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.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.PART I - INTRODUCTION AND MOTIVATION OF MODELS1. Introduction and Motivation2. Linear Dispersive Wave Equations3. Nonlinear Dispersive Wave EquationsPART II - KORTEWEG-DE VRIES (KDV) EQUATION4. The Korteweg-de Vries (KdV) Equation5. From Boussinesq to KdV - Boussinesq Solitons as KdV Solitons6. Traveling Wave Reduction, Elliptic Functions, and Connections to KdV7. Burgers and KdV-Burgers (KdVB) Equations - Regularized ShockWaves8. A Final Touch From KdV: Invariances and Self-Similar Solutions9. Spectral Methods10. B?cklund Transformation for the KdV11. Inverse Scattering Transform I - the KdV equation*12. Directls+
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