Preprint des Projektes: SFB 701: Spektrale Strukturen und Topologische Methoden in der Mathematik - Projekt B3

Numerische Analyse äquivarianter Evolutionsgleichungen

09-030 Wolf-Jürgen Beyn, Vera Thümmler.
Dynamics of Patterns in Nonlinear Equivariant PDEs

Many solutions of nonlinear time dependent partial differential equations show particular spatio-temporal patterns, such as traveling waves in one space dimension or spiral and scroll waves in higher space dimensions. The purpose of this paper is to review some recent progress on the analytical and the numerical treatment of such patterns. Particular emphasis is put on symmetries and on the dynamical systems viewpoint that goes beyond existence, uniqueness and numerical simulation of solutions for single initial value problems. The nonlinear asymptotic stability of dynamic patterns is discussed and a numerical approach (the freezing method) is presented that allows to compute co-moving frames in which solutions converging to the patterns become stationary. The results are related to the theory of relative equilibria for equivariant evolution equations. We discuss several applications to parabolic systems with nonlinearities of FitzHugh-Nagumo and Ginzburg-Landau type.