + λ0, and

∆ = −[|ψ1|2 + |ψ2|2

].

Define a transformation as follows:

(q, r1, r2) → (Q,R1, R2) ,φ → Φ ;

Q = q − 2b ,

R1 = r1 + 2(D+a) ,(9.12)

R2 = r2 − 2(D−d) ,

Φ = Γφ ;

where φ is any solution to the Lax pair (9.3, 9.4) at (q, q, r1, r2;λ), D+ and D− are

defined in (9.5), we have the following theorem [62].


Theorem 9.1. The transformation (9.12) is a Backlund-Darboux transforma-tion. That is, the function Q defined through the transformation (9.12) is alsoa solution to the DSII (9.1). The function Φ defined through the transformation(9.12) solves the Lax pair (9.3, 9.4) at (Q, Q,R1, R2;λ).

9.4. Homoclinic Orbits

Since DSII is a higher dimensional generalization of NLS, a nontrivial homo-clinic orbit should depend on both x and y. Consider the spatially independentsolution,

(9.13) qc = η exp−2i[η2 − ω2]t+ iγ .

As shown in Section 9.2, the dispersion relation for the linearized DSII at qc is

Ω = ± |ξ21 − ξ22 |√ξ21 + ξ22

√4η2 − (ξ21 + ξ22) , for δq ∼ qc expi(ξ1x+ ξ2y) + Ωt ,

where ξ1 = k1κ1, ξ2 = k2κ2, and k1 and k2 are integers. We restrict κ1 and κ2 asfollows to have only two unstable modes (±κ1, 0) and (0,±κ2),

κ2 < κ1 < 2κ2 , κ21 < 4η2 < minκ2

1 + κ22, 4κ

22 ,

or

κ1 < κ2 < 2κ1 , κ22 < 4η2 < minκ2

1 + κ22, 4κ

21 .

The Bloch eigenfunction of the Lax pair (9.3) and (9.4) is given as,

(9.14) ψ = c(t)

[−qcχ

]exp i(ξ1x+ ξ2y) ,

where

c(t) = c0 exp [2ξ1(iαξ2 − λ) + ir2] t ,r2 − r1 = 2(|qc|2 − ω2) ,

χ = (iαξ2 − λ) − iξ1 ,

(iαξ2 − λ)2 + ξ21 = η2 .

For the iteration of the Backlund-Darboux transformations, one needs two sets of

eigenfunctions. First, we choose ξ1 = ± 12κ1, ξ2 = 0, λ0 =

√η2 − 1

4κ21 (for a fixed

branch),

ψ± = c±

−qc

χ±

exp

±i1

2κ1x

,(9.15)

where

c± = c±0 exp [∓κ1λ0 + ir2] t ,

χ± = −λ0 ∓ i1

2κ1 = ηe∓i(

π2 +ϑ1) .

We apply the Backlund-Darboux transformations with ψ = ψ+ +ψ−, which gener-ates the unstable foliation associated with the (κ1, 0) and (−κ1, 0) linearly unstable

9.4. HOMOCLINIC ORBITS 73

modes. Then, we choose ξ2 = ± 12κ2, λ = 0, ξ01 =

√η2 − 1

4κ22 (for a fixed branch),

(9.16) φ± = c±

−qc

χ±

exp

i(ξ01x± 1

2κ2y)

,

where

c± = c0± exp[±iακ2ξ

01 + ir2

]t,

χ± = ±iα1

2κ2 − iξ01 = ±ηe∓iϑ2 .

We start from these eigenfunctions φ± to generate Γφ± through Backlund-Darbouxtransformations, and then iterate the Backlund-Darboux transformations with Γφ++Γφ− to generate the unstable foliation associated with all the linearly unstablemodes (±κ1, 0) and (0,±κ2). It turns out that the following representations areconvenient,

ψ± =

√c+0 c

−0 e

ir2t

(v±1v±2

),(9.17)

φ± =√c0+c

0−e

iξ01x+ir2t

(w±

1

w±2

),(9.18)

where

v±1 = −qce∓τ2 ±ix , v±2 = ηe∓

τ2 ±iz ,

w±1 = −qce±

τ2 ±iy , w±

2 = ±ηe± τ2 ±iz ,

and

c+0 /c−0 = eρ+iϑ , τ = 2κ1λ0t− ρ , x =

1

2κ1x+

ϑ

2, z = x− π

2− ϑ1 ,

c0+/c0− = eρ+iϑ , τ = 2iακ2ξ

01t+ ρ , y =

1

2κ2y +

ϑ

2, z = y − ϑ2 .

The following representations are also very useful,

ψ = ψ+ + ψ− = 2

√c+0 c

−0 e

ir2t

(v1v2

),(9.19)

φ = φ+ + φ− = 2√c0+c

0−e

iξ01x+ir2t

(w1

w2

),(9.20)

where

v1 = −qc[coshτ

2cos x− i sinh

τ

2sin x] , v2 = η[cosh

τ

2cos z − i sinh

τ

2sin z] ,

w1 = −qc[coshτ

2cos y + i sinh

τ

2sin y] , w2 = η[sinh

τ

2cos z + i cosh

τ

2sin z] .


Applying the Backlund-Darboux transformations (9.12) with ψ given in (9.19), wehave the representations,

a = −λ0 sech τ sin(x+ z) sin(x− z)(9.21)

×[1 + sech τ cos(x+ z) cos(x− z)

]−1

,

b = −qcb = −λ0qcη

[cos(x− z) − i tanh τ sin(x− z)(9.22)

+ sech τ cos(x+ z)

][1 + sech τ cos(x+ z) cos(x− z)

]−1

,

c = b , d = −a = −a .(9.23)

The evenness of b in x is enforced by the requirement that ϑ− ϑ1 = ±π2 , and

a± = ∓λ0 sech τ cosϑ1 sin(κ1x)(9.24)

×[1 ∓ sech τ sinϑ1 cos(κ1x)

]−1

,

b± = −qcb± = −λ0qcη

[− sinϑ1 − i tanh τ cosϑ1(9.25)

± sech τ cos(κ1x)

][1 ∓ sech τ sinϑ1 cos(κ1x)

]−1

,

c = b , d = −a = −a .(9.26)

Notice also that a± is an odd function in x. Under the above Backlund-Darbouxtransformations, the eigenfunctions φ± (9.16) are transformed into

(9.27) ϕ± = Γφ± ,

where

Γ =

Λ + a b

b Λ − a

,

and Λ = α∂y − λ with λ evaluated at 0. Let ϕ = ϕ+ +ϕ− (the arbitrary constantsc0± are already included in ϕ±), ϕ has the representation,

(9.28) ϕ = 2√c0+c

0−e

iξ01x+ir2t

−qcW1

ηW2

,

where

W1 = (α∂yw1) + aw1 + ηbw2 , W2 = (α∂yw2) − aw2 + ηbw1 .

9.4. HOMOCLINIC ORBITS 75

We generate the coefficients in the Backlund-Darboux transformations (9.12) withϕ (the iteration of the Backlund-Darboux transformations),

a(I) = −[W2(α∂yW2) +W1(α∂yW1)

][|W1|2 + |W2|2

]−1

,(9.29)

b(I) =qcη

[W2(α∂yW1) −W1(α∂yW2)

][|W1|2 + |W2|2

]−1

,(9.30)

c(I) = b(I) , d(I) = −a(I) ,(9.31)

where

W2(α∂yW2) +W1(α∂yW1)

=1

2ακ2

cosh τ

[− ακ2a+ iaη(b+ b) cosϑ2

]

+

[1

4κ2

2 − a2 − η2|b|2]

cos(y + z) sinϑ2 + sinh τ

[aη(b − b) sinϑ2

],

|W1|2 + |W2|2

= cosh τ

[a2 +

1

4κ2

2 + η2|b|2 + iακ2η1

2(b+ b) cosϑ2

]

+

[1

4κ2

2 − a2 − η2|b|2]

sin(y + z) sinϑ2 + sinh τ

[ακ2η

1

2(b − b) sinϑ2

],

W2(α∂yW1) −W1(α∂yW2)

=1

2ακ2

cosh τ

[− ακ2ηb + i(−a2 +

1

4κ2

2 + η2b2) cosϑ2

]

+ sinh τ(a2 − 1

4κ2

2 + η2b2) sinϑ2

.

The new solution to the focusing Davey-Stewartson II equation (9.1) is given by

(9.32) Q = qc − 2b− 2b(I) .

The evenness of b(I) in y is enforced by the requirement that ϑ−ϑ2 = ±π2 . In fact,

we have

Lemma 9.2. Under the requirements that ϑ− ϑ1 = ±π2 , and ϑ− ϑ2 = ±π

2 ,

(9.33) b(−x) = b(x) , b(I)(−x, y) = b(I)(x, y) = b(I)(x,−y) ,and Q = qc − 2b− 2b(I) is even in both x and y.

Proof: It is a direct verification by noticing that under the requirements, a isan odd function in x. Q.E.D.

The asymptotic behavior of Q can be computed directly. In fact, we have theasymptotic phase shift lemma.

Lemma 9.3 (Asymptotic Phase Shift Lemma). For λ0 > 0, ξ01 > 0, and iα = 1,as t→ ±∞,

(9.34) Q = qc − 2b− 2b(I) → qceiπe∓i2(ϑ1−ϑ2) .


In comparison, the asymptotic phase shift of the first application of the Backlund-Darboux transformations is given by

qc − 2b→ qce∓i2ϑ1 .


The DSII (9.1) can be written in the Hamiltonian form,

(9.35)

iqt = δH/δq ,iqt = −δH/δq ,

where

H =

∫ 2π

0

∫ 2π

0

[|qy|2 − |qx|2 +1

2(r2 − r1) |q|2] dx dy .

We have the lemma [62].

Lemma 9.4. The inner product of the vector

U =

(ψ2ψ2

ψ1ψ1

)−

+ S

(ψ2ψ2

ψ1ψ1

),

where ψ = (ψ1, ψ2) is an eigenfunction solving the Lax pair (9.3, 9.4), and ψ =

(ψ1, ψ2) is an eigenfunction solving the corresponding congruent Lax pair (9.10,

9.11), and S =

(0 11 0

), with the vector field J∇H given by the right hand side

of (9.35) vanishes,

〈U , J∇H〉 = 0 ,

where J =

(0 1−1 0

).

If we only consider even functions, i.e., q and u = r2 − r1 are even functions inboth x and y, then we can split U into its even and odd parts,

U = U (e,x)(e,y) + U (e,x)

(o,y) + U (o,x)(e,y) + U (o,x)

(o,y) ,

where

U (e,x)(e,y) =

1

4

[U(x, y) + U(−x, y) + U(x,−y) + U(−x,−y)

],

U (e,x)(o,y) =

1

4

[U(x, y) + U(−x, y) − U(x,−y) − U(−x,−y)

],

U (o,x)(e,y) =

1

4

[U(x, y) − U(−x, y) + U(x,−y) − U(−x,−y)

],

U (o,x)(o,y) =

1

4

[U(x, y) − U(−x, y) − U(x,−y) + U(−x,−y)

].

Then we have the lemma [62].

Lemma 9.5. When q and u = r2 − r1 are even functions in both x and y, wehave

〈U (e,x)(e,y) , J∇H〉 = 0 .


9.5.1. Melnikov Integrals. Consider the perturbed DSII equation,

(9.36)

i∂tq = [∂2

x − ∂2y ]q + [2(|q|2 − ω2) + uy]q + ǫif ,

[∂2x + ∂2

y ]u = −4∂y |q|2 ,where q and u are respectively complex-valued and real-valued functions of threevariables (t, x, y), and G = (f, f) are the perturbation terms which can depend onq and q and their derivatives and t, x and y. The Melnikov integral is given by [62],

M =

∫ ∞

−∞

〈U , G〉 dt

= 2

∫ ∞

−∞

∫ 2π

0

∫ 2π

0

Re

(ψ2ψ2)f + (ψ1ψ1)f

dx dy dt ,(9.37)

where the integrand is evaluated on an unperturbed homoclinic orbit in certaincenter-unstable (= center-stable) manifold, and such orbit can be obtained throughthe Backlund-Darboux transformations given in Theorem 9.1. When we only con-sider even functions, i.e., q and u are even functions in both x and y, the corre-sponding Melnikov function is given by [62],

M (e) =

∫ ∞

−∞

〈U (e,x)(e,y) ,

~G〉 dt

=

∫ ∞

−∞

〈U , ~G〉 dt ,(9.38)

which is the same as expression (9.37).

9.5.2. An Example . We continue from the example in Section 9.4. Wegenerate the following eigenfunctions corresponding to the potential Q given in(9.32) through the iterated Backlund-Darboux transformations,

Ψ± = Γ(I)Γψ± , at λ = λ0 =

√η2 − 1

4κ2

1 ,(9.39)

Φ± = Γ(I)Γφ± , at λ = 0 ,(9.40)

where

Γ =

Λ + a b

b Λ − a

, Γ(I) =

Λ + a(I) b(I)

b(I) Λ − a(I)

,

where Λ = α∂y − λ for general λ.

Lemma 9.6 (see [62] [72]). The eigenfunctions Ψ± and Φ± defined in (9.39)and (9.40) have the representations,

Ψ± =±2λ0W (ψ+, ψ−)

∆

(−λ0 + a(I))ψ2 − b(I)ψ1

b(I)ψ2 + (λ0 + a(I))ψ1

,(9.41)

Φ± =∓iακ2

∆(I)

[Ξ1

Ξ2

],(9.42)


where

W (ψ+, ψ−) =

∣∣∣∣∣∣

ψ+1 ψ−

1

ψ+2 ψ−

2

∣∣∣∣∣∣= −iκ1c

+0 c

−0 qc exp i2r2t ,

∆ = −[|ψ1|2 + |ψ2|2

],

ψ = ψ+ + ψ− ,

∆(I) = −[|ϕ1|2 + |ϕ2|2

],

ϕ = ϕ+ + ϕ− ,

ϕ± = Γφ± at λ = 0 ,

Ξ1 = ϕ1(ϕ+1 ϕ

−1 ) + ϕ+

2 (ϕ+1 ϕ

−2 ) + ϕ−

2 (ϕ−1 ϕ

+2 ) ,

Ξ2 = ϕ2(ϕ+2 ϕ

−2 ) + ϕ+

1 (ϕ−1 ϕ

+2 ) + ϕ−

1 (ϕ+1 ϕ

−2 ) .

If we take r2 to be real (in the Melnikov vectors, r2 appears in the form r2 − r1 =

2(|qc|2 − ω2)), then

(9.43) Ψ± → 0 , Φ± → 0 , as t→ ±∞ .

Next we generate eigenfunctions solving the corresponding congruent Lax pair(9.10, 9.11) with the potential Q, through the iterated Backlund-Darboux trans-formations and the symmetry transformation (9.9) [62] [72].

Lemma 9.7. Under the replacements

α −→ −α (ϑ2 −→ π − ϑ2), ϑ −→ ϑ+ π − 2ϑ2 , ρ −→ −ρ ,

the coefficients in the iterated Backlund-Darboux transformations are transformedas follows,

a(I) −→ a(I) , b(I) −→ b(I) ,

(c(I) = b(I)

)−→

(c(I) = b(I)

),

(d(I) = −a(I)

)−→

(d(I) = −a(I)

).

Lemma 9.8 (see [62] [72]). Under the replacements

α 7→ −α (ϑ2 −→ π − ϑ2), r1 7→ −r2 ,(9.44)

r2 7→ −r1 , ϑ −→ ϑ+ π − 2ϑ2 , ρ −→ −ρ ,

the potentials are transformed as follows,

Q −→ Q ,

(R = Q) −→ (R = Q) ,

R1 −→ −R2 ,

R2 −→ −R1 .


The eigenfunctions Ψ± and Φ± given in (9.41) and (9.42) depend on the vari-ables in the replacement (9.44):

Ψ± = Ψ±(α, r1, r2, ϑ, ρ) ,

Φ± = Φ±(α, r1, r2, ϑ, ρ) .

Under replacement (9.44), Ψ± and Φ± are transformed into

Ψ± = Ψ±(−α,−r2,−r1, ϑ+ π − 2ϑ2,−ρ) ,(9.45)

Φ± = Φ±(−α,−r2,−r1, ϑ+ π − 2ϑ2,−ρ) .(9.46)

Corollary 9.9 (see [62] [72]). Ψ± and Φ± solve the congruent Lax pair (9.10,9.11) at (Q,Q,R1, R2;λ0) and (Q,Q,R1, R2; 0), respectively.

