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

Numerische Analyse äquivarianter Evolutionsgleichungen

11-016 Jens Rottmann-Matthes.
Stability and freezing of nonlinear waves in first order hyperbolic PDEs

It is a well-known problem to derive nonlinear stability of a traveling wave from the spectral stability of a linearization. In this paper we prove such a result for a large class of hyperbolic systems. To cope with the unknown asymptotic phase, the problem is reformulated as a partial differential algebraic equation for which asymptotic stability becomes the usual Lyapunov stability. The stability proof is then based on linear estimates from a previous paper and a careful analysis of the nonlinear terms. Moreover, we show that the freezing method is well-suited for the long time simulation and numerical approximation of the asymptotic behavior. The theory is illustrated by two numerical examples, including a hyperbolic version of the Hodgkin-Huxley equations.