New Approaches to Nonlinear Waves

Front Cover
Elena Tobisch
Springer, Aug 19, 2015 - Science - 298 pages

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.

 

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Contents

1 Introduction
1
2 The Effective Equation Method
21
3 On the Discovery of the SteadyState Resonant Water Waves
43
4 Modulational Instability in Equations of KdV Type
83
5 Modulational Instability and Rogue Waves in Shallow Water Models
135
6 Hamiltonian Framework for Short Optical Pulses
153
7 Modeling Water Waves Beyond Perturbations
197
A Perspective of an Experimentalist
211
Index
294
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