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 Schr?dinger (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.
Author(s): Vasily Y. Belashov, Sergey V. Vladimirov
Series: Springer Series in Solid-State Sciences
Edition: Softcover reprint of hardcover 1st ed. 2005
Publisher: Springer
Year: 2010
Language: English
Pages: 305
Table of Contents......Page 10
Preface......Page 7
Introduction......Page 13
1.1.1 Derivation of the KdV Equation......Page 29
1.1.2 Universality of the KdV Model. Scaling Transformations and Similarity Principle......Page 33
1.1.3 Other (1+1)-Dimensional KdV-Class Equations......Page 36
1.2.1 Fundamentals of the Inverse Scattering Theory......Page 38
1.2.2 Integration of the KdV Equation Using the IST Method......Page 45
1.2.3 Generalization of the GLM Equation......Page 47
1.2.4 The Variational Principle......Page 52
1.3 Numerical Integration of (1+1)-Dimensional KdV-Class Equations......Page 53
1.3.1 Explicit Difference Schemes......Page 54
1.3.2 Implicit Difference Schemes......Page 55
1.3.3 Remarks on Numerical Integration......Page 60
1.3.4 Test of Numerical Methods, Their Comparative Characteristics, and Use......Page 61
1.3.5 Numerical Solutions of Some KdV-Class Equations......Page 62
1.4.1 The Ion-Acoustic Waves......Page 70
1.4.2 Nonrelativistic Approximation......Page 71
1.4.3 Weakly-Relativistic Effects......Page 73
2.1.1 The KdV–Burgers Equation. Some Applications......Page 75
2.1.2 Higher Order Dispersion Corrections......Page 79
2.1.3 Modified KdV Equations......Page 81
2.1.4 Higher Order Dispersive Nonlinearity......Page 85
2.2.1 Evolution of Solitons of Generalized KdV Equations......Page 92
2.2.2 Soliton Evolution in Media with Stochastic Fluctuations of the Wave Field......Page 99
2.2.3 Qualitative Analysis and Asymptotics of Solutions of Generalized KdV-Class Equations......Page 104
2.3.1 Derivation of the NLS equation......Page 116
2.3.2 IST for the NLS Equation. NLS Solitons......Page 119
2.3.3 Zakharov System of Equations......Page 123
2.3.4 Langmuir Solitons......Page 125
2.3.5 Near-Sonic Solitons......Page 128
2.4.1 Origin of the DNLS Equation......Page 132
2.4.2 DNLS Equation as an Integrability Condition for Two Linear Differential Equations......Page 135
2.4.3 Stability of DNLS Solitons......Page 137
2.4.4 Numerical Approaches to Study Dynamics of Alfvén Solitons......Page 139
2.4.5 Results of Numerical Simulations......Page 146
3.1.1 Generalization of the KdV Equation on Weakly Non-One-Dimensional Case......Page 149
3.1.2 The KP Equation and its Solutions......Page 152
3.1.3 Stability of Two- and Three-Dimensional KP Solitons......Page 158
3.1.4 Numerical Approaches to Integration......Page 159
3.2.1 Analytical Integration. "Dressing" Method......Page 162
3.2.2 Three-Dimensional Inverse Scattering Problem......Page 170
3.2.3 Dynamics of Three-Dimensional Nonlinear Waves in the KP Model. Wave Collapse and Self-Focusing......Page 177
4.1.1 Generalized KP Equation......Page 186
4.1.2 3-DNLS Equation......Page 188
4.1.3 Stability of Two-Dimensional and Three-Dimensional Solutions of GKP and 3-DNLS Equations......Page 190
4.2.1 Basic Equations......Page 198
4.2.2 Generalization to Multidimensional Cases......Page 200
4.2.3 Concluding Remarks......Page 204
4.3 Approaches to Numerical Integration of Equations of GKP-Class and 3-DNLS-Class......Page 205
4.3.1 Groups of Explicit and Implicit Difference Schemes......Page 206
4.3.2 Boundary Conditions and Diffraction Terms......Page 213
4.3.3 Dynamic Spectral Method......Page 215
4.3.4 Comparative Characteristics of Different Schemes and Their Use in Numerical Simulation......Page 219
4.4.1 Structure of Two-Dimensional Solutions of GKP-Class Equations......Page 223
4.4.2 Interactions of Two-Dimensional Solitons......Page 229
4.4.3 Influence of Dissipation on Evolution of Two-Dimensional Solitons......Page 233
4.4.4 Evolution of Two-Dimensional Solitons in Media with Stochastic Fluctuations of the Wave Field......Page 234
4.4.5 Structure and Evolution of Two-Dimensional Solitons in Media with Variable Dispersion......Page 240
4.5.1 Structure and Evolution of Three-Dimensional Solutions of GKP-Class Equations......Page 245
4.5.2 Structure and Evolution of Three-Dimensional Solutions of 3-DNLS-Class Equations......Page 253
4.5.3 Influence of Dissipation on Evolution of Three-Dimensional Nonlinear Waves......Page 257
4.6 Applications......Page 260
4.6.1 Nonlinear Ion-Acoustic Waves in a Plasma......Page 261
4.6.2 Nonlinear Effects in Propagation of FMS Waves in a Magnetized Plasma......Page 266
4.6.3 Solitary Internal Gravity Waves in the F-layer of Earth's Ionosphere......Page 274
4.6.4 Two-Dimensional Solitons in Shallow Water......Page 282
5. Appendices......Page 286
References......Page 290
F......Page 300
M......Page 301
W......Page 302
Z......Page 303
