Computation and stability of waves in equivariant evolution equations

TL;DR

This paper investigates the computation and stability of travelling and rotating waves in equivariant PDEs, focusing on the freezing method, spectral analysis, and numerical techniques for wave stability and interactions.

Contribution

It provides a comprehensive analysis of stability, discretization effects, spectral structures, and numerical methods for waves in equivariant evolution equations.

Findings

Linear stability implies non-linear asymptotic stability

Persistence of stability under discretization is established

Spectral analysis and numerical methods for wave interactions are developed

Abstract

Travelling and rotating waves are ubiquitous phenomena observed in time dependent PDEs modelling the combined effect of dissipation and non-linear interaction. From an abstract viewpoint they appear as relative equilibria of an equivariant evolution equa- tion. 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 degree by the project: linear stability implies non-linear (asymp- totic) 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 non-linear eigenvalue problems via a contour method.