0909/0909.0910.md

Chaos in Partial Differential Equations

Y. Charles Li

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MISSOURI, COLUMBIA,
MO 65211# Contents

Preface	xi
Chapter 1. General Setup and Concepts	1
1.1. Cauchy Problems of Partial Differential Equations	1
1.2. Phase Spaces and Flows	2
1.3. Invariant Submanifolds	3
1.4. Poincaré Sections and Poincaré Maps	4
Chapter 2. Soliton Equations as Integrable Hamiltonian PDEs	5
2.1. A Brief Summary	5
2.2. A Physical Application of the Nonlinear Schrödinger Equation	7
Chapter 3. Figure-Eight Structures	11
3.1. 1D Cubic Nonlinear Schrödinger (NLS) Equation	11
3.2. Discrete Cubic Nonlinear Schrödinger Equation	16
3.3. Davey-Stewartson II (DSII) Equations	19
3.4. Other Soliton Equations	25
Chapter 4. Melnikov Vectors	27
4.1. 1D Cubic Nonlinear Schrödinger Equation	27
4.2. Discrete Cubic Nonlinear Schrödinger Equation	32
4.3. Davey-Stewartson II Equations	34
Chapter 5. Invariant Manifolds	39
5.1. Nonlinear Schrödinger Equation Under Regular Perturbations	39
5.2. Nonlinear Schrödinger Equation Under Singular Perturbations	41
5.3. Proof of the Unstable Fiber Theorem 5.3	42
5.4. Proof of the Center-Stable Manifold Theorem 5.4	50
5.5. Perturbed Davey-Stewartson II Equations	53
5.6. General Overview	54
Chapter 6. Homoclinic Orbits	55
6.1. Silnikov Homoclinic Orbits in NLS Under Regular Perturbations	55
6.2. Silnikov Homoclinic Orbits in NLS Under Singular Perturbations	57
6.3. The Melnikov Measurement	57
6.4. The Second Measurement	61
6.5. Silnikov Homoclinic Orbits in Vector NLS Under Perturbations	64
6.6. Silnikov Homoclinic Orbits in Discrete NLS Under Perturbations	65
6.7. Comments on DSII Under Perturbations	65
6.8. Normal Form Transforms	66
6.9. Transversal Homoclinic Orbits in a Periodically Perturbed SG	69

6.10. Transversal Homoclinic Orbits in a Derivative NLS	69
Chapter 7. Existence of Chaos	71
7.1. Horseshoes and Chaos	71
7.2. Nonlinear Schrödinger Equation Under Singular Perturbations	76
7.3. Nonlinear Schrödinger Equation Under Regular Perturbations	76
7.4. Discrete Nonlinear Schrödinger Equation Under Perturbations	76
7.5. Numerical Simulation of Chaos	77
7.6. Shadowing Lemma and Chaos in Finite-D Periodic Systems	77
7.7. Shadowing Lemma and Chaos in Infinite-D Periodic Systems	81
7.8. Periodically Perturbed Sine-Gordon (SG) Equation	81
7.9. Shadowing Lemma and Chaos in Finite-D Autonomous Systems	82
7.10. Shadowing Lemma and Chaos in Infinite-D Autonomous Systems	82
7.11. A Derivative Nonlinear Schrödinger Equation	83
7.12. $\backslash lambda$-Lemma	83
7.13. Homoclinic Tubes and Chaos Cascades	83
Chapter 8. Stabilities of Soliton Equations in $\backslash mathbb\{R\}^n$	85
8.1. Traveling Wave Reduction	85
8.2. Stabilities of the Traveling-Wave Solutions	86
8.3. Breathers	87
Chapter 9. Lax Pairs of Euler Equations of Inviscid Fluids	91
9.1. A Lax Pair for 2D Euler Equation	91
9.2. A Darboux Transformation for 2D Euler Equation	92
9.3. A Lax Pair for Rossby Wave Equation	93
9.4. Lax Pairs for 3D Euler Equation	93
Chapter 10. Linearized 2D Euler Equation at a Fixed Point	95
10.1. Hamiltonian Structure of 2D Euler Equation	95
10.2. Linearized 2D Euler Equation at a Unimodal Fixed Point	96
Chapter 11. Arnold's Liapunov Stability Theory	103
11.1. A Brief Summary	103
11.2. Miscellaneous Remarks	104
Chapter 12. Miscellaneous Topics	105
12.1. KAM Theory	105
12.2. Gibbs Measure	105
12.3. Inertial Manifolds and Global Attractors	106
12.4. Zero-Dispersion Limit	106
12.5. Zero-Viscosity Limit	106
12.6. Finite Time Blowup	107
12.7. Slow Collapse	107
12.8. Burgers Equation	108
12.9. Other Model Equations	108
12.10. Kolmogorov Spectra and An Old Theory of Hopf	108
12.11. Onsager Conjecture	108
12.12. Weak Turbulence	109
12.13. Renormalization Idea	109

CONTENTS

ix

12.14. Random Forcing	109
12.15. Strange Attractors and SBR Invariant Measure	109
12.16. Arnold Diffusions	109
12.17. Averaging Technique	110
Bibliography	111
Index	119

## Preface

The area: Chaos in Partial Differential Equations, is at its fast developing stage. Notable results have been obtained in recent years. The present book aims at an overall survey on the existing results. On the other hand, we shall try to make the presentations introductory, so that beginners can benefit more from the book.

It is well-known that the theory of chaos in finite-dimensional dynamical systems has been well-developed. That includes both discrete maps and systems of ordinary differential equations. Such a theory has produced important mathematical theorems and led to important applications in physics, chemistry, biology, and engineering etc.. For a long period of time, there was no theory on chaos in partial differential equations. On the other hand, the demand for such a theory is much stronger than for finite-dimensional systems. Mathematically, studies on infinite-dimensional systems pose much more challenging problems. For example, as phase spaces, Banach spaces possess much more structures than Euclidean spaces. In terms of applications, most of important natural phenomena are described by partial differential equations – nonlinear wave equations, Maxwell equations, Yang-Mills equations, and Navier-Stokes equations, to name a few. Recently, the author and collaborators have established a systematic theory on chaos in nonlinear wave equations.

Nonlinear wave equations are the most important class of equations in natural sciences. They describe a wide spectrum of phenomena – motion of plasma, nonlinear optics (laser), water waves, vortex motion, to name a few. Among these nonlinear wave equations, there is a class of equations called soliton equations. This class of equations describes a variety of phenomena. In particular, the same soliton equation describes several different phenomena. Mathematical theories on soliton equations have been well developed. Their Cauchy problems are completely solved through inverse scattering transforms. Soliton equations are integrable Hamiltonian partial differential equations which are the natural counterparts of finite-dimensional integrable Hamiltonian systems. We have established a standard program for proving the existence of chaos in perturbed soliton equations, with the machineries: 1. Darboux transformations for soliton equations, 2. isospectral theory for soliton equations under periodic boundary condition, 3. persistence of invariant manifolds and Fenichel fibers, 4. Melnikov analysis, 5. Smale horseshoes and symbolic dynamics, 6. shadowing lemma and symbolic dynamics.

The most important implication of the theory on chaos in partial differential equations in theoretical physics will be on the study of turbulence. For that goal, we chose the 2D Navier-Stokes equations under periodic boundary conditions to begin a dynamical system study on 2D turbulence. Since they possess Lax pair and Darboux transformation, the 2D Euler equations are the starting point for ananalytical study. The high Reynolds number 2D Navier-Stokes equations are viewed as a singular perturbation of the 2D Euler equations through the perturbation parameter $\epsilon = 1/Re$ which is the inverse of the Reynolds number.

Our focus will be on nonlinear wave equations. New results on shadowing lemma and novel results related to Euler equations of inviscid fluids will also be presented. The chapters on figure-eight structures and Melnikov vectors are written in great details. The readers can learn these machineries without resorting to other references. In other chapters, details of proofs are often omitted. Chapters 3 to 7 illustrate how to prove the existence of chaos in perturbed soliton equations. Chapter 9 contains the most recent results on Lax pair structures of Euler equations of inviscid fluids. In chapter 12, we give brief comments on other related topics.

The monograph will be of interest to researchers in mathematics, physics, engineering, chemistry, biology, and science in general. Researchers who are interested in chaos in high dimensions, will find the book of particularly valuable. The book is also accessible to graduate students, and can be taken as a textbook for advanced graduate courses.

I started writing this book in 1997 when I was at MIT. This project continued at Institute for Advanced Study during the year 1998-1999, and at University of Missouri - Columbia since 1999. In the Fall of 2001, I started to rewrite from the old manuscript. Most of the work was done in the summer of 2002. The work was partially supported by an AMS centennial fellowship in 1998, and a Guggenheim fellowship in 1999.

Finally, I would like to thank my wife Sherry and my son Brandon for their strong support and appreciation.## CHAPTER 1

General Setup and Concepts

We are mainly concerned with the Cauchy problems of partial differential equations, and view them as defining flows in certain Banach spaces. Unlike the Euclidean space $\mathbb{R}^n$ , such Banach spaces admit a variety of norms which make the structures in infinite dimensional dynamical systems more abundant. The main difficulty in studying infinite dimensional dynamical systems often comes from the fact that the evolution operators for the partial differential equations are usually at best $C^0$ in time, in contrast to finite dimensional dynamical systems where the evolution operators are $C^1$ smooth in time. The well-known concepts for finite dimensional dynamical systems can be generalized to infinite dimensional dynamical systems, and this is the main task of this chapter.

1.1. Cauchy Problems of Partial Differential Equations

The types of evolution equations studied in this book can be casted into the general form,

$$(1.1){\mathrm{\partial }}_{t}Q=G(Q,{\mathrm{\partial }}_{x}Q,\dots ,{\mathrm{\partial }}_x^{\mathrm{\ell }}Q),(1.1)\; \backslash quad\; \backslash partial\_t\; Q\; =\; G(Q,\; \backslash partial\_x\; Q,\; \backslash dots,\; \backslash partial\_x^\backslash ell\; Q)\; ,$$

where $t \in \mathbb{R}^1$ (time), $x = (x_1, \dots, x_n) \in \mathbb{R}^n$ , $Q = (Q_1, \dots, Q_m)$ and $G = (G_1, \dots, G_m)$ are either real or complex valued functions, and $\ell$ , $m$ and $n$ are integers. The equation (1.1) is studied under certain boundary conditions, for example,

• • periodic boundary conditions, e.g. $Q$ is periodic in each component of $x$ with period $2\pi$ ,
• • decay boundary conditions, e.g. $Q \rightarrow 0$ as $x \rightarrow \infty$ .

Thus we have Cauchy problems for the equation (1.1), and we would like to pose the Cauchy problems in some Banach spaces $\mathcal{H}$ , for example,

• • $\mathcal{H}$ can be a Sobolev space $H^k$ ,
• • $\mathcal{H}$ can be a Sobolev space $H_{\epsilon,p}^k$ of even periodic functions.

We require that the problem is well-posed in $\mathcal{H}$ , for example,

• • for any $Q_0 \in \mathcal{H}$ , there exists a unique solution $Q = Q(t, Q_0) \in C^0([-\infty, \infty); \mathcal{H})$ or $C^0([0, \infty), \mathcal{H})$ to the equation (1.1) such that $Q(0, Q_0) = Q_0$ ,
• • for any fixed $t_0 \in (-\infty, \infty)$ or $[0, \infty)$ , $Q(t_0, Q_0)$ is a $C^r$ function of $Q_0$ , for $Q_0 \in \mathcal{H}$ and some integer $r \geq 0$ .

