Freezing Solutions of Equivariant Evolution Equations

In this paper we develop numerical methods for integrating general evolution equations ut = F (u), u(0) = u0, where F is defined on a dense subspace of some Banach space (generally infinite dimensional) and is equivariant with respect to the action of a finite dimensional (not necessarily compact) Lie group. Such equations typically arise from autonomous PDE’s on unbounded domains that are invariant under the action of the Euclidean group or one of its subgroups. In our approach we write the solution u(t) as a composition of the action of a time dependent group element with a ’frozen solution’ in the given Banach space. We keep the ’frozen solution’ as constant as possible by introducing a set of algebraic constraints (phase conditions) the number of which is given by the dimension of the Lie group. The resulting PDAE (Partial Differential Algebraic Equation) is then solved by combining classical numerical methods, such as restriction to a bounded domain with asymptotic boundary conditions, half-explicit Euler methods in time and finite differences in space. We provide applications to reaction diffusion systems that have traveling wave or spiral solutions in one and two space dimensions. AMS classification. 65M99, 35K57