1 Introduction 1
1.1 Historical remarks and applications 1
1.2 Physical Derivation of the Kadomtsev-Petviashvili equation 8
1.3 Travelling wave solutions of the Korteweg-de Vries equation 13
1.4 The discovery of the soliton 17
1.5 An infinite number of conserved quantities 19
1.6 Fourier transforms 21
1.7 The associated linear scattering problem and inverse scattering 24
1.7.1 The inverse scattering method 24
1.7.2 Reflectionless potentials 27
1.8 Lax's generalization 32
1.9 Linear scattering problems and associated nonlinear evolution equations 34
1.10 Generalizations of the I.S.T.in one spatial dimension 42
1.11 Classes of integrable equations 48
1.11.1 Ordinary differential equations 48
1.11.2 Partial differential equations in one spatial dimension 49
1.11.3 Differential-difference equations 55
1.11.4 Singular integro-differential equations 57
1.11.5 Partial differential equations in two spatial dimensions 59
1.11.6 Multidimensional scattering equations 65
1.11.7 Multidimensional differential geometric equations 67
1.11.8 The Self-dual Yang-Mills equations 68
2 Inverse Scattering for the Korteweg-de Vries Equation 70
2.1 Introduction 70
2.2 The direct scattering problem 70
2.3 The inverse scattering problem 79
2.4 The time dependence 81
2.5 Further remarks 83
2.5.1 Soliton solutions 83
2.5.2 Delta-function initial profile 83
2.5.4 The Gel'fand-Levitan-Marchenko integral equation 84
2.5.3 A general class of solutions of the Korteweg-de Vries equation 85
2.6 Properties of completely integrable equations 88
2.6.1 Solitons 88
2.6.2 Infinite number of conservation laws 89
2.6.3 Compatibility of linear operators 89
2.6.4 Completely integrable Hamiltonian system and action-angle variables 90
2.6.5 Bilinear representation 94
2.6.6 B？ckland transformations 96
2.6.7 Painlevé property 98
2.6.8 Prolongation structure 100
3 General Inverse Scattering in One Dimension 105
3.1 Inverse scattering and Riemann-Hilbert problems for N×N matrix systems 105
3.1.1 The direct and inverse scattering problems:2nd order case 105
3.1.2 The direct and inverse scattering problems:Nth order case 111
3.1.3 The time dependence 115
3.1.4 Hamiltonian system and action-angle variables for the nonlinear Schr？dinger equation 117
3.1.5 Riemann-Hilbert problems for Nth order Sturm-Liouville scattering problems 119
3.2 Riemann-Hilbert problems for discrete scattering problems 121
3.2.1 Differential-difference equations:discrete Schr？dinger scattering problem 121
3.2.2 Differential-difference equations:discrete 2×2 scattering problem 123
3.2.3 Partial-difference equations 125
3.3 Homoclinic structure and numerically induced chaos for the nonlinear Schr？dinger equation 127
3.3.1 Introduction 127
3.3.2 A linearized stability analysis 130
3.3.3 Hirota's method for the single homoclinic orbit 131
3.3.4 Combination homoclinic orbits 134
3.3.5 Numerical homoclinic instability 137
3.3.6 Duffing's equations and Mel'nikov analysis 150
3.4 Cellular Automata 152
4 Inverse Scattering for Integro-Differential Equations 163
4.1 Introduction 163
4.2 The intermediate long wave equation 164
4.2.1 The direct scattering problem 164
4.2.2 The inverse scattering problem 168
4.2.3 The time dependence 171
4.2.4 Further remarks 171
4.3 The Benjamin-Ono equation 173
4.3.1 The direct scattering problem 173
4.3.2 The inverse scattering problem 175
4.3.3 The time dependence 179
4.3.4 Further remarks 180
4.4 Classes of integrable integro-differential equations 182
4.4.1 Introduction 182
4.4.2 The Sine-Hilbert equation 187
4.4.3 Further examples 192
5 Inverse Scattering in Two Dimensions 195
5.1 Introduction 195
5.2 The Kadomtsev-Petviashvili Ⅰ equation 199
5.2.1 The direct scattering problem 199
5.2.2 The inverse scattering problem 206
5.2.3 The time dependence 207
5.2.4 Further remarks 208
5.3 The Kadomtsev-Petviashvili Ⅱ equation 212
5.3.1 The direct scattering problem 212
5.3.2 The inverse scattering problem 215
5.3.3 The time dependence 217
5.3.4 Comments on rigorous analysis 218
