Phase Conditions, Symmetries and PDE Continuation

In this chapter we discuss the usefulness of phase conditions for the numerical analysis of finite- and infinite-dimensional dynamical systems that have continuous symmetries. Our main topic is the general approach known as the freezing method, which was developed in [33] and [7]. It will be presented in an abstract framework for evolution equations that are equivariant with respect to the action of a (not necessarily compact) Lie group. Specifically, we introduce an extra parameter (an element in the associated Lie algebra) that determines the position on the group orbit and impose further constraints or phase conditions such that the point in phase space (e.g., the spatial profile in case of a PDE) varies as little as possible. We show particular applications of phase conditions to periodic, heteroclinic and homoclinic orbits in ODEs, to relative equilibria and relative periodic orbits in PDEs, as well as to time integration of equivariant PDEs.