Example: Consider the integrable cubic nonlinear Schrödinger (NLS) equation,

$$(1.2)i{q}_t=q_{xx}+2[\mathrm{\mid }q{\mathrm{\mid }}^2-{\omega }^2]q,(1.2)\; \backslash quad\; iq\_t\; =\; q\_\{xx\}\; +\; 2\; [|q|^2\; -\; \backslash omega^2]\; q\; ,$$where $i = \sqrt{-1}$ , $t \in \mathbb{R}^1$ , $x \in \mathbb{R}^1$ , $q$ is a complex-valued function of $(t, x)$ , and $\omega$ is a real constant. We pose the periodic boundary condition,

$$q(t,x+1)=q(t,x)\mathrm{.}q(t,\; x\; +\; 1)\; =\; q(t,\; x).$$

The Cauchy problem for equation (1.2) is posed in the Sobolev space $H^1$ of periodic functions,

$$\mathcal{H}\equiv \{Q=(q,\bar{q})\mid q(x+1)=q(x),q\in H_{[0,1]}^1:\text{the}\text{Sobolev space }{H}^1\text{ over the period interval }[0,1]\},\backslash mathcal\{H\}\; \backslash equiv\; \backslash left\backslash \{\; Q\; =\; (q,\; \backslash bar\{q\})\; \backslash mid\; q(x\; +\; 1)\; =\; q(x),\; q\; \backslash in\; H^1\_\{[0,1]\}\; :\; \backslash text\{the\}\; \backslash right.\; \backslash \backslash \; \backslash left.\; \backslash text\{Sobolev\; space\; \}\; H^1\; \backslash text\{\; over\; the\; period\; interval\; \}\; [0,\; 1]\; \backslash right\backslash \},$$

and is well-posed [38] [32] [33].

Fact 1: For any $Q_0 \in \mathcal{H}$ , there exists a unique solution $Q = Q(t, Q_0) \in C^0([-\infty, \infty), \mathcal{H})$ to the equation (1.2) such that $Q(0, Q_0) = Q_0$ .

Fact 2: For any fixed $t_0 \in (-\infty, \infty)$ , $Q(t_0, Q_0)$ is a $C^2$ function of $Q_0$ , for $Q_0 \in \mathcal{H}$ .

1.2. Phase Spaces and Flows

For finite dimensional dynamical systems, the phase spaces are often $\mathbb{R}^n$ or $\mathbb{C}^n$ . For infinite dimensional dynamical systems, we take the Banach space $\mathcal{H}$ discussed in the previous section as the counterpart.

DEFINITION 1.1. We call the Banach space $\mathcal{H}$ in which the Cauchy problem for (1.1) is well-posed, a phase space. Define an operator $F^t$ labeled by $t$ as

$$Q(t,Q_0)=F^t(Q_0);Q(t,\; Q\_0)\; =\; F^t(Q\_0);$$

then $F^t : \mathcal{H} \rightarrow \mathcal{H}$ is called the evolution operator (or flow) for the system (1.1).

A point $p \in \mathcal{H}$ is called a fixed point if $F^t(p) = p$ for any $t$ . Notice that here the fixed point $p$ is in fact a function of $x$ , which is the so-called stationary solution of (1.1). Let $q \in \mathcal{H}$ be a point; then $\ell_q \equiv {F^t(q), \text{ for all } t}$ is called the orbit with initial point $q$ . An orbit $\ell_q$ is called a periodic orbit if there exists a $T \in (-\infty, \infty)$ such that $F^T(q) = q$ . An orbit $\ell_q$ is called a homoclinic orbit if there exists a point $q_* \in \mathcal{H}$ such that $F^t(q) \rightarrow q_*$ , as $|t| \rightarrow \infty$ , and $q_*$ is called the asymptotic point of the homoclinic orbit. An orbit $\ell_q$ is called a heteroclinic orbit if there exist two different points $q_{\pm} \in \mathcal{H}$ such that $F^t(q) \rightarrow q_{\pm}$ , as $t \rightarrow \pm\infty$ , and $q_{\pm}$ are called the asymptotic points of the heteroclinic orbit. An orbit $\ell_q$ is said to be homoclinic to a submanifold $W$ of $\mathcal{H}$ if $\inf_{Q \in W} |F^t(q) - Q| \rightarrow 0$ , as $|t| \rightarrow \infty$ .

Example 1: Consider the same Cauchy problem for the system (1.2). The fixed points of (1.2) satisfy the second order ordinary differential equation

$$(1.3)q_{xx}+2[\mathrm{\mid }q{\mathrm{\mid }}^2-{\omega }^2]q=0.(1.3)\; \backslash quad\; q\_\{xx\}\; +\; 2\; [|q|^2\; -\; \backslash omega^2]\; q\; =\; 0.$$

In particular, there exists a circle of fixed points $q = \omega e^{i\gamma}$ , where $\gamma \in [0, 2\pi]$ . For simple periodic solutions, we have

$$(1.4)q=a{e}^{i\theta (t)},\theta (t)=-[2(a^2-{\omega }^2)t-\gamma ];(1.4)\; \backslash quad\; q\; =\; ae^\{i\backslash theta(t)\},\; \backslash quad\; \backslash theta(t)\; =\; -\; [2(a^2\; -\; \backslash omega^2)t\; -\; \backslash gamma];$$where $a > 0$ , and $\gamma \in [0, 2\pi]$ . For orbits homoclinic to the circles (1.4), we have

$$(1.5)q=\frac{1}{\mathrm{\Lambda }}[\cos 2p-\sin p\mathrm{sech}\tau \cos 2\pi x-i\sin 2p\tanh \tau ]a{e}^{i\theta (t)},(1.5)\; \backslash quad\; q\; =\; \backslash frac\{1\}\{\backslash Lambda\}\; \backslash left[\; \backslash cos\; 2p\; -\; \backslash sin\; p\; \backslash operatorname\{sech\}\; \backslash tau\; \backslash cos\; 2\backslash pi\; x\; -\; i\; \backslash sin\; 2p\; \backslash tanh\; \backslash tau\; \backslash right]\; a\; e^\{i\backslash theta(t)\},$$

$$\mathrm{\Lambda }=1+\sin p\mathrm{sech}\tau \cos 2\pi x,\backslash Lambda\; =\; 1\; +\; \backslash sin\; p\; \backslash operatorname\{sech\}\; \backslash tau\; \backslash cos\; 2\backslash pi\; x,$$

where $\tau = 4\pi\sqrt{a^2 - \pi^2} t + \rho$ , $p = \arctan \left[ \frac{\sqrt{a^2 - \pi^2}}{\pi} \right]$ , $\rho \in (-\infty, \infty)$ is the Bäcklund parameter. Setting $a = \omega$ in (1.5), we have heteroclinic orbits asymptotic to points on the circle of fixed points. The expression (1.5) is generated from (1.4) through a Bäcklund-Darboux transformation [137].

Example 2: Consider the sine-Gordon equation,

$$u_{tt}-u_{xx}+\sin u=0,u\_\{tt\}\; -\; u\_\{xx\}\; +\; \backslash sin\; u\; =\; 0,$$

under the decay boundary condition that $u$ belongs to the Schwartz class in $x$ . The well-known “breather” solution,

$$(1.6)u(t,x)=4\arctan \left[\frac{\tan \nu \cos [(\cos \nu )t]}{\cosh [(\sin \nu )x]}\right],(1.6)\; \backslash quad\; u(t,\; x)\; =\; 4\; \backslash arctan\; \backslash left[\; \backslash frac\{\backslash tan\; \backslash nu\; \backslash cos[(\backslash cos\; \backslash nu)t]\}\{\backslash cosh[(\backslash sin\; \backslash nu)x]\}\; \backslash right],$$

where $\nu$ is a parameter, is a periodic orbit. The expression (1.6) is generated from trivial solutions through a Bäcklund-Darboux transformation [59].

1.3. Invariant Submanifolds

Invariant submanifolds are the main objects in studying phase spaces. In phase spaces for partial differential equations, invariant submanifolds are often submanifolds with boundaries. Therefore, the following concepts on invariance are important.

DEFINITION 1.2 (Overflowing and Inflowing Invariance). A submanifold $W$ with boundary $\partial W$ is

• • overflowing invariant if for any $t > 0$ , $\bar{W} \subset F^t \circ W$ , where $\bar{W} = W \cup \partial W$ ,
• • inflowing invariant if any $t > 0$ , $F^t \circ \bar{W} \subset W$ ,
• • invariant if for any $t > 0$ , $F^t \circ \bar{W} = \bar{W}$ .

DEFINITION 1.3 (Local Invariance). A submanifold $W$ with boundary $\partial W$ is locally invariant if for any point $q \in W$ , if $\bigcup_{t \in [0, \infty)} F^t(q) \not\subset W$ , then there exists $T \in (0, \infty)$ such that $\bigcup_{t \in [0, T)} F^t(q) \subset W$ , and $F^T(q) \in \partial W$ ; and if $\bigcup_{t \in (-\infty, 0]} F^t(q) \not\subset W$ , then there exists $T \in (-\infty, 0)$ such that $\bigcup_{t \in (T, 0]} F^t(q) \subset W$ , and $F^T(q) \in \partial W$ .

Intuitively speaking, a submanifold with boundary is locally invariant if any orbit starting from a point inside the submanifold can only leave the submanifold through its boundary in both forward and backward time.

Example: Consider the linear equation,

$$(1.7)i{q}_t=(1+i)q_{xx}+iq,(1.7)\; \backslash quad\; iq\_t\; =\; (1\; +\; i)q\_\{xx\}\; +\; iq,$$

where $i = \sqrt{-1}$ , $t \in \mathbb{R}^1$ , $x \in \mathbb{R}^1$ , and $q$ is a complex-valued function of $(t, x)$ , under periodic boundary condition,

$$q(x+1)=q(x)\mathrm{.}q(x\; +\; 1)\; =\; q(x).$$Let $q = e^{\Omega_j t + i k_j x}$ ; then

$${\mathrm{\Omega }}_j=(1-k_j^2)+i{k}_j^2,\backslash Omega\_j\; =\; (1\; -\; k\_j^2)\; +\; i\; k\_j^2,$$

where $k_j = 2j\pi$ , ( $j \in \mathbb{Z}$ ). $\Omega_0 = 1$ , and when $|j| > 0$ , $\operatorname{Re}{\Omega_j} < 0$ . We take the $H^1$ space of periodic functions of period 1 to be the phase space. Then the submanifold

is an outflowing invariant submanifold, the submanifold

is an inflowing invariant submanifold, and the submanifold

is a locally invariant submanifold. The unstable subspace is given by

$$W^{(u)}=\{q\in H^1\mid q=c_0,c_0\text{ is complex}\},W^\{(u)\}\; =\; \backslash left\backslash \{\; q\; \backslash in\; H^1\; \backslash mid\; q\; =\; c\_0,\; c\_0\; \backslash text\{\; is\; complex\}\; \backslash right\backslash \},$$

and the stable subspace is given by

$$W^{(s)}=\{q\in H^1\mid q=\sum _{j\in \mathbb{Z}\setminus \{0\}}{c}_j{e}^{i{k}_{j}x},c_j\text{’s are complex}\}\mathrm{.}W^\{(s)\}\; =\; \backslash left\backslash \{\; q\; \backslash in\; H^1\; \backslash mid\; q\; =\; \backslash sum\_\{j\; \backslash in\; \backslash mathbb\{Z\}\; \backslash setminus\; \backslash \{0\backslash \}\}\; c\_j\; e^\{i\; k\_j\; x\},\; c\_j\; \backslash text\{\text{'}s\; are\; complex\}\; \backslash right\backslash \}.$$

Actually, a good way to view the partial differential equation (1.7) as defining an infinite dimensional dynamical system is through Fourier transform, let

$$q(t,x)=\sum _{j\in \mathbb{Z}}{c}_j(t)e^{i{k}_{j}x};q(t,\; x)\; =\; \backslash sum\_\{j\; \backslash in\; \backslash mathbb\{Z\}\}\; c\_j(t)\; e^\{i\; k\_j\; x\};$$

then $c_j(t)$ satisfy

$${\dot{c}}_j=[(1-k_j^2)+i{k}_j^2]c_j,j\in \mathbb{Z};\backslash dot\{c\}\_j\; =\; [(1\; -\; k\_j^2)\; +\; i\; k\_j^2]\; c\_j,\; \backslash quad\; j\; \backslash in\; \backslash mathbb\{Z\};$$

which is a system of infinitely many ordinary differential equations.

