Nonlinear Waves & Hamiltonian Systems

Ricardo Carretero-González holds a Summa Cum Laude B.Sc. in Physics from the Universidad Nacional Autónoma de México, and a Ph.D. in Applied Mathematics and Computation from Queen Mary College, University of London. He previously held a post-doctoral post at University College London and at Simon Fraser University, Canada, and is currently a Professor of Applied Mathematics at San Diego State University. His research focuses on spatio-temporal dynamical systems, nonlinear waves, and their applications. He is an active advocate of the dissemination of science and he regularly delivers engaging presentations at local high schools and science festivals.Dimitrios J. Frantzeskakis holds an Electrical Engineering Diploma from the University of Patras, and a PhD in Engineering from the National Technical University of Athens. He was an Instructor in the Naval Academy of Greece, a Research Associate at NTUA, and is currently a Professor in the Department of Physics at the National and Kapodistrian University of Athens. His research focuses on nonlinear waves and solitons, with applications to various physical contexts. Many of his theoretical results initiated important experimental projects, which verified a number of theoretical predictions reported in his papers; he is a co-author of several works reporting these experimental results.Panayotis G. Kevrekidis holds a Distinguished University Professorship at the University of Massachusetts Amherst since 2015. Previously, he was an Assistant (2001-2005), Associate (2005-2010) and Full Professor (2010-2015) at UMass. His doctoral studies were conducted at Rutgers University, where he earned an M.S., an M. Phil. and a Ph.D. He subsequently spent a post-doctoral year between Princeton's Program in Applied and Computational Mathematics and Los Alamos' Center for Nonlinear Studies.

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 primarypurpose 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 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