Notice that as a function of η, ξ01 has two (plus and minus) branches. Inorder to construct Melnikov vectors, we need to study the effect of the replacementξ01 −→ −ξ01 .

Lemma 9.10 (see [62] [72]). Under the replacements

(9.47) ξ01 −→ −ξ01 (ϑ2 −→ −ϑ2), ϑ −→ ϑ+ π − 2ϑ2, ρ −→ −ρ,the coefficients in the iterated Backlund-Darboux transformations are invariant,

a(I) 7→ a(I) , b(I) 7→ b(I) ,

(c(I) = b(I)) 7→ (c(I) = b(I)) , (d(I) = −a(I)) 7→ (d(I) = −a(I)) ;

thus the potentials are also invariant,

Q −→ Q , (R = Q) −→ (R = Q) ,

R1 −→ R1 , R2 −→ R2 .

The eigenfunction Φ± given in (9.42) depends on the variables in the replace-ment (9.47):

Φ± = Φ±(ξ01 , ϑ, ρ) .

Under the replacement (9.47), Φ± is transformed into

(9.48) Φ± = Φ±(−ξ01 , ϑ+ π − 2ϑ2,−ρ) .

Corollary 9.11. Φ± solves the Lax pair (9.3,9.4) at (Q,Q,R1, R2 ; 0).

In the construction of the Melnikov vectors, we need to replace Φ± by Φ± toguarantee the periodicity in x of period L1 = 2π

κ1.

The Melnikov vectors for the Davey-Stewartson II equations are given by,

U± =

(Ψ±

2 Ψ±2

Ψ±1 Ψ±

1

)−

+ S

(Ψ±

2 Ψ±2

Ψ±1 Ψ±

1

),(9.49)

U± =

(Φ

(2)± Φ

(2)±

Φ(1)± Φ

(1)±

)−

+ S

(Φ

(2)± Φ

(2)±

Φ(1)± Φ

(1)±

),(9.50)


where S =

(0 11 0

). The corresponding Melnikov functions (9.37) are given by,

M± =

∫ ∞

−∞

〈U± , ~G〉 dt

= 2

∫ ∞

−∞

∫ 2π

0

∫ 2π

0

Re

[Ψ±

2 Ψ±2 ]f(Q,Q)

+[Ψ±1 Ψ±

1 ]f(Q,Q)

dx dy dt ,(9.51)

M± =

∫ ∞

−∞

〈U± , ~G〉 dt

= 2

∫ ∞

−∞

∫ 2π

0

∫ 2π

0

Re

[Φ

(2)± Φ

(2)± ]f(Q,Q)

+[Φ(1)± Φ

(1)± ]f(Q,Q)

dx dy dt ,(9.52)

where Q is given in (9.32), Ψ± is given in (9.41), Φ± is given in (9.42) and (9.48),

Ψ± is given in (9.41) and (9.45), and Φ± is given in (9.42) and (9.46). As given in(9.38), the above formulas also apply when we consider even function Q in both xand y.

9.6. Extra Comments

In general, one can classify soliton equations into two categories. Category Iconsists of those equations possessing instabilities, under periodic boundary condi-tion. In their phase space, figure-eight structures (i.e. separatrices) exist. CategoryII consists of those equations possessing no instability, under periodic boundary con-dition. In their phase space, no figure-eight structure (i.e. separatrix) exists. Typ-ical Category I soliton equations are for example, focusing nonlinear Schrodingerequation, sine-Gordon equation, modified KdV equation, Heisenberg ferromagnetequation, derivative nonlinear Schrodinger equation, discrete nonlinear Schrodingerequation, Davey-Stewartson II equation etc.. Typical Category II soliton equationsare for example, KdV equation, defocusing nonlinear Schrodinger equation, sinh-Gordon equation, Toda lattice, Boussinesq equation, KP equations etc.. In prin-ciple, figure-eight structures for Category I soliton equations can be constructedthrough Darboux transformations, as illustrated in previous chapters. It should beremarked that Darboux transformations still exist for Category II soliton equations,but do not produce any figure-eight structure.

CHAPTER 10

Acoustic Spectral Problem

This chapter deals with the Darboux transformations for the acoustic problemand the related Harry Dym (HD) equation. The acoustic problem and the linearSchrodinger operator are closely connected. This connection is the key for studyingthe acoustic problem and the HD equation, see e.g. [44] [31] [32] [110]. However,as discussed below, the connection is not simple. This makes the acoustic probleminteresting in its own right.


The acoustic problem describes wave propagation in an inhomogeneous media.The acoustic problem on the real line is described by the following ODE:

(10.1) ψxx =λ

u2(x)ψ ,

where ψ is complex-valued, u(x) is real-valued, and λ is a complex parameter ingeneral. Here is one way how this equation arises in physics. Consider the Maxwellequations in a medium without external sources, with the standard notations (E,H)for the electromagnetic field and D = ǫE, B = κH. Assume that the mediumis isotropic but inhomogeneous, i.e. ǫ, κ are scalar quantities; and κ ≡ 1 andǫ = ǫ(x, y, z), then

(10.2) [∇,B] =1

c

∂D

∂t, [∇,E] = −1

c

∂B

∂t, (∇,D) = (∇,B) = 0,

where [·, ·] and (·, ·) denote the vector and dot product in R3, respectively. Elimi-nating B from (10.2) leads to an equation connecting the quantities E and D:

(10.3) [∇, [∇,E]] = − 1

c2∂2D

∂t2.

For the electric field E, one seeks E(t;x, y, z) = eiωtψ(x, y, z) and takes into accountthe last equation of (10.2) to obtain

(10.4) ∇(

(ψ,∇)ǫ

ǫ

)+ ∆ψ = −ω

2

c2ǫψ,

where ∆ is a three dimensional Laplacian. Equation (10.1) follows if one letsǫ = ǫ(x), ψ = (0, 0, ψ(x)), λ = −ω2/c2, u−2(x) = ǫ(x). The dielectric permitivityǫ(x) will be henceforth referred to as a potential. Alternatively, one can chooseǫ = ǫ(x, y) as well as ψ = (0, 0, ψ(x, y)). If this is the case, (10.4) is reduced to alinear PDE

(10.5) ∆ψ = λǫψ,

where ∆ is a two dimensional Laplacian.

81

82 10. ACOUSTIC SPECTRAL PROBLEM

10.2. Connection with Linear Schrodinger Operator

Starting from (10.1), let [31], [32]

(10.6) u(x) = vy(y), x = v(y).

This reduces equation (10.1) to

(10.7) ψyy = Uψy + λψ,

with U = vyy/vy. Finally, a formal substitution

ψ → √vyψ,

transforms (10.7) into the stationary Schrodinger equation

(10.8) ψyy = (λ+ V (y))ψ,

where the potential V (y) is related to the potential U(y) via

(10.9) V =U2 − 2Uy

4.

Notice also that with a substitution U = −2py/p, p(y) in turn satisfies (10.8) withλ = 0.

10.3. Discrete Symmetries of the Acoustic Problem

Following Shabat [97], we seek elementary discrete symmetries of equation(10.7) in the form

(10.10) ψ → ψ(1) = fψy + gψ,

for some λ-independent functions f and g of y. One easily verifies that there arethree discrete symmetries of the type (10.10) for equation (10.7). They are

(10.11) ψ → ψ(1) = ψv , v → v(1) = 1

v ; ψ → ψ(1) =ψy

vy, v → v(1) =

∫dyvy

;

and

(10.12)

ψ → ψ(1) =ψ1ψy

ψ1,y− ψ,

vy → v(1)y = vy

(ψ1

ψ1,y

)2

, U → U (1) = U + 2D ln ψ1

ψ1,y.

In the latter equation ψ1 = ψ1(y, λ1) is a particular solution of (10.7) with thespectral parameter value λ1, further referred to as a prop solution, D = ∂y and

ψ1,y = Dψ1. The former two symmetries (10.11) define the new quantity U (1) =

v(1)yy /v

(1)y in a way independent of any solution ψ(y, λ) of (10.7). These symmetries

arise as a particular case of (10.10) as the result of gauging corresponding to thechoice of f or g alternatively zero. These symmetries have a trivial kernel in the so-lution space of (10.7). According to the terminology of [97], we call the symmetries(10.11) T-symmetries, sometimes referred to as Schlesinger transforms1. On theother hand, the transformation (10.12) alias the Darboux transformation, which[97] calls an S-symmetry, does have a non-trivial kernel on the solution space of(10.7). This property is essential. We use the common term dressing for the appli-cation of the transformation (10.12) to a triplet (ψ, v, U), and the resulting triplet(ψ(1), v(1), U (1)) is referred to as the dressed one.

1In the context of soliton solutions, the T-symmetries play the part of explicitly invertibleBacklund transforms, see e.g. [101].

10.4. CRUM FORMULAE AND DRESSING CHAINS FOR THE ACOUSTIC PROBLEM 83

10.4. Crum Formulae and Dressing Chains for the Acoustic Problem

In this section, we present the formulae describing an n-step dressing procedurefor any n ∈ N, analogues of which are known for the Schrodinger equation as theCrum formulae [28] [108]. A single act of dressing (10.12) can be iterated n timesto yield a triple (ψ(n),v(n), U (n)). One starts by dressing a triple (ψ, v, U) =

(ψ(0), v(0), U (0)) with a prop function ψ(0)1 , which solves (10.7) at (λ1, U

(0)). The

resulting solution ψ(1) solves (10.7) with the dressed potential U (1). In the jth

step (j = 1, · · · , n), one uses a prop solution ψ(j−1)j , which solves (10.7) at (λj ,

U (j−1)), to produce the j-times dressed solution ψ(j) and potential U (j) (as well as

the function v(j) with U (j) = v(j)yy /v

(j)y ). Note that the spectral parameter λ in the

dressed equations for ψ(j) is the same for all j = 1, · · · , n. It is easy to see that then times dressed solution ψ(n) shall have the form

(10.13) ψ(n) =n∑

j=1

ajDjψ + (−1)nψ,

with the function-coefficients aj to be found, which of course will depend on the

choice of the prop solutions ψ(j−1)j . It follows from (10.12) that

(10.14) U (n) = U + 2D ln an,

thus

ψ(n) =

n∏

j=1

ψ(j−1)j

Dψ(j−1)j

Dnψ(0) + . . .+ (−1)nψ(0), U (n) = U (0)+2D ln

n∏

j=1

ψ(j−1)j

Dψ(j−1)j

.

So far the choice of the prop solutions ψ(i−1)i has been quite arbitrary. But sup-

pose now that the original equation (10.7) possesses n distinct formal solutions ψj ,

corresponding to spectral parameter values λj , j = 1, . . . , n. Let ψj ≡ ψ(0)j and

consider the following dressing procedure:

(10.15)

ψ(0) ψ(0)1 ψ

(0)2 · · · ψ

(0)n−1 ψ

(0)n U (0)

ψ(1) 0 ψ(1)2 · · · ψ

(1)n−1 ψ

(1)n U (1)

.

.

.

ψ(n−1) 0 0 · · · 0 ψ(n−1)n U (n−1)

ψ(n) 0 0 · · · 0 0 U (n)

That is, for j = 1, . . . , n on the above diagram (10.15), every new line j + 1 isobtained by dressing the functions from the preceding line j by (10.12) with a prop

solution ψ(j−1)j .

Zeroes, proliferating as one moves down the diagram stem from the non-trivialkernel property of the S-symmetry, and it is this property that now enables one

to find the unknown functions aj . Indeed, substitution of any ψj = ψ(0)j for ψ in

the right hand side of (10.13) shall yield zero. Hence, the coefficients aj satisfy asystem of n independent linear algebraic equations, namely

n∑

k=1

akDkψj + (−1)nψj = 0, j = 1, .., n.


Solving it by the Kramer rule and substituting the result into (10.13) and (10.14),we obtain

U (n) = U + 2D ln∆n

∆n,

(10.16) v(n)y = vy

(∆n

∆n

)2

, ψ(n) =∆n+1

∆n,

where ∆n, ∆n are determinants of square n × n matrices, whereas ∆n+1 – of an(n+ 1) × (n+ 1) matrix as follows:

∆n =

∣∣∣∣∣∣∣

Dψ1 . . . Dnψ1

......

Dψn . . . Dnψn

∣∣∣∣∣∣∣, ∆n =

∣∣∣∣∣∣∣

ψ1 . . . Dn−1ψ1

......

ψn . . . Dn−1ψn

∣∣∣∣∣∣∣,

(10.17) ∆n+1 =

∣∣∣∣∣∣∣∣∣

ψ . . . Dnψψ1 . . . Dnψ1

......

ψn ... Dnψn

∣∣∣∣∣∣∣∣∣

.

Notice that the linearity of (10.7) makes the choice of the sign before ψ(n) irrelevant.The formulae (10.16) and (10.17) make it possible to find rich families of exactsolutions of (10.1).

Example 10.1 (reflexionless potentials). The easiest case is dressing from thebirthday suit, or on the vacuum background, assuming U = 0 (hence x = c1y+ c2,u(x) = c1, where c1 and c2 are arbitrary constants). Such a natural rigging yieldsgratis all the B-potentials reported by Novikov [93], moreover the formulae for theircomputation derived therein turn out to be particular cases of (10.12) and (10.13)with merely vy = 1. For instance ψ1 = sinh ξ1, ψ2 = cosh ξ2 (here n = 2, yj areconstants, ξi = kj(y − yj), j = 1, 2) yield a reflexionless B-potential with a power

2/3 singularity: ǫ(2) = 1[u(2)]2

, where2

u(2) =

(k2 sinh ξ1 sinh ξ2 − k1 cosh ξ1 cosh ξ2k1 sinh ξ1 sinh ξ2 − k2 cosh ξ1 cosh ξ2

)2

.

The same ψ1 and ψ2 = sinh ξ2 yield another potential with a power 4/5 singularity:

u(2) =

(k2 sinh ξ1 cosh ξ2 − k1 sinh ξ2 cosh ξ1k1 sinh ξ1 cosh ξ2 − k2 sinh ξ2 cosh ξ1

)2

.

By construction, these potentials have only two levels λ1,2.

All the regular reflexionless potentials can be also built by the formulae (10.12)and (10.13) once again by dressing U = 0. Matveev and Salle [84] find super-reflexionless potentials for the KdV equation, alias positons. In the same vein onecan operate on equations (10.1, 10.7). In order to do so, one should use the formulae

2Note that the expressions for u(2) are parametric. In order to interpret the formulae cor-rectly, the reader is referred back to (10.1), (10.6), (10.7). The orders of the singularities pertainto the potential ǫ(x), which is a zero of the function u(x) and a singularity of the potentialU(y) = vyy/vy , where the function v(y) solves the equation u[v(y)] = vy(y).

10.4. CRUM FORMULAE AND DRESSING CHAINS FOR THE ACOUSTIC PROBLEM 85

(10.12) and (10.13) with n = 2 choosing the prop solutions ψ1,2 respectively asψ1(y, λ1) and ψ1(y, λ1 + δ), and then letting δ → 0. If with U = 0 and ψ1 generatesa single soliton potential, then (10.16) defines a single positon potential. In additionto B-potentials, various other interesting ones can be produced. For instance, onecan construct soluble potentials with a finite equidistant spectrum3. One can alsoinvestigate potentials which change in a specific simple way under the Darbouxtransform, e.g. such that U → U+ constant or U → constant ×U . For theSchrodinger equation, the former transformation is shape-invariant and results inthe harmonic oscillator potential.

Example 10.2 (shape-invariant potential). Let

U (1) = U +2

ω2,

for a constant ω. Then we can obtain parametrically the function u(x) from (10.1)as follows:

u(x) = αz exp

(−ω2z − κ2

z

), x = x0 − αω2

∫dz exp

(−ω2z − κ2

z

),

where, κ, x0, α are real constants and z = exp(−y/ω2). The prop function ψ1

rendering the potential U (1) from U has the countenance ψ1 = exp(−ω2z) andsolves (10.7) with an eigenvalue λ1 = b2/ω2. It’s easy to verify that the dielectricpermitivity ǫ(x) = 1/u2(x) has a second order pole at x = x0.

