Computation and Stability of Patterns in Hyperbolic-Parabolic Systems

This work is devoted to the stability analysis and numerical long time-simulation of relative equilibria in partial differential equations. The simplest examples of nontrivial relative equilibria - or patterns - are traveling waves. These arise in a wide range of applications like nerve axon equations or reaction-diffusion models. Due to their relevance, the stability of traveling waves has been considered by many authors. Important contributions are made by [Evans, 1972-1975], [Sattinger, 1976], [Henry, 1981], and [Bates and Jones, 1989], to name just a few.A major difficulty with the numericallong-time simulation of relative equilibria arises from the fact that the relevant part of the solution typically leaves the computational domain. Here the freezing method is a possibility to counter this problem. It is presented in a very general setting and justified for a large class of equations. Originally the method was introduced independently by [Beyn and Thümmler, 2004] and [Rowleyet al., 2003]. Its basic idea is to split the evolution into a symmetry and a shape part. In practice, the method leads to a partial differential algebraic equation (PDAE).