孤立子、非线性发展方程和逆散射【2025|PDF下载-Epub版本|mobi电子书|kindle百度云盘下载】

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1 Introduction1

1.1 Historical remarks and applications1

1.2 Physical Derivation of the Kadomtsev-Petviashvili equation8

1.3 Travelling wave solutions of the Korteweg-de Vries equation13

1.4 The discovery of the soliton17

1.5 An infinite number of conserved quantities19

1.6 Fourier transforms21

1.7 The associated linear scattering problem and inverse scattering24

1.7.1 The inverse scattering method24

1.7.2 Reflectionless potentials27

1.8 Lax's generalization32

1.9 Linear scattering problems and associated nonlinear evolution equations34

1.10 Generalizations of the I.S.T.in one spatial dimension42

1.11 Classes of integrable equations48

1.11.1 Ordinary differential equations48

1.11.2 Partial differential equations in one spatial dimension49

1.11.3 Differential-difference equations55

1.11.4 Singular integro-differential equations57

1.11.5 Partial differential equations in two spatial dimensions59

1.11.6 Multidimensional scattering equations65

1.11.7 Multidimensional differential geometric equations67

1.11.8 The Self-dual Yang-Mills equations68

2 Inverse Scattering for the Korteweg-de Vries Equation70

2.1 Introduction70

2.2 The direct scattering problem70

2.3 The inverse scattering problem79

2.4 The time dependence81

2.5 Further remarks83

2.5.1 Soliton solutions83

2.5.2 Delta-function initial profile83

2.5.4 The Gel'fand-Levitan-Marchenko integral equation84

2.5.3 A general class of solutions of the Korteweg-de Vries equation85

2.6 Properties of completely integrable equations88

2.6.1 Solitons88

2.6.2 Infinite number of conservation laws89

2.6.3 Compatibility of linear operators89

2.6.4 Completely integrable Hamiltonian system and action-angle variables90

2.6.5 Bilinear representation94

2.6.6 B？ckland transformations96

2.6.7 Painlevé property98

2.6.8 Prolongation structure100

3 General Inverse Scattering in One Dimension105

3.1 Inverse scattering and Riemann-Hilbert problems for N×N matrix systems105

3.1.1 The direct and inverse scattering problems:2nd order case105

3.1.2 The direct and inverse scattering problems:Nth order case111

3.1.3 The time dependence115

3.1.4 Hamiltonian system and action-angle variables for the nonlinear Schr？dinger equation117

3.1.5 Riemann-Hilbert problems for Nth order Sturm-Liouville scattering problems119

3.2 Riemann-Hilbert problems for discrete scattering problems121

3.2.1 Differential-difference equations:discrete Schr？dinger scattering problem121

3.2.2 Differential-difference equations:discrete 2×2 scattering problem123

3.2.3 Partial-difference equations125

3.3 Homoclinic structure and numerically induced chaos for the nonlinear Schr？dinger equation127

3.3.1 Introduction127

3.3.2 A linearized stability analysis130

3.3.3 Hirota's method for the single homoclinic orbit131

3.3.4 Combination homoclinic orbits134

3.3.5 Numerical homoclinic instability137

3.3.6 Duffing's equations and Mel'nikov analysis150

3.4 Cellular Automata152

4 Inverse Scattering for Integro-Differential Equations163

4.1 Introduction163

4.2 The intermediate long wave equation164

4.2.1 The direct scattering problem164

4.2.2 The inverse scattering problem168

4.2.3 The time dependence171

4.2.4 Further remarks171

4.3 The Benjamin-Ono equation173

4.3.1 The direct scattering problem173

4.3.2 The inverse scattering problem175

4.3.3 The time dependence179

4.3.4 Further remarks180

4.4 Classes of integrable integro-differential equations182

4.4.1 Introduction182

4.4.2 The Sine-Hilbert equation187

4.4.3 Further examples192

5 Inverse Scattering in Two Dimensions195

5.1 Introduction195

5.2 The Kadomtsev-Petviashvili Ⅰ equation199

5.2.1 The direct scattering problem199

5.2.2 The inverse scattering problem206

5.2.3 The time dependence207

5.2.4 Further remarks208

5.3 The Kadomtsev-Petviashvili Ⅱ equation212

5.3.1 The direct scattering problem212

5.3.2 The inverse scattering problem215

