Solitary Waves in Dispersive Complex Media

This book is devoted to one of the most interesting and rapidly developing areas of modern nonlinear physics and mathematics - the theoretical, analytical and advanced numerical study of the structure and dynamics of one-dimensional as well as two- and three-dimensional solitons and nonlinear waves described by Korteweg-de Vries (KdV), Kadomtsev-Petviashvili (KP), nonlinear Schrödinger (NLS) and derivative NLS (DNLS) classes of equations. Special attention is paid to generalizations (relevant to various complex physical media) of these equations, accounting for higher-order dispersion corrections, influence of dissipation, instabilities, and stochastic fluctuations of the wave fields. The book addresses researchers working in the theory and numerical simulations of dispersive complex media in such fields as hydrodynamics, plasma physics, and aerodynamics. It will also be useful as a reference work for graduate students in physics and mathematics.

Preface

1. Introduction

           1.1 Korteweg–de Vries Equation and KdV-Class Equations
                 1.1.1 Derivation of the KdV Equation
                 1.1.2 Universality of the KdV Model. Scaling Transformations and Similarity Principle
                 1.1.3 Other (1+1)-Dimensional KdV-Class Equations

           1.2 Inverse Scattering Transform and Analytical Integration
                 1.2.1 Fundamentals of the Inverse Scattering Theory
                 1.2.2 Integration of the KdV Equation Using the IST Method
                 1.2.3 Generalization of the GLM Equation
                 1.2.4 The Variational Principle

           1.3 Numerical Integration of (1+1)-Dimensional KdV-Class Equations
                 1.3.1 Explicit Difference Schemes
                 1.3.2 Implicit Difference Schemes
                 1.3.3 Remarks on Numerical Integration
                 1.3.4 Test of Numerical Methods, Their Comparative Characteristics, and Use
                 1.3.5 Numerical Solutions of Some KdV-Class Equations

           1.4 Ion-Acoustic Waves in Plasmas
                 1.4.1 The Ion-Acoustic Waves
                 1.4.2 Nonrelativistic Approximation
                 1.4.3 Weakly-Relativistic Effects

2. Generalized KdV Equations. NLS and DNLS Equations

           2.1 Generalized KdV Equations
                  2.1.1 The KdV–Burgers Equation. Some Applications
                  2.1.2 Higher Order Dispersion Corrections
                  2.1.3 Modified KdV Equations
                  2.1.4 Higher Order Dispersive Nonlinearity

           2.2 Structure and Evolution of Solutions of Generalized KdV Equations
                  2.2.1 Evolution of Solitons of Generalized KdV Equations
                  2.2.2 Soliton Evolution in Media with Stochastic Fluctuations of the Wave Field
                  2.2.3 Qualitative Analysis and Asymptotics of Solutions of Generalized KdV-Class Equations
 

           2.3 Nonlinear Schr¨odinger and Zakharov Equations
                 2.3.1 Derivation of the NLS equation
                 2.3.2 IST for the NLS Equation. NLS Solitons
                 2.3.3 Zakharov System of Equations
                 2.3.4 Langmuir Solitons
                 2.3.5 Near-Sonic Solitons

           2.4 Derivative Nonlinear Schr¨odinger Equation
                 2.4.1 Origin of the DNLS Equation
                 2.4.2 DNLS Equation as an Integrability Condition for Two Linear Differential Equations
                 2.4.3 Stability of DNLS Solitons
                 2.4.4 Numerical Approaches to Study Dynamics of Alfv´en Solitons
                 2.4.5 Results of Numerical Simulations

3. Classic Two- and Three-Dimensional KP Models and Their Applications

          3.2 KP Equation: Analytical Integration and Dynamics of Waves
                3.2.1 Analytical Integration. “Dressing” Method
                3.2.2 Three-Dimensional Inverse Scattering Problem
                3.2.3 Dynamics of 3D Nonlinear Waves in the KP Model. Wave Collapse and Self- Focusing

4. Generalized Two- and Three-Dimensional Models and Their Applications

          4.2 Asymptotic and Qualitative Analysis of Solutions of GKP Equation and 3-DNLS Equation
                 4.2.1 Basic Equations
                 4.2.2 Generalization to Multidimensional Cases
                 4.2.3 Concluding Remarks

           4.3 Approaches to Numerical Integration of Equations of GKPClass and 3-DNLS-Class
                 4.3.1 Groups of Explicit and Implicit Difference Schemes
                 4.3.2 Boundary Conditions and Diffraction Terms
                 4.3.3 Dynamic Spectral Method
                 4.3.4 Comparative Characteristics of Different Schemes and Their Use in Numerical Simulation

           4.4 Dynamics of Two-Dimensional Solitons in Dispersive Media
                 4.4.1 Structure of Two-Dimensional Solutions of GKP-Class Equations
                 4.4.2 Interactions of Two-Dimensional Solitons
                 4.4.3 Influence of Dissipation on Evolution of Two-Dimensional Solitons
                 4.4.4 Evolution of Two-Dimensional Solitons in Media with Stochastic Fluctuations of the Wave Field
                 4.4.5 Structure and Evolution of Two-Dimensional Solitons in Media with Variable Dispersion

           4.5 Evolution of Three-Dimensional Nonlinear Waves in Dispersive Media
                 4.5.1 Structure and Evolution of Three-Dimensional Solutions of GKP-Class Equations
                 4.5.2 Structure and Evolution of Three-Dimensional Solutions of 3-DNLS-Class Equations :
                 4.5.3 Influence of Dissipation on Evolution of Three-Dimensional Nonlinear Waves

           4.6 Applications
                 4.6.1 Nonlinear Ion-Acoustic Waves in a Plasma
                 4.6.2 Nonlinear Effects in Propagation of FMS Waves in a Magnetized Plasma
                 4.6.3 Solitary Internal GravityWaves in the F-layer of Earth’s Ionosphere
                 4.6.4 Two-Dimensional Solitons in Shallow Water

5. Appendices

References