Computation and stability of waves in equivariant evolution equations
For researchers studying wave phenomena in PDEs, this work provides a rigorous framework and numerical tools for analyzing stability and computing waves in equivariant systems.
The project addressed the computation and stability of travelling and rotating waves in equivariant evolution equations, showing that linear stability implies nonlinear asymptotic stability and that stability persists under discretisation. It also developed numerical methods for wave interactions and nonlinear eigenvalue problems via a contour method.
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.