The theory of Darboux transformations for the Schrodinger equation utilizesthe concept of dressing chains of discrete symmetries and their closing. The work ofVeselov and Shabat [102] elucidates how the dressing chain closing method can beused in order to obtain various potentials with meaningful mathematical physics.Namely, a simple closing procedure leads one to the harmonic oscillator potential(resulting also in a shape-invariant change of potential). A more complicated closingscheme results in finite-gap potentials as well as the fourth and the fifth Painleveequations, see [102].

Dressing chains can be written out for equation (10.7) as well. Let us introducea sequence fnn≥1 of functions as follows:

fn = D lnψ(n−1)n ,

with the quantity ψ(n−1)n as it has been introduced in the diagram (10.15) above.

In particular, it corresponds to the pre-chosen value λn of the spectral parameter.One can verify, starting from n = 1, that

U (n) = U − 2D ln

n∏

j=1

fj .

Besides, direct substitution shows that fn satisfies the equation

f ′n + f2

n − U (n)fn = λn,

where f ′ = Df . The two latter relations imply the recursion connecting fn andfn+1 as follows:

(10.18) (fnfn+1)′ = fnfn+1(fn − fn+1) + λn+1fn − λnfn+1.

3The same statement applies to the (stationary) Schrodinger equation.


This equation (10.18) represents a dressing chain for the acoustic problem.In a way analogous to the theory of dressing chains for the Schrodinger equation

[102], we are interested in T-periodic chain closing, namely imposing the conditionfn+T = fn for an integer T ≥ 1. We shall consider here the easiest case T = 1.

Example 10.3 (regular potential). Given the spectral parameter values λ1,2,one obtains a one-parameter family of potentials, indexed by a constant c:

U =−(λ1 − λ2)

2y2 + 2c(λ2 − λ1)y + 6λ1 − 2λ2 − c2

2[(λ1 − λ2)y + c].

If λ2 = 3λ1 > 0 and c = 0, we can express the function u(x) parametrically:

x(y) =

√π

2αErf(αy), u(y) = exp

(−α2y

),

where α2 = −λ/2 > 0.

It is known that for the Schrodinger equation, a nontrivial chain closing oper-ation with T > 1 results in finite gap potentials [102]. Such potentials for the HDequation are due to Dmitrieva [32]. A close analogy between the Schrodinger equa-tion and the acoustic problem (see [44], [31], [24]) suggests that one can expectresults similar to those of [102] apropos of the analysis of higher order chain clos-ing for T > 1. We expect that potentials built in such a way can have interestingphysical applications, such as for instance a model of wave propagation in mediawhose dielectric permitivity is a periodic function of a single spatial variable.

10.5. Harry-Dym Equation

The Harry-Dym (HD) equation

(10.19) ut = u3uxxx + βux,

with some real constant β, has been studied quite extensively since late 70’s, seee.g. [44], [31], [32] and references therein. The principal approach to it has beenbased on its relation to the KdV, mKdV and other more classical hierarchies ofintegrable PDEs [31], [32]. However, as mentioned before, this relation is notstraightforward, and the direct approach developed herein in principle enables oneto produce a wider range of solutions of the HD equation.

The acoustic problem (10.1) is the first equation in the Lax pair for the HDequation (10.19), the full pair is

(10.20) ψxx =λ

u2ψ, ψt = (4λu+ β)ψx − 2λuxψ.

The coordinate change (10.6) in the presence of time dependence becomes

t→ t, x→ v(y, t),

thus

∂x → 1

vy∂y, ∂t → ∂t −

vtvy∂y.

After this change, (10.20) becomes

(10.21) ψyy =vyyvyψy + λψ, ψt =

(vt + β

vy+ 4λ

)ψy −

2λvyyvy

ψ,

10.5. HARRY-DYM EQUATION 87

and the HD equation (10.19) is transformed into

(10.22) vy(vtvyy − vytvy) + 3v2yy + vy(v4yvy − 4v3yvyy + βvyy) = 0.

with the notations v3y, v4y for the partial derivatives in y of order 3 and 4 respec-tively.

The goal now is to extend the Darboux transformation (10.12) for equation(10.7) alias the first equation in (10.21), so that it agrees with the second equationin the Lax pair. At this point, it only provides the value of the partial derivative

v(1)y =

(ψ1

ψ1,y

)2

≡ A(y, t),

rather than the dressed quantity v(1) of interest.

Lemma 10.4. The Darboux transformation is an S-symmetry for the Lax pair(10.21), and therefore for the HD equation (10.22).

Proof. Let v(1)t = B(y, t) be an unknown and let’s assume that the dressed

function ψ(1) according to (10.12) satisfies the second equation of the pair (10.21)with v(1), and the quantities (λ, β) remain the same. One can express the unknownquantity B as follows:

B =

(ψ1

ψ1,y

)2

(β + 4µvy + vt − 2vyUy) +4ψ1vyyψ1,y

− 4vy − β,

and verify that By = At. It follows that

(10.23) v(1)(y, t) =

∫Ady +Bdt,

with a closed 1-form under the integral.

Example 10.5 (single soliton solution and positon solution). : Let v = y,λ1 = k2, ψ1(y, t, k) = sinh[φ(y, t, k)] with φ = k

(y + (4k2 + β)t

), then by (10.23)

one has

v(1) =1

k3(φ− tanhφ) − (4 + β)t.

The function v(1)(y, t, k) determines a single soliton B-potential U (1) = v(1)yy /v

(1)y .

The single positon potential is obtained from two distinct soliton solutions ψ1(y, t, k)and ψ1(y, t, k + δ), using them as the prop functions ψ1,2 in the formulae (10.12)and (10.13) with n = 2 and taking the limit as δ → 0. Namely,

v(2)y = vy

(ψ1,yykψ1,y − ψ1,yyψ1,yk

ψ1,ykψ1 − ψ1,yψ1,k

)2

,

where the subscript k means differentiation in k. Taking ψ1 explicitly as the hy-perbolic sine in the previous example results in

v(2)y = k4

(sinh(2φ) + 2φ

sinh(2φ) − 2φ

)2

,

with φ = k(y + (12k2 + β)t

).


10.6. Modified Harry-Dym Equation

It is well known that the dressing formalism enables one to produce hierarchiesof integrable PDEs. Borisov and Zykov [22] proposed a technique for proliferationof integrable equations, which they applied to the KdV and the Sine-Gordon (SG)equations. The technique is based on dressing chain closing. The main idea of theapproach is as follows: Take KdV equation for example, rewrite it as a compatibilitycondition of a pair of equations denoted by L1 and A1. Using invariance of thepair with respect to the Darboux transformation (which is viewed as a discretesymmetry), a second pair L2 and A2 is built. Eliminating the potentials from L1

and L2, and A1 and A2, it is possible to obtain two equations called an x and at chains [22] denoted by Cx and Ct. If a potential is eliminated from L1 and A1,one ends up with a modified KdV. This procedure can be repeated, producing newequations with their Lax pairs, e.g. m2KdV, m3KdV etc.. The former becomes theexponential Calogero-Degasperis equation [24] after an exponential change, and thelatter contains an elliptic equation of the same authors.

Using the technique described, it was shown [88] that the mNKdV equationswith N = 0, ..., 3 together with the Krichever-Novikov equation, exhaust (mod-ulo a contact transformation) all the integrable equations of the form ut + uxxx +f (uxx, ux, u) = 0. Using the same approach to the sine-Gordon equation, Borisovand Zykov [22] obtained a new nonlinear equation in the second step. This equa-tion has a non-trivial Backlund transform, admitting an interesting 2π-kink-shelfsolution. The same technique was also shown to be applicable to the study ofmore difficult (1+2)-dimensional nonlinear PDEs. For instance, in [109], the pro-liferation procedure was successfully adapted to the Kadomtsev-Petviashvili andBoiti-Leon-Pempinelli equations.

Let us apply this formalism to the HD equation. First note that the Lax pairfor (10.22) can be written as a system of two Ricatti equations:

(10.24)gy = −λg2 − Ug + 1,

gt = λ(2Uy − vt+β

vy− 4λ

)g2 −

(vty

vy+ 4λU

)g + vt+β

vy+ 4λ,

where the function g = g(y, t) is connected with the solution ψ of (10.21) as g =ψy/ψ. Eliminating the v from (10.24) and returning to the old variables via x = g,u = gy, we obtain a modified Harry Dym (mHD) equation:

(10.25) ut = u3u3x+3u2uxuxx−3λ2xu2− 3u2(uuxx + u2x)

x+

6u3uxx2

+3u2(1 − u2)

x3.

Note that (10.25) can be rewritten quite nicely in the new variables x = 1/z,

u(x, t) =√θ(z, t):

(10.26)(θ−1/2

)

t=

3λ2

z+ z3

(1

2z3θ3z +

9

2z2θzz + 9zθz + 3θ − 3

),

which can be further simplified as[(e−2ξη + 1

)−1/2]

t= 3λ2e−ξ +

1

2eξ(η

3ξ− η

ξ

),

by another change of variables

θ = e−2ξη(ξ, t) + 1, ξ = log z.

10.7. MOUTARD TRANSFORMATIONS 89

The dressing chain method produces not only equation (10.26), but also its Laxpair. It is constructed as follows: Return to the chain (10.18) and let fn = 1/gn,fn+1 = Ψ, λn = λ, λn+1 = µ. Regarding µ as a spectral parameter, one can seethat (10.18) can be viewed as an L-equation of the Lax pair for equation (10.26).One can also define the second non-stationary chain Ct for the functions gn interms of the temporal equation in (10.21) and build the A-equation. Omittingthe lengthy but straightforward computation, we present the Lax pair for equation(10.26), written in the variables t, z:

(10.27)Ψz =

µ

z2√θΨ2 +

(1

z+

1

z√θ− µ

z3√θ

)Ψ − 1

z2√θ,

Ψt = µaΨ2 + bΨ + c,

where

a = −4µ+ 2λ− λ2

z2 − 2λ√θ + (θ − 1) z2 + 2z3θz + 1

2z4θzz,

b = 4(λz − z − z

√θ)µ+ 1

2z5θzz + 2z4θz +

(θ − λ

2 θzz − 1)z3−

−3λz2θz + 3λ(1 − θ)z − 3λ2

z +(λz

)3,

c = 4µ− 12z

4θzz − 3z3θz +(1 − 3θ − 2

√θ)z2 − 2λ+

(λz

)2.

Note that the spectral parameter in (10.27) is µ, whereas λ enters the non-linearequation (10.26). As one can see, the Lax pair for (10.26) also has the form of apair of Ricatti equations. These equations can be simultaneously linearized in orderto represent (10.26) as a compatibility condition for two linear equations, as it isdone in the theory of solitons.

We will not proceed further with the mHD equation. To conclude this section,we would like to repeat the statement that the exact solutions of (10.26) can beeasily found via the dressing technique, and this procedure can be extended in orderto produce the mHD equation and its Lax pair. It is worthy of emphasizing that HDand (10.26) are members of different hierarchies. Discrete symmetries enable one toestablish connections between different integrable hierarchies in a way apparentlymore systematic than trying to guess the Miura transformation.

10.7. Moutard Transformations

In this section, we return to the PDE (10.5) which was obtained from theMaxwell equations (10.2) in the case of an isotropic but inhomogeneous two-dimensional(x, y) medium. One can obtain the counterpart of Darboux transformation for thePDE (10.5), known also as the Moutard transformation [84].

Lemma 10.6. Let ψ = ψ(x, y) and φ = φ(x, y) be two particular solutions of(10.5), i.e.

(10.28) ∆ψ − λǫψ = ∆φ− λǫφ = 0.

The following transformation is the Moutard transformation:

(10.29) ψ → ψ(1) =θ[ψ, φ]

φ, ǫ→ ǫ(1) = ǫ− 2λ∆lnφ,

where

(10.30) θ[ψ, φ] =

∫

Γ

dxµεµν (φ∂νψ − ψ∂νφ) ,


with the following tensor notations: µ ∈ 1, 2, xµ ∈ x, y, ∂µ = ∂/∂xµ, εµνis a fully antisymmetric tensor with ε12 = 1, summation is implied over repeatedindices.

The Moutard transformation (10.29) can be iterated several times, and theresult can be expressed via Pfaffian forms [84]. In terms of the original variables

of the Maxwell equations (10.2). A straightforward computation (recall, λ = − c2

ω2 )

yields the expressions for the dressed electric and magnetic fields E(1), B(1):

(10.31) E(1) = eiωt(0, 0, ψ(1)

),B(1) =

c

ωeiωt

(−ψ(1)

y , ψ(1)x , 0

),D(1) = ǫ(1)E(1).

On the basis of (10.31) one can build a variety of exact solutions of the Maxwellequations.

Example 10.7. (singular potentials): As a simple example, let’s dress ǫ =0. This isn’t quite a medium, but one can easily proceed with formal calcula-tions (10.29)-(10.31) which result in a new “medium” whose dielectric permitivityǫ(1)(x, y) and the stationary component of the field ψ(1) are as follows:

(10.32) ǫ(1) = −8c2

ω2

a′(z)b′(z)

(a(z) + b(z))2 , ψ(1) =

a(z)β(z) − α(z)b(z) + ξ(z, z)

a(z) + b(z),

where

ξ(z, z) =

∫dz (α(z)a′(z) − a(z)α′(z)) +

∫dz (β′(z)b(z) − b′(z)β(z)) ,

a(z), α(z), b(z), β(z) are arbitrary functions of z = x + iy, z = x − iy. Note thatthe function ψ(1) from (10.29) and (10.31) provides in fact a general solution of thedressed equation, for it is described in terms of two arbitrary functions α(z) andβ(z). To ensure that the quantities found correspond to a physical non-absorbingmedium, one should require that the dressed dielectric permitivity function ǫ(1) tobe real. This imposes an extra restriction to the quantities a(z) and b(z), namely

b(z) = a(z). Generally speaking, the functions ǫ(1) and ψ(1) from (10.32) will havesingularities along certain curves in the (x, y)-plane.

The reflexionless B-potentials for the one-dimensional problem (10.1) above (seeExample 10.1) possess point singularities on the real line (corresponding to zeroes ofthe function u(x)). Clearly, their 2D-analogues, such as (10.32) for equation (10.5)allow a much more diverse structure of singularities on the real plane. On the otherhand, not requiring that the quantity ǫ(1) be real, one obtains an absorbing mediumwhich may not be devoid of interest for physical applications.

Finally, let us study a dressing chain generated by the Moutard transformations(10.29). A simple periodic closing of the dressing chain results in a regular dielectricpermitivity, similar to the 1D case studied above. Denote by fn = lnφ, fn+1 =lnψ(1). Then after a straightforward computation

(10.33) ∆ (fn + fn+1) = ‖∇fn‖2 − ‖∇fn+1‖2,

where ‖ · ‖ is the Euclidean norm. The chain (10.33) is closely related to that ofVeselov and Shabat [102] for the Schrodinger equation. Choosing fn specifically as

fn =√λny +

∫dx gn(x),

10.7. MOUTARD TRANSFORMATIONS 91

and substituting it into (10.33) (λn being constant), we obtain for the quantitiesgn(x) the following expression

(gn + gn+1)′= g2

n − g2n+1 + λn − λn+1,

matching the corresponding formula of [102].

Example 10.8. (regular dielectric permitivity): The simplest periodic closingof the dressing chain (10.33) is fn+1 = fn = F (x, y), which implies that the latterfunction F is harmonic, and that the regular dielectric permitivity function in thecorresponding medium is given by the formula

ǫ(x, y) =c2

ω2

(F 2x + F 2

y

).

CHAPTER 11

SUSY and Spectrum Reconstructions

Supersymmetric quantum mechanics realizes the quantum description of sys-tems with double degeneracy of energy levels. In one dimension, the supersymmetryis intrinsically connected with the Darboux transformation (DT) [47] [29]. The DTconnects two Hamiltonians h0 and h1 with equivalent spectra:

(11.1) h0 = q+q + E0, h1 = qq+ + E0, q ≡ d

dx− (lnϕ)′,

where q+ is the Hermitian conjugate of the q operator, and ϕ is a solution to theequation h0ϕ = E0ϕ. It is easy to see that h0 and h1 are intertwined by q and q+

(11.2) q h0 = h1 q, h0 q+ = q+ h1,

and therefore

(11.3) ψ(1) = q ψ, ψ = q+ ψ(1),

if

(11.4) h0ψ = E1ψ, h1ψ(1) = E1ψ

(1),

and ϕ(1) = ϕ−1 solves h1ϕ(1) = E0ϕ

(1). For brevity, the normalizing multipliers in(11.3) are omitted.

