Contents Preface Chapter 1 Introduction 1.1 The Origin of Solitons 1.2 KdV Equation and Its Soliton Solutions 1.3 Soliton Solutions for Nonlinear Schr?dinger Equations and Other Nonlinear Evolutionary Equations 1.4 Experimental Observation and Application of Solitons 1.5 Research on the Problem of Soliton Theory References Chapter 2 Inverse Scattering Method 2.1 Introduction 2.2 The KdV Equation and Inverse Scattering Method 2.3 Lax Operator and Generalization of Zakharov, Shabat, AKNS 2.4 More General Evolutionary Equation (AKNS Equation) 2.5 Solution of the Inverse Scattering Problem for AKNS Equation 2.6 Asymptotic Solution of the Evolution Equation (t → ∞) 2.6.1 Discrete spectrum 2.6.2 Continuous spectrum 2.6.3 Estimation of discrete spectrum 2.7 Mathematical Theory Basis of Inverse Scattering Method 2.8 High-Order and Multidimensional Scattering Inversion Problems References Chapter 3 Interaction of Solitons and Its Asymptotic Properties 3.1 Interaction of Solitons and Asymptotic Properties of t → ∞ 3.2 Behaviour State of the Solution to KdV Equation Under Weak Dispersion and WKB Method 3.3 Stability Problem of Soliton 3.4 Wave Equation under Water Wave and Weak Nonlinear Effect References Chapter 4 Hirota Method 4.1 Introduction 4.2 Some Properties of the D Operator 4.3 Solutions to Bilinear Differential Equations 4.4 Applications in Sine-Gordon Equation and MKdV Equation 4.5 B?cklund Transform in Bilinear Form References Chapter 5 B?cklund Transformation and Infinite Conservation Law 5.1 Sine-Gordon Equation and B?cklund Transformation 5.2 B?cklund Transformation of a Class of Nonlinear Evolution Equation 5.3 B Transformation Commutability of the KdV Equation 5.4 B?cklund Transformations for High-Order KdV Equation and High-Dimensional Sine-Gordon Equation 5.5 B?cklund Transformation of Benjamin-Ono Equation 5.6 Infinite Conserved Laws for the KdV Equation 5.7 Infinite Conservation Quantities of AKNS Equation References Chapter 6 Multidimensional Solitons and Their Stability 6.1 Introduction 6.2 The Existence Problem of Multidimensional Solitons 6.3 Stability and Collapse of Multidimensional Solitons References Chapter 7 Numerical Calculation Methods for Some Nonlinear Evolution Equations
7.1 Introduction 7.2 The Finite Difference Method and Galerkin Finite Element Method for the KdV Equations 7.3 The Finite Difference Method for Nonlinear Schr?dinger Equations 7.4 Numerical Calculation of the RLW Equation 7.5 Numerical Computation of the Nonlinear Klein-Gordon Equation 7.6 Numerical Computation of a Class of Nonlinear Wave Stability Problems References Chapter 8 The Geometric Theory of Solitons 8.1 B?cklund Transform and Surface with Total Curvature K = -1 8.2 Lie Group and Nonlinear Evolution Equations 8.3 The Prolongation Structure of Nonlinear Equations References Chapter 9 The Global Solution and 「Blow up」 Problem of Nonlinear Evolution Equations 9.1 Nonlinear Evolutionary Equations and the Integral Estimation Method 9.2 The Periodic Initial Value Problem and Initial Value Problem of the KdV Equation 9.3 Periodic Initial Value Problem for a Class of Nonlinear Schr?dinger Equations 9.4 Initial Value Problem of Nonlinear Klein-Gordon Equation 9.5 The RLW Equation and the Galerkin Method 9.6 The Asymptotic Behavior of Solutions and「Blow up」 Problem for t → ∞ 9.7 Well-Posedness Problems for the Zakharov System and Other Coupled Nonlinear Evolutionary Systems References Chapter 10 Topological Solitons and Non-topological Solitons 10.1 Solitons and Elementary Particles 10.2 Preliminary Topological and Homotopy Theory 10.3 Topological Solitons in One-Dimensional Space 10.4 Topological Solitons in Two-Dimensional 10.5 Three-Dimensional Magnetic Monopole Solution 10.6 Topological Solitons in Four-Dimensional Space—Instantons 10.7 Non-topological Solitons 10.8 Quantization of Solitons References Chapter 11 Solitons in Condensed Matter Physics 11.1 Soliton Motion in Superconductors 11.2 Soliton Motion in Ferroelectrics 11.3 Solitons in Coupled Systems in Solids 11.4 Statistical Mechanics of Toda Lattice Solitons References Chapter 12 Rogue Wave and Wave Turbulence 12.1 Rogue Wave 12.2 Formation of Rogue Wave 12.3 Wave Turbulence 12.4 Soliton and Quasi Soliton 12.4.1 The Instability and Blow-up of Solitons 12.4.2 The Case of Quasi-Solitons References
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