New Approaches to Nonlinear Waves

The book details a few of the novel methods developed in the last few years for studying various aspects of nonlinear wave systems. The introductory chapter provides a general overview, thematically linking the objects described in the book.

Two chapters are devoted to wave systems possessing resonances with linear frequencies (Chapter 2) and with nonlinear frequencies (Chapter 3).

In the next two chapters modulation instability in the KdV-type of equations is studied using rigorous mathematical methods (Chapter 4) and its possible connection to freak waves is investigated (Chapter 5).

The book goes on to demonstrate how the choice of the Hamiltonian (Chapter 6) or the Lagrangian (Chapter 7) framework allows us to gain a deeper insight into the properties of a specific wave system.

The final chapter discusses problems encountered when attempting to verify the theoretical predictions using numerical or laboratory experiments.

All the chapters are illustrated by ample constructive examples demonstrating the applicability of these novel methods and approaches to a wide class of evolutionary dispersive PDEs, e.g. equations from Benjamin-Oro, Boussinesq, Hasegawa-Mima, KdV-type, Klein-Gordon, NLS-type, Serre, Shamel , Whitham and Zakharov.

This makes the book interesting for professionals in the fields of nonlinear physics, applied mathematics and fluid mechanics as well as students who are studying these subjects. The book can also be used as a basis for a one-semester lecture course in applied mathematics or mathematical physics.

Inhalt

Introduction (E. Tobisch).- Brief historical overview.- Main notions.- Resonant interactions.- Modulation instability.- Frameworks.- Reality check.- References.- The effective equation method (Sergei Kuksin and Alberto Maiocchi).- Introduction.- How to construct the effective equation.- Structure of resonances.- CHM: resonance clustering.- Concluding remarks.- References.- On the discovery of the steady-state resonant water waves (Shijun Liao, Dali Xu and Zeng Liu).- Introduction.- Basic ideas of homotopy analysis method.- Steady-state resonant waves in constant-depth water.- Experimental observation.- Concluding remarks.- References.- Modulational instability in equations of KdV type (Jared C. Bronski, Vera Mikyoung Hur and Mathew A. Johnson).- Introduction.- Periodic traveling waves of generalized KdV equations.- Formal asymptotics and Whitham's modulation theory.- Rigorous theory of modulational instability.- Applications.- Concluding remarks.- References.- Modulational instability and rogue waves in shallow water models (R. Grimshaw, K. W. Chow and H. N. Chan).- Introduction.- Korteweg-de Vries equations.- Boussinesq model.- Hirota-Satsuma model.- Discussion.- References.- Hamiltonian framework for short optical pulses (Shalva Amiranashvili).- Introduction.- Poisson brackets.- Pulses in optical fibers.- Hamiltonian description of pulses.- Concluding remarks.- References.- Modeling water waves beyond perturbations (Didier Clamond and Denys Dutykh).- Introduction.- Preliminaries.- Variational formulations.- Examples.- Discussion.- References.- Quantitative Analysis of Nonlinear Water-Waves: a Perspective of an Experimentalist (Lev Shemer).- Introduction.- The experimental facilities.- The Nonlinear Schrödinger Equation.- The Modified Nonlinear Schrödinger (Dysthe) Equation.- The Spatial Zakharov Equation.- Statistics of nonlinear unidirectional water waves.- Discussion and Conclusions.- References.