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Full Description
Nonlinear waves are of significant scientific interest across many diverse contexts, ranging from mathematics and physics to engineering, biosciences, chemistry, and finance. The study of nonlinear waves is relevant to Bose-Einstein condensates, the interaction of electromagnetic waves with matter, optical fibers and waveguides, acoustics, water waves, atmospheric and planetary scales, and even galaxy formation.
The aim of this book is to provide a self-contained introduction to the continuously developing field of nonlinear waves, that offers the background, the basic ideas, and mathematical, as well as computational methods, while also presenting an overview of associated physical applications.
Originated from the authors' own research activity in the field for almost three decades and shaped over many years of teaching on relevant courses, the primary purpose of this book is to serve as a textbook. However, the selection and exposition of the material will be useful to anyone who is curious to explore the fascinating world of nonlinear waves.
Contents
PART I - INTRODUCTION AND MOTIVATION OF MODELS
1: Introduction and Motivation
2: Linear Dispersive Wave Equations
3: Nonlinear Dispersive Wave Equations
PART II - KORTEWEG-DE VRIES (KDV) EQUATION
4: The Korteweg-de Vries (KdV) Equation
5: From Boussinesq to KdV - Boussinesq Solitons as KdV Solitons
6: Traveling Wave Reduction, Elliptic Functions, and Connections to KdV
7: Burgers and KdV-Burgers (KdVB) Equations - Regularized ShockWaves
8: A Final Touch From KdV: Invariances and Self-Similar Solutions
9: Spectral Methods
10: Bäcklund Transformation for the KdV
11: Inverse Scattering Transform I - the KdV equation*
12: Direct Perturbation Theory for Solitons*
13: The Kadomtsev-Petviashvili Equation*
PART III - KLEIN-GORDON, SINE-GORDON, AND PHI-4 MODELS
14: Another Class of Models: Nonlinear Klein-Gordon Equations
15: Additional Tools/Results for Klein-Gordon Equations
16: Klein-Gordon to NLS Connection - Breathers as NLS Solitons
17: Interlude: Numerical Considerations for Nonlinear Wave Equations
PART IV - THE NONLINEAR SCHRÖDINGER EQUATIONS
18: The Nonlinear Schrödinger (NLS) Equation
19: NLS to KdV Connection - Dark Solitons as KdV Solitons
20: Actions, Symmetries, Conservation Laws, Noether's Theorem, and all that
21: Applications of Conservation Laws - Adiabatic Perturbation Method
22: Numerical Techniques for NLS
23: Inverse Scattering Transform II - the NLS Equation*
24: The Gross-Pitaevskii (GP) Equation
25: Variational Approximation for the NLS and GP Equations
26: Stability Analysis in 1D
27: Multi-Component Systems
28: Transverse Instability of Solitons Stripes - Perturbative Approach
29: Transverse Instability of Dark Stripes - Adiabatic Invariant Approach
30: Vortices in the 2D Defocusing NLS
PART V - DISCRETE MODELS
31: The Discrete Klein-Gordon model
32: Discrete Models of the Nonlinear Schrödinger Type
33: From Toda to FPUT and Beyond