DT allows us to construct one-dimensional potentials with arbitrary preas-signed discrete spectrum. For instance, if the support function ϕ(E0;x) is a wavefunction (w.f.) of the ground state h0, then the discrete spectrum of h1 coincideswith the spectrum of h0 without lower level E0 [28]. It is possible to add the levelE0 (missing in the spectrum of h0) to the spectrum of h1. To achieve this, it issufficient to exploit such ϕ that

ϕ→ +∞, x→ ±∞,

and ϕ is a positive definite function of x. It is convenient to choose ϕ as:

(11.5) ϕ = λϕ+ + (1 − λ)ϕ−,

where ϕ+ and ϕ− are positive definite functions with the following asymptoticbehavior:

(11.6) ϕ± →

+∞, for x→ ±∞0, for x→ ∓∞,

and λ is a real parameter in the interval [0, 1]. If 0 < λ < 1, then the level E0 is onthe lower boundary of the spectrum of h1. If λ = 0 or λ = 1, the level E0 is not inthe spectra of h0 and h1 and the spectra are the same.

This relates to the one-dimensional supersymmetric quantum mechanics basedon the following commutation relations:

(11.7) [Q,H ] = [Q+, H ] = 0, Q,Q+ = H,

93

94 11. SUSY AND SPECTRUM RECONSTRUCTIONS

where

Q = qσ+, Q+ = q+σ−, H = diagh0 − E0, h1 − E0,and σ± = (σ1 ± iσ2)/2, and σ1,2 are Pauli matrices. Notice that q and q+ arebosonic, and σ− and σ+ are fermionic, operators of creation-annihilation. If thespectra of h0 and h1 differ by the only level, then (11.7) corresponds to the exactsupersymmetry. If the level E0 is absent in the spectra of both operators, then thesupersymmetry is broken. It is easy to see that the level E0 cannot simultaneouslyappear in the spectra of h0 and h1.

11.1. SUSY in Two Dimensions

In this section, we consider the case of two-dimensional supersymmetric quan-tum mechanics. The explicit form of operators that satisfy the algebraic relation(11.7) in 2D is determined by the expressions:

(11.8) Q =

0 0 0 0q1 0 0 0q2 0 0 00 q2 −q1 0

, Q+ =

0 q+1 q+2 00 0 0 q+20 0 0 −q+10 0 0 0

,

(11.9) H = diag(h0 − E0, hlm − 2δlmE0, h1 − E0),

where

(11.10) h0 = q+mqm + E0, h1 = qmq+m + E0, hlm ≡ hlm +Hlm − E0δlm,

and

(11.11) hlm = qlq+m + E0δlm, Hlm = plp

+m + E0δlm,

and ql = ∂l−∂l(lnϕ), pl = εlkq+k , where εlk is the antisymmetric tensor, ∂l ≡ ∂/∂xl,

(l = 1, 2), and the sum by repeated indices is implied.In contrast to 1D, there is no association between the spectra of h0 and h1 here.

In 2D, there is no formula expressing wave functions of h1 via wave functions of h0.The existence of such a formula implies the connection between the spectra of h0

and h1. However, for special potentials, such a connection can still exist. Moreover,there could be expressions that connect wave functions of h0 and h1, that are notrelated to a physical spectrum.

The general coupling between the spectra exists for h0 and hlm or h1 and Hlm.Notice that h1 may be represented as h1 = p+

mpm + E0, it is easy to verify therelations:

(11.12) qlh0 = hlmqm, plh1 = Hlmpm, h0q+l = q+mhml, h1p

+l = p+

mHml.

The same formulae apply to the operator, hlm, h0, and h1, perhaps without thelevel E0.

The supersymmetry defined by the operators (11.8)-(11.9), was studied in [28][7] [10], with the assumption that ϕ is a wave function of the basic state of theHamiltonian h0. It was shown that such a choice of ϕ leads to the assertion thatthe level E0 is either absent from the physical spectra of hlm and h1, or relatedto the unbroken supersymmetry. In the following sections, we study the inverseproblem, i.e. addition of the level E0, that is absent from the spectrum of h0, tothe spectra of hlm and h1.

11.2. THE LEVEL ADDITION 95

11.2. The Level Addition

Let u = u(x, y) be an integrable potential, i.e. one can solve the Schrodingerequation h0ψ = Eψ explicitly for any spectral parameter value E, h0 = −∆ + u.Unlike the one-dimensional case, the potential

(11.13) u(1) = u− 2∆ lnϕ,

where ϕ is a support function, is not integrable here. Assume that the spectralparameter value E0 lies below the basic state energy of the Hamiltonian h0, howshould one choose the support function ϕ such that the level E0 appears in thephysical spectra of h1 and hlm?

For the scalar Hamiltonian h1, it is easy to verify that the function ϕ−1 satisfiesthe equation

(11.14) h11

ϕ= E0

1

ϕ,

therefore, it is enough to choose ϕ as a positive function of x and y, which growsexponentially in all directions on the plane.

For the matrix Hamiltonian hlm, if the function ψ is a solution of the Schrodinger equation with E0, then the function

(11.15) ψm = qmψ,

satisfies the equation

(11.16) hlmψ = E0ψ.

One can show that the level E0 belongs to the spectrum of hlm, iff ψm is normableand satisfies the condition:

(11.17) hlmψm = Hlmψm = E0ψm.

In fact, let ψm be such that

(11.18) hlmψm = E0ψm, (ψm, ψm) = 1.

Define the functions ρm and σm by

(11.19) ρm ≡ hlmψm, σm ≡ Hlmψm.

From (11.10)-(11.11), it follows that σ + ρ = 2E0ψ (indices omitted), i.e.

(11.20) (ρ+ σ, ρ+ σ) = 4E20 ,

if ρ and σ are normable. Otherwise one may check that

(11.21) hmkHkl = Hmkhkl = E0hml.

From (11.21), it follows that

(11.22) hlmσm = Hlmρm = E20 ψl.

Hence

(11.23) (ψm, hlmσl) = (hlmψm, σl) = (ρm, σm) = E20 .

Combining with (11.20), we obtain (ρ− σ, ρ− σ) = 0, therefore σm = ρm = E0ψm.Finally from (11.19), we get (11.17). Thus for the level E0 lying in the physical


spectrum of hlm, it is necessary to find a normable solution of (11.17). Let ψm be

such a solution, applying hlm, one gets

(11.24) q+mψm = p+mψm = 0.

This means that there exist two functions ψ and ψ(1) such that

(11.25) ψm = qmψ = pmψ(1),

where

(11.26) h0ψ = E0ψ, h1ψ(1) = E0ψ

(1).

Solving (11.25) with respect to ψ(1), we get the important relation:

(11.27) ψ(1) =1

ϕ

∫dxkεkm(ϕ∂mψ − ψ∂mϕ),

that is known as a Moutard transformation [16]. From the relations among ψm,ρm, and σm, the formula (11.15) obviously follows.

11.3. Potentials with Cylindrical Symmetry

Let ψ and ϕ > 0 be the solutions described at the end of the previous section.For the construction of matrix potentials with the level E0, it is convenient tointroduce an auxiliary function

(11.28) f ≡ ψ

ϕ

that satisfies the equation

(11.29) ∂m(ϕ2∂mf) = 0.

Then

(11.30) ψm = ϕ∂mf.

Consider the case when the seed potential possess the cylindrical symmetry u =u(r). Integrating (11.29) and substituting in (11.30), one gets

(11.31) ψm =xmr2ϕ

.

The normalizing integral of (11.31) converges if ϕ grows at infinity as a powerfunction, and at the vicinity of zero, it behaves as r−k, k > 0. If one requires thatthe asymptotic behavior of ϕ is determined by the conditions:

(11.32) ϕ→ra, for x2 + y2 → ∞rb, for x2 + y2 → 0,

where a > 1, b < 1, then the normalizing integral of ϕ−1 should converge as well.This means that the level E0 will be present in both spectra of the operators h1

and hlm simultaneously.The spectrum of the supersymmetrical Hamiltonian (11.9) consists of the levels

Ei − E0, E(1)i − E0,

11.4. EXTENDED SUPERSYMMETRY 97

where Ei, E(1)i are the levels of the discrete spectra of h0 and h1, respectively. If

the condition (11.32) is satisfied, then in the spectrum of (11.9), there is a doublydegenerated level E = 0 to which the following eigenfunctions correspond,

(11.33) Ψ1 =

0001ϕ

, Ψ2 =

0ϕ∂1fϕ∂2f

0

.

Using the explicit form of the odd supersymmetric operators (11.8), one can provethe relations for wave functions of the zero level

QΨ1,2 = Q+Ψ1,2 = 0.

As an example, choose ϕ = exp(br)/rk, where b, k > 0. It satisfies the necessaryasymptotic (11.32). As a result, we obtain two scalar potentials of the Hamiltoniansh0 and h1:

(11.34) u =k2

r2− b(2k − 1)

r, u(1) =

k2

r2− b(2k + 1)

r.

The added level corresponds to the energy E0 = −b2. The discrete spectrum isdetermined by the formula:

(11.35) EN

= − b2(2k ∓ 1)2

(1 + 2[N +√m2 + k2])2

,

where the sign “-” corresponds to u, and “+” to u(1), and N is the principal andm – magnetic quantum numbers. The constructed potentials exhibit the differencebetween DT in multidimensions and one dimension. Specifically, the comparisonof the spectra of the Hamiltonians h0 and h1 shows that the addition of the lowestlevel shifts all the spectrum. For the potential (11.34), it can be shown that when

k =(N + 1)2 −m2

2(N + 1),

the addition of the level E0 = −b2, does not move the exited level with the numberN for fixed m. In general, the levels of the Hamiltonian h1 go down with respectto the levels of h0. The maximal displacement happens in the lowest part of thespectrum. The spectrum of the supersymmetric Hamiltonian (11.9) is doubly de-generated at the level E = 0. Its normable vacuum wave functions are given by theexpressions (11.33), and

(11.36)1

ϕ= rk exp(−br), ∂mf =

xm(rϕ)2

.

11.4. Extended Supersymmetry

In the previous section, we had demonstrated how to build a two-dimensionalHamiltonian with doubly degenerated level E = 0. One may obtain models withall degenerated levels with extended symmetry [9] [92]:

(11.37) Qi, Qk = δikH, [Qi, H ] = 0, i, k = 1, ..., N.

First one can define two Hamiltonians: H1 is determined by (11.9), and H2 differsfrom it by the permutation of h0 and h1. Then one can define three operators: Q1


is defined by (11.8), and(11.38)

Q2 =

0 0 0 0−q+2 0 0 0q+1 0 0 00 −q+1 −q+2 0

, B =

0 q2 −q1 0−q1 0 0 q+2−q2 0 0 −q+10 −q+1 −q+2 0

,

that satisfy the commutation relations

(11.39) Q+2 B +BQ+

1 = Q2B +BQ1 = BH1 −H2B = 0,

(11.40) H1 = Q+1 , Q1 = B+B, H2 = Q+

2 , Q2 = BB+.

One can verify that

(11.41) H(2) = diag(H1, H2), Q1(2) = diag(Q1, Q2),

(11.42) Q2(2) =

(0 0B 0

),

form the algebra (11.37), when N = 2. The introduced operators are the buildingblocks for the construction of the extended supersymmetry matrices for any N .It is possible to demonstrate that at every step there is an operator factorizing asuperhamiltonian as in (11.40). At the step N , the superhamiltonian

(11.43) H(N) =

(B+

NB

N0

0 BNB+

N

),

is factorized by the operator BN+1

= diag(BN, B+

N). In turn, H(N+1), Bk(N +1),

(k ≤ N) are defined by the substitution BN

→ BN+1

and the addition of the newoperator Q

N+1(N + 1) in the new matrix. At every step the dimension of matrices

duplicates, therefore, the corresponding algebra is realized by 2N+1×2N+1 matrices.For example, at N = 4, there would be four operators Qi of dimension 32× 32 andthe superhamiltonian H :

(11.44) H ≡ H(4) = diag(H1, H2, H2, H1, H2, H1, H1, H2),

(11.45) Q1(4) = diag(Q1, Q2, Q2, Q1, Q2, Q1, Q1, Q2),

(11.46) Qs(4) =

0 0 0 0 0 0 0 0B 0 0 0 0 0 0 0B 0 0 0 0 0 0 00 B+ −B+ 0 0 0 0 0B 0 0 0 0 0 0 00 B+ 0 0 −B+ 0 0 00 0 B+ 0 −B+ 0 0 00 0 0 B 0 −B B 0

,

and s = 2, 3, 4. The nonzero elements of the three operators in (11.46) are underthe diagonal. If as a basic scalar model, one takes the potential considered in thelast section, it is obvious that all levels including zero, are degenerate with themultiplicity 2N .

CHAPTER 12

Darboux Transformation for Dirac Equation

12.1. Dirac Equation

Let us consider the four-dimensional Dirac equation

(12.1) (iγµ∂µ − γµAµ(t, x) +m)Ψ = 0.

We use the γ–matrix representation

γ0 =

(0 σ2

σ2 0

), γ1 =

(0 −iσ1

−iσ1 0

),

γ2 =

(−iI 00 iI

), γ3 =

(0 −iσ3

−iσ3 0

),

and for Aµ(t, x) we have either:

(12.2) Aµ(t, x) = (0, 0, A2(t, x), A3(t, x))T ,

or

(12.3) Aµ(t, x) = (0, 0, Q(t, x), cQ(t, x))T , c = const.

One can easily see that (12.2) and (12.3) are solutions of Maxwell equations. LetΨ have the form:

(12.4) Ψ = Φ(t, x) exp(i(py + qz)), Im p = Im q = 0.

Then Φ satisfies the Zakharov–Shabat equation:

(12.5) Φt = JΦx + UΦ.

For the case (12.2):

U =

0 A 0 m− B

A 0 B −m 0

0 B +m 0 A

−B −m 0 A 0

, J =

1 0 0 00 −1 0 00 0 1 00 0 0 −1

,

where A = i(A3 − q), B = i(p−A2).For the case (12.3) we get:

(12.6) Φ =

(AΓBΓ

),

where

A =

(0 αµ 0

), B =

(0 βµρ 0

), Γ =

(ψ(t, x)φ(t, x)

),

(12.7) c =iq(α2 + β2) + 2mαβ

ip(α2 + β2) +m(α2 − β2), ρ =

i(αp+ βq) + αm

i(αq − βp) + βm.

99

100 12. DARBOUX TRANSFORMATION FOR DIRAC EQUATION

The condition c∗ = c gives us for α ≡ αR

+ iαI, β ≡ β

R+ iβ

I

(12.8)

βI

αI

=qα

R−pβ

R

pαR

+qβR

,

(pαR

+ qβR)2(α

R

2 − αI

2 + βR

2) − αI

2(qαR− pβ

R)2

+2mαI(β

R

2 + αR

2)(pαR

+ qβR) = 0.

One of the nontrivial solutions of (12.8) is

(12.9) βI

= 0, βR

=qα

R

p, α

I

± =α

R

p(m±

√m2 + p2 + q2).

Substituting (12.6), (12.7) into (12.1) and taking (12.8) into account, we get a2 × 2 equation (12.5) for Γ where J = σ3,

U =

(0 uv 0

),

u(t, x) = λ1 + λ2Q(t, x), v(t, x) = ν1 + ν2Q(t, x),

λ1 =mβ + i(qα− pβ)

µ, λ2 =

1

µ

(α2 + β2)(qα− pβ − imβ)

ip(α2 + β2) +m(α2 − β2),

(12.10) ν1 =µ(m2 + q2 + p2)

i(pβ − qα) −mβ, ν2 =

iµ

β(1 − cρ).

Let χ1 and χ2 be 4×4 (for the case (12.2)) or 2×2 (for the case (12.3)) matrixsolutions of equation (12.5). We define a matrix function τ

1≡ χ1,xχ

−11. It easy to

see that τ1

satisfies the following nonlinear equations:

τ1,t= σ3τ1,x

+ [U, τ1 ] + [σ3, τ1 ] + Ux.

