《SOLITON（孤立子）》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*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。

Chapter 1
　　Introduction
　　1.1 The Origin of Solitons
　　In 1834， British scientist Scott Russell accidentally observed a wonderful water wave. In 1844， he vividly described this phenomenon in his article On Waves published in the Report of the 14th Conference of the British Association for the Advancement of Science: “I observed the movement of a ship， which was pulled by two horses and rapidly advanced along a narrow canal. Suddenly， the ship came to a stop， but the large amount of water pushed by the ship did not stop. They accumulated around the bow of the ship and violently disturbed. Then， the waves suddenly appeared as a round， smooth， and well-defined huge isolated peak， rolling forward at a huge speed and rapidly leaving the bow. Its shape and speed did not change significantly during the journey. I rode on a horse and followed closely to observe that it rolled forward at a speed of about eight to nine miles per hour and maintained its original shape of about 30 feet long and 1 to 1.5 feet high. Gradually， its height decreased. When I tracked for 1 to 2 miles， it finally disappeared into the meandering river channel.” This was a peculiar phenomenon observed by Russell， who then believed that this isolated wave was el stable solution to fluid motion and called it “solitary wave”. Russell was unable to successfully prove and convince physicists of his argument at the time， thus blaming mathematicians for not being able to predict this phenomenon from known fluid motion equations. Subsequently， the issue of solitary waves sparked widespread debate among many physicists of the time. Until 1895， 60 years later， Korteweg de Vries studied the motion of shallow water waves and established the following shallow water wave motion equation for unidirectional motion under the assumption of long wave approximation and small amplitude
　　here， rj is the wave height， I is the water depth， g is the gravitational acceleration， a and a are constants. They conducted a relatively complete analysis of the solitary wave phenomenon and derived a pulse like solitary wave solution with shape invariance
　　from Eq.(l.l)，which is consistent with Russell’s description， thus theoretically proving the existence of solitary waves. However， is this wave stable? Can two solitary waves deform after collision? These questions have not been answered yet. Some people even suspect that Eq.(l.l) is a nonlinear partial differential equation， and the superposition principle of solutions does not satisfy it. After collision， the shape of the two solitary waves may be completely destroyed. This viewpoint has led many people to believe that this type of wave is “unstable”，and solitary waves remain buried for a long time until new discoveries are made.
　　Another question is， do solitary waves like what Russell said appear in other physical fields besides fluid mechanics? In the early 20th century， this was an elusive issue. It was not until the 1950s that a new situation emerged due to the work of Fermi， Pasta and Ulam. They connected 64 particles with nonlinear springs to form a nonlinear vibrating string. Initially， all the energy of these resonators was concentrated in one， while the initial energy of the other 63 was zero. According to classical theory， as long as nonlinear effects exist， there will be phenomena such as energy equalization and ergodicity of states， that is， any weak nonlinear interaction can cause the system to transition from a non-equilibrium state to an equilibrium state. But the actual calculation results surprised them greatly， that the concept of achieving energy balance mentioned above was incorrect. In fact， from Fig. 1-1，it can be seen that after a long period of time， almost all the energy returns to its original initial distribution， which is the famous FPU problem. At that time， due to their only examining the frequency space， solitary wave solutions could not be found， so the problem was not properly explained. Later， people regarded the crystal as a chain of elastic tubes with a mass scene and approximately simulated this situation. Toda studied the nonlinear vibration of this mode and obtained solitary wave solutions， which further aroused people’s interest in solitary wave research.
　　Subsequently， in 1962，Perring and Skyrme applied the sine-Gordon equation to the study of elementary particles， and numerical calculations revealed that such solitary waves did not disperse， retaining their original shape and velocity even after a collision.
　　In 1965， the renowned American scientists Zabusky and Kruskal used numerical simulation methods to investigate in detail the nonlinear interaction process of soliton collisions in plasma， obtaining relatively complete and rich results， and further confirming the theory that soliton interactions do not change the waveform， which surprised people.
　　Due to the above results and the fact that stable solitary waves with unchanged waveforms after coll

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 T