Computation and stability of waves in equivariant evolution equations

Abstract

Travelling and rotating waves are ubiquitous phenomena observed in time dependent PDEs that model the combined effect of dissipation and nonlinear interaction. From an abstract viewpoint, they appear as relative equilibria of an equivariant evolution equation. In numerical computations, the freezing method takes advantage of this structure by splitting the evolution of the PDE into the dynamics on the underlying Lie group and on some reduced phase space. The approach raises a series of questions which were answered to a certain extent: linear stability implies nonlinear (asymptotic) stability, persistence of stability under discretisation, analysis and computation of spectral structures, first versus second order evolution systems, well-posedness of partial differential algebraic equations, spatial decay of wave profiles and truncation to bounded domains, analytical and numerical treatment of wave interactions, relation to connecting orbits in dynamical systems. A further numerical problem related to this topic will be discussed, namely the solution of nonlinear eigenvalue problems via a contour method.