Spectral and dynamical stability of nonlinear waves
type: Item
instance of: scholarly article
title: Spectral and dynamical stability of nonlinear waves
zbMATH Open document ID: 1297.37001
DOI: 10.1007/978-1-4614-6995-7
author: Todd Kapitula
author: Keith Promislow
published in: Applied Mathematical Sciences
publication date: 2013-04-08T00:00:00Z
review text: The book under review focuses on stability of the equilibria of evolution equations. It is a well-known fact that the dynamics of a nonlinear evolution equation mainly depends on (and can be understood by) the dynamics of the linear system obtained by linealization, and that this linear dynamics is fully dependent on the spectrum of the corresponding linear operator. The spectral analysis of general linear operators is provided by functional analysis, while the specific results for differential operators are given by the theory of dynamical systems. Both theoretical viewpoints are involved in this presentation. The book consists of two parts. The first part, Chapters 3--7, is about essential, absolute, and point spectra and develops nonlinear stability theory for dissipative and Hamiltonian systems. The theory is exemplified by the cases of reaction-diffusion equations, scalar viscous conservation law, nonlinear Schrödinger type equations, generalized Korteweg-de Vries equation, forced Ginzburg-Landau equation, Hamiltonian and non-Hamiltonian perturbations of the nonlinear Schrödinger equation. Particularly, the stability theory for Hamiltonian systems is generously developed and includes Krein signature and other stability indices. The second part of the book, Chapters 8--10, grants it maximum of originality compared to everything that has been written on the subject. It offers for the first time in a book format, the theory based on the Evans function, a complex-valued function of the spectral parameter whose zeros are the eigenvalues of the corresponding linear operator. The construction of the Evans function, akin in some sense to that of the Wronskian, is first given for the Sturm-Liouville problem and then, for a series of cases of increasing complexity. The connection of the Evans function with different concepts from dynamical systems, such as winding number, orientation index, or edge bifurcation, is presented in a general framework or on some particular ca…
Mathematics Subject Classification ID: 37-02
Mathematics Subject Classification ID: 37L15
Mathematics Subject Classification ID: 37K45
Mathematics Subject Classification ID: 34L05
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 35Q55
zbMATH DE Number: 6151577
zbMATH Keywords: stability
zbMATH Keywords: spectrum
zbMATH Keywords: waves
zbMATH Keywords: dissipative system
zbMATH Keywords: dynamical system
zbMATH Keywords: Hamiltonian system
zbMATH Keywords: evolution equation
reviewed by: Radu Precup
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-1-4614-6995-7
OpenAlex ID: W360456325
description: scientific article; zbMATH DE number 6151577