本书主要对孤立子的由来，基本问题以及它的数学物理方法做了简要的介绍，在此基础上，增加了怪波和波湍流等比较重要的最新研究成果。孤立子理论是重要的数学和物理理论，它揭示了非线性波动现象中的一种特殊行为，即孤立波在碰撞后能够保持形状、大小和方向不变。这一发现不仅在数学和物理领域产生了深远的影响，还推动了非线性科学的发展，使其成为非线性科学的三大普适类之一。此外，孤立子理论在多个学科领域都有广泛的应用。例如，在物理学中，孤立子理论被用于解释和预测各种非线性波动现象，如光学孤子、声学孤子等。在生物学、医学、海洋学、经济学和人口问题等领域，孤立子理论也发挥着重要作用，为解决这些领域中的非线性问题提供了新的思路和方法。
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本科：复旦大学1958年9月—1963年1月 复旦大学 助教
1963年2月—1982年10月 第二机械工业部第九研究所 助理研究员
1982年11月—1987年10月 北京应用物理与计算数学研究所 副研究员
1987年10月—至今 北京应用物理与计算数学研究所 研究员非线性发展方程及其数值解、孤立子解以及无穷维动力系统等离子体物理某些非线性发展方程整体解及数值的研究，国防科工委科技进步一等奖;
中国光华科技二等奖;
无穷维动力系统的理论研究及其应用，国防科工委科技进步一等奖;
何梁何利基金科学与技术进步奖;1.曾任中国数学会理事，国家自然科学基金会数学评审组成员，北京市数学会常务理事、副理事长；
2.担任中南大学名誉教授、长江师范学院双聘院士、湖北文理学院“隆中学者”特聘教授、华南理工大学双聘院士、河南理工大学特聘教授、南宁师范大学特聘教授、广州大学数学与信息科学学院教授、广西壮族自治区主席院士顾问团成员、国家自然科学基金委重大项目咨询委员会委员；
3.美国“数学评论”评议员、美国数学会成员；
4.《偏微分方程杂志》《计算数学》《数学研究》《北京数学》等杂志编委、副主编
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
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