1.4. Poincaré Sections and Poincaré Maps

In the infinite dimensional phase space $\mathcal{H}$ , Poincaré sections can be defined in a similar fashion as in a finite dimensional phase space. Let $l_q$ be a periodic or homoclinic orbit in $\mathcal{H}$ under a flow $F^t$ , and $q_*$ be a point on $l_q$ , then the Poincaré section $\Sigma$ can be defined to be any codimension 1 subspace which has a transversal intersection with $l_q$ at $q_*$ . Then the flow $F^t$ will induce a Poincaré map $P$ in the neighborhood of $q_*$ in $\Sigma_0$ . Phase blocks, e.g. Smale horseshoes, can be defined using the norm.## CHAPTER 2

Soliton Equations as Integrable Hamiltonian PDEs

2.1. A Brief Summary

Soliton equations are integrable Hamiltonian partial differential equations. For example, the Korteweg-de Vries (KdV) equation

$$u_t=-6u{u}_x-u_{xxx},u\_t\; =\; -6uu\_x\; -\; u\_\{xxx\}\; ,$$

where $u$ is a real-valued function of two variables $t$ and $x$ , can be rewritten in the Hamiltonian form

$$u_t={\mathrm{\partial }}_x\frac{\delta H}{\delta u},u\_t\; =\; \backslash partial\_x\; \backslash frac\{\backslash delta\; H\}\{\backslash delta\; u\}\; ,$$

where

$$H=\int [\frac{1}{2}{u}_x^2-u^3]dx,H\; =\; \backslash int\; \backslash left[\; \backslash frac\{1\}\{2\}\; u\_x^2\; -\; u^3\; \backslash right]\; dx\; ,$$

under either periodic or decay boundary conditions. It is integrable in the classical Liouville sense, i.e., there exist enough functionally independent constants of motion. These constants of motion can be generated through isospectral theory or Bäcklund transformations [8]. The level sets of these constants of motion are elliptic tori [178] [154] [153] [68].

There exist soliton equations which possess level sets which are normally hyperbolic, for example, the focusing cubic nonlinear Schrödinger equation [137],

$$i{q}_t=q_{xx}+2\mathrm{\mid }q{\mathrm{\mid }}^{2}q,iq\_t\; =\; q\_\{xx\}\; +\; 2|q|^2\; q\; ,$$

where $i = \sqrt{-1}$ and $q$ is a complex-valued function of two variables $t$ and $x$ ; the sine-Gordon equation [157],

$$u_{tt}=u_{xx}+\sin u,u\_\{tt\}\; =\; u\_\{xx\}\; +\; \backslash sin\; u\; ,$$

where $u$ is a real-valued function of two variables $t$ and $x$ , etc.

Hyperbolic foliations are very important since they are the sources of chaos when the integrable systems are under perturbations. We will investigate the hyperbolic foliations of three typical types of soliton equations: (i). (1+1)-dimensional soliton equations represented by the focusing cubic nonlinear Schrödinger equation, (ii). soliton lattices represented by the focusing cubic nonlinear Schrödinger lattice, (iii). (1+2)-dimensional soliton equations represented by the Davey-Stewartson II equation.

REMARK 2.1. For those soliton equations which have only elliptic level sets, the corresponding representatives can be chosen to be the KdV equation for (1+1)-dimensional soliton equations, the Toda lattice for soliton lattices, and the KP equation for (1+2)-dimensional soliton equations.Soliton equations are canonical equations which model a variety of physical phenomena, for example, nonlinear wave motions, nonlinear optics, plasmas, vortex dynamics, etc. [5] [1]. Other typical examples of such integrable Hamiltonian partial differential equations are, e.g., the defocusing cubic nonlinear Schrödinger equation,

$$i{q}_t=q_{xx}-2\mathrm{\mid }q{\mathrm{\mid }}^{2}q,iq\_t\; =\; q\_\{xx\}\; -\; 2|q|^2q\; ,$$

where $i = \sqrt{-1}$ and $q$ is a complex-valued function of two variables $t$ and $x$ ; the modified KdV equation,

$$u_t=\pm 6u^2{u}_x-u_{xxx},u\_t\; =\; \backslash pm\; 6u^2u\_x\; -\; u\_\{xxx\}\; ,$$

where $u$ is a real-valued function of two variables $t$ and $x$ ; the sinh-Gordon equation,

$$u_{tt}=u_{xx}+\sinh u,u\_\{tt\}\; =\; u\_\{xx\}\; +\; \backslash sinh\; u\; ,$$

where $u$ is a real-valued function of two variables $t$ and $x$ ; the three-wave interaction equations,

$$\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }t}+a_i\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }x}=b_i{\bar{u}}_j{\bar{u}}_k,\backslash frac\{\backslash partial\; u\_i\}\{\backslash partial\; t\}\; +\; a\_i\; \backslash frac\{\backslash partial\; u\_i\}\{\backslash partial\; x\}\; =\; b\_i\; \backslash bar\{u\}\_j\; \backslash bar\{u\}\_k\; ,$$

where $i, j, k = 1, 2, 3$ are cyclically permuted, $a_i$ and $b_i$ are real constants, $u_i$ are complex-valued functions of $t$ and $x$ ; the Boussinesq equation,

$$u_{tt}-u_{xx}+(u^2{)}_{xx}\pm u_{xxxx}=0,u\_\{tt\}\; -\; u\_\{xx\}\; +\; (u^2)\_\{xx\}\; \backslash pm\; u\_\{xxxx\}\; =\; 0\; ,$$

where $u$ is a real-valued function of two variables $t$ and $x$ ; the Toda lattice,

$${\mathrm{\partial }}^2{u}_n\mathrm{/}\mathrm{\partial }{t}^2=\exp \{-(u_n-u_{n-1})\}-\exp \{-(u_{n+1}-u_n)\},\backslash partial^2\; u\_n\; /\; \backslash partial\; t^2\; =\; \backslash exp\; \backslash \{\; -(u\_n\; -\; u\_\{n-1\})\; \backslash \}\; -\; \backslash exp\; \backslash \{\; -(u\_\{n+1\}\; -\; u\_n)\; \backslash \}\; ,$$

where $u_n$ 's are real variables; the focusing cubic nonlinear Schrödinger lattice,

$$i\frac{\mathrm{\partial }{q}_n}{\mathrm{\partial }t}=(q_{n+1}-2q_n+q_{n-1})+\mathrm{\mid }{q}_n{\mathrm{\mid }}^2(q_{n+1}+q_{n-1}),i\; \backslash frac\{\backslash partial\; q\_n\}\{\backslash partial\; t\}\; =\; (q\_\{n+1\}\; -\; 2q\_n\; +\; q\_\{n-1\})\; +\; |q\_n|^2\; (q\_\{n+1\}\; +\; q\_\{n-1\})\; ,$$

where $q_n$ 's are complex variables; the Kadomtsev-Petviashvili (KP) equation,

$$(u_t+6u{u}_x+u_{xxx}{)}_x=\pm 3u_{yy},(u\_t\; +\; 6uu\_x\; +\; u\_\{xxx\})\_x\; =\; \backslash pm\; 3u\_\{yy\}\; ,$$

where $u$ is a real-valued function of three variables $t, x$ and $y$ ; the Davey-Stewartson II equation,

$$\{\begin{array}{c}i{\mathrm{\partial }}_{t}q=[{\mathrm{\partial }}_x^2-{\mathrm{\partial }}_y^2]q+[2\mathrm{\mid }q{\mathrm{\mid }}^2+u_y]q,\\ [{\mathrm{\partial }}_x^2+{\mathrm{\partial }}_y^2]u=-4{\mathrm{\partial }}_y\mathrm{\mid }q{\mathrm{\mid }}^2,\end{array}\backslash begin\{cases\}\; i\backslash partial\_t\; q\; =\; [\backslash partial\_x^2\; -\; \backslash partial\_y^2]q\; +\; [2|q|^2\; +\; u\_y]q\; ,\; \backslash \backslash \; [\backslash partial\_x^2\; +\; \backslash partial\_y^2]u\; =\; -4\backslash partial\_y|q|^2\; ,\; \backslash end\{cases\}$$

where $i = \sqrt{-1}$ , $q$ is a complex-valued function of three variables $t, x$ and $y$ ; and $u$ is a real-valued function of three variables $t, x$ and $y$ . For more complete list of soliton equations, see e.g. [5] [1].

The cubic nonlinear Schrödinger equation is one of our main focuses in this book, which can be written in the Hamiltonian form,

$$i{q}_t=\frac{\delta H}{\delta \bar{q}},iq\_t\; =\; \backslash frac\{\backslash delta\; H\}\{\backslash delta\; \backslash bar\{q\}\}\; ,$$

where

$$H=\int [-\mathrm{\mid }{q}_x{\mathrm{\mid }}^2\pm \mathrm{\mid }q{\mathrm{\mid }}^4]dx,H\; =\; \backslash int\; [-|q\_x|^2\; \backslash pm\; |q|^4]\; dx\; ,$$under periodic boundary conditions. Its phase space is defined as

$${\mathcal{H}}^k\equiv \{\vec{q}=\left(\begin{array}{c}q\\ r\end{array}\right)\mid r=-\bar{q},q(x+1)=q(x),q\in H_{[0,1]}^k:\text{the Sobolev space }{H}^k\text{ over the period interval }[0,1]\}\mathrm{.}\backslash mathcal\{H\}^k\; \backslash equiv\; \backslash left\backslash \{\; \backslash vec\{q\}\; =\; \backslash begin\{pmatrix\}\; q\; \backslash \backslash \; r\; \backslash end\{pmatrix\}\; \backslash mid\; r\; =\; -\backslash bar\{q\},\; q(x+1)\; =\; q(x),\; \backslash right.\; \backslash \backslash \; \backslash left.\; q\; \backslash in\; H\_\{[0,1]\}^k\; :\; \backslash text\{the\; Sobolev\; space\; \}\; H^k\; \backslash text\{\; over\; the\; period\; interval\; \}\; [0,\; 1]\; \backslash right\backslash \}.$$

REMARK 2.2. It is interesting to notice that the cubic nonlinear Schrödinger equation can also be written in Hamiltonian form in spatial variable, i.e.,

$$q_{xx}=i{q}_t\pm 2\mathrm{\mid }q{\mathrm{\mid }}^{2}q,q\_\{xx\}\; =\; iq\_t\; \backslash pm\; 2|q|^2\; q,$$

can be written in Hamiltonian form. Let $p = q_x$ ; then

$$\frac{\mathrm{\partial }}{\mathrm{\partial }x}\left(\begin{array}{c}q\\ \bar{q}\\ p\\ \bar{p}\end{array}\right)=J\left(\begin{array}{c}\frac{\delta H}{\delta q}\\ \frac{\delta H}{\delta \bar{q}}\\ \frac{\delta H}{\delta p}\\ \frac{\delta H}{\delta \bar{p}}\end{array}\right),\backslash frac\{\backslash partial\}\{\backslash partial\; x\}\; \backslash begin\{pmatrix\}\; q\; \backslash \backslash \; \backslash bar\{q\}\; \backslash \backslash \; p\; \backslash \backslash \; \backslash bar\{p\}\; \backslash end\{pmatrix\}\; =\; J\; \backslash begin\{pmatrix\}\; \backslash frac\{\backslash delta\; H\}\{\backslash delta\; q\}\; \backslash \backslash \; \backslash frac\{\backslash delta\; H\}\{\backslash delta\; \backslash bar\{q\}\}\; \backslash \backslash \; \backslash frac\{\backslash delta\; H\}\{\backslash delta\; p\}\; \backslash \backslash \; \backslash frac\{\backslash delta\; H\}\{\backslash delta\; \backslash bar\{p\}\}\; \backslash end\{pmatrix\},$$

where

$$H=\int [\mathrm{\mid }p{\mathrm{\mid }}^2\mp \mathrm{\mid }q{\mathrm{\mid }}^4-\frac{i}{2}(q_t\bar{q}-{\bar{q}}_{t}q)]dt,H\; =\; \backslash int\; [|p|^2\; \backslash mp\; |q|^4\; -\; \backslash frac\{i\}\{2\}(q\_t\; \backslash bar\{q\}\; -\; \backslash bar\{q\}\_t\; q)]\; dt,$$

under decay or periodic boundary conditions. We do not know whether or not other soliton equations have this property.