Equation (12.5) is covariant with respect to DT:

(12.11) χ2[1] = χ2,x − τ

1χ

2, U [1] = U + [σ3, τ1 ].

It is necessary to choose the function χ1

in such a way that the structure ofthe matrix U [1] be the same as the structure of the matrix U . This is the conditionthat we call the reduction restriction.

The transformation (12.11) allows us to construct a superalgebra, in just thesame way as the DT for the steady–state Shrodinger equation in the one-dimensionalcase [8]. In order to do this we introduce the following operators:

G(+) =∂

∂x− τ1 , G(−) =

∂

∂x+ τ+

1.

Let us define several new operators as follow:

h ≡ G(−)G(+), h[1] ≡ G(+)G(−),

(12.12) T ≡ ∂

∂t− J

∂

∂x− U, T [1] ≡ ∂

∂t− J

∂

∂x− U [1].

It easy to see that

G(+)T = T [1]G(+), TG(−) = G(−)T [1], [h, T ] = [h[1], T [1]] = 0.

12.2. CRUM LAWS 101

The operators h and h[1] are the typical one–dimensional matrix Hamiltonians:

h =∂2

∂x2+ ρ

D

∂

∂x+ V, h[1] =

∂2

∂x2+ ρ

D

∂

∂x+ V [1],

ρD

= (τ+1− τ1)D

, V = −(τ1,x+ τ+

1τ1), V [1] = τ+

1,x− τ1τ

+1,

where (τ1)

Dis the diagonal part of τ

1. It is easy to see that the operators q(±), H

q(+) =

(0 0

G(+) 0

), q(−) =

(0 G(−)

0 0

), H =

(h 00 h[1]

)

generate a supersymmetry algebra:

q(±), q(±) = [q(±), H ] = 0, q(+), q(−) = H.

12.2. Crum Laws

Let us consider 2N + 1 particular solutions of (12.5) with Γk ≡ (ψk, φk)T ,

k ≤ 2N, Φ ≡ (ψ, φ):

(12.13) ψk,t + ψk,x + u(t, x)φk = 0, φk,t − φk,x + v(t, x)ψk = 0.

The following theorem can be established:

Theorem 12.1. The functions ψ[N ] and φ[N ] defined below satisfy (12.13)with potentials u[N ] and v[N ] also defined below:

(12.14) ψ[N ] =∆1

D, φ[N ] =

∆2

D, u[N ] = u+ 2

D1

D, v[N ] = v − 2

D2

D,

where ∆1,2, D1,2 and D are the following determinants (ψ(N) = ∂Nψ(t,x)∂xN ) :

D =

∣∣∣∣∣∣∣∣∣∣∣∣

ψ(N−1)1

... ψ1

φ(N−1)1

... φ1

ψ(N−1)2

... ψ2

φ(N−1)2

... φ2

.

.

.

ψ(N−1)2N ... ψ2N φ

(N−1)2N ... φ2N

∣∣∣∣∣∣∣∣∣∣∣∣

,

D1 =

∣∣∣∣∣∣∣∣∣∣∣∣

ψ(N)1

... ψ1 φ(N−2)1

... φ1

ψ(N)2

... ψ2

φ(N−2)2

... φ2

.

.

.

ψ(N)2N ... ψ2N φ

(N−2)2N ... φ2N

∣∣∣∣∣∣∣∣∣∣∣∣

,

D2 =

∣∣∣∣∣∣∣∣∣∣∣∣

ψ(N−2)1

... ψ1

φ(N)1

... φ1

ψ(N−2)2

... ψ2

φ(N)2

... φ2

.

.

.

ψ(N−2)2N ... ψ2N φ

(N)2N ... φ2N

∣∣∣∣∣∣∣∣∣∣∣∣

,


∆1 =

∣∣∣∣∣∣∣∣∣∣∣∣

ψ(N) ... ψ φ(N−1) ... φψ(N)

1... ψ

1φ(N−1)

1... φ

1

.

.

.

ψ(N)2N ... ψ2N φ

(N−1)2N ... φ2N

∣∣∣∣∣∣∣∣∣∣∣∣

,

∆2 =

∣∣∣∣∣∣∣∣∣∣∣∣

φ(N) ψ(N−1) ... ψ φ(N−1) ... φ

φ(N)1

ψ(N−1)1

... ψ1

φ(N−1)1

... φ1

.

.

.

φ(N)2N ψ

(N−12N ) ... ψ2N φ

(N−1)2N ... φ2N

∣∣∣∣∣∣∣∣∣∣∣∣

.

Proof. We constructN 2×2 matrix functitons χk = (Φ2k−1,Φ2k), 1 ≤ k ≤ N :

χk =

(ψ2k−1 ψ2k

φ2k−1 φ2k

).

These functions satisfy the matrix equation (12.5) with J = σ3. After N–times DT(12.11), we get χ[N ] and U [N ] which satisfy (12.5). Let us write χ[N ] as a series:

(12.15) χ[N ] = χ(N) −N∑

i=1

Ai(t, x)χ(N−i), Ai =

(ai bici di

).

Substituting (12.15) into (12.13) we get (here CkN = N !k! (N−k)! ):

N∑

k=0

CkNU(k)χ(N−k) +

N∑

k=1

[(Ak,t − σ3Ak,x)χ(N−k) − 2σ3(Ak)F

χ(N−k+1)]+

+

N∑

k=1

N−k∑

i=0

CiNAkU(i)χ(N−k−i) − U [N ](χ(N) +

N∑

k=1

Akχ(N−k)) = 0,

where (Ak)Fis the off–diagonal part of Ak. Therefore

u[N ] = u+ 2b1, v[N ] = v − 2c1.

To compute b1 and c1, we take into account that χk[N ] = 0 if k ≤ N , therefore weget a system of 2N equations as follows:

ψ(N)i =

N∑

n=1

(anψ(N−n)i + bnφ

(N−n)i ), φ

(N)i =

N∑

n=1

(cnψ(N−n)i + dnφ

(N−n)i ),

for i = 1, ..., N. Using Kramer’s formulae, we get (12.14).

In (12.10), if µ = i(β∗ − cα∗)−1, then ν2 and λ2 are real. The constants λ1

and ν1 may be annihilated by the standard gauge U(1) transformations. Using thefreedom in our choice of parameters, let us assume that v(t, x) = κu(t, x), κ = ±1.Now we can supplement (12.14) with the transformations law for χ

1(N = 1):

ψ1 [1] =φ

2

∆, ψ2 [1] =

κφ1

∆, φ1 [1] =

κψ2

∆,

(12.16) φ2[1] =

ψ1

∆, ∆ =

∣∣∣∣ψ

1ψ

2

φ1

φ2

∣∣∣∣ .

12.2. CRUM LAWS 103

It is necessary to require that after N–times DT, the following reduction restrictionwill be true:

(12.17) v[N ] = κu[N ].

In the general case, we do not have an algorithm allowing us to keep (12.17) in allthe steps of DT. However, it is possible by the introduction of the so called binaryDT which allows one to preserve the reduction restriction (12.17) [57].

Let us consider a closed 1–form

dΩ = dx ζχ+ dt ζσ3χ, Ω ≡∫dΩ

where a 2 × 2 matrix function ζ solves the equation:

(12.18) ζt = ζxσ3 − ζU.

We shall apply the DT of (12.5). We can verify by immediate substitution that(12.18) is covariant with respect to the transform if

ζ[+1] = Ω(ζ, χ)χ−1.

Now we can transform U as

(12.19) U [+1,−1] = U + [σ3, χΩ−1ζ].

It can be shown that

χ[+N,−N ] = χ−N∑

k=1

θkΩ(ζk, χ), ζ[+N,−N ] = ζ −N∑

k=1

Ω(ζ, χk)sk,

where θk and sk can be found from the following equations:

N∑

k=1

θkΩ(ζk, χi) = χi,

N∑

k=1

Ω(ζi, χk)sk = ζi.

The transformation:

U [+N,−N ] = U +

N∑

i,k=1

[σ3, θiΩ(ζk, χi)sk]

is the forementioned binary DT.Let v = κu∗, then U [+N,−N ] will satisfy the reduction restriction if we choose

ζk and χk such that:

ζk = χkR, R = diag(1,−κ).The solution that follows from (12.19) has the form

u[+1,−1] = u+2κ(ψ

2φ∗

1θ∗12 + ψ

1φ∗

2θ12 − ψ

1φ∗

1θ22 − ψ

2φ∗

2θ11)

θ11θ22− | θ12 |2 ,

θik =

∫dx (ψ∗

i ψk − κφ∗iφk) + dt (ψ∗i ψk + κφ∗i φk).

Note that the square of the absolute value of u[+1,−1] can be expressed by theconcise formula:

| u[+1,−1] |2=| u |2 −κ( ∂2

∂t2− ∂2

∂x2) ln(θ11θ22− | θ12 |2).


DT allows us to construct a rich set of exact solutions of the Dirac equation with(1+1) potentials. Finally, we consider fields decreasing as t, x → ±∞ (adiabaticengaging and turning-off). Let u = v = 0. The particular solutions of (12.13) are:

ψk = Ak eω(t−x) + Bk e

ω(x−t), φk = Ck eλ(t+x) +Dk e

−λ(t+x),

where k = 1, 2; A, B, C, D, ω and λ are real constants. After DT (12.11) undercondition (12.10), we get:

(12.20) Q(t, x) =1

µ1 cosh(at+ bx+ δ1) + µ2 cosh(bt+ ax+ δ2),

u(t, x) = ξ Q(t, x), v(t, x) = −1 + c2

ξ2u(t, x),

ξ =| αc− β |2= 4ω(A1B2 −A2B1), a = ω + λ, b = λ− ω, µ(β∗ − cα∗) = i,

1 + c2 = 4λξ(C2D1 − C1D2), ω(A1B2 −A2B1) = λ(D1C2 −D2C1),

µ1 =√

(A1C2 −A2C1)(B1D2 −B2D1), µ2 =√

(B1C2 −B2C1)(A1D2 −A2D1),

δ1 =1

2lnA1C2 −A2C1

B1D2 −B2D1, δ2 =

1

2lnB1C2 −B2C1

A1D2 −A2D1,

where α and β satisfy (12.9). The potential (12.20) is a localized impulse whichdecreases as t→ ±∞. A particular solution of the Dirac equation may be calculatedfrom (ψ1,2[1],φ1,2[1])T , using (12.4) and (12.6), where ψ1,2[1] and φ1,2[1] are definedin (12.16). This bispinor describes the quasi-stationary state of a fermion. Thefermion is free along (y, z)-axes and restrained along x-axis.

CHAPTER 13

Moutard Transformations for the 2D and 3D

Schrodinger Equations

Darboux transformations (DT) for 1D Schrodinger equation is well-known. Inthis chapter, we will investigate DT for 2D and 3D Schrodinger equations. Forinstance, the Novikov-Veselov equation (NVE) can be represented as a compatibilitycondition of the linear system

(13.1) Hψ = −ψxx − ψyy + Uψ = 0, ψt = Aψ,

where ψ = ψ(x, y, t), U = U(x, y, t), and A is a linear operator. The system (13.1)is covariant with respect to the action of Darboux type transformation which shouldbe more properly called a Moutard transformation (MT) since both the prop andthe dressed eigenfunctions are at the same spectral value (here it is zero). The MTmay not be as powerfull as DT in the quantum theory (i.e. we can not use it tofind spectrum), but it is very useful for constructing exact solutions of the NVE.

13.1. A 2D Moutard Transformation

Let’s consider the 2D Schrodinger (13.1) Hψ = 0 with U = U(x, y), ψ =

ψ(x, y). Let ϕ = ϕ(x, y) be another solution: Hϕ = 0 such that ψ 6= ϕ. We defineoperators qi = ∂i − ∂i logϕ and q+i = −∂i − ∂i logϕ, with i = 1, 2. Then

H = q+mqm = −∆(2) + U,

and summation is implied over repeated indices. Let’s define the “dressed” Hamil-

tonian ˆHˆH = qmq

+m = −∆(2) + U ,

where

(13.2) U = U − 2∆(2) logϕ.

Then one needs to find such ψ that

(13.3) ˆHψ = −∆(2)ψ + U ψ = 0.

A solution is known:

(13.4) ψ =θ

ϕ,

where θ =∫dθ,

(13.5) dθ = dxmεmkϕ∂kψ = dx1Ω1 + dx2Ω2.

105

10613. MOUTARD TRANSFORMATIONS FOR THE 2D AND 3D SCHRODINGER EQUATIONS

The map U → U , ψ → ψ is the MT. Another way to see this MT is as follows.Notice that ψ satisfies

(13.6) ˆHψ = qmq+mψ = 0,

Notice also that

(13.7) εmk qmqk = 0 =⇒ εmkqmqkψ = 0.

Thus

qmq+mψ = εmk qmqkψ =⇒ qm(q+mψ − εmkqkψ) = 0 .

One can chooses

q+mψ = εmkqkψ.

This equation can be write in another way:

(13.8) ∂m(ψϕ) = εkm(ϕ∂kψ − ψ∂kϕ).

Equation (13.8) is true only if the curl of the right is zero. Then one gets

(13.9) (εkm∂l − εkl∂m)[ϕqkψ] = 0 .

It’s easy to show that the last equation is true if and only if, ψ and ϕ are solutions

of the equations Hψ = 0 and Hϕ = 0 correspondingly. If we integrate equation(13.8), then we get the MT (13.4) and (13.5).

13.2. A 3D Moutard Transformation

The Hamiltonian in 3D has the form:

(13.10) H = −∆(3) + U(x, y, z) = −∂21 − ∂2

2 − ∂23 + U(x1, x2, x3)

As before, one can write

(13.11) Hψ = 0

with

H = q+mqm, qm ≡ ∂m − ∂m lnϕ,

where Hϕ = 0, m = 1, 2, 3. The dressed Hamiltonian will be

ˆH ≡ qmq+m = −∆(3) + U = −∆(3) + U − 2∆(3) lnϕ.

Now one needs to find such ψ that

ˆHψ = 0.

Using the simple method above for 2D, one gets instead of (13.8)

(13.12) q+mψ = εmklqk q+l ψ,

and instead of (13.9)

(13.13) (εlkm∂s − εskm∂l)[ϕqk q+mψ] = 0

Unfortunately, in contrast to (13.9), equation (13.13) is not true if ψ and ϕ aresolution of the (13.11). This is the principal problem in getting MT in 3D (andD > 3 as well). One can obtain MT in 3D by introducing an external magnetic field

with magnetic induction ~B = rot~κ, where ~κ is the vectorial potential. Of course,this field is a solution of the Maxwell equation

(13.14) div ~B = 0,

13.2. A 3D MOUTARD TRANSFORMATION 107

The introduction of the magnetic field can be described by the usual rule

∂m → Dm = ∂m + iκm.

Define the superpotential χ = χ(x, y, z) ≡ − logϕ. Then one gets the followingHamiltonians:

H = Q+mQm,

ˆH = QmQ+m,

where:

(13.15)Qm = ∂m + θm, Q+

m = −∂m + θ∗m,θm = iκm + ∂mχ, θ∗m = −iκm + ∂mχ,

It is easy to show that

εmklqmqkq+l = i( ~B,−~∇− i~κ+ gradχ).

Thus, it is identically zero if

(13.16) ~B = rot~κ = ~0.

Below we will consider the Aaronov-Bohm magnetic field: ~B = 0, ~κ 6= 0. We have

(13.17) H = −∆(3) − idiv~κ− 2i(~κ, ~∇) + ~κ2 +∆ϕ

ϕ,

and

(13.18) ˆH = −∆ − idiv~κ− 2i(~κ, ~∇) + ~κ2 − ∆ϕ

ϕ+ 2

(~∇ϕϕ

)2

.

Notice also that H = −∆(3) + U and ˆH = −∆(3) + U . We have

(13.19) U = U − 2∆(3) logϕ.

When ~B = ~0, we get from (13.12)

Q+mψ = εmklQkQ

+l ψ,

By analogy with the 2D case, we shall find ψ as

(13.20) ψ =f

ϕ= eχf,

where f satisfies

(13.21) ~∇f + i~κf = 2~∇ψ ∧ ~∇ϕ+ 2i~κ ∧ ~∇ϕψ.Define p as

(13.22) ~κ = −i~∇p,then (13.21) has the form

(13.23) ~∇f + f ~∇p = 2(~∇ψ + ψ~∇p

)∧ ~∇ϕ.

