Numerical Analysis of equivariant evolution equations
13-020 Wolf-Jürgen Beyn, Denny Otten, Jens Rottmann-Matthes.
Stability and Computation of Dynamic Patterns in PDEs
Nonlinear waves are a common feature in
many applications such as the spread of epidemics, electric
signaling in nerve cells, and excitable chemical reactions.
Mathematical models of such systems lead to time-dependent PDEs
of parabolic, hyperbolic or mixed type. Common types of such
waves are fronts and pulses in one, rotating and spiral waves in
two, and scroll waves in three space dimensions. These patterns
may be viewed as relative equilibria of an equivariant evolution
equation where equivariance is caused by the action of a Lie
group. Typical examples of such actions are rotations,
translations or gauge transformations. The aim of the lectures is
to give an overview of problems related to the theoretical and
numerical analysis of such dynamic patterns. One major
theoretical topic is to prove nonlinear stability and relate it
to linearized stability determined by the spectral behavior of
linearized operators. The numerical part focusses on the freezing
method which uses equivariance to transform the given PDE into a
partial differential algebraic equation (PDAE). Solving