Nonlinear Waves and Hamiltonian Systems From One to Many Degrees of Freedom, from Discrete to Continuum

Cover -- Title page -- Copyright page -- Preface -- Acknowledgments -- Contents -- PART I INTRODUCTION AND MOTIVATION OF MODELS -- 1 Introduction and Motivation -- 1.1 Few degrees of freedom -- A linear example: Hooke's law -- A nonlinear example: the pendulum -- 1.2 Many degrees of freedom -- 1.3 A nonlinear variant: the FPUT lattice -- Exercises -- 2 Linear Dispersive Wave Equations -- 2.1 Dispersion relations and relevant notions -- 2.2 Examples of linear dispersive wave equations -- The transport (unidirectional wave) equation -- The (bidirectional) wave equation -- The linear Schrödinger equation -- 2.3 Wavepackets and group velocity -- 2.4 Dissipation, instability, and diffusion -- Exercises -- 3 Nonlinear Dispersive Wave Equations -- 3.1 Dispersion relations, linear and nonlinear equations -- 3.2 Unidirectional propagation: KdV, KP, and NLS -- The Korteweg-de Vries (KdV) equation -- Other versions of the KdV model -- The Kadomtsev-Petviashvili (KP) equation -- The nonlinear Schrödinger equation -- 3.3 Bidirectional propagation: KG and Boussinesq -- The Klein-Gordon equation -- The Boussinesq equation -- Exercises -- PART II KORTEWEG-DE VRIES (KDV) EQUATION -- 4 The Korteweg-de Vries (KdV) Equation -- 4.1 Obtaining KdV as a limit of FPUT -- 4.2 Obtaining KdV for shallow water waves -- 4.3 The effects of dispersion and nonlinearity -- The effect of dispersion - linearized KdV equation -- The effect of nonlinearity - Hopf equation, method of characteristics and shock waves -- 4.4 Putting it all together: the Zabusky-Kruskal numerical experiments -- 4.5 A cute twist: conservation laws -- Exercises -- 5 From Boussinesq to KdV - Boussinesq Solitons as KdV Solitons -- 5.1 Boussinesq to a single KdV (right-going waves) -- 5.2 Boussinesq to a KdV pair (right- and left-going waves) -- 5.3 Connecting the Boussinesq soliton with the KdV soliton.