Solving this equation, one gets

(13.24) ψ =f

ϕ=e−p

ϕ

(2

∫ ~r

~r0

(~∇(epψ), ~∇ϕ, d~r

)+A

),

10813. MOUTARD TRANSFORMATIONS FOR THE 2D AND 3D SCHRODINGER EQUATIONS

where A is an arbitrary constant. The integrand has to be a gradient, therefore,we get the additional requirement on p (and on κ as well):

(13.25) ~∇ ∧(~∇(epψ) ∧ ~∇ϕ

)= ~0.

Direct substitutin shows that such ψ is a solution to the equation ˆHψ = 0, withˆH given by (13.18). The transformations (13.19) and (13.24) are the 3D Moutardtransformations.

CHAPTER 14

BLP Equation

Like Davey-Stewartson (DS) and Kadomtsev-Petviashvili (KP) equations, theBoiti-Leon-Pempinelli (BLP) equation is another (1+2)-dimensional integrable sys-tem [20] [37]. A somewhat peculiar feature of the BLP equation is that a reductionis the viscous Burgers equation. Since BLP equation has to be a Hamiltonian sys-tem and viscous Burgers equation is a dissipative system, it seems to be paradoxical.The answer to the puzzle is that the Hamiltonian of the BLP equation reduces to0 by this reduction.

14.1. The Darboux Transformation of the BLP Equation

The BLP equation can be written in the form

aty + (a2 − ax)xy + 2bxx = 0,

(14.1)

bt + (bx + 2ab)x = 0,

where a = a(t, x, y) and b = b(t, x, y) are real-valued functions of three variables.One can rewrite (14.1) in the following form with the change of variable b = py,

at + 2aax + (2p− a)xx = 0,

(14.2)

pyt + (pyx + 2apy)x = 0.

The reduction p = 0 reduces the BLP equation (14.2) to the viscous Burgersequation

(14.3) at + 2aax − axx = 0.

Another reduction p = a reduces the BLP equation (14.2) to the ”anti-Burgersequation”.

The Lax pair of the BLP equation is

(14.4) ψxy + aψy + pyψ = 0, ψt = ψxx + 2pxψ.

Theorem 14.1. Let φ and ψ be two solutions of (14.4). Define two functions

τ = ∂x(lnφ) and ρ = (∂y(lnφ))−1

. The Lax pair (14.4) is covariant with respect tothe two types of Darboux transformations,

(14.5) ψ → ψ[1] = ρψy − ψ, a→ a[1] = a− ∂x ln ρ, p→ p[1] = p− ρpy,

and

(14.6) ψ → ψ[1] = ψx − τψ, a→ a[1] = a− ∂x ln(a+ τ), p→ p[1] = p+ τ.

109

110 14. BLP EQUATION

In addition, it is also covariant with respect to the Laplace transformation,

(14.7) ψ → ψ[1] =ψypy, a→ a[1] = a+ ∂x ln py, p→ p[1] = p+ a+ ∂x ln py,

and the Laplace inverse transformation is given by

(14.8) ψ → ψ[−1] = ∂xψ+aψ, a→ a[−1] = a−∂x ln (p− a)y , p→ p[−1] = p−a.Let us define three functions u, v, χ by,

a = −∂x lnu, b = py = −uv, χ =∂xψ

u

and cone variables ξ and η,

∂y = ∂η − ∂ξ, ∂x = ∂η + ∂ξ,

then the spatial part of the Lax pair (14.4) can be rewritten in the Zakharov-Shabatform

(14.9) ∂ηΦ = σ3∂ξΦ + UΦ,

where

Φ =

(ψχ

), U =

(0 uv 0

), σ3 =

(1 00 −1

).

Equation (14.9) is the spectral problem for the DS equations. The above Laplacetransformation then produces an explicitly invertible Backlund autotransformationfor the DS equations, which can be used to construct solutions to the DS equationsthat fall off in all directions in the plane according to exponential and algebraiclaw.

14.2. Crum Law

Consider the DT (14.5) and define QNψ = ∂N+1

y ∂xψ. It is easy to check thatif ψ is a solution of (14.4), then Q

Ncan be written as

(14.10) QN

= −a∂N+1y − (p+Na)y∂

Ny +

N−1∑

k=1

cN,k∂ky − ∂N+1

y p,

where cN,k

= cN,k

(a, py; ax, pxy; ay, pyy; ...). Let ψ, ψ1, ... ψN be N+1 particular

solutions of (14.4). After N times of DT (14.5), we get

(14.11) ψ[N ] = −ψ +

N∑

k=1

αN,k∂kyψ.

Substituting (14.11) into

(14.12) ψxy[N ] + a[N ]ψy[N ] + py[N ]ψ[N ] = 0

and using (14.10), we get

a[N ] = a− ∂x lnαN,N

, p[N ] = p−N∑

k=1

αN,k∂kyp.

To compute αN,k

, we take into account that

N∑

k=1

αN,k∂kyψj = ψj , j = 1, ..., N.

14.2. CRUM LAW 111

Thus

(14.13)

ψ[N ] =D

N(1 | 1)

DN

(1, N + 2 | 1, 2), a[N ] = a− ∂x ln

DN

(1, 2 | 1, 2)

DN

(1, N + 2 | 1, 2),

p[N ] = p+ (−1)ND

N(1 | 2)

DN

(1, N + 2 | 1, 2),

where DN

(i, j | k,m) are determinants which can be obtained by the deletion ofcolumns i, j and rows k, m from the (N + 2) × (N + 2) determinant D

N

(14.14) DN

=

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

∂N+1p ∂Np ∂N−1p ... ∂p 0∂N+1ψ ∂Nψ ∂N−1ψ ... ∂ψ ψ∂N+1ψ1 ∂Nψ1 ∂N−1ψ1 ... ∂ψ1 ψ1

. .

. .

. .∂N+1ψ

N∂Nψ

N∂N−1ψ

N... ∂ψ

Nψ

N

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

,

where ∂ = ∂y.Next we consider the Crum law for the DT (14.6). Define P

Nψ = ∂N+1

x ∂yψ,then

(14.15) PN

= −py∂Nx +N−1∑

k=0

βN,k∂kx + λ

N∂y.

Since PN+1

= ∂xPN,

(14.16)β

N+1,N= β

N,N−1− pxy, β

N+1,k= ∂xβN,k

+ βN,k−1

, 1 ≤ k ≤ N − 1,

βN+1,0

= ∂xβN,0− pyλN

, λN+1

= ∂xλN− aλ

N.

After N times of DT (14.6), we get

(14.17) ψ[N ] = ∂Nx ψ +N−1∑

k=0

αN,N−k

∂kxψ.

Substituting (14.17) into (14.12) and taking (14.15)-(14.16) into account, we get

(14.18)a[N ]py[N ] = apy + ∂y

(∂x(αN,1

−Np) + αN,2

− 12α

2N,1

),

p[N ] = p− αN,1.

To compute αN,k

, we notice that (j = 1, ..., N)

(14.19) ∂Nx ψj +N−1∑

k=0

αN,N−k

∂kxψj = 0,

therefore

(14.20) ψ[N ] =W

N

WN

(1 | 1), α

N,1= −WN

(2 | 1)

WN

(1 | 1), α

N,2=W

N(3 | 1)

WN

(1 | 1),


where

(14.21) WN

=

∣∣∣∣∣∣∣∣∣∣∣∣

∂Nψ ∂N−1ψ ... ψ∂Nψ1 ∂N−1ψ1 ... ψ1

. .

. .

. .∂Nψ

N∂N−1ψ

N... ψ

N

∣∣∣∣∣∣∣∣∣∣∣∣

and ∂ = ∂x.Starting from (14.7) and (14.8), we can also find the N -fold LT a[N ] (a[−N ])

and b[N ] (b[−N ]). (it is more convenient to use field b instead of field p in thiscase.) After N times of LT (14.7), we get

(14.22) ψ[N ] =

N∑

k=1

αN,k∂kyψ.

For simplicity, we denote a[N ] by aN and b[N ] by bN . Substituting (14.22) into(14.12), we get

(14.23) aN = a− ∂x lnαN,N

, bN = b+ ∂y(Na− ∂x ln(α

N−1,N−1α

N,N))

with the supplementary condition

(14.24)

N∑

k=1

αN,k∂k−1y b = 0.

Thus

(14.25) αN,N

=

N∏

j=1

bj−1

−1

,

and (14.23) is the nonlinear superposition formula for the BLP equations. Tocompute ψ[N ] we must obtain the rest coefficients α

N,k. We need to introduce the

operators

fj =1

bj∂y, F = f

N−1f

N−2...f ,

where j = 0, ..., N − 1, b0 = b, f0 = f . It’s clear that

(14.26) ψ[N ] =

N∑

k=1

αN,k∂kyψ = Fψ.

The coefficients αN,k

from (14.26) can be calculated by finding N functions θj(j = 1, ..., N) such that Fθj = 0. It is easy to obtain these functions in thefollowing way: For an arbitrary index j we construct the sequence of equations for

the j functions θ(k)j (k = 1, ..., j),

(14.27) fk−1θ(k−1)j = θ

(k)j , θ

(0)j = θj , θ

(j−1)j = 1,

The system (14.27) can be solved and we get N functions θj ,

θj =< b < b1 < ... < bj−1 > ... >y,

where and in the following < ... >x,y denotes the integration in the x, y-variables

< S >x=1

2

∫dz sgn(x− z)S(z, y, t), < S >y=

1

2

∫dz sgn(y − z)S(x, z, t).

14.3. EXACT SOLUTIONS 113

Using (14.24) to express αN,1

, then

(14.28) Fθj =

N∑

k=2

MkjαN,k= 0, Mkj = ∂ky θj −

∂k−1y b

b∂yθj .

It is obvious that Mk1 = Mk2 = 0. Substituting (14.25) into (14.28), we get thenonhomogeneous system of linear equations for the desired coefficients, which canbe solved by the Kramer’s formulae.

The N -fold LT (14.8) can be obtained in analogy with the LT (14.7). It is easyto see that ψ[−N ] has the form (14.17) with new coefficients α

N,k. This means

that b[−N ] and a[−N ] can be computed by (14.18). Then we need to find the newcoefficients α

N,k. These coefficients may be expressed in terms of the determinants

(14.20)-(14.21) with the change ψk → θk where the functions (θk) satisfy equation(14.19). To find θk, we rewrite (14.19) as

(14.29)(∂x + a

N−1

) (∂x + a

N−2

)... (∂x + a) θj = 0, j = 1, ..., N

where we denote a[−k] by ak and b[−k] by bk. Then (14.29) can be rewritten as,

(14.30)(∂x + aj) θ

(j)m = θ(j+1)

m , j = 0, ...,m− 2,

(∂x + am−1) θ(m−1)m = 0,

for any θm. In (14.30), θ(0)m = θm, m = 2, ..., N , and

θ1 = exp (− < a >x) .

By solving (14.30), we get the required functions

θm = θ1 Φm, Φm =< b1 < b2 < ... < bm−1 > ... >x .

14.3. Exact Solutions

In [37], Backlund transformations were applied to construct singular solutionsthat decay according to rational laws in all directions on the plane with a → 0and b → constant as x2 + y2 → ∞. Here We shall consider two examples: (i).the localized nonsingular solution b decaying according to the exponential law as afunction of x and according to the rational law as a function of y; (ii). the “blow-up”solution which is nonsingular when t > 0 and singular when t < 0.

Let a = b = 0. To construct the solution (i), we choose the solution of (14.4)in the form

φ = exp(µ2t)cosh(µx) +B(y),

where B = B(y) is an arbitrary differentiable function. Taking (14.6) into account,we get

b[1] = − µB′ sinh(µx) exp(µ2t)

(B + cosh(µx) exp(µ2t))2,

a[1] = − µ(exp(µ2t)

+B cosh(µx))

sinh(µx) (B + cosh(µx) exp(µ2t)).(14.31)

If we choose the function B(y) to be increasing according to the exponential law asy → ±∞, then the solution (14.31) has nonclosed level curves. To avoid this, wechoose B(y) to be an even polynomial (the property of being even guarantees the


nonsinular behavior of (14.31)). Let B(y) = ν2y2N +λ2 (ν and λ are constants andℑν = ℑλ = 0), we get the desired solution

(14.32) b[1] = − 2µNν2y2N−1 sinh(µx) exp(µ2t)

(λ2 + ν2y2N + cosh(µx) exp(µ2t))2 .

The regular solution (14.32) decays according to the exponential law as x → ±∞,and the rational law (1/y2N+1) as y → ±∞. On the other hand, the singularsolution a[1] has level curves along y-lines.

To construct blow-up solutions of BLP, we shall apply two DTs, (14.5) and(14.6), on the vacuum background (a = b = 0). As a result, we get

(14.33) b[2] =W2(1 | 1)D2(1, 4 | 1, 2)

D22(1, 1 | 1, 1)

.

The functions ψk (k = 1, 2) in the determinants D and W [see (14.14) and (14.21)]are solutions of (14.4). We choose them to be

ψ1 = C1 exp(λ2t)cosh(λx) + cosh(αy), ψ2 = C2 exp

(−µ2t

)sin(µx) + sinh(βy).

Then

D2(1, 1 | 1, 1) = D1(y) +D2(x, y, t)

where

D1(y) = β cosh(βy) cosh(αy) − α sinh(βy) sinh(αy),D2(x, y, t) = βC1 exp

(λ2t)cosh(λx) cosh(βy) − αC2 exp

(−µ2t

)sinh(αy) sin(µx).

If β > α > 0, then D1(y) > 0 for y ∈ (−∞,+∞). If C1 > 0, then D2(x, 0, t) >0. So the solution (14.33) is nonsingular at y = 0. A singularity can appear ifD2(x, y, t) < 0. It is possible that D2(x

′, y′, t′) = 0, i.e.

(14.34) ρ exp((λ2 + µ2)t′)

)cosh(βy′) cosh(λx′) = sinh(αy′) sin(µx′),

when ρ ≡ βC1(αC2)−1. We can rewrite (14.34) as,

(14.35) ρ exp((λ2 + µ2)t′)

)| θ1(y′) |=| θ2(x′) |,

with

θ1(y) =cosh(βy)

sinh(αy), θ2(x) =

sin(µx)

cosh(λx).

Let y0 and x0 be solutions of the equations

tanh(βy0) tanh(αy0) =α

β, tanh(λx0) tan(µx0) =

µ

λ.

Since exp((λ2 + µ2)t′)

)≥ 1 when t′ ≥ 0, for such values of t we can to choose

(14.36) ρ >θ2(x0)

θ1(y0).

Then the condition (14.36) guarantees the nonsingular behavior of (14.33) whent ≥ 0. It is possible to choose parameters such that (14.33) is singular when t < 0.We choose ψ2 as

ψ2 = C2 exp(µ2t)sinh(µx) + sinh(βy).

The solution (14.33) will be regular when t ≥ 0, and singular when t < 0, if λ > µand

ρ >θ2(x0)

θ1(y0), θ2(x) =

sinh(µx)

cosh(λx),

14.4. DRESSING FROM BURGERS EQUATION 115

where x0 > 0 is maximum point of the function θ2(x).

14.4. Dressing From Burgers Equation

Starting from an arbitrary solution of the Burgers equation, one can constructexact solutions of (14.2) via LT (14.8). It is convenient to redefine variables in thefollowing way

x =ξ√ν, y =

η

ν√ν, a =

A(ξ, η, t)

2√ν

, p =P (ξ, η, t)

ν√ν

,

where ν > 0 and µ are arbitrary constants. As a result, the BLP equations takethe form

(14.37)At +AAξ − νAξξ = −4Pξξ,Ptη + (APη + νPξη)ξ = 0,

where ν is the parameter that may be called the coefficient of viscosity. Assumethat P = 0 and A is a solution of the Burgers equation, then (14.8) allows us toconstruct solutions of the BLP equations (14.37),

(14.38) P [−1] = −ν2A, A[−1] = A− 2ν∂ξ ln (Aη) .

Let us consider a simple example: The shock wave solution of the Burgers equation[105],

A = A1 +A2 −A1

1 + expA2−A1

2ν (ξ − Ut) ,

where

A1 = A1(η), A2 = A2(η), U =A1 +A2

2.