2.2. A Physical Application of the Nonlinear Schrödinger Equation

The cubic nonlinear Schrödinger (NLS) equation has many different applications, i.e. it describes many different physical phenomena, and that is why it is called a canonical equation. Here, as an example, we show how the NLS equation describes the motion of a vortex filament – the beautiful Hasimoto derivation [82]. Vortex filaments in an inviscid fluid are known to preserve their identities. The motion of a very thin isolated vortex filament $\vec{X} = \vec{X}(s, t)$ of radius $\epsilon$ in an incompressible inviscid unbounded fluid by its own induction is described asymptotically by

$$(2.1)\mathrm{\partial }\vec{X}\mathrm{/}\mathrm{\partial }t=G\kappa \vec{b},(2.1)\; \backslash quad\; \backslash partial\; \backslash vec\{X\}\; /\; \backslash partial\; t\; =\; G\; \backslash kappa\; \backslash vec\{b\},$$

where $s$ is the length measured along the filament, $t$ is the time, $\kappa$ is the curvature, $\vec{b}$ is the unit vector in the direction of the binormal and $G$ is the coefficient of local induction,

$$G=\frac{\mathrm{\Gamma }}{4\pi }[\ln (1\mathrm{/}ϵ)+O(1)],G\; =\; \backslash frac\{\backslash Gamma\}\{4\backslash pi\}\; [\backslash ln(1/\backslash epsilon)\; +\; O(1)],$$which is proportional to the circulation $\Gamma$ of the filament and may be regarded as a constant if we neglect the second order term. Then a suitable choice of the units of time and length reduces (2.1) to the nondimensional form,

$$(2.2)\mathrm{\partial }\vec{X}\mathrm{/}\mathrm{\partial }t=\kappa \vec{b}\mathrm{.}(2.2)\; \backslash quad\; \backslash partial\; \backslash vec\{X\}\; /\; \backslash partial\; t\; =\; \backslash kappa\; \backslash vec\{b\}\; .$$

Equation (2.2) should be supplemented by the equations of differential geometry (the Frenet-Serret formulae)

$$(2.3)\mathrm{\partial }\vec{X}\mathrm{/}\mathrm{\partial }s=\vec{t},\mathrm{\partial }\vec{t}\mathrm{/}\mathrm{\partial }s=\kappa \vec{n},\mathrm{\partial }\vec{n}\mathrm{/}\mathrm{\partial }s=\tau \vec{b}-\kappa \vec{t},\mathrm{\partial }\vec{b}\mathrm{/}\mathrm{\partial }s=-\tau \vec{n},(2.3)\; \backslash quad\; \backslash partial\; \backslash vec\{X\}\; /\; \backslash partial\; s\; =\; \backslash vec\{t\}\; ,\; \backslash quad\; \backslash partial\; \backslash vec\{t\}\; /\; \backslash partial\; s\; =\; \backslash kappa\; \backslash vec\{n\}\; ,\; \backslash quad\; \backslash partial\; \backslash vec\{n\}\; /\; \backslash partial\; s\; =\; \backslash tau\; \backslash vec\{b\}\; -\; \backslash kappa\; \backslash vec\{t\}\; ,\; \backslash quad\; \backslash partial\; \backslash vec\{b\}\; /\; \backslash partial\; s\; =\; -\backslash tau\; \backslash vec\{n\}\; ,$$

where $\tau$ is the torsion and $\vec{t}$ , $\vec{n}$ and $\vec{b}$ are the tangent, the principal normal and the binormal unit vectors. The last two equations imply that

$$(2.4)\mathrm{\partial }(\vec{n}+i\vec{b})\mathrm{/}\mathrm{\partial }s=-i\tau (\vec{n}+i\vec{b})-\kappa \vec{t},(2.4)\; \backslash quad\; \backslash partial(\backslash vec\{n\}\; +\; i\backslash vec\{b\})\; /\; \backslash partial\; s\; =\; -i\backslash tau(\backslash vec\{n\}\; +\; i\backslash vec\{b\})\; -\; \backslash kappa\; \backslash vec\{t\}\; ,$$

which suggests the introduction of new variables

$$(2.5)\vec{N}=(\vec{n}+i\vec{b})\exp \left\{i{\int }_0^s\tau ds\right\},(2.5)\; \backslash quad\; \backslash vec\{N\}\; =\; (\backslash vec\{n\}\; +\; i\backslash vec\{b\})\; \backslash exp\; \backslash left\backslash \{\; i\; \backslash int\_0^s\; \backslash tau\; ds\; \backslash right\backslash \}\; ,$$

and

$$(2.6)q=\kappa \exp \left\{i{\int }_0^s\tau ds\right\}\mathrm{.}(2.6)\; \backslash quad\; q\; =\; \backslash kappa\; \backslash exp\; \backslash left\backslash \{\; i\; \backslash int\_0^s\; \backslash tau\; ds\; \backslash right\backslash \}\; .$$

Then from (2.3) and (2.4), we have

$$(2.7)\mathrm{\partial }\vec{N}\mathrm{/}\mathrm{\partial }s=-q\vec{t},\mathrm{\partial }\vec{t}\mathrm{/}\mathrm{\partial }s=\text{Re}\{q\vec{N}\}=\frac{1}{2}(\bar{q}\vec{N}+q\overline{\stackrel{⃗}{N}})\mathrm{.}(2.7)\; \backslash quad\; \backslash partial\; \backslash vec\{N\}\; /\; \backslash partial\; s\; =\; -q\; \backslash vec\{t\}\; ,\; \backslash quad\; \backslash partial\; \backslash vec\{t\}\; /\; \backslash partial\; s\; =\; \backslash text\{Re\}\backslash \{q\; \backslash vec\{N\}\backslash \}\; =\; \backslash frac\{1\}\{2\}(\backslash bar\{q\}\; \backslash vec\{N\}\; +\; q\; \backslash overline\{\backslash vec\{N\}\})\; .$$

We are going to use the relation $\frac{\partial^2 \vec{N}}{\partial s \partial t} = \frac{\partial^2 \vec{N}}{\partial t \partial s}$ to derive an equation for $q$ . For this we need to know $\partial \vec{t} / \partial t$ and $\partial \vec{N} / \partial t$ besides equations (2.7). From (2.2) and (2.3), we have

i.e.

$$(2.8)\mathrm{\partial }\vec{t}\mathrm{/}\mathrm{\partial }t=\text{Re}\{i(\mathrm{\partial }q\mathrm{/}\mathrm{\partial }s)\overline{\stackrel{⃗}{N}}\}=\frac{1}{2}i[(\mathrm{\partial }q\mathrm{/}\mathrm{\partial }s)\overline{\stackrel{⃗}{N}}-(\mathrm{\partial }q\mathrm{/}\mathrm{\partial }s{)}^{-}\vec{N}]\mathrm{.}(2.8)\; \backslash quad\; \backslash partial\; \backslash vec\{t\}\; /\; \backslash partial\; t\; =\; \backslash text\{Re\}\backslash \{i(\backslash partial\; q\; /\; \backslash partial\; s)\; \backslash overline\{\backslash vec\{N\}\}\backslash \}\; =\; \backslash frac\{1\}\{2\}i[(\backslash partial\; q\; /\; \backslash partial\; s)\; \backslash overline\{\backslash vec\{N\}\}\; -\; (\backslash partial\; q\; /\; \backslash partial\; s)^-\; \backslash vec\{N\}]\; .$$

We can write the equation for $\partial \vec{N} / \partial t$ in the following form:

$$(2.9)\mathrm{\partial }\vec{N}\mathrm{/}\mathrm{\partial }t=\alpha \vec{N}+\beta \overline{\stackrel{⃗}{N}}+\gamma \vec{t},(2.9)\; \backslash quad\; \backslash partial\; \backslash vec\{N\}\; /\; \backslash partial\; t\; =\; \backslash alpha\; \backslash vec\{N\}\; +\; \backslash beta\; \backslash overline\{\backslash vec\{N\}\}\; +\; \backslash gamma\; \backslash vec\{t\}\; ,$$

where $\alpha$ , $\beta$ and $\gamma$ are complex coefficients to be determined.

i.e. $\alpha = iR$ where $R$ is an unknown real function.

Thus

$$(2.10)\mathrm{\partial }\vec{N}\mathrm{/}\mathrm{\partial }t=i[R\vec{N}-(\mathrm{\partial }q\mathrm{/}\mathrm{\partial }s)\vec{t}]\mathrm{.}(2.10)\; \backslash quad\; \backslash partial\; \backslash vec\{N\}\; /\; \backslash partial\; t\; =\; i[R\; \backslash vec\{N\}\; -\; (\backslash partial\; q\; /\; \backslash partial\; s)\; \backslash vec\{t\}]\; .$$

From (2.7), (2.10) and (2.8), we have

Thus, we have

$$(2.11)\mathrm{\partial }q\mathrm{/}\mathrm{\partial }t=i[{\mathrm{\partial }}^{2}q\mathrm{/}\mathrm{\partial }{s}^2+Rq],(2.11)\; \backslash quad\; \backslash partial\; q\; /\; \backslash partial\; t\; =\; i[\backslash partial^2\; q\; /\; \backslash partial\; s^2\; +\; R\; q]\; ,$$

and

$$(2.12)\frac{1}{2}q\mathrm{\partial }\bar{q}\mathrm{/}\mathrm{\partial }s=\mathrm{\partial }R\mathrm{/}\mathrm{\partial }s-\frac{1}{2}(\mathrm{\partial }q\mathrm{/}\mathrm{\partial }s)\bar{q}\mathrm{.}(2.12)\; \backslash quad\; \backslash frac\{1\}\{2\}\; q\; \backslash partial\; \backslash bar\{q\}\; /\; \backslash partial\; s\; =\; \backslash partial\; R\; /\; \backslash partial\; s\; -\; \backslash frac\{1\}\{2\}\; (\backslash partial\; q\; /\; \backslash partial\; s)\; \backslash bar\{q\}\; .$$

The comparison of expressions for $\frac{\partial^2 \vec{t}}{\partial s \partial t}$ from (2.7) and (2.8) leads only to (2.11). Solving (2.12), we have

$$(2.13)R=\frac{1}{2}(\mathrm{\mid }q{\mathrm{\mid }}^2+A),(2.13)\; \backslash quad\; R\; =\; \backslash frac\{1\}\{2\}\; (|q|^2\; +\; A)\; ,$$

where $A$ is a real-valued function of $t$ only. Thus we have the cubic nonlinear Schrödinger equation for $q$ :

$$-i\mathrm{\partial }q\mathrm{/}\mathrm{\partial }t={\mathrm{\partial }}^{2}q\mathrm{/}\mathrm{\partial }{s}^2+\frac{1}{2}(\mathrm{\mid }q{\mathrm{\mid }}^2+A)q\mathrm{.}-i\; \backslash partial\; q\; /\; \backslash partial\; t\; =\; \backslash partial^2\; q\; /\; \backslash partial\; s^2\; +\; \backslash frac\{1\}\{2\}\; (|q|^2\; +\; A)\; q\; .$$

The term $Aq$ can be transformed away by defining the new variable

$$\tilde{q}=q\exp [-\frac{1}{2}i{\int }_0^{t}A(t)dt]\mathrm{.}\backslash tilde\{q\}\; =\; q\; \backslash exp[-\backslash frac\{1\}\{2\}\; i\; \backslash int\_0^t\; A(t)\; dt]\; .$$## CHAPTER 3

Figure-Eight Structures

For finite-dimensional Hamiltonian systems, figure-eight structures are often given by singular level sets. These singular level sets are also called separatrices. Expressions for such figure-eight structures can be obtained by setting the Hamiltonian and/or other constants of motion to special values. For partial differential equations, such an approach is not feasible. For soliton equations, expressions for figure-eight structures can be obtained via Bäcklund-Darboux transformations [137] [118] [121].

