  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.
