Stability properties of space periodic standing waves
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.
scientific article; zbMATH DE number 4067449