Assuming that A1 = 0 and taking (14.38) into account, we get

A[−1] =A2

[G2A2(ξ −A2t) − 2ν(2G2 + 3G+ 1)

]

(1 +G) [GA2(ξ −A2t) − 2ν(1 +G)],

P [−1] = − νA2

2(1 +G), G = exp

A2(2ξ −A2t)

4ν

.(14.39)

It is interesting to consider the behavior of (14.39) as ν → 0. Assume that A2 > 0for all values of η. Then for 2ξ 6= A2t, we get

P [−1] → 0, A[−1] → A2,

and for 2ξ = A2t,

P [−1] → 0, A[−1] → A2

2.

Using other well known solutions of the Burgers equation (see [105]), it iseasy to construct a rich set of exact solutions for the BLP equations via the abovedressing procedure. For example, we consider the solution of the Burgers equation,

(14.40) A = −2ν∂ξ ln f, f = α+

√β

texp

(− ξ2

4νt

), P = 0,

where α and β are some arbitrary constants. This is the so called N-waves solutionof the dissipative Burgers equation [105]. Substituting (14.40) into (14.38), we get

A[−1] = −2ν

ξ+αξ

tf, P [−1] = −νξ(f − α)

2tf.


It easy to see that if t→ +∞ then

P [−1] → 0, A[−1] → AB = −2ν

ξ+ξ

t,

and AB is the solution of the Burgers equation.

CHAPTER 15

Goursat Equation

The Goursat equation (GE) has the form [39]:

(15.1) ζxy = 2√λ ζxζy,

where ζ = ζ(x, y), λ = λ(x, y). We call λ a potential function. This equation canbe linearized by the substitution: ψ =

√ζx, and χ =

√ζy. We have

ψy =√λχ, χx =

√λψ

or

(15.2) ψxy =1

2(lnλ)xψy + λψ,

and a similar equation for χ. The equation (15.2) is a particular case of the followingequation studied in the previous chapter,

(15.3) ψxy + aψy + bψ = 0,

which has the Laplace transformations (LT)

(15.4) a→ a−1 = a−∂x ln(b−ay), b→ b−1 = b−ay, ψ → ψ−1 = ψx+aψ,

(15.5) a→ a1 = a+ ∂x ln b, b→ b1 = b+ ∂y (a+ ∂x ln b) , ψ → ψ1 =ψyb

;

and the Darboux transformations (DT)

(15.6) a→ a1 = a− ∂x ln(a+ τ), b→ b1 = b+ τy , ψ → ψ1 = ψx − τψ,

(15.7) a→1a = −(τ + bρ), b→

1b = b− (bρ)y, ψ →

1ψ = ρψy − ψ.

where τ = φx/φ, ρ = φ/φy, ψ and φ are particular solutions of (15.3).The aim of this chapter is to study the validity of LT and DT for the GE. It is

clear that after single DT or LT, the reduction restriction

(15.8) a = −∂x ln b,

will be true only for a special class of potentials. GE has two major applications:(1). Let x be a complex variable, y = −x,

√λ is a real-valued function, and ψ

and χ in (15.3) are complex-valued functions. Then one defines three real-valued

117

118 15. GOURSAT EQUATION

functions Xi, i = 1, 2, 3 which are the coordinates of a surface in R3 [52]:

(15.9)

X1 + iX2 = 2i

∫

Γ

(ψ2dy′ − χ2dx′

),

X1 − iX2 = −2i

∫

Γ

(ψ2dy′ − χ2dx′

),

X3 = −2

∫

Γ

(ψχdy′ + χψdx′

),

where Γ is an arbitrary path of integration in the complex plane. The correspondingfirst fundamental form, the Gaussian curvature K and the mean curvature H yield:

ds2 = 4U2dxdy, K =1

U2∂x∂y lnU, H =

√λ

U,

where

U =| ψ |2 + | χ |2,and any analytic surface in R3 can be globally represented by (15.9) (see [54]).

(2). The 2D-MKdV equation has the form

(15.10)4λ2(λt −Aλx +Bλy − λ3x − λ3y) + 4λ3 [(2λ+B)y + (2λ−A)x] ++6λ(λyλyy + λxλxx) − 3(λ3

x + λ3y) = 0,

Bx = 3λy − λx, Ay = λy − 3λx;

where λ = λ(x, y, t), A = A(x, y, t), B = B(x, y, t). If we introduce the function

u =√λ, then we can rewrite (15.10) in the more customary form (see [21]):

(15.11)ut + 2u2(ux + uy) + 1

2 (By − Ax)u+Buy −Aux − u3y − u3x = 0Bx = (3∂y − ∂x)u

2, Ay = (∂y − 3∂x)u2.

The reduction condition

A = −B = −2u2, uy = ux,

lead to the MKdV equation,

ut + 12u2ux − 2u3x.

The 2D-MKdV equation (15.10) has a Lax pair which is given by (15.2) and

ψt = ψ3x + ψ3y −3

2

λyλψyy +

[3

4

(λyλ

)2

− λ−B

]ψy + (A− λ)ψx +

1

2(Ax − λx)ψ.

15.1. The Reduction Restriction

The reduction restriction (15.8) is valid only for a special type of potentials.The invariance of the reduction restriction under the Laplace transformations (15.4)is guaranteed by

(15.12) λ−1 = λ− 1

2∂x∂y lnλ =

C

2λ,

which is the well-known sinh-Gordon equation

(15.13) ∂x∂y lnλ = 2λ− C

λ,

15.1. THE REDUCTION RESTRICTION 119

where C is a constant, and the new potential λ−1 is a solution of (15.13) too. Inthis case, the GE may be integrated and

λ =f ′g′

(f + g)2, ζ = − 1

C2∂y ln(f + g) + V,

where f = f(x) and g = g(y) are arbitrary differentiable functions, and V = V (y)is the function such that

V ′ =

[1

2C(ln g′)

′]2

=1

4C2

(g′′

g′

)2

,

and

ψ =

√f ′g′

C(f + g), χ =

1

2C∂y ln

(−∂y

1

f + g

).

Next we consider the DT (15.6). Inserting both transforms into the reductioncondition (15.8) yields

(15.14) λ1 = λ− τy = λ

(τ − λx

2λ

).

Using the new notations α = lnφ, Λ = lnλ, and noticing that

λ− τy =

(−1

2Λx + αx

)αy,

and τ = αx, one gets from the transform (15.14) the condition for Λ:

(15.15)

(αx −

1

2Λx

)[αy − exp(Λ)

(αx −

1

2Λx

)]= 0.

Setting the first parentheses to zero yields:

Λxy = 2 exp(Λ),

and α = Λ/2− c(y), where c(y) is arbitrary function. Setting the square brakets in(15.15) to zero yields: relevant equation

(15.16) (exp(−2α)λ)x = (exp(−2α))y.

Thus

θx = ψ2 =1

Fx + C2, λ =

Fy + C1

Fx + C2,

where F = F (x, y) is any differentiable function, and C1,2 are constants. Substi-tuting (15.16) into (15.2), we get

2(C2 + Fx)C21 + [(Fyxx + 4Fy)C2 + FxFyxx + 4FyFx − FxxFyx]C1

+

(FyxxFy −

1

2F 2yx + 2F 2

y

)C2 + 2F 2

yFx −1

2F 2yxFx

−FyFxxFyx + FxFyFyxx = 0.(15.17)

Introducing the new variables

Fx = P − C2, Fy = Q− C1,

then (15.17) can be splitted into the system

(15.18) 2QxQPx − (2QxxQ−Q2x + 4Q2)P = 0, Py = Qx.


After an integration of the first equation, we get

P =CQx√Q

exp(G), Gx = 2Q

Qx,

where C is the third constant of integration. Let

Q = n2, G = lnm,

where m = m(x, y) and n = n(x, y). The reduction equation then takes the simpleform

(15.19)(n2)x

= 2C (mnx)y , mxnx = mn.

This system can be rewritten in a more convenient form. Let

nx = n exp(S), mx = m exp(−S),

where S = S(x, y). After substituting into (15.19), we get

Sy =1

C

n

m− ∂y ln(mn),

therefore

(15.20) Sxy = 4 sinhS ∂y∂−1x coshS,

which is a 2D generalization of the sinh-Gordon equation. Equation (15.20) hasthe Lax pair:

Kψ = 0, K1Dψ = 0

where

K = ∂x∂y −1

2

λxλ∂y − λ, K1 = ∂x∂y −

1

2

λ1,x

λ1∂y − λ1, D = ∂x − τ,

the variables λ and λ1 are defined by

(15.21) λ =(Sx + 2 coshS)y

4 sinhSexp(−S), λ1 =

(Sx + 2 coshS)y4 sinhS

exp(S),

and

τy ≡ λ− λ1.

We can study the reduction for the DT (15.7) analogously. As a result we get

(15.22) λ = C1φy exp(F ), 1λ = −C1C2φ2

φyexp(F ),

where φ is the support function of the DT (15.7) and the reduction equation canbe written as

φxy = φy [Fx + 2C1φ exp(F )], Fyφy = C2φ.

15.2. BINARY DARBOUX TRANSFORMATION 121

15.2. Binary Darboux Transformation

In [36], Ganzha studied the one analogue of the Moutard transformation forthe Goursat equation, which is valid without a reduction restriction. Here we willpresent a binary Darboux transformation for the GE with the same property.

We introduce the new variables ξ and η,

∂y = ∂η − ∂ξ, ∂x = ∂η + ∂ξ,

and rewrite (15.2) in the matrix form

(15.23) Ψη = σ3Ψξ + UΨ,

where

(15.24) Ψ =

(ψ1 ψ2

χ1 χ2

), U =

√λσ1,

where ψk = ψk(ξ, η) and χk = χk(ξ, η) (k = 1, 2) are particular solutions of (15.3)for some λ(ξ, η), and σ1,3 are the Pauli matrices. Let Ψ1 be a solution of theequation (15.23), Ψ 6= Ψ1; we define a matrix function τ = Ψ1,ξΨ

−11. The equation

(15.23) is covariant with respect to the DT

(15.25) Φ[1] = Φξ − τΦ, U [1] = U + [σ3, τ ].

Remark 15.1. It is not difficult to check that the DT (15.25) is the superposi-tion formula for the two simpler Darboux transformations given by formulas (15.6)and (15.7). Eq. (15.23) is the spectral problem for the Davey-Stewartson (DS)equations.

Let us consider a closed 1-form

dΩ = dξΦΨ + dηΦσ3Ψ, Ω ≡∫dΩ,

where a 2 × 2 matrix function Φ solves the equation:

(15.26) Φη = Φξσ3 − ΦU.

One can verify that (15.26) is covariant with respect to the transform (15.25) if

Φ[+1] = Ω(Φ,Ψ1)Ψ−11 .

Now we can transform U by

U [+1,−1] = U + [σ3,Ψ1Ω−1Φ].

A particular solution of the equation (15.26) has the form

(15.27) Φ1 =

(s1ψ1 + s2ψ2 −s1χ1 − s2χ2

s3ψ1 + s4ψ2 −s3χ1 − s4χ2

),

where sk (k = 1, .., 4) are constants. It is convenient to choose

(15.28) Φ1 = ΨT1 σ3,

where ΨT1 is the transposed matrix Ψ1. In this case

(15.29) U [+1,−1] = U − 2AF,

where AF

is the off–diagonal part of the matrix A = Ψ1Ω−1ΨT

1 , Ω = Ω(Φ1,Ψ1),and

(15.30) ATF

= AF

= fσ1


where f = f(ξ, η) is an arbitrary function. Using (15.24), (15.29) and (15.30), wecan see that U [+1,−1] has the same form as the initial matrix U :

U [+1,−1] ≡(

0√λ[+1,−1]√

λ[+1,−1] 0

)=

(0

√λ− 2f√

λ− 2f 0

),

thus the reduction restriction is valid. The new function Φ[+1,−1] has the form

(15.31) Φ[+1,−1] = Φ − Ω(Φ,Ψ1) (Ω(Φ1,Ψ1))−1 Φ1,

where Φ is an arbitrary solution of the equation (15.26).

Theorem 15.2. Let

ψk,y =√λχk, χk,x =

√λψk,

αk,y = −√λβk, βk,x = −

√λαk,

where k = 1, 2; then the new functions:

α′1 = α1 −

A1ψ1 +A2ψ2

D, β′

1 = β1 +A1χ1 +A2χ2

D,

are solutions of the equations

α′1,y =

√λ′ β′

1, β′1,x =

√λ′ α′

1,

where√λ′ = −

√λ+

ψ1χ1Ω22 + ψ2χ2Ω11 − (ψ1χ2 + ψ2χ1)Ω12

D,

and

Ω11 =

∫dxψ2

1 + dyχ21, Ω12 = Ω21 =

∫dxψ1ψ2 + dyχ1χ2,

Ω22 =

∫dxψ2

2 + dyχ22, D = Ω11Ω22 − Ω2

12,

Λ11 =

∫dxα1ψ1 + dyβ1χ1, Λ12 =

∫dxα1ψ2 + dyβ1χ2,

Λ21 =

∫dxα2ψ1 + dyβ2χ1, Λ22 =

∫dxα2ψ2 + β2χ2.

A1 = Λ11Ω22 − Λ12Ω12, A2 = Λ12Ω11 − Λ11Ω12,

where∫

=∫Γ, and Γ is an arbitrary path of integration on the plane.

15.3. Moutard Transformation for 2D-MKdV Equation

Recall that the Lax pair of the 2D-MKdV equation (15.11) has the form,

ψxy =uxuψy + u2ψ,

ψt = ψ3x + ψ3y − 3uyuψyy +

[3(uyu

)2

− u2 −B

]ψy

+(A− u2

)ψx +

1

2

(A− u2

)xψ.(15.32)

15.3. MOUTARD TRANSFORMATION FOR 2D-MKDV EQUATION 123

Let φ be a solution of (15.32) (the support function), then we have a closed 1-form,

dθ = dx θ1 + dy θ2 + dt θ3, θ =

∫dθ,

where

θ1 = φ2, θ2 =

(φyu

)2

, θ3 = (A− u2)φ2 − φ2y − φ2

x + 2φφxx+

+(2φ3yφy − φ2

yy −Bφ2y)u

2 − 2uφy(uyφy)y + 3 (uyφy)2

u4.

We define the generalized Moutard transformation by

(15.33)

u→ u = u−√

(ln θ)x(ln θ)y, A→ A = A− (∂x∂y − 3∂2x) ln θ,

B → B = B + (∂x∂y − 3∂2y) ln θ, ψ → ψ =

φQ

θ,

where

Q ≡∫dQ, dQ = dxQ1 + dy Q2 + dtQ3,

and (w = ψ/φ),

Q1 = θwx, Q2 = −θ3(1/θ)xywy

θxy,

Q3 = θw3x + c1w3y + c2wxx + c3wyy + c4wx + c5wy,

with

c1 = − θxy2u2

+ θ, c2 =3

2θ(ln θx)x − θx, c4 =

(3φxxφ

+A− u2

)θ − θxx

2,

c3 =uyθxy2u3

+φφyyu2

− 3uyθ

u+ 3

(θ

2(ln θx)y − θy

), c5 = −

3u2yθxy

2u4+

+1

u3(θxyuyy + uyφφyy) +

1

u2

(3θu2

y − φφ3y +1

2

[B − φyy

φ

]θxy

)

+

(3φyyφ

−B

)θ + +

uyu

(2θy −

3θθxyθx

)+θxy2

− u2θ.

The 1-form dQ is closed,

Q1,y = Q2,x, Q1,t = Q3,x, Q2,t = Q3,y.

It is easy to verify that the Lax pair (15.32) is covariant with respect to the gener-alized Moutard transformation (15.33).

Now we use these transformations to construct exact solutions of the 2D-MKdVequation (15.11). Let u=const., A = B = 0. We will consider two examles.

(1). If we choose the solution of (15.32) as φ = sinh ξ, where

(15.34) ξ = ax+u2

ay +

(u2 − a2)(u4 − a4)

a3t,


with the real constant a =, then using (15.33) we get new solutions of the 2D-MKdVequation,

u =u[2η − a3 sinh(2ξ)

]

2η + a3 sinh(2ξ), A =

16a3 sinh ξ[3a5 sinh ξ − (u2 − 3a2)η cosh ξ

]

(2η + a3 sinh(2ξ))2 ,

B =16av2 cosh ξ

[3a3u2 cosh ξ − (3u2 − a2)η sinh ξ

]

(2η + a3 sinh(2ξ))2 ,

where

(15.35) η = a2(u2y − a2x) + (u2 − a2)(3u4 + 3a4 + 2a2u2)t.

