Nonlinear Waves & Hamiltonian Systems

Abstract Wave phenomena are ubiquitous in the world around us, from art (e.g. the “Great Wave off Kanagawa” of K. Hokusai) to science (e.g. arterial pressure in our veins) and from the world around us (the Draupner wave off the coast of Norway, to the Morning Glory cloud in the Gulf of Carpentaria in Australia) to planetary (Giant Red Spot of Jupiter) or quantum (ultracold atoms) and even cosmological scales. Therefore, the study of waves has an intrinsic and broad scientific interest in many diverse contexts, ranging from mathematics and physics to engineering, biosciences, chemistry and finance just to name a few. 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. This book summarizes some of the central models (such as the Korteweg-de Vries equation, the Nonlinear Shrödinger equation and Klein-Gordon equations) that describe such waves, including solitons, kinks, breathers, and vortices, in their continuum (partial differential equations) and in their discrete (differential-difference equations) form. Although the primary purpose of this book is to serve as a textbook, the selection and exposition of the material should also be useful to anyone who wishes to be introduced in the field of nonlinear waves.