3.1. 1D Cubic Nonlinear Schrödinger (NLS) Equation

We take the focusing nonlinear Schrödinger equation (NLS) as our first example to show how to construct figure-eight structures. If one starts from the conservation laws of the NLS, it turns out that it is very elusive to get the separatrices. On the contrary, starting from the Bäcklund-Darboux transformation to be presented, one can find the separatrices rather easily. We consider the NLS

$$(3.1)i{q}_t=q_{xx}+2\mathrm{\mid }q{\mathrm{\mid }}^{2}q,(3.1)\; \backslash quad\; iq\_t\; =\; q\_\{xx\}\; +\; 2|q|^2q\; ,$$

under periodic boundary condition $q(x + 2\pi) = q(x)$ . The NLS is an integrable system by virtue of the Lax pair [212],

$$(3.2){\phi }_x=U\phi ,(3.2)\; \backslash quad\; \backslash varphi\_x\; =\; U\backslash varphi\; ,$$

$$(3.3){\phi }_t=V\phi ,(3.3)\; \backslash quad\; \backslash varphi\_t\; =\; V\backslash varphi\; ,$$

where

where $\sigma_3$ denotes the third Pauli matrix $\sigma_3 = \text{diag}(1, -1)$ , $r = -\bar{q}$ , and $\lambda$ is the spectral parameter. If $q$ satisfies the NLS, then the compatibility of the over determined system (3.2, 3.3) is guaranteed. Let $M = M(x)$ be the fundamental matrix solution to the ODE (3.2), $M(0)$ is the $2 \times 2$ identity matrix. We introduce the so-called transfer matrix $T = T(\lambda, \vec{q})$ where $\vec{q} = (q, -\bar{q})$ , $T = M(2\pi)$ .

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

$$Y(2n\pi )=T^{n}Y(0)\mathrm{.}Y(2n\backslash pi)\; =\; T^n\; Y(0)\; .$$

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

$$Y(x)=M(x)Y(0)\mathrm{.}Y(x)\; =\; M(x)\; Y(0)\; .$$Thus,

$$Y(2\pi )=TY(0)\mathrm{.}Y(2\backslash pi)\; =\; T\; Y(0)\; .$$

Assume that

$$Y(2l\pi )=T^{l}Y(0)\mathrm{.}Y(2l\backslash pi)\; =\; T^l\; Y(0)\; .$$

Notice that $Y(x + 2l\pi)$ also solves the ODE (3.2); then

$$Y(x+2l\pi )=M(x)Y(2l\pi );Y(x\; +\; 2l\backslash pi)\; =\; M(x)\; Y(2l\backslash pi)\; ;$$

thus,

$$Y(2(l+1)\pi )=TY(2l\pi )=T^{l+1}Y(0)\mathrm{.}Y(2(l+1)\backslash pi)\; =\; T\; Y(2l\backslash pi)\; =\; T^\{l+1\}\; Y(0)\; .$$

The lemma is proved. Q.E.D.

DEFINITION 3.2. We define the Floquet discriminant $\Delta$ as,

$$\mathrm{\Delta }(\lambda ,\vec{q})=\text{trace}\{T(\lambda ,\vec{q})\}\mathrm{.}\backslash Delta(\backslash lambda,\; \backslash vec\{q\})\; =\; \backslash text\{trace\}\; \backslash \{T(\backslash lambda,\; \backslash vec\{q\})\backslash \}\; .$$

We define the periodic and anti-periodic points $\lambda^{(p)}$ by the condition

$$\mathrm{\mid }\mathrm{\Delta }({\lambda }^{(p)},\vec{q})\mathrm{\mid }=2.|\backslash Delta(\backslash lambda^\{(p)\},\; \backslash vec\{q\})|\; =\; 2\; .$$

We define the critical points $\lambda^{(c)}$ by the condition

$${\frac{\mathrm{\partial }\mathrm{\Delta }(\lambda ,\vec{q})}{\mathrm{\partial }\lambda }\mid }_{\lambda ={\lambda }^{(c)}}=0.\backslash left.\; \backslash frac\{\backslash partial\; \backslash Delta(\backslash lambda,\; \backslash vec\{q\})\}\{\backslash partial\; \backslash lambda\}\; \backslash right|\_\{\backslash lambda=\backslash lambda^\{(c)\}\}\; =\; 0\; .$$

A multiple point, denoted $\lambda^{(m)}$ , is a critical point for which

$$\mathrm{\mid }\mathrm{\Delta }({\lambda }^{(m)},\vec{q})\mathrm{\mid }=2.|\backslash Delta(\backslash lambda^\{(m)\},\; \backslash vec\{q\})|\; =\; 2\; .$$

The algebraic multiplicity of $\lambda^{(m)}$ is defined as the order of the zero of $\Delta(\lambda) \pm 2$ . Usually it is 2, but it can exceed 2; when it does equal 2, we call the multiple point a double point, and denote it by $\lambda^{(d)}$ . The geometric multiplicity of $\lambda^{(m)}$ is defined as the maximum number of linearly independent solutions to the ODE (3.2), and is either 1 or 2.

Let $q(x, t)$ be a solution to the NLS (3.1) for which the linear system (3.2) has a complex double point $\nu$ of geometric multiplicity 2. We denote two linearly independent solutions of the Lax pair (3.2,3.3) at $\lambda = \nu$ by $(\phi^+, \phi^-)$ . Thus, a general solution of the linear systems at $(q, \nu)$ is given by

$$(3.4)\varphi (x,t)=c_{+}{\varphi }^{+}+c_{-}{\varphi }^{-}\mathrm{.}(3.4)\; \backslash quad\; \backslash phi(x,\; t)\; =\; c\_+\; \backslash phi^+\; +\; c\_-\; \backslash phi^-\; .$$

We use $\phi$ to define a Gauge matrix [180] $G$ by

where

Then we define $Q$ and $\Psi$ by

$$(3.7)Q(x,t)=q(x,t)+2(\nu -\bar{\nu })\frac{{\varphi }_1{\bar{\varphi }}_2}{{\varphi }_1{\bar{\varphi }}_1+{\varphi }_2{\bar{\varphi }}_2}(3.7)\; \backslash quad\; Q(x,\; t)\; =\; q(x,\; t)\; +\; 2(\backslash nu\; -\; \backslash bar\{\backslash nu\})\; \backslash frac\{\backslash phi\_1\; \backslash bar\{\backslash phi\}\_2\}\{\backslash phi\_1\; \backslash bar\{\backslash phi\}\_1\; +\; \backslash phi\_2\; \backslash bar\{\backslash phi\}\_2\}$$

and

$$(3.8)\mathrm{\Psi }(x,t;\lambda )=G(\lambda ;\nu ;\varphi )\psi (x,t;\lambda )(3.8)\; \backslash quad\; \backslash Psi(x,\; t;\; \backslash lambda)\; =\; G(\backslash lambda;\; \backslash nu;\; \backslash phi)\; \backslash psi(x,\; t;\; \backslash lambda)$$where $\psi$ solves the Lax pair (3.2,3.3) at $(q, \nu)$ . Formulas (3.7) and (3.8) are the Bäcklund-Darboux transformations for the potential and eigenfunctions, respectively. We have the following [180] [137],

THEOREM 3.3. Let $q(x, t)$ be a solution to the NLS equation (3.1), for which the linear system (3.2) has a complex double point $\nu$ of geometric multiplicity 2, with eigenbasis $(\phi^+, \phi^-)$ for the Lax pair (3.2,3.3), and define $Q(x, t)$ and $\Psi(x, t; \lambda)$ by (3.7) and (3.8). Then

1. (1) $Q(x, t)$ is an solution of NLS, with spatial period $2\pi$ ,
2. (2) $Q$ and $q$ have the same Floquet spectrum,
3. (3) $Q(x, t)$ is homoclinic to $q(x, t)$ in the sense that $Q(x, t) \rightarrow q_{\theta_{\pm}}(x, t)$ , exponentially as $\exp(-\sigma_{\nu}|t|)$ , as $t \rightarrow \pm\infty$ , where $q_{\theta_{\pm}}$ is a “torus translate” of $q$ , $\sigma_{\nu}$ is the nonvanishing growth rate associated to the complex double point $\nu$ , and explicit formulas exist for this growth rate and for the translation parameters $\theta_{\pm}$ ,
4. (4) $\Psi(x, t; \lambda)$ solves the Lax pair (3.2,3.3) at $(Q, \lambda)$ .

This theorem is quite general, constructing homoclinic solutions from a wide class of starting solutions $q(x, t)$ . Its proof is one of direct verification [118].

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 temporal structure, and is not just a fixed point. (ii) Nevertheless, the homoclinic orbit typically 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 Bäcklund-Darboux transformations to generate more complicated homoclinic orbits. (iv) The number of complex double points with nonvanishing growth rates counts the dimension of the unstable manifold of the critical torus in that two unstable directions are coordinatized by the complex ratio $c_+/c_-$ . Under even symmetry only one real dimension satisfies the constraint of evenness, as will be clearly illustrated in the following example. (v) These Bäcklund-Darboux formulas provide global expressions for the stable and unstable manifolds of the critical tori, which represent figure-eight structures.

