Breathers for a relativistic nonlinear wave equation
type: Item
instance of: scholarly article
title: Breathers for a relativistic nonlinear wave equation
zbMATH Open document ID: 1024.35076
author: Alain Bensoussan
author: Cyrill Iliine
author: A. Komech
published in: Archive for Rational Mechanics and Analysis
publication date: 2003-06-02T00:00:00Z
review text: The authors of this interesting paper study real-valued solutions to a relativistic nonlinear wave equation of the type \[ \partial^2 u/\partial t^2(x,t)=\partial^2 u/\partial x^2(x,t)+F(u(x,t)), \] \(x\in \mathbb R\). It is assumed that \(u\in C(\mathbb R ^2)\) and the nonlinear term \(F(u)\) is a distribution of the type \[ F(u)= \sum\limits_{k\in\mathbb Z}F_k\delta (u-z_k), \] where \(u\in \mathbb R\), \(\mathbb Z\) is a discrete closed subset of \(\mathbb R\), \(\delta (u-z_k) \) is \(\delta \)-function of Dirac. Then there exists a piecewise constant potential \(V(u)\) such that \(F(u)=-V^{\prime }(u)\). The sense of the equation is defined according to the least-action principle. The energy and momentum conservation is established. The main results are the existence of time-periodic finite-energy solutions, the existence of global solutions and soliton-type asymptotics for a class of finite-energy initial data.
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35L15
Mathematics Subject Classification ID: 82D75
Mathematics Subject Classification ID: 35Q51
Mathematics Subject Classification ID: 35Q75
zbMATH DE Number: 1922364
zbMATH Keywords: time-periodic finite-energy solutions
zbMATH Keywords: finite-energy initial data
zbMATH Keywords: least-action principle
zbMATH Keywords: soliton-type asymptotics
reviewed by: Dimitar A. Kolev
MaRDI profile type: Publication
full work available at URL: s00205-002-0226-5
OpenAlex ID: W2143686473
DOI: 10.1007/S00205-002-0226-5
description: scientific article; zbMATH DE number 1922364