(2). To construct the algebraic solutions of (15.11), we choose the solution of(15.32) as,

φ = (−1)n∫ β

α

dkζ(k) exp(ξ(k))dn

dknδ(k − k0),

with ξ(k) given by (15.34), a = a(k), β > k0 > α > 0, ζ(k) is an arbitrarydifferentiable function. Choosing n = 1, ζ = 1, we get

(15.36)

u =u(a6 − 2η2 − 2a3η)

2η2 + 2a3η + a6, A = −8a6(u2 + 3a2)η(η + a3)

(2η2 + 2a3η + a6)2,

B =8u2a4(3u2 + a2)η(η + a3)

(2η2 + 2a3η + a6)2,

with the η given by (15.35), and a = a(k0). (15.36) is the simple nonsingularalgebraic solution of the 2D-MKdV equation.

CHAPTER 16

Links Among Integrable Systems

An interesting, and rather philosophical, question is whether or not there is ahidden link among all integrable systems. This is obviously a very difficult questionwhich is still open. Partial progress was made via Darboux transformations. Forinstance, Korteweg-de Vries equation (KdV), sine-Gordon equation (SG), Calogero-Degasperis equation (CD), and higher order modified KdV are linked [22]. AlsoKdV, mnKdV (n = 1, 2, 3), and Krichever-Novikov equation exhaust all the inte-grable equations of the form ut + uxxx + f(uxx, ux, u) = 0 [22] [88].

16.1. Borisov-Zykov’s Method

Consider the KdV equation

(16.1) ut − 6uux + uxxx = 0,

with its Lax pair

(16.2) ψxx = (u− λ)ψ, ψt = 2(u+ 2λ)ψx − uxψ.

Setting τ = φx/φ, where φ is a solution of (16.2) at λ = µ, then

(16.3) τx = −τ2 + u− µ, τt = [2(u+ 2µ)τ − ux]x .

Eliminating u from (16.3), we get

(16.4) τt = 6τ2τx − τxxx + 6µτx.

If µ = 0, then we have the well-known mKdV equation. To construct the Lax pairfor mKdV, we use the DT

u1 = u− 2τx, ψ1 = ψx − τψ.

Setting σ = ψ1,x/ψ1, we get the x-chain

(16.5) (σ + τ)x = −σ2 + τ2 − λ+ µ.

and the t-chain,

(σ + τ)t =[2(−τx + τ2 + µ+ 2λ)σ − 2ττx + 6µτ + 2τ3

]x.

Let

σ + τ = Ψ,

we obtain the Lax pair for mKdV (16.4),

(16.6) Ψx = −Ψ2 + 2τΨ − λ+ µ, Ψt = 2[(τ2 − τx + µ+ 2λ)Ψ + 2(µ− λ)τ

]x.

Continuing the above procedure, one can find higher order modified KdV, m2KdVand m3KdV. m2KdV is the so-called Calogero-Degasperis equation (CD). For m3KdV,one can not rewrite its Lax pair as a pair of Riccati equations anymore. So theabove procedure stops at m3KdV. Nevertheless, as mentioned above, KdV, mnKdV

125

126 16. LINKS AMONG INTEGRABLE SYSTEMS

(n = 1, 2, 3), and Krichever-Novikov equation exhaust all the integrable equationsof the form ut + uxxx + f(uxx, ux, u) = 0 [22] [88].

In [6], the lower order equations in the KdV hierarchy were constructed. Let

L = −∂2x + u(x, t),

and the nonlinear equation be given by

Lt = [L,AN

].

where the operators AN

have the form

AN

=

N∑

m=−1

Km(L)m,

where N = −1, −2, ... and Km are some operators. The first lower order KdV(N = −1) equation has the form,

KdV−1(σ) ≡ (σ2x + σxx)t −

(e2σ)x

= 0,

where σ is connected with u by

u = −σxx − σ2x.

Let σ = iq/2, then

(16.7) KdV−1(σ) =1

2(i∂x − qx) (qxt − 2 sin q) = 0.

Thus we have a Miura transformation (u → q) between the KdV−1 equation andthe sine-Gordon equation. The Lax pair of (16.7) has the form,

(16.8) ψxx =

(iqxx2

− q2x4

+ λ2

)ψ, ψt =

1

2λ2eiq(ψx −

iqx2ψ

).

Starting out from (16.8), we can find the Lax pair for the sine-Gordon equation, by

introducing the new function ψ

(16.9) ψ =1

λ

(ψx −

i

2qxψ

).

Then

(16.10) ψx = λψ − i

2qxψ, ψt =

1

2λeiqψ, ψt = − 1

2λ

(iqxt − eiq

)ψ.

Replacing qxt by 2 sin q in the equation for ψt, we see that (16.9)-(16.10) are thewell known Lax pair for the sine-Gordon equation.

16.2. Higher Dimensional Systems

The Borisov-Zykov’s method can be applied to higher dimensional systems too.Here we consider the Kadomtsev-Petviashvili (KP) equation

(16.11) ut + 6uux + uxxx = −3α2 vy, vx = uy,

where u = u(x, y, t), v = v(x, y, t), and α2 = ±1. The Lax pair for KP is

(16.12) αψy + ψxx + uψ = 0, ψt + 4ψxxx + 6uψx + 3(ux − αv)ψ = 0.

16.2. HIGHER DIMENSIONAL SYSTEMS 127

Setting f = logφ, τ = fx, where φ is a solution of (16.12), we can rewrite the Laxpair (16.12) as

(16.13)ατy +

(τx + τ2 + u

)x

= 0,

τt +(4τxx + 12ττx + 4τ3 + 6uτ + 3ux − 3αv

)x

= 0.

It follows from the first equation in (16.13) that

(16.14) u = −τx − τ2 − αF, Fx = τy.

Substituting (16.14) into the second equation in (16.13), we obtain the well-knownmKP equation [53]

(16.15) τt − 6τ2τx + τxxx = 3α(2τxF − αFy), Fx − τy = 0.

The KP equation (16.11) admits the Darboux transformation [84]

(16.16) ψ1 = ψ − τψ, u1 = u+ 2τx, v1 = v + 2τy.

Setting s = logψ1, σ = sx, we see that σ satisfies the system of equations obtainedfrom (16.13) by the replacements u→ u1 and v → v1. Comparing these equationswith (16.13) and eliminating the potentials u and v, we obtain

(16.17)α(s − f)y + (s+ f)xx + s2x − f2

x = 0,(s− f)t +

[2(2s+ f)xx + 6s2x − 3f2

x

]x

+ 4s3x+6(fxx − f2

x − αfy)sx − 6α(fxy − fxfy) + 2f3

x = 0.

Let Ψ = s− f , we obtain

(16.18)

αΨy + (2f + Ψ)xx + 2fxΨx + Ψ2x = 0,

Ψt + 2(2Ψxx + 6fxΨx + 3Ψ2

x + 3θ)x

+ 6θΨx + 4 (3f + Ψ)xΨ2x = 0,

θ ≡ fxx + f2x − αfy.

The compatibility condition for these equations is(ft − 2f3

x + fxxx)x

= 3α (2fxxfy − αfyy) ,

which is merely another form of the mKP equation (with τ = fx and F = fy).This process can be continued. In particular, the system (16.18) contains a

new nonlinear equation obtained by eliminating the potential f . Let

(16.19) f = −1

2(Ψ + αξ) ,

where ξ = ξ(x, y, t). Substituting (16.19) into the second equation in (16.18),differentiating it with respect to x, and introducing S = Ψx, we obtain the system

(16.20)

St + Sxxx − 32S

2Sx = −3α2[(

12ξ

2x + ξy

)S +

(12ξ

2x + ξy

)x

]x,

Sy = (ξxx + ξxS)x .

Although the system (16.20) looks like a two-dimensional generalization of themKdV equation, its one-dimensional limit (∂y = 0)

gt +

(gxx −

α2

2g3 − 3

2

g2x

g

)

x

= 0,

(where g(x, t) = ξx = exp(−Ψ)) reduces to the exponential Calogero-Degasperisequation (CD).


16.3. Modified Nonlinear Schrodinger Equations

Consider the nonlinear Schrodinger equations (NLS)

(16.21) iut + uxx + 2u2v = 0, −ivt + vxx + 2v2u = 0.

It has the Lax pair

(16.22) Ψt = −2iσ3ΨΛ2 + 2iUΨΛ + VΨ, Ψx = −iσ3ΨΛ + iUΨ,

where

U =

(0 uv 0

), σ3 =

(1 00 −1

), V = σ3

(iU2 − Ux

),

and Λ is arbitrary constant matrix.Let Φ and Ψ be solutions of (16.22) with

Λ = diag((λ + µ)/2, (λ− µ)/2), and Λ1 = diag((λ1 + µ1)/2, (λ1 − µ1)/2),

respectively, and τ = ΦΛΦ−1. Then

(16.23) τx = i[τ, σ3]τ + i[U, τ ], τt = 2τxτ + [V, τ ].

The Darboux transformation has the form

(16.24) Ψ → Ψ1 = ΨΛ1 − τΨ, U → U1 = U + [τ, σ3].

Thus

(16.25) u→ u1 = u− 2b, v → v1 = v + 2c,

where

(16.26) τ =

(a bc d

).

Substituting (16.26) into (16.23), we get

ax = −dx = −i(2bc− uc+ vb),

bx = −i(2bd+ (a− d)u),(16.27)

cx = i(2ac+ (a− d)v),

and

at = (a2)x + 2bxc− bvx − cux,

dt = (d2)x + 2bcx + bvx + cux,

bt = 2(ibuv + axb+ bxd) + (a− d)ux,(16.28)

ct = 2(−icuv + acx + cdx) + (a− d)vx.

Calculating the determinant and trace of matrix τ we get

ad− bc =λ2 − µ2

4, a+ d = λ.

Eliminating u and v from the last two equations in (16.27), we get the mNLSequation

(µ2 − 4bc

) (ibt + bxx − 2b2c

)+ 2λ (λc+ 2icx) b

2 + 2(bxc+ 2bcx)bx = 0,

(µ2 − 4bc

) (−ict + cxx − 2c2b

)+ 2λ (λb− 2ibx) c

2 + 2(bcx + 2bxc)cx = 0.

16.4. NLS AND TODA LATTICE 129

16.4. NLS and Toda Lattice

In the Lax pair (16.22) of NLS, let

Λ =

(λ 00 µ

),

where λ and µ are spectral parameters. Denote the components of the first columnof the matrix Ψ by ψn and φn. Let u = un and v = vn. Then, the spatial part of(16.22) has the form

(16.29) ψn,x = −iλψn + iunφn, φn,x = iλφn + ivnψn.

The Schlesinger transformation of the NLS is given by

un → un+1 = un [unvv + (log un)xx] , vn → vn+1 =1

un,(16.30)

ψn → ψn+1 = (−2λ+ i(log un)x)ψn + unφn, φn → φn+1 =ψnun,(16.31)

and its inverse is given by

vn → vn−1 = vn [unvv + (log vn)xx] , un → un−1 =1

vn,(16.32)

φn → φn−1 = (2λ+ i(log vn)x)φn + vnψn, ψn → ψn−1 =φnvn.(16.33)

Denote by

(16.34) qn = log(un), pn = qn,x, Un =unun−1

= eqn−qn−1 .

Then qn satisfies the Toda lattice (TL) equation [101] [4]

(16.35) qn,xx = eqn+1−qn − eqn−qn−1 .

One can also derive the Lax pair and Darboux transformations for the Todalattice from those of NLS. Introduce the shift-operator T ,

Tun = un+1.

Let T−1 act on the second equation in (16.31), then substitute the expression forφn into the first equation in (16.29), we obtain the first equation of the Lax pairfor TL

(16.36) ψn,x = −iλψn + iUnψn−1.

To obtain the second equation of the Lax pair, let T act on the second equationin (16.29), together with the expressions of φn+1 from (16.31), vn+1 from (16.30),and ψn,x from the (16.36), we get the second equation of the Lax pair for TL

(16.37) ψn+1 = (2λ+ ipn)ψn + Unψn−1.

Let ψ1 and φ1 be the components of the first column of the matrix Ψ, where Ψ is asolution of (16.22) with λ = λ1. Then one can write two DTs for NLS (the indicesare omitted):

(16.38)

ψ → ψ(1) =

[2(λ− λ1) +

φ1

ψ1u

]ψ − uφ, φ→ φ(1) = φ− φ1

ψ1ψ,

u→ u(1) = iux −(

2λ1 −φ1

ψ1u

)u, v → v(1) =

φ1

ψ1,


and

(16.39)

ψ → (1)ψ = ψ − ψ1

φ1φ, φ→ (1)φ =

[2(λ1 − λ) +

ψ1

φ1v

]φ− vψ,

u→ (1)u =ψ1

φ1, v → (1)v = ivx +

(2λ1 +

ψ1

φ1v

)v.

Let ψ2 and φ2 be the components of the second column of the matrix Ψ with µ = λ2.Then one can iterate the Darboux transformations (16.38) and (16.39) to obtain

u→ u(1) → (2)u(1), v → v(1) → (2)v(1),

ψ → ψ(1) → (2)ψ(1), φ→ φ(1) → (2)φ(1).(16.40)

Note that the DT transformations are commutative:(2)u(1) = (1)u(2), (2)v(1) = (1)v(2).

Then it is easy to find the DT for TL. Let ψ1,n be the solution of the Lax pair(16.36)-(16.37) with λ = λ1. Then we have two DTs for TL equations:

(16.41) ψn → ψ(1)n = −ψn+1 +

ψ1,n+1

ψ1,nψn, qn → q(1)n = qn + log

ψ1,n+1

ψ1,n;

and

(16.42) ψn → (1)ψn = ψn − ψ1,n

ψ1,n−1ψn−1, qn → (1)qn = qn−1 + log

ψ1,n

ψ1,n−1.

Using (16.36)-(16.37), it is also easy to find the first modified Toda latticeequation. Define τn and ξn as follows

τn =ψ1,n

ψ1,n+1=

1

∂ξn,

where ∂ ≡ −i∂/∂x, Vn ≡ ipn. Then the Lax pair for TL can be rewritten as

(16.43) ∂τn = τn (τn−1Un − τnUn+1) , τn+1 =1

2λ1 + Vn+1 + τnUn+1.

Eliminating the potentials Un, Un+1 and Vn+1, one get the modified Toda latticeequation m1TL, which is just a Volterra equation,

∂2ξn = ∂ξn(eξn−ξn+1 − eξn−1−ξn

).

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Index

Anti-periodic point, 9

Backlund transfromationfor KdV equation, 4, 5for nonlinear Schrodinger equation, 5for sine-Gordon equation, 4

Category I (II) soliton equations, 80Congruent Lax pair, 70Continuum limit, 61Counting lemma, 16Crum rule, 83

Darboux transfromationfor acoustic problem, 82for BLP equation, 109for Davey-Stewartson II equations, 71for derivative nonlinear Schrodinger

equation, 48for Dirac equation, 100for discrete nonlinear Schrodinger

equation, 64for Goursat equation, 117for Harry-Dym equation, 86for Heisenberg ferromagnet equation, 30for KP equation, 127for Modified nonlinear Schrodinger

equation, 128for nonlinear Schrodinger equation, 9for sine-Gordon equation, 20

for Toda lattice, 130for vector nonlinear Schrodinger

equation, 43Domain wall, 31Dressing, 83

Floquet discriminant, 9

Jordan form, 12

L’Hospital’s rule, 18

Melnikov vectorfor Davey-Stewartson II equations, 77, 80

for derivative nonlinear Schrodingerequation, 53, 56

for discrete nonlinear Schrodingerequation, 62, 66

for Heisenberg ferromagnet equation, 36,38

for nonlinear Schrodinger equation, 13,18

for sine-Gordon equation, 22, 28for vector nonlinear Schrodinger

equation, 45Moutard transformation, 89Multiple point, 9

Periodic point, 9Phase shift, 11

135

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