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

$$(3.9)q_c=c\exp \{-i[2c^{2}t+\gamma ]\}\mathrm{.}(3.9)\; \backslash quad\; q\_c\; =\; c\; \backslash exp\; \backslash \{-i[2c^2\; t\; +\; \backslash gamma]\backslash \}\; .$$

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

$$(3.10){\varphi }^{(\pm )}(x,t;\lambda )=e^{\pm i\kappa (x+2\lambda t)}\left(\begin{array}{c}c{e}^{-i(2c^{2}t+\gamma )\mathrm{/}2}\\ (\pm \kappa -\lambda )e^{i(2c^{2}t+\gamma )\mathrm{/}2}\end{array}\right),(3.10)\; \backslash quad\; \backslash phi^\{(\backslash pm)\}(x,\; t;\; \backslash lambda)\; =\; e^\{\backslash pm\; i\backslash kappa(x+2\backslash lambda\; t)\}\; \backslash begin\{pmatrix\}\; ce^\{-i(2c^2\; t\; +\; \backslash gamma)/2\}\; \backslash \backslash \; (\backslash pm\backslash kappa\; -\; \backslash lambda)e^\{i(2c^2\; t\; +\; \backslash gamma)/2\}\; \backslash end\{pmatrix\}\; ,$$

where

$$\kappa =\kappa (\lambda )=\sqrt{c^2+{\lambda }^2}\mathrm{.}\backslash kappa\; =\; \backslash kappa(\backslash lambda)\; =\; \backslash sqrt\{c^2\; +\; \backslash lambda^2\}\; .$$

With these solutions one can construct the fundamental matrix

from which the Floquet discriminant can be computed:

$$(3.12)\mathrm{\Delta }(\lambda ;q_c)=2\cos (2\kappa \pi )\mathrm{.}(3.12)\; \backslash quad\; \backslash Delta(\backslash lambda;\; q\_c)\; =\; 2\; \backslash cos(2\backslash kappa\backslash pi)\; .$$

From $\Delta$ , spectral quantities can be computed:- (1) simple periodic points: $\lambda^\pm = \pm i c$ ,

• (2) double points: $\kappa(\lambda_j^{(d)}) = j/2$ , $j \in \mathbb{Z}$ , $j \neq 0$ ,
• (3) critical points: $\lambda_j^{(c)} = \lambda_j^{(d)}$ , $j \in \mathbb{Z}$ , $j \neq 0$ ,
• (4) simple periodic points: $\lambda_0^{(c)} = 0$ .

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

$$(3.13)({\lambda }_j^{(d)}{)}^2=j^2\mathrm{/}4-c^2,j=1,2,\dots ,N;(3.13)\; \backslash quad\; (\backslash lambda\_j^\{(d)\})^2\; =\; j^2/4\; -\; c^2,\; \backslash quad\; j\; =\; 1,\; 2,\; \backslash dots,\; N;$$

where

From this spectral data, the homoclinic orbits can be explicitly computed through Bäcklund-Darboux transformation. Notice that to have temporal growth (and decay) in the eigenfunctions (3.10), one needs $\lambda$ to be complex. Notice also that the Bäcklund-Darboux transformation is built with quadratic products in $\phi$ , thus choosing $\nu = \lambda_j^{(d)}$ will guarantee periodicity of $Q$ in $x$ . When $N = 1$ , the Bäcklund-Darboux transformation at one purely imaginary double point $\lambda_1^{(d)}$ yields $Q = Q(x, t; c, \gamma; c_+/c_-)$ [137]:

where $c_+/c_- \equiv \exp(\rho + i\beta)$ and $p$ is defined by $1/2 + i\sqrt{c^2 - 1/4} = c \exp(i\rho)$ , $\tau \equiv \sigma t - \rho$ , and $\vartheta = p - (\beta + \pi/2)$ .

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 $-4p$ occurs when one compares the asymptotic behavior of the orbit as $t \rightarrow -\infty$ with its behavior as $t \rightarrow +\infty$ .
• (3) For small $p$ , the formula for $Q$ becomes more transparent:

$$Q\simeq [(\cos 2p-i\sin 2p\tanh \tau )-2\sin p\mathrm{sech}\tau \cos (x+\vartheta )]c{e}^{-i(2c^{2}t+\gamma )}\mathrm{.}Q\; \backslash simeq\; \backslash left[\; (\backslash cos\; 2p\; -\; i\; \backslash sin\; 2p\; \backslash tanh\; \backslash tau)\; -\; 2\; \backslash sin\; p\; \backslash operatorname\{sech\}\; \backslash tau\; \backslash cos(x\; +\; \backslash vartheta)\; \backslash right]\; c\; e^\{-i(2c^2\; t\; +\; \backslash gamma)\}.$$

• (4) An evenness constraint on $Q$ in $x$ can be enforced by restricting the phase $\phi$ to be one of two values

$$\varphi =0,\pi \mathrm{.}(\text{evenness})\backslash phi\; =\; 0,\; \backslash pi.\; \backslash quad\; (\backslash text\{evenness\})$$

In this manner, the even symmetry disconnects the level set. Each component constitutes one loop of the figure eight. While the target $q$ is independent 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 spatial excitation located near $x = 0$ , while the second loop has the dependence $\cos(x - \pi)$ , which we interpret as spatial structure located near $x = \pi$ . In this example, the disconnected nature of the level set is clearly relatedFIGURE 1. An illustration of the figure-eight structure.

to distinct spatial structures on the individual loops. See Figure 1 for an illustration.

(5) Direct calculation shows that the transformation matrix $M(1; \lambda_1^{(d)}; Q)$ is similar to a Jordan form when $t \in (-\infty, \infty)$ ,

and when $t \rightarrow \pm\infty$ , $M(1; \lambda_1^{(d)}; Q) \rightarrow -I$ (the negative of the 2x2 identity matrix). Thus, when $t$ is finite, the algebraic multiplicity ( $= 2$ ) of $\lambda = \lambda_1^{(d)}$ 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 determined by the number of purely imaginary double points which in turn is controlled by the amplitude $c$ of the plane wave target and by the spatial period. (The dimension of the loops increases linearly with the spatial period.) When there are several complex double points, Bäcklund-Darboux transformations must be iterated to produce complete representations. Thus, Bäcklund-Darboux transformations give global representations of the figure-eight structures.

3.1.1. Linear Instability. The above figure-eight structure corresponds to the following linear instability of Benjamin-Feir type. Consider the uniform solution to the NLS (3.1),

$$q_c=c{e}^{i\theta (t)},\theta (t)=-[2c^{2}t+\gamma ]\mathrm{.}q\_c\; =\; ce^\{i\backslash theta(t)\},\; \backslash quad\; \backslash theta(t)\; =\; -[2c^2t\; +\; \backslash gamma].$$

Let

$$q=[c+\tilde{q}]e^{i\theta (t)},q\; =\; [c\; +\; \backslash tilde\{q\}]e^\{i\backslash theta(t)\},$$

and linearize equation (3.1) at $q_c$ , we have

$$i{\tilde{q}}_t={\tilde{q}}_{xx}+2c^2[\tilde{q}+\overline{\stackrel{~}{q}}]\mathrm{.}i\backslash tilde\{q\}\_t\; =\; \backslash tilde\{q\}\_\{xx\}\; +\; 2c^2[\backslash tilde\{q\}\; +\; \backslash bar\{\backslash tilde\{q\}\}].$$

Assume that $\tilde{q}$ takes the form,

$$\tilde{q}=[A_j{e}^{{\mathrm{\Omega }}_{j}t}+B_j{e}^{{\bar{\mathrm{\Omega }}}_{j}t}]\cos k_{j}x,\backslash tilde\{q\}\; =\; \backslash left[\; A\_j\; e^\{\backslash Omega\_j\; t\}\; +\; B\_j\; e^\{\backslash bar\{\backslash Omega\}\_j\; t\}\; \backslash right]\; \backslash cos\; k\_j\; x,$$

where $k_j = 2j\pi$ , ( $j = 0, 1, 2, \dots$ ), $A_j$ and $B_j$ are complex constants. Then,

$${\mathrm{\Omega }}_j^{(\pm )}=\pm k_j\sqrt{4c^2-k_j^2}\mathrm{.}\backslash Omega\_j^\{(\backslash pm)\}\; =\; \backslash pm\; k\_j\; \backslash sqrt\{4c^2\; -\; k\_j^2\}.$$

Thus, we have instabilities when $c > 1/2$ .3.1.2. Quadratic Products of Eigenfunctions. Quadratic products of eigenfunctions play a crucial role in characterizing the hyperbolic structures of soliton equations. Its importance lies in the following aspects: (i). Certain quadratic products of eigenfunctions solve the linearized soliton equation. (ii). Thus, they are the perfect candidates for building a basis to the invariant linear subbundles. (iii). Also, they signify the instability of the soliton equation. (iv). Most importantly, quadratic products of eigenfunctions can serve as Melnikov vectors, e.g., for Davey-Stewartson equation [121].

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

we have the following lemma [137].

LEMMA 3.4. Let $\varphi^{(j)} = \varphi^{(j)}(t, x; \lambda, q)$ ( $j = 1, 2$ ) be any two eigenfunctions solving the Lax pair (3.2, 3.3) at an arbitrary $\lambda$ . Then

solve the same equation (3.15); thus

$$\mathrm{\Phi }=\left(\begin{array}{c}{\phi }_1^{(1)}{\phi }_1^{(2)}\\ {\phi }_2^{(1)}{\phi }_2^{(2)}\end{array}\right)+S{\left(\begin{array}{c}{\phi }_1^{(1)}{\phi }_1^{(2)}\\ {\phi }_2^{(1)}{\phi }_2^{(2)}\end{array}\right)}^{-}\backslash Phi\; =\; \backslash begin\{pmatrix\}\; \backslash varphi\_1^\{(1)\}\backslash varphi\_1^\{(2)\}\; \backslash \backslash \; \backslash varphi\_2^\{(1)\}\backslash varphi\_2^\{(2)\}\; \backslash end\{pmatrix\}\; +\; S\; \backslash begin\{pmatrix\}\; \backslash varphi\_1^\{(1)\}\backslash varphi\_1^\{(2)\}\; \backslash \backslash \; \backslash varphi\_2^\{(1)\}\backslash varphi\_2^\{(2)\}\; \backslash end\{pmatrix\}^\{-\}$$

solves the equation (3.15) and satisfies the reality condition $\Phi_2 = \overline{\Phi}_1$ .

Proof: Direct calculation leads to the conclusion. Q.E.D.

The periodicity condition $\Phi(x + 2\pi) = \Phi(x)$ can be easily accomplished. For example, we can take $\varphi^{(j)}$ ( $j = 1, 2$ ) to be two linearly independent Bloch functions $\varphi^{(j)} = e^{\sigma_j x}\psi^{(j)}$ ( $j = 1, 2$ ), where $\sigma_2 = -\sigma_1$ and $\psi^{(j)}$ are periodic functions $\psi^{(j)}(x + 2\pi) = \psi^{(j)}(x)$ . Often we choose $\lambda$ to be a double point of geometric multiplicity 2, so that $\varphi^{(j)}$ are already periodic or antiperiodic functions.

