Numerical Analysis of equivariant evolution equations
09-030 Wolf-Jürgen Beyn, Vera Thümmler.
Dynamics of Patterns in Nonlinear Equivariant PDEs
Many solutions of nonlinear time dependent
partial differential equations show particular spatio-temporal
patterns, such as traveling waves in one space dimension or spiral and
scroll waves in higher space dimensions. The purpose of this paper is
to review some recent progress on the analytical and the numerical
treatment of such patterns. Particular emphasis is put on symmetries
and on the dynamical systems viewpoint that goes beyond existence,
uniqueness and numerical simulation of solutions for single initial
value problems. The nonlinear asymptotic stability of dynamic patterns
is discussed and a numerical approach (the freezing method) is
presented that allows to compute co-moving frames in which solutions
converging to the patterns become stationary. The results are related
to the theory of relative equilibria for equivariant evolution
equations. We discuss several applications to parabolic systems with
nonlinearities of FitzHugh-Nagumo and Ginzburg-Landau type.
