Synopses & Reviews
"This monograph is devoted to the nonlinear Schrödinger equation (and some of its generalizations) which governs the envelope dynamics of a weakly nonlinear quasi-monochromatic wave-packet propagating in a dispersive medium. Special attention is paid to the phenomenon of self-focusing and wave collapse. Various approaches ranging from rigorous mathematical analysis to formal asymptotic expansions and numerical simulations are presented. The aim it to relate more closely the theoretical understanding of singularity formation to applications in domains such as nonlinear optics, or plasma physics, where this effect provides an efficient mechanism for small-scale formation and wave dissipation. An extended and up-to-date bibliography is also included. This book, which fills the gap between mathematical literature and phenomenological modeling, should promote the transfer of information between the various communities concerned with nonlinear waves. Graduate students and researchers in the fields of pure and applied mathematics, nonlinear optics, plasma physics, hydrodynamics and magnetohydrodynamics will find this book useful."
Synopsis
This monograph aims to fill the gap between the mathematical literature which significantly contributed during the last decade to the understanding of the collapse phenomenon, and applications to domains like plasma physics and nonlinear optics where this process provides a fundamental mechanism for small scale formation and wave dissipation. This results in a localized heating of the medium and in the case of propagation in a dielectric to possible degradation of the material. For this purpose, the authors have chosen to address the problem of wave collapse by several methods ranging from rigorous mathematical analysis to formal asymptotic expansions and numerical simulations.
Synopsis
Filling the gap between the mathematical literature and applications to domains, the authors have chosen to address the problem of wave collapse by several methods ranging from rigorous mathematical analysis to formal aymptotic expansions and numerical simulations.
Description
Includes bibliographical references (p. [309]-338) and indexes.
Table of Contents
x I Basic framework
x 1 The physical context
x 2 Structural properties
x II Rigorous theory
x 3 Existence and long-time behavior
x 4 Standing wave solutions
x 5 Blowup solutions
x III Asymptotic analysis near collapse
x 6 Numerical observations
x 7 Supercritical collapse
x 8 Critical collapse
x 9 Perturbations of focusing NLS
x IV Coupling to a mean field
x 10 Mean field generation
x 11 Gravity-capillary surface waves
x 12 The Davey-Stewartson system
x V Coupling to acoustic type waves
x 13 Langmuir oscillations
x 14 The scalar model
x 15 Progressive waves in plasmas
x Synopsis
x References
x Index