5.3.3 The time dependence217

5.3.4 Comments on rigorous analysis218

5.3.5 Boundary conditions and the choice of the operator？221

5.3.6 Hamiltonian formalism and action-angle variables225

5.4 Hyperbolic and elliptic systems in the plane227

5.4.1 Hyperbolic systems228

5.4.2 Elliptic systems234

5.4.3 The n-wave interaction equations236

5.4.5 Comments on rigorous analysis for the elliptic scattering problem238

5.5 The Davey-Stewartson Equations240

5.5.1 Introduction240

5.5.2 Inverse scattering for the DSⅠ equations242

5.5.3 Inverse scattering for the DSⅡ equations244

5.5.4 The strong coupling limit246

5.5.5 The ？-limit case248

5.5.6 Hamiltonian formalism for the DSⅡ equations254

5.5.7 Localized solitons of the DSⅠ equations260

5.5.8 On the physical derivation of the boundary conditions for the Davey-Stewartson Equations264

5.6 Further Examples267

5.6.1 Equations related to the Davey-Stewartson equation267

5.6.2 Multidimensional isospectral flows associated with second order scalar operators268

6 Inverse Scattering in Multidimensions272

6.1 Introduction272

6.2 Multidimensional inverse scattering associated with the"time"-dependent and"time:-independent Schr？dinger equation274

6.2.1 The direct scattering problem274

6.2.2 The inverse scattering problem276

6.2.3 The characterization problem278

6.2.4 The"time"-dependent Schr？dinger equation281

6.2.5 The"time"-independent Schr？dinger equation284

6.2.6 The relationship between the inverse data and the scattering data287

6.2.7 Further remarks290

6.3 Multidimensional inverse scattering for first order systems291

6.3.1 The direct and inverse scattering problems291

6.3.2 The characterization problem294

6.3.3 The hyperbolic limit298

6.3.4 The N-wave interaction equations302

6.4 The Generalized Wave and Generalized Sine-Gordon equations304

6.4.1 Introduction304

6.4.2 The direct and inverse scattering problems for the Generalized Wave Equation308

6.4.3 The direct and inverse scattering problems for the Generalized Sine-Gordon Equation312

6.4.4 Further remarks315

6.5 The Self-dual Yang-Mills equations316

6.5.1 Introduction316

6.5.2 Reductions to 2+1-dimensional equations320

6.5.3 Reductions to l+1-dimensional equations328

6.5.4 Reductions to ordinary differential equations332

6.5.5 The SDYM hierarchy344

7 The Painlevé Equations347

7.1 Historical origins and physical applications347

7.1.1 Singularities of ordinary differential equations347

7.1.2 First order ordinary differential equations349

7.1.3 The work of Sophie Kowalevski349

7.1.4 Second order ordinary differential equations352

7.1.5 Third and higher order ordinary differential equations354

7.1.6 Physical applications358

7.2 The Painlevétests359

7.2.1 The relationship between the Painlevé equations and inverse scattering359

7.2.2 The Painlevé ODE test362

7.2.3 Applications of the Painlevé ODE test365

7.2.4 The Painlevé PDE test370

7.2.5 Applications of the PainlevéPDE test373

7.2.6 Quasilinear partial differential equations and the Painlevé tests386

7.3 Inverse Problems for the Painlevéequations390

7.3.1 Inverse scattering for the Modified KdV equation390

7.3.2 Gel'fand-Levitan-Marchenko integral equation method393

7.3.3 The Inverse Monodromy Transform method:introduction395

7.3.4 The Inverse Monodromy Transform method:direct problem398

7.3.5 The Inverse Monodromy Transform method:inverse problem401

7.4 Connection formulae for the Painlevé equations404

7.4.1 Introduction404

7.4.2 The Gel'fand-Levitan-Marchenko integral equation approach406

7.4.3 The Inverse Monodromy Transform approach414

7.5 Properties of the Painlevé equations420

8 Further Remarks and Open Problems424

8.1 Multidimensional equations425

8.2 Boundary value problems426

8.3 Ordinary differential equations430

8.4 Functional analysis and 2+1-dimensions432

8.5 Quantum inverse scattering and statistical mechanics435

8.6 Complete integrability438

Appendix A:Remarks on Riemann-Hilbert problems440

Appendix B:Remarks on ？ problems453

References459

Subject Index513