3.2. Discrete Cubic Nonlinear Schrödinger Equation

Consider the discrete focusing cubic nonlinear Schrödinger equation (DNLS)

$$(3.16)i{\dot{q}}_n=\frac{1}{h^2}[q_{n+1}-2q_n+q_{n-1}]+\mathrm{\mid }{q}_n{\mathrm{\mid }}^2(q_{n+1}+q_{n-1})-2{\omega }^2{q}_n,(3.16)\; \backslash quad\; i\backslash dot\{q\}\_n\; =\; \backslash frac\{1\}\{h^2\}[q\_\{n+1\}\; -\; 2q\_n\; +\; q\_\{n-1\}]\; +\; |q\_n|^2(q\_\{n+1\}\; +\; q\_\{n-1\})\; -\; 2\backslash omega^2\; q\_n,$$

under periodic and even boundary conditions,

$$q_{n+N}=q_n,q_{-n}=q_n,q\_\{n+N\}\; =\; q\_n,\; \backslash quad\; q\_\{-n\}\; =\; q\_n,$$

where $i = \sqrt{-1}$ , $q_n$ 's are complex variables, $n \in \mathbb{Z}$ , $\omega$ is a positive parameter, $h = 1/N$ , and $N$ is a positive integer $N \geq 3$ . The DNLS is integrable by virtue of the Lax pair [2]:

$$(3.17){\phi }_{n+1}=L_n^{(z)}{\phi }_n,(3.17)\; \backslash quad\; \backslash varphi\_\{n+1\}\; =\; L\_n^\{(z)\}\backslash varphi\_n,$$

$$(3.18){\dot{\phi }}_n=B_n^{(z)}{\phi }_n,(3.18)\; \backslash quad\; \backslash dot\{\backslash varphi\}\_n\; =\; B\_n^\{(z)\}\backslash varphi\_n,$$where

and where $z = \exp(i\lambda h)$ . Compatibility of the over-determined system (3.17,3.18) gives the “Lax representation”

$${\dot{L}}_n=B_{n+1}{L}_n-L_n{B}_n\backslash dot\{L\}\_n\; =\; B\_\{n+1\}L\_n\; -\; L\_nB\_n$$

of the DNLS (3.16). Let $M(n)$ be the fundamental matrix solution to (3.17), the Floquet discriminant is defined as

$$\mathrm{\Delta }=\text{trace}\{M(N)\}\mathrm{.}\backslash Delta\; =\; \backslash text\{trace\}\; \backslash \{M(N)\backslash \}.$$

Let $\psi^+$ and $\psi^-$ be any two solutions to (3.17), and let $W_n(\psi^+, \psi^-)$ be the Wronskian

$$W_n({\psi }^{+},{\psi }^{-})={\psi }_n^{(+,1)}{\psi }_n^{(-,2)}-{\psi }_n^{(+,2)}{\psi }_n^{(-,1)}\mathrm{.}W\_n(\backslash psi^+,\; \backslash psi^-)\; =\; \backslash psi\_n^\{(+,1)\}\backslash psi\_n^\{(-,2)\}\; -\; \backslash psi\_n^\{(+,2)\}\backslash psi\_n^\{(-,1)\}.$$

One has

$$W_{n+1}({\psi }^{+},{\psi }^{-})={\rho }_n{W}_n({\psi }^{+},{\psi }^{-}),W\_\{n+1\}(\backslash psi^+,\; \backslash psi^-)\; =\; \backslash rho\_n\; W\_n(\backslash psi^+,\; \backslash psi^-),$$

where $\rho_n = 1 + h^2|q_n|^2$ , and

$$W_N({\psi }^{+},{\psi }^{-})=D^2{W}_0({\psi }^{+},{\psi }^{-}),W\_N(\backslash psi^+,\; \backslash psi^-)\; =\; D^2\; W\_0(\backslash psi^+,\; \backslash psi^-),$$

where $D^2 = \prod_{n=0}^{N-1} \rho_n$ . Periodic and antiperiodic points $z^{(p)}$ are defined by

$$\mathrm{\Delta }(z^{(p)})=\pm 2D\mathrm{.}\backslash Delta(z^\{(p)\})\; =\; \backslash pm\; 2D.$$

A critical point $z^{(c)}$ is defined by the condition

$${\frac{d\mathrm{\Delta }}{dz}\mid }_{z=z^{(c)}}=0.\backslash left.\; \backslash frac\{d\backslash Delta\}\{dz\}\; \backslash right|\_\{z=z^\{(c)\}\}\; =\; 0.$$

A multiple point $z^{(m)}$ is a critical point which is also a periodic or antiperiodic point. The algebraic multiplicity of $z^{(m)}$ is defined as the order of the zero of $\Delta(z) \pm 2D$ . Usually it is 2, but it can exceed 2; when it does equal 2, we call the multiple point a double point, and denote it by $z^{(d)}$ . The geometric multiplicity of $z^{(m)}$ is defined as the dimension of the periodic (or antiperiodic) eigenspace of (3.17) at $z^{(m)}$ , and is either 1 or 2.

Fix a solution $q_n(t)$ of the DNLS (3.16), for which (3.17) has a double point $z^{(d)}$ of geometric multiplicity 2, which is not on the unit circle. We denote two linearly independent solutions (Bloch functions) of the discrete Lax pair (3.17,3.18) at $z = z^{(d)}$ by $(\phi_n^+, \phi_n^-)$ . Thus, a general solution of the discrete Lax pair (3.17,3.18) at $(q_n(t), z^{(d)})$ is given by

$${\varphi }_n=c^{+}{\varphi }_n^{+}+c^{-}{\varphi }_n^{-},\backslash phi\_n\; =\; c^+\; \backslash phi\_n^+\; +\; c^-\; \backslash phi\_n^-,$$

where $c^+$ and $c^-$ are complex parameters. We use $\phi_n$ to define a transformation matrix $\Gamma_n$ by

where,

From these formulae, we see that

$${\bar{a}}_n=-d_n,{\bar{b}}_n=c_n\mathrm{.}\backslash bar\{a\}\_n\; =\; -d\_n,\; \backslash quad\; \backslash bar\{b\}\_n\; =\; c\_n.$$

Then we define $Q_n$ and $\Psi_n$ by

$$(3.19)Q_n\equiv \frac{i}{h}{b}_{n+1}-a_{n+1}{q}_n(3.19)\; \backslash quad\; Q\_n\; \backslash equiv\; \backslash frac\{i\}\{h\}\; b\_\{n+1\}\; -\; a\_\{n+1\}\; q\_n$$

and

$$(3.20){\mathrm{\Psi }}_n(t;z)\equiv {\mathrm{\Gamma }}_n(z;z^{(d)};{\varphi }_n){\psi }_n(t;z)(3.20)\; \backslash quad\; \backslash Psi\_n(t;\; z)\; \backslash equiv\; \backslash Gamma\_n(z;\; z^\{(d)\};\; \backslash phi\_n)\; \backslash psi\_n(t;\; z)$$

where $\psi_n$ solves the discrete Lax pair (3.17, 3.18) at $(q_n(t), z)$ . Formulas (3.19) and (3.20) are the Bäcklund-Darboux transformations for the potential and eigenfunctions, respectively. We have the following theorem [118].

THEOREM 3.5. Let $q_n(t)$ denote a solution of the DNLS (3.16), for which (3.17) has a double point $z^{(d)}$ of geometric multiplicity 2, which is not on the unit circle. We denote two linearly independent solutions of the discrete Lax pair (3.17, 3.18) at $(q_n, z^{(d)})$ by $(\phi_n^+, \phi_n^-)$ . We define $Q_n(t)$ and $\Psi_n(t; z)$ by (3.19) and (3.20). Then

1. (1) $Q_n(t)$ is also a solution of the DNLS (3.16). (The evenness of $Q_n$ can be obtained by choosing the complex Bäcklund parameter $c^+/c^-$ to lie on a certain curve, as shown in the example below.)
2. (2) $\Psi_n(t; z)$ solves the discrete Lax pair (3.17, 3.18) at $(Q_n(t), z)$ .
3. (3) $\Delta(z; Q_n) = \Delta(z; q_n)$ , for all $z \in C$ .
4. (4) $Q_n(t)$ is homoclinic to $q_n(t)$ in the sense that $Q_n(t) \rightarrow e^{i\theta_{\pm}} q_n(t)$ , exponentially as $\exp(-\sigma|t|)$ as $t \rightarrow \pm\infty$ . Here $\theta_{\pm}$ are the phase shifts, $\sigma$ is a nonvanishing growth rate associated to the double point $z^{(d)}$ , and explicit formulas can be developed for this growth rate and for the phase shifts $\theta_{\pm}$ .

Example: We start with the uniform solution of (3.16)

$$(3.21)q_n=q_c,\mathrm{\forall }n;q_c=a\exp \{-i[2(a^2-{\omega }^2)t-\gamma ]\}\mathrm{.}(3.21)\; \backslash quad\; q\_n\; =\; q\_c,\; \backslash quad\; \backslash forall\; n;\; \backslash quad\; q\_c\; =\; a\; \backslash exp\; \backslash left\backslash \{\; -i[2(a^2\; -\; \backslash omega^2)t\; -\; \backslash gamma]\; \backslash right\backslash \}.$$

We choose the amplitude $a$ in the range

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 $z = z_1^{(d)} = z_1^{(c)}$ which corresponds to $\beta = \pi/N$ . The homoclinic orbit $Q_n$ is given by

$$(3.23)Q_n=q_c({\widehat{E}}_{n+1}{)}^{-1}[{\widehat{A}}_{n+1}-2\cos \beta \sqrt{\rho {\cos }^2\beta -1}{\widehat{B}}_{n+1}],(3.23)\; \backslash quad\; Q\_n\; =\; q\_c(\backslash hat\{E\}\_\{n+1\})^\{-1\}\; \backslash left[\; \backslash hat\{A\}\_\{n+1\}\; -\; 2\; \backslash cos\; \backslash beta\; \backslash sqrt\{\backslash rho\; \backslash cos^2\; \backslash beta\; -\; 1\}\; \backslash hat\{B\}\_\{n+1\}\; \backslash right],$$

where

where $\varphi = \sin^{-1}[\sqrt{\rho}(ha)^{-1} \sin \beta]$ , $\varphi \in (0, \pi/2)$ .

Next we study the “evenness” condition: $Q_{-n} = Q_n$ . It turns out that the choices $\vartheta = -\beta$ , $-\beta + \pi$ in the formula of $Q_n$ lead to the evenness of $Q_n$ in $n$ . In terms of figure eight structure of $Q_n$ , $\vartheta = -\beta$ corresponds to one ear of the figure eight, and $\vartheta = -\beta + \pi$ corresponds to the other ear. The even formula for $Q_n$ is given by,

$$(3.24)Q_n=q_c[\mathrm{\Gamma }\mathrm{/}{\mathrm{\Lambda }}_n-1],(3.24)\; \backslash quad\; Q\_n\; =\; q\_c\; \backslash left[\; \backslash Gamma\; /\; \backslash Lambda\_n\; -\; 1\; \backslash right],$$

where

where ‘+’ corresponds to $\vartheta = -\beta$ .

The heteroclinic orbit (3.24) represents the figure eight structure. If we denote by $S$ the circle, we have the topological identification:

$$(\text{figure 8})\otimes S=\bigcup _{p\in (-\mathrm{\infty },\mathrm{\infty }),\gamma \in [0,2\pi ]}{Q}_n(p,\gamma ,a,\omega ,\pm ,N)\mathrm{.}(\backslash text\{figure\; 8\})\; \backslash otimes\; S\; =\; \backslash bigcup\_\{p\; \backslash in\; (-\backslash infty,\; \backslash infty),\; \backslash gamma\; \backslash in\; [0,\; 2\backslash pi]\}\; Q\_n(p,\; \backslash gamma,\; a,\; \backslash omega,\; \backslash pm,\; N).$$

