近可积无穷维动力系统 郭柏灵,张隽,李景 等 编 专业科技 文轩网

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 : Ⅱ0 →Ⅱ0 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 Ⅱc 192
5.3 The equations in a neighborhood of the circle of fixed points 197
5.3.1 Basic equations 197
5.3.2 Normal forms 200
5.3.3 Local equations 205
5.4 Theory of invariant manifolds 206
5.4.1 Existence of local invariant manifolds 206
5.4.2 The fibration for invariant manifolds 216
5.4.3 Stable manifold to Q in M" 224
5.5 Global integrable theory 231
5.5.1 Lax pair 231
5.5.2 Zakharov-Shabat spectral problem 231
5.5.3 The basic example 234
5.5.4 Homoclinic orbits and whiskered tori 235
5.5.5 An important invariant 239
5.5.6 F0(qh) 241
5.6 Persistent homoclinic orbit 242
5.6.1 The first measurement 243
5.6.2 The second measurement 250
5.6.3 Existence of a homoclinic orbit 254
Chapter 6 Homoclinic Orbits and Chaos for the Discrete Disturbed Nonlinear Schr.odinger Equation 257
6.1 Integrable case 257
6.1.1 Spectral theory of Ln 259
6.1.2 Hyperbolic structure and homoclinic orbits 260
6.2 Persistent invariant manifolds 263
6.2.1 Persistent invariant plane 264
6.2.2 Persistent invariant manifold theorem 265
6.2.3 The proof of the local persistent invariant manifold theorem 267
6.3 Fenichel fibers 272
6.3.1 An example showing fenichel fibers 272
6.3.2 Fiber theorem 273
6.3.3 The unique explicit fenichel fiber for \figure 8 - A" 275
6.4 Melnikov measurement: Wu(q") \Wcs " 276
6.4.1 Main argument 276
6.4.2 Derivation of Melnikov integral 280
6.4.3 Approximation 290
6.4.4 Computation for ^M~ F1 292
6.4.5 The intersection between Wu(q") and Ws(M") Wcs " 294
6.5 Existence of orbits homoclinic to q": the second measurement 295
6.6 General theory of symbolic dynamics 302
6.6.1 General framework 302
6.6.2 Smooth normal form reduction 304
6.6.3 Some definitions 305
6.6.4 Poincar.e map P1 0 310
6.6.5 Poincar.e map P0 1 310
6.6.6 Fixed point of Poincar.e map P ′ P0 1 ± P10 313
6.6.7 Smale horseshoes 322
6.6.8 Symbol dynamics 333
6.7 Application to discrete NLS systems 337
6.7.1 Transformation of (6.6.1) to the form (6.1.3) 337
6.7.2 The Generic assumptions 338
6.7.3 Smale horseshoes and chaos created by a pair of homoclinic orbits in the discrete nonlinear Schr.oinger systems 339
Chapter 7 Persistent Homoclinic Orbits for the Perturbed Sine-Gordon Equation 348
7.1 Persistent homoclinic orbits for a kind of Sine-Gordon equation under dissipative perurbation 348
7.2 Persistent homoclinic orbits for another kind of Sine-Gordon equation under dissipative perturbation 355
7.3 Persistent homoclinic orbits for a kind of Klein-Gordon equation under small perturbation 373
Chapter 8 Persistent Homoclinic Orbits of Perturbed High-order Nonlinear Schr.odinger Equtions 381
8.1 Persistent homoclinic orbits of a perturbed cubic-quintic NLS equation 381
8.1.1 Some fundamental results 381
8.1.2 The equations in a neighborhood of C! 387
8.1.3 Invariant manifolds 390
8.1.4 Persistent homoclinic orbit 399
8.2 Homoclinic orbits in a six dimensional model of derivative nonlinear Schr.odinger equation 405
8.2.1 The Fourier truncation of a perturbed derivative NLS equation 406
8.2.2 Persistence of the normally hyperbolic invariant manifold 413
8.2.3 Persistence of the homoclinic orbits 416
8.3 Persistent homoclinic orbits for a perturbed coupled nonlinear Schr.odinger system 420
8.3.1 The preliminary results 420
8.3.2 An equation in a neighborhood of Sw 426
8.3.3 Existence of local invariant manifolds 432
8.3.4 Homoclinic orbit of unperturbed system 441
8.3.5 Persistent homoclinic orbit 442
8.4 Persistent homoclinic orbits for a perturbed nonlinear Schr.odingerequation with derivation term under a small perturbation 448
8.4.1 The preliminary results 448
8.4.2 Analysis of space-independent solutions 449
8.4.3 Equation in a neighborhood of C! 452
8.4.4 Invariant manifolds 454
8.4.5 Persistent homoclinic orbit 461
Chapter 9 Homoclinic Orbits of a Perturbed Nonlinear Schr.odinger Equation 468
9.1 Main theorems and establishment of basic equations 468
9.2 Invariant manifolds and invariant foliations 472
9.3 Homoclinic orbits 509
9.3.1 Homoclinic orbits for unperturbed NLS 510
9.3.2 The first measurement 512
9.3.3 Second measurement 519
9.3.4 Existence of a Homoclinic Orbit 522
Chapter 10 Morse Functions and Floquet Theory 525
10.1 Morse and Melnikov functions for nonlinear Schr.odinger equation 525
10.1.1 Floquet Spectral Theory 526
10.1.2 Critical Structure of Fj 532
10.1.3 The Morse description of the isospectral stratification 536
10.1.4 A Melnikov vector 547
10.2 Hill equation 548
10.3 Topological classification of integrable partial di.erential equations 556
Bibliography 561