5.3.5 Boundary conditions and the choice of the operator？ 221
5.3.6 Hamiltonian formalism and action-angle variables 225
5.4 Hyperbolic and elliptic systems in the plane 227
5.4.1 Hyperbolic systems 228
5.4.2 Elliptic systems 234
5.4.3 The n-wave interaction equations 236
5.4.5 Comments on rigorous analysis for the elliptic scattering problem 238
5.5 The Davey-Stewartson Equations 240
5.5.1 Introduction 240
5.5.2 Inverse scattering for the DSⅠ equations 242
5.5.3 Inverse scattering for the DSⅡ equations 244
5.5.4 The strong coupling limit 246
5.5.5 The ？-limit case 248
5.5.6 Hamiltonian formalism for the DSⅡ equations 254
5.5.7 Localized solitons of the DSⅠ equations 260
5.5.8 On the physical derivation of the boundary conditions for the Davey-Stewartson Equations 264
5.6 Further Examples 267
5.6.1 Equations related to the Davey-Stewartson equation 267
5.6.2 Multidimensional isospectral flows associated with second order scalar operators 268
6 Inverse Scattering in Multidimensions 272
6.1 Introduction 272
6.2 Multidimensional inverse scattering associated with the"time"-dependent and"time:-independent Schr？dinger equation 274
6.2.1 The direct scattering problem 274
6.2.2 The inverse scattering problem 276
6.2.3 The characterization problem 278
6.2.4 The"time"-dependent Schr？dinger equation 281
6.2.5 The"time"-independent Schr？dinger equation 284
6.2.6 The relationship between the inverse data and the scattering data 287
6.2.7 Further remarks 290
6.3 Multidimensional inverse scattering for first order systems 291
6.3.1 The direct and inverse scattering problems 291
6.3.2 The characterization problem 294
6.3.3 The hyperbolic limit 298
6.3.4 The N-wave interaction equations 302
6.4 The Generalized Wave and Generalized Sine-Gordon equations 304
6.4.1 Introduction 304
6.4.2 The direct and inverse scattering problems for the Generalized Wave Equation 308
6.4.3 The direct and inverse scattering problems for the Generalized Sine-Gordon Equation 312
6.4.4 Further remarks 315
6.5 The Self-dual Yang-Mills equations 316
6.5.1 Introduction 316
6.5.2 Reductions to 2+1-dimensional equations 320
6.5.3 Reductions to l+1-dimensional equations 328
6.5.4 Reductions to ordinary differential equations 332
6.5.5 The SDYM hierarchy 344
7 The Painlevé Equations 347
7.1 Historical origins and physical applications 347
7.1.1 Singularities of ordinary differential equations 347
7.1.2 First order ordinary differential equations 349
7.1.3 The work of Sophie Kowalevski 349
7.1.4 Second order ordinary differential equations 352
7.1.5 Third and higher order ordinary differential equations 354
7.1.6 Physical applications 358
7.2 The Painlevétests 359
7.2.1 The relationship between the Painlevé equations and inverse scattering 359
7.2.2 The Painlevé ODE test 362
7.2.3 Applications of the Painlevé ODE test 365
7.2.4 The Painlevé PDE test 370
7.2.5 Applications of the PainlevéPDE test 373
7.2.6 Quasilinear partial differential equations and the Painlevé tests 386
7.3 Inverse Problems for the Painlevéequations 390
7.3.1 Inverse scattering for the Modified KdV equation 390
7.3.2 Gel'fand-Levitan-Marchenko integral equation method 393
7.3.3 The Inverse Monodromy Transform method:introduction 395
7.3.4 The Inverse Monodromy Transform method:direct problem 398
7.3.5 The Inverse Monodromy Transform method:inverse problem 401
7.4 Connection formulae for the Painlevé equations 404
7.4.1 Introduction 404
7.4.2 The Gel'fand-Levitan-Marchenko integral equation approach 406
7.4.3 The Inverse Monodromy Transform approach 414
7.5 Properties of the Painlevé equations 420
8 Further Remarks and Open Problems 424
8.1 Multidimensional equations 425
8.2 Boundary value problems 426
8.3 Ordinary differential equations 430
8.4 Functional analysis and 2+1-dimensions 432
8.5 Quantum inverse scattering and statistical mechanics 435
8.6 Complete integrability 438
Appendix A:Remarks on Riemann-Hilbert problems 440
Appendix B:Remarks on ？ problems 453
References 459
Subject Index 513