โ Scribed by M. A. Ablowitz, P. A. Clarkson
No coin nor oath required. For personal study only.
This book brings together several aspects of soliton theory currently available only in research papers. Emphasis is given to the multi-dimensional problems which arise and includes inverse scattering in multi-dimensions, integrable nonlinear evolution equations in multi-dimensions and the dbar method.
Contents......Page 5
Preface......Page 11
1.1 Historical remarks and applications......Page 13
1.2 Physical Derivation of the Kadomtsev-Petviashvili equation......Page 20
1.3 Travelling wave solutions of the Korteweg-de Vries equation......Page 25
1.4 The discovery of the soliton......Page 29
1.5 An infinite number of conserved quantities......Page 31
1.6 Fourier transforms......Page 33
1.7.1 The inverse scattering method......Page 36
1.7.2 Reflectionless potentials......Page 39
1.8 Lax's generalization......Page 44
1.9 Linear scattering problems and associated nonlinear evolution equations......Page 46
1.10 Generalizations of the I.S.T. in one spatial dimension......Page 54
1.11.1 Ordinary differential equations......Page 60
1.11.2 Partial differential equations in one spatial dimension......Page 61
1.11.3 Differential-difference equations......Page 67
1.11.4 Singular integro-differential equations......Page 69
1.11.5 Partial differential equations in two spatial dimensions......Page 71
1.11.6 Multidimensional scattering equations......Page 77
1.11.7 Multidimensional differential geometric equations......Page 79
1.11.8 The Self-dual Yang-Mills equations......Page 80
2.2 The direct scattering problem......Page 82
2.3 The inverse scattering problem......Page 91
2.4 The time dependence......Page 93
2.5.2 Delta-function initial profile......Page 95
2.5.4 The Gel'fand-Levitan-Marchenko integral equation......Page 96
2.5.3 A general class of solutions of the Korteweg-de Vries equation......Page 97
2.6.1 Solitons......Page 100
2.6.3 Compatibility of linear operators......Page 101
2.6.4 Completely integrable Hamiltonian system and action-angle variables......Page 102
2.6.5 Bilinear representation......Page 106
2.6.6 Backland transformations......Page 108
2.6.7 Painleve property......Page 110
2.6.8 Prolongation structure......Page 112
3.1.1 The direct and inverse scattering problems: 2nd order case......Page 117
3.1.2 The direct and inverse scattering problems: Nth order case......Page 123
3.1.3 The time dependence......Page 127
3.1.4 Hamiltonian system and action-angle variables for the nonlinear Schrodinger equation......Page 129
3.1.5 Riemann-Hilbert problems for Nth order Sturm-Liouville scattering problems......Page 131
3.2.1 Differential-difference equations: discrete Schrodinger scattering problem......Page 133
3.2.2 Differential-difference equations: discrete 2 x 2 scattering problem......Page 135
3.2.3 Partial-difference equations......Page 137
3.3.1 Introduction......Page 139
3.3.2 A linearized stability analysis......Page 142
3.3.3 Hirota's method for the single homoclinic orbit......Page 143
3.3.4 Combination homoclinic orbits......Page 146
3.3.5 Numerical homoclinic instability......Page 149
3.3.6 Duffing's equations and Mel'nikov analysis......Page 162
3.4 Cellular Automata......Page 164
4.1 Introduction......Page 175
4.2.1 The direct scattering problem......Page 176
4.2.2 The inverse scattering problem......Page 180
4.2.4 Further remarks......Page 183
4.3.1 The direct scattering problem......Page 185
4.3.2 The inverse scattering problem......Page 187
4.3.3 The time dependence......Page 191
4.3.4 Further remarks......Page 192
4.4.1 Introduction......Page 194
4.4.2 The Sine-Hilbert equation......Page 199
4.4.3 Further examples......Page 204
5.1 Introduction......Page 207
5.2.1 The direct scattering problem......Page 211
5.2.2 The inverse scattering problem......Page 218
5.2.3 The time dependence......Page 219
5.2.4 Further remarks......Page 220
5.3.1 The direct scattering problem......Page 224
5.3.2 The inverse scattering problem......Page 227
5.3.3 The time dependence......Page 229
5.3.4 Comments on rigorous analysis......Page 230
