Periodic soliton solutions to nonlinear Klein-Gordon equations on \({\mathbb{R}}^ 2\)
type: Item
instance of: scholarly article
title: Periodic soliton solutions to nonlinear Klein-Gordon equations on \({\mathbb{R}}^ 2\)
zbMATH Open document ID: 0731.35065
published in: Differential and Integral Equations
publication date: 1990-01-01T00:00:00Z
review text: The paper deals with nonlinear one-dimensional Klein-Gordon equations of the form \[ (1)\quad u_{tt}(x,t)=u_{xx}(x,t)-g(u(x,t)) \] where \((x,t)\in {\mathbb{R}}^ 2\), g: \({\mathbb{R}}\to {\mathbb{R}}\), \(g(u_ 0)=0\) and \(g'(u_ 0)>0\) for some \(u_ 0\in {\mathbb{R}}\). The author proves the existence and the regularity of time-periodic spatially-decaying towards \(u_ 0\) (as \(x\to \pm \infty)\) small-amplitude solutions to equations of the form (1) restricted to the half-spaces \({\mathbb{R}}_ 0^{\pm}\times {\mathbb{R}}\). The author also presents nonexistence results for small- amplitude homoclinic orbits of a certain dynamical system associated with equation (1) on \({\mathbb{R}}^ 2\), which demonstrate the generic impossibility to continue the above-mentioned solutions defined on \({\mathbb{R}}_ 0^{\pm}\times {\mathbb{R}}\) to the whole of \({\mathbb{R}}^ 2\). The obtained existence and nonexistence results are applied to several semilinear wave equations of current interest in theoretical physics.
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 81Q05
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 35Q40
zbMATH DE Number: 4208670
zbMATH Keywords: Klein-Gordon equations
zbMATH Keywords: existence
zbMATH Keywords: nonexistence
zbMATH Keywords: semilinear wave equations
author: Pierre-A. Vuillermot
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4208670