Nonlinear Dispersive Equations: Existence and Stability of Solitary and Periodic Travelling Wave Solutions

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American Mathematical Soc., 2009 - Mathematics - 256 pages
This book provides a self-contained presentation of classical and new methods for studying wave phenomena that are related to the existence and stability of solitary and periodic travelling wave solutions for nonlinear dispersive evolution equations. Simplicity, concrete examples, and applications are emphasized throughout in order to make the material easily accessible. The list of classical nonlinear dispersive equations studied includes Korteweg-de Vries, Benjamin-Ono, and Schrodinger equations. Many special Jacobian elliptic functions play a role in these examples. The author brings the reader to the forefront of knowledge about some aspects of the theory and motivates future developments in this fascinating and rapidly growing field. The book can be used as an instructive study guide as well as a reference by students and mature scientists interested in nonlinear wave phenomena.
 

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Contents

Basic Models
17
Solitary and Periodic Travelling Wave Solutions
25
Initial Value Problem
49
Definition of Stability
61
Orbital Stabilitythe Classical Method
69
GrillakisShatahStrausss Stability Approach
91
Existence and Stability of Solitary Waves for the GBO
105
More about the ConcentrationCompactness Principle
127
Sobolev Spaces of Periodic Type
206
The Symmetric Decreasing Rearrangement
207
The Jacobian Elliptic Functions
208
Appendix B Operator Theory
211
PseudoDifferential Operators and heir Spectrum T
229
Spectrum of Linear Operators Associated to Solitary Waves
231
SturmLiouville Theory
237
Floquet Theory
240

Instability of Solitary Wave Solutions
137
Stability of Cnoidal Waves
161
Appendix A Sobolev Spaces and Elliptic Functions
201
Sobolev Spaces
204

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