Contents
Preface
Chapter 1 Introduction 1
1.1 The Origin of Solitons 1
1.2 KdV Equation and Its Soliton Solutions 4
1.3 Soliton Solutions for Nonlinear Schr.dinger Equations and Other Nonlinear Evolutionary Equations 6
1.4 Experimental Observation and Application of Solitons 10
1.5 Research on the Problem of Soliton Theory 10
References 11
Chapter 2 Inverse Scattering Method 12
2.1 Introduction 12
2.2 The KdV Equation and Inverse Scattering Method 12
2.3 Lax Operator and Generalization of Zakharov, Shabat, AKNS21
2.4 More General Evolutionary Equation (AKNS Equation) 28
2.5 Solution of the Inverse Scattering Problem for AKNS Equation 35
2.6 Asymptotic Solution of the Evolution Equation (t→∞) 46
2.6.1 Discrete spectrum 46
2.6.2 Continuous spectrum 49
2.6.3 Estimation of discrete spectrum.52
2.7 Mathematical Theory Basis of Inverse Scattering Method.56
2.8 High-Order and Multidimensional Scattering Inversion Problems 74
References 83
Chapter 3 Interaction of Solitons and Its Asymptotic Properties 85
3.1 Interaction of Solitons and Asymptotic Properties of t→ ∞ 85
3.2 Behaviour State of the Solution to KdV Equation Under Weak
Dispersion and WKB Method 94
3.3 Stability Problem of Soliton .100
3.4 Wave Equation under Water Wave and Weak Nonlinear Effect 102
References 109
Chapter 4 Hirota Method 111
4.1 Introduction 111
4.2 Some Properties of the D Operator 113
4.3 Solutions to Bilinear Differential Equations.115
4.4 Applications in Sine-Gordon Equation and MKdV Equation 117
4.5 B.cklund Transform in Bilinear Form 125
References 127
Chapter 5 B.cklund Transformation and Infinite Conservation Law 129
5.1 Sine-Gordon Equation and B.cklund Transformation 129
5.2 B.cklund Transformation of a Class of Nonlinear Evolution Equation 134
5.3 B Transformation Commutability of the KdV Equation 141
5.4 B.cklund Transformations for High-Order KdV Equation and High-Dimensional Sine-Gordon Equation 143
5.5 B.cklund Transformation of Benjamin-Ono Equation 145
5.6 Infinite Conservation Laws for the KdV Equation 151
5.7 Infinite Conserved Quantities of AKNS Equation 154
References 157
Chapter 6 Multidimensional Solitons and Their Stability 159
6.1 Introduction 159
6.2 The Existence Problem of Multidimensional Solitons 160
6.3 Stability and Collapse of Multidimensional Solitons 174
References 180
Chapter 7 Numerical Calculation Methods for Some Nonlinear Evolution Equations 182
7.1 Introduction 182
7.2 The Finite Difference Method and Galerkin Finite Element Method for the KdV Equations 184
7.3 The Finite Difference Method for Nonlinear Schr.dinger Equations 189
7.4 Numerical Calculation of the RLW Equation 194
7.5 Numerical Computation of the Nonlinear Klein–Gordon Equation 195
7.6 Numerical Computation of a Class of Nonlinear Wave Stability Problems 197
References 202
Chapter 8 The Geometric Theory of Solitons.204
8.1 B.cklund Transform and Surface with Total Curvature K = .1 204
8.2 Lie Group and Nonlinear Evolution Equations 207
8.3 The Prolongation Structure of Nonlinear Equations 211
References 217
Chapter 9 The Global Solution and “Blow up” Problem of Nonlinear Evolution Equations.219
9.1 Nonlinear Evolutionary Equations and the Integral Estimation Method 219
9.2 The Periodic Initial Value Problem and Initial Value Problem of the KdV Equation 221
9.3 Periodic Initial Value Problem for a Class of Nonlinear Schr.dinger Equations 229
9.4 Initial Value Problem of Nonlinear Klein-Gordon Equation 235
9.5 The RLW Equation and the Galerkin Method 243
9.6 The Asymptotic Behavior of Solutions and “Blow up” Problem for t→∞ 251
9.7 Well-Posedness Problems for the Zakharov System and Other Coupled Nonlinear Evolutionary Systems 256
References 258
Chapter 10 Topological Solitons and Non-topological Solitons 261
10.1 Solitons and Elementary Particles 261
10.2 Preliminary Topological and Homotopy Theory 265
10.3 Topological Solitons in One-Dimensional Space 270
10.4 Topological Solitons in Two-Dimensional 276
10.5 Three-Dimensional Magnetic Monopole Solution 282
10.6 Topological Solitons in Four-Dimensional Space—Instantons 288
10.7 Nontopological Solitons 292
10.8 Quantization of Solitons 296
References 301
Chapter 11 Solitons in Condensed Matter Physics.303
11.1 Soliton Motion in Superconductors 304
11.2 Soliton Motion in Ferroelectrics 315
11.3 Solitons of Coupled Systems in Solids 318
11.4 Statistical Mechanics of Toda Lattice Solitons 322
References 327
Chapter 12 Rogue Wave and Wave Turbulence 329
12.1 Rogue Wave 329
12.2 Formation of Rogue Wave 329
12.3 Wave Turbulence 333
12.4 Soliton and Quasi Soliton 336
12.4.1 The Instability and Blow-up of Solitons 338
12.4.2 The Case of Quasi-Solitons 339
References 341