3.3. Davey-Stewartson II (DSII) Equations

Consider the Davey-Stewartson II equations (DSII),

$$(3.25)\{\begin{array}{c}i{\mathrm{\partial }}_{t}q=[{\mathrm{\partial }}_x^2-{\mathrm{\partial }}_y^2]q+[2(\mathrm{\mid }q{\mathrm{\mid }}^2-{\omega }^2)+u_y]q,\\ [{\mathrm{\partial }}_x^2+{\mathrm{\partial }}_y^2]u=-4{\mathrm{\partial }}_y\mathrm{\mid }q{\mathrm{\mid }}^2,\end{array}(3.25)\; \backslash quad\; \backslash begin\{cases\}\; i\backslash partial\_t\; q\; =\; [\backslash partial\_x^2\; -\; \backslash partial\_y^2]q\; +\; [2(|q|^2\; -\; \backslash omega^2)\; +\; u\_y]q,\; \backslash \backslash \; [\backslash partial\_x^2\; +\; \backslash partial\_y^2]u\; =\; -4\backslash partial\_y|q|^2,\; \backslash end\{cases\}$$

where $q$ and $u$ are respectively complex-valued and real-valued functions of three variables $(t, x, y)$ , and $\omega$ is a positive constant. We pose periodic boundary conditions,

and the even constraint,

Its Lax pair is defined as:

$$(3.26)L\psi =\lambda \psi ,(3.26)\; \backslash quad\; L\backslash psi\; =\; \backslash lambda\backslash psi,$$

$$(3.27){\mathrm{\partial }}_t\psi =A\psi ,(3.27)\; \backslash quad\; \backslash partial\_t\backslash psi\; =\; A\backslash psi,$$

where $\psi = (\psi_1, \psi_2)$ , and

$$(3.28)D^{+}=\alpha {\mathrm{\partial }}_y+{\mathrm{\partial }}_x,D^{-}=\alpha {\mathrm{\partial }}_y-{\mathrm{\partial }}_x,{\alpha }^2=-1.(3.28)\; \backslash quad\; D^+\; =\; \backslash alpha\backslash partial\_y\; +\; \backslash partial\_x,\; \backslash quad\; D^-\; =\; \backslash alpha\backslash partial\_y\; -\; \backslash partial\_x,\; \backslash quad\; \backslash alpha^2\; =\; -1.$$

$r_1$ and $r_2$ have the expressions,

$$(3.29)r_1=\frac{1}{2}[-w+iv],r_2=\frac{1}{2}[w+iv],(3.29)\; \backslash quad\; r\_1\; =\; \backslash frac\{1\}\{2\}[-w\; +\; iv],\; \backslash quad\; r\_2\; =\; \backslash frac\{1\}\{2\}[w\; +\; iv],$$

where $u$ and $v$ are real-valued functions satisfying

$$(3.30)[{\mathrm{\partial }}_x^2+{\mathrm{\partial }}_y^2]w=2[{\mathrm{\partial }}_x^2-{\mathrm{\partial }}_y^2]\mathrm{\mid }q{\mathrm{\mid }}^2,(3.30)\; \backslash quad\; [\backslash partial\_x^2\; +\; \backslash partial\_y^2]w\; =\; 2[\backslash partial\_x^2\; -\; \backslash partial\_y^2]|q|^2,$$

$$(3.31)[{\mathrm{\partial }}_x^2+{\mathrm{\partial }}_y^2]v=i4\alpha {\mathrm{\partial }}_x{\mathrm{\partial }}_y\mathrm{\mid }q{\mathrm{\mid }}^2,(3.31)\; \backslash quad\; [\backslash partial\_x^2\; +\; \backslash partial\_y^2]v\; =\; i4\backslash alpha\backslash partial\_x\backslash partial\_y|q|^2,$$

and $w = 2(|q|^2 - \omega^2) + u_y$ . Notice that DSII (3.25) is invariant under the transformation $\sigma$ :

$$(3.32)\sigma \circ (q,\bar{q},r_1,r_2;\alpha )=(q,\bar{q},-r_2,-r_1;-\alpha )\mathrm{.}(3.32)\; \backslash quad\; \backslash sigma\; \backslash circ\; (q,\; \backslash bar\{q\},\; r\_1,\; r\_2;\; \backslash alpha)\; =\; (q,\; \backslash bar\{q\},\; -r\_2,\; -r\_1;\; -\backslash alpha).$$

Applying the transformation $\sigma$ (3.32) to the Lax pair (3.26, 3.27), we have a congruent Lax pair for which the compatibility condition gives the same DSII. The congruent Lax pair is given as:

$$(3.33)\widehat{L}\widehat{\psi }=\lambda \widehat{\psi },(3.33)\; \backslash quad\; \backslash hat\{L\}\backslash hat\{\backslash psi\}\; =\; \backslash lambda\backslash hat\{\backslash psi\},$$

$$(3.34){\mathrm{\partial }}_t\widehat{\psi }=\widehat{A}\widehat{\psi },(3.34)\; \backslash quad\; \backslash partial\_t\backslash hat\{\backslash psi\}\; =\; \backslash hat\{A\}\backslash hat\{\backslash psi\},$$

where $\hat{\psi} = (\hat{\psi}_1, \hat{\psi}_2)$ , and

The compatibility condition of the Lax pair (3.26, 3.27),

$${\mathrm{\partial }}_{t}L=[A,L],\backslash partial\_t\; L\; =\; [A,\; L],$$

where $[A, L] = AL - LA$ , and the compatibility condition of the congruent Lax pair (3.33, 3.34),

$${\mathrm{\partial }}_t\widehat{L}=[\widehat{A},\widehat{L}]\backslash partial\_t\; \backslash hat\{L\}\; =\; [\backslash hat\{A\},\; \backslash hat\{L\}]$$give the same DSII (3.25). Let $(q, u)$ be a solution to the DSII (3.25), and let $\lambda_0$ be any value of $\lambda$ . Let $\psi = (\psi_1, \psi_2)$ be a solution to the Lax pair (3.26, 3.27) at $(q, \bar{q}, r_1, r_2; \lambda_0)$ . Define the matrix operator:

where $\Lambda = \alpha \partial_y - \lambda$ , and $a, b, c, d$ are functions defined as:

in which $\Lambda_1 = \alpha \partial_y - \lambda_0$ , $\Lambda_2 = \alpha \partial_y + \bar{\lambda}_0$ , and

$$\mathrm{\Delta }=-[\mathrm{\mid }{\psi }_1{\mathrm{\mid }}^2+\mathrm{\mid }{\psi }_2{\mathrm{\mid }}^2]\mathrm{.}\backslash Delta\; =\; -[|\backslash psi\_1|^2\; +\; |\backslash psi\_2|^2].$$

Define a transformation as follows:

where $\phi$ is any solution to the Lax pair (3.26, 3.27) at $(q, \bar{q}, r_1, r_2; \lambda)$ , $D^+$ and $D^-$ are defined in (3.28), we have the following theorem [121].

THEOREM 3.6. The transformation (3.35) is a Bäcklund-Darboux transformation. That is, the function $Q$ defined through the transformation (3.35) is also a solution to the DSII (3.25). The function $\Phi$ defined through the transformation (3.35) solves the Lax pair (3.26, 3.27) at $(Q, \bar{Q}, R_1, R_2; \lambda)$ .

3.3.1. An Example . Instead of using $L_1$ and $L_2$ to describe the periods of the periodic boundary condition, one can introduce $\kappa_1$ and $\kappa_2$ as $L_1 = \frac{2\pi}{\kappa_1}$ and $L_2 = \frac{2\pi}{\kappa_2}$ . Consider the spatially independent solution,

$$(3.36)q_c=\eta \exp \{-2i[{\eta }^2-{\omega }^2]t+i\gamma \}\mathrm{.}(3.36)\; \backslash quad\; q\_c\; =\; \backslash eta\; \backslash exp\backslash \{-2i[\backslash eta^2\; -\; \backslash omega^2]t\; +\; i\backslash gamma\backslash \}.$$

The dispersion relation for the linearized DSII at $q_c$ is

$$\mathrm{\Omega }=\pm \frac{\mathrm{\mid }{\xi }_1^2-{\xi }_2^2\mathrm{\mid }}{\sqrt{{\xi }_1^2+{\xi }_2^2}}\sqrt{4{\eta }^2-({\xi }_1^2+{\xi }_2^2)},\text{for }\delta q\sim q_c\exp \{i({\xi }_{1}x+{\xi }_{2}y)+\mathrm{\Omega }t\},\backslash Omega\; =\; \backslash pm\; \backslash frac\{|\backslash xi\_1^2\; -\; \backslash xi\_2^2|\}\{\backslash sqrt\{\backslash xi\_1^2\; +\; \backslash xi\_2^2\}\}\; \backslash sqrt\{4\backslash eta^2\; -\; (\backslash xi\_1^2\; +\; \backslash xi\_2^2)\},\; \backslash quad\; \backslash text\{for\; \}\; \backslash delta\; q\; \backslash sim\; q\_c\; \backslash exp\backslash \{i(\backslash xi\_1\; x\; +\; \backslash xi\_2\; y)\; +\; \backslash Omega\; t\backslash \},$$where $\xi_1 = k_1 \kappa_1$ , $\xi_2 = k_2 \kappa_2$ , and $k_1$ and $k_2$ are integers. We restrict $\kappa_1$ and $\kappa_2$ as follows to have only two unstable modes $(\pm \kappa_1, 0)$ and $(0, \pm \kappa_2)$ ,

or

The Bloch eigenfunction of the Lax pair (3.26) and (3.27) is given as,

$$(3.37)\psi =c(t)\left[\begin{array}{c}-q_c\\ \chi \end{array}\right]\exp \{i({\xi }_{1}x+{\xi }_{2}y)\},(3.37)\; \backslash quad\; \backslash psi\; =\; c(t)\; \backslash begin\{bmatrix\}\; -q\_c\; \backslash \backslash \; \backslash chi\; \backslash end\{bmatrix\}\; \backslash exp\backslash \{i(\backslash xi\_1\; x\; +\; \backslash xi\_2\; y)\backslash \},$$

where

For the iteration of the Bäcklund-Darboux transformations, one needs two sets of eigenfunctions. First, we choose $\xi_1 = \pm \frac{1}{2}\kappa_1$ , $\xi_2 = 0$ , $\lambda_0 = \sqrt{\eta^2 - \frac{1}{4}\kappa_1^2}$ (for a fixed branch),

$$(3.38){\psi }^{\pm }=c^{\pm }\left[\begin{array}{c}-q_c\\ {\chi }^{\pm }\end{array}\right]\exp \{\pm i\frac{1}{2}{\kappa }_{1}x\},(3.38)\; \backslash quad\; \backslash psi^\backslash pm\; =\; c^\backslash pm\; \backslash begin\{bmatrix\}\; -q\_c\; \backslash \backslash \; \backslash chi^\backslash pm\; \backslash end\{bmatrix\}\; \backslash exp\backslash left\backslash \{\backslash pm\; i\backslash frac\{1\}\{2\}\backslash kappa\_1\; x\backslash right\backslash \},$$

where

We apply the Bäcklund-Darboux transformations with $\psi = \psi^+ + \psi^-$ , which generates the unstable foliation associated with the $(\kappa_1, 0)$ and $(-\kappa_1, 0)$ linearly unstable modes. Then, we choose $\xi_2 = \pm \frac{1}{2}\kappa_2$ , $\lambda = 0$ , $\xi_1^0 = \sqrt{\eta^2 - \frac{1}{4}\kappa_2^2}$ (for a fixed branch),

$$(3.39){\varphi }_{\pm }=c_{\pm }\left[\begin{array}{c}-q_c\\ {\chi }_{\pm }\end{array}\right]\exp \{i({\xi }_1^{0}x\pm \frac{1}{2}{\kappa }_{2}y)\},(3.39)\; \backslash quad\; \backslash phi\_\backslash pm\; =\; c\_\backslash pm\; \backslash begin\{bmatrix\}\; -q\_c\; \backslash \backslash \; \backslash chi\_\backslash pm\; \backslash end\{bmatrix\}\; \backslash exp\backslash left\backslash \{i(\backslash xi\_1^0\; x\; \backslash pm\; \backslash frac\{1\}\{2\}\backslash kappa\_2\; y)\backslash right\backslash \},$$

where

We start from these eigenfunctions $\phi_\pm$ to generate $\Gamma\phi_\pm$ through Bäcklund-Darboux transformations, and then iterate the Bäcklund-Darboux transformations with $\Gamma\phi_+ + \Gamma\phi_-$ to generate the unstable foliation associated with all the linearly unstable modes $(\pm \kappa_1, 0)$ and $(0, \pm \kappa_2)$ . It turns out that the following representations are