Contents
Chapter 1 Chaos for Nearly Integrable Systems 1
1.1 Direct methods of perturbation theory for solitons 1
1.2 Perturbation theory based on the inverse scattering transform 4
1.3 Motion of a soliton in a driven Sine-Gordon equation 8
1.3.1 Soliton motion of Sine-Gordon equation 8
1.3.2 Motion of a SG soliton in the fields of two waves 10
1.3.3 Stochastic dynamics of a three-dimensional bubble in a driven SG equation 11
1.3.4 SG soliton similar to the Fermi-Pasta-Ulam problem 13
1.3.5 Dynamical chaos of a breather under the action of an external field 14
1.3.6 Dynamical chaos in the SG system with parametric excitation 16
1.3.7 Stochastization of soliton lattices in the perturbed SG equation 18
1.4 Motion of the soliton of nonlinear Schr.odinger equation with damping under the action of an external field 20
1.4.1 Nonlinear Schr.odinger equation 20
1.4.2 Stochastic dynamics of NLS solitons in a periodic potential 20
1.5 Dynamical chaos of the KdV equation and the perturbation equations 23
1.5.1 Chaotic state of the cnoidal wave in the periodic inhomogeneous medium 23
1.5.2 Karamoto-Sivashinsky equation 24
Chapter 2 Some Numerical Results and Their Analysis 26
2.1 Coherent structure and numerical calculation results 27
2.2 Fundamental analysis 54
2.2.1 Connections between NLS equation and Sine-Gordon equation 54
2.2.2 Space independent fixed point 55
2.2.3 Space dependent fixed point 57
2.2.4 Integrable structure of nonlinear Schr.odinger equation 59
2.2.5 The Whisker ring of focusing nonlinear Schr.odinger equation 75
Chapter 3 Homoclinic Orbits in a Four Dimensional Model of a Perturbed Nonlinear Schr.odinger Equation 91
3.1 Dynamics and geometric structure for the unperturbed systerm 91
3.1.1 M0 and Ws(M0)TWu(M0) 93
3.1.2 The dynamics on M0 95
3.1.3 The unperturbed homoclinic orbits and their relationship to the dynamics on M0 and Ws(M0) \Wu(M0) 95
3.2 Geometric structure of the perturbed systerm 98
3.2.1 The persistence of M0;Ws(M0) and Wu(M0) under perturbation 99
3.2.2 The dynamics on M" near resonance 99
3.3 Fiber representations of stable and unstable manifolds 103
3.3.1 Representation of Ws(M0) and Wu(M0) through homoclinic orbits 103
3.3.2 An intuitive introduction to fibrations of stable and unstable manifolds 104
3.3.3 A second example 107
3.3.4 Fibers for Ws(M0) and Wu(M0) of the two mode equations 111
3.3.5 Properties and characteristics of the fibers 112
3.3.6 Fibers representations for the subset of Wu(q") and Ws loc(A M") 113
3.4 Homoclinic orbits for q" 114
3.4.1 Homoclinic coordinates and the hyperplane § 115
3.4.2 The Melnikov function for Ws(A M")TWu(q") 117
3.4.3 Explicit expression of the Melnikov function at I = 1 121
3.4.4 The existence of orbits homoclinic to q" 124
3.5 Numerical results of orbits homoclinic to q" 130
3.5.1 Numerical computation for periodic solution 130
3.5.2 Computation for homoclinic manifolds 131
3.6 The dynamical consequences of orbits homoclinic to q": the existence and property of chaos 137
3.6.1 Construction of the domains for the maps 139
3.6.2 Construction of the map P0 near the origin 140
3.6.3 Construction of the map along the homoclinic orbits outside a neighborhood of the origin 143
3.6.4 The full map�� P �� P0 ± P1 : ��0 ����0 145
3.6.5 Verification of the hypotheses of the theorem for the two-mode truncation 146
Chapter 4 Homoclinic Orbits of a Damped and Forced Sine-Gordon Equation 150
4.1 Structure of the unperturbed system 151
4.1.1 The normally hyperbolic invariant manifold M 151
4.1.2 The dynamics on M 152
4.1.3 Ws(M);Wu(M) and the homoclinic manifold 152
4.1.4 The dynamics on and its relation to the dynamics in M 153
4.2 Structure of the perturbed system 154
4.2.1 The persistence of M�� Ws(M) and Wu(M) under perturbation 154
4.2.2 The dynamics on M" 156
4.2.3 The fibering of Ws(A") and Wu(A"): the singular perturbation nature 160
4.3 The existence of a homoclinic connection to p" 163
4.3.1 Wu(p") Ws(A"): The higher dimensional Melnikov theory 164
4.3.2 Wu(p") \Ws(p"): a homoclinic orbit to p" 166
4.4 Chaos: Silnikov's theorem 170
4.5 An application:model dynamics of the damped�� driven�� nonlinear Schr.odinger equation 171
4.5.1 The unperturbed integrable structure 173
4.5.2 Dynamics near the resonance on A" 178
4.5.3 Calculation of the Melnikov function 180
4.5.4 The existence of an orbit homoclinic to p" 183
4.5.5 The geometrical interpretation of chaos in phase space 185
Chapter 5 Persistent Homoclinic Orbits for a Perturbed Nonlinear Schr.odinger Equation 189
5.1 Introduction 189
5.2 Analysis of space-independent solutions and motion on the invariant plane 190
5.2.1 Motion on the invariant plane 190
5.2.2 The stable manifolds at Q in ��