Stability properties of space periodic standing waves
type: Item
instance of: scholarly article
title: Stability properties of space periodic standing waves
zbMATH Open document ID: 0654.35046
DOI: 10.4171/ZAA/267
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1987-01-01T00:00:00Z
review text: We investigate equilibrium solutions of parabolic systems of the form \((*)\quad u_ t=D\Delta u+F(\alpha,u)\) where the state vector \(u(x,y)=(v(x,y),w(x,y))\) is subject to periodic boundary conditions, the period L being taken as a parameter. \(F(\alpha,u)\) is a nonlinearity, polynomial in \(u=(v,w)\) and depending on the bifurcation parameter \(\alpha\) ; D is a \(2\times 2\) diagonal matrix with positive entries. A trivial solution branch \(u(\delta)\in R^ 2\), \(\delta \in (- \epsilon,+\epsilon)\) is given, i.e. such that \(F(\alpha_ 0+\delta,u(\delta))=0\) for \(\delta \in (-\epsilon,+\epsilon)\). Under suitable genericity assumptions, bifurcating branches of space periodic standing waves are constructed. It is shown that these branches become generically unstable as \(L\uparrow \infty\). Under the assumption \(d_{uu}F(\alpha_ 0,u(0))=0\) however they will remain stable against nL-periodic perturbations for \(1\leq n\leq N\) provided that the trivial solution branch \(u(\delta)\) has this property for small \(\delta\). Examples from physics which fall under this scope are Landau-Ginzburg type equations with periodic boundary conditions.
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35B10
zbMATH DE Number: 4067449
zbMATH Keywords: equilibrium solutions
zbMATH Keywords: periodic boundary conditions
zbMATH Keywords: bifurcation
zbMATH Keywords: solution branch
zbMATH Keywords: genericity
zbMATH Keywords: space periodic standing waves
zbMATH Keywords: generically unstable
zbMATH Keywords: Landau-Ginzburg type equations
author: Bruno Scarpellini
reviewed by: Bruno Scarpellini
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/zaa/267
OpenAlex ID: W2564830445
description: scientific article; zbMATH DE number 4067449