5.3.5 Boundary conditions and the choice of the operator 8z 1......Page 233
5.3.6 Hamiltonian formalism and action-angle variables......Page 237
5.4 Hyperbolic and elliptic systems in the plane......Page 239
5.4.1 Hyperbolic systems......Page 240
5.4.2 Elliptic systems......Page 246
5.4.3 The n-wave interaction equations......Page 248
5.4.5 Comments on rigorous analysis for the elliptic scattering problem......Page 250
5.5.1 Introduction......Page 252
5.5.2 Inverse scattering for the DSI equations......Page 254
5.5.3 Inverse scattering for the DSII equations......Page 256
5.5.4 The strong coupling limit......Page 258
5.5.5 The i-limit case......Page 260
5.5.6 Hamiltonian formalism for the DSII equations......Page 266
5.5.7 Localized solitons of the DSI equations......Page 272
5.5.8 On the physical derivation of the boundary conditions for the Davey-Stewartson Equations......Page 276
5.6.1 Equations related to the Davey-Stewartson equation......Page 279
5.6.2 Multidimensional isospectral flows associated with second order scalar operators......Page 280
6.1 Introduction......Page 284
6.2.1 The direct scattering problem......Page 286
6.2.2 The inverse scattering problem......Page 288
6.2.3 The characterization problem......Page 290
6.2.4 The "time"-dependent Schrodinger equation......Page 293
6.2.5 The "time"-independent Schrodinger equation......Page 296
6.2.6 The relationship between the inverse data and the scattering data......Page 299
6.2.7 Further remarks......Page 302
6.3.1 The direct and inverse scattering problems......Page 303
6.3.2 The characterization problem......Page 306
6.3.3 The hyperbolic limit......Page 310
6.3.4 The N-wave interaction equations......Page 314
6.4.1 Introduction......Page 316
6.4.2 The direct and inverse scattering problems for the Generalized Wave Equation......Page 320
6.4.3 The direct and inverse scattering problems for the Generalized Sine-Gordon Equation......Page 324
6.4.4 Further remarks......Page 327
6.5.1 Introduction......Page 328
6.5.2 Reductions to 2 + 1-dimensional equations......Page 332
6.5.3 Reductions to 1 + 1-dimensional equations......Page 340
6.5.4 Reductions to ordinary differential equations......Page 344
6.5.5 The SDYM hierarchy......Page 356
7.1.1 Singularities of ordinary differential equations......Page 359
7.1.3 The work of Sophie Kowalevski......Page 361
7.1.4 Second order ordinary differential equations......Page 364
7.1.5 Third and higher order ordinary differential equations......Page 366
7.1.6 Physical applications......Page 370
7.2.1 The relationship between the Painleve equations and inverse scattering......Page 371
7.2.2 The Painleve ODE test......Page 374
7.2.3 Applications of the Painleve ODE test......Page 377
7.2.4 The Painleve PDE test......Page 382
7.2.5 Applications of the Painleve PDE test......Page 385
7.2.6 Quasilinear partial differential equations and the Painleve tests......Page 398
7.3.1 Inverse scattering for the Modified KdV equation......Page 402
7.3.2 Gel'fand-Levitan- Ma.rchenko integral equation method......Page 405
7.3.3 The Inverse Monodromy Transform method: introduction......Page 407
7.3.4 The Inverse Monodromy Transform method: direct problem......Page 410
7.3.5 The Inverse Monodromy Transform method: inverse problem......Page 413
7.4.1 Introduction......Page 416
7.4.2 The Gel'fa.nd-Levitan-Marchenko integral equation approach......Page 418
7.4.3 The Inverse Monodromy Transform approach......Page 426
7.5 Properties of the Painleve equations......Page 432
8 Further Remarks and Open Problems......Page 436
8.1 Multidimensional equations......Page 437
8.2 Boundary value problems......Page 438
8.3 Ordinary differential equations......Page 442
8.4 Functional analysis and 2 + 1-dimensions......Page 444
8.5 Quantum inverse scattering and statistical mechanics......Page 447
8.6 Complete integrability......Page 450
Appendix A: Remarks on Riemann-Hilbert problems......Page 452
Appendix B: Remarks on 0 problems......Page 465
References......Page 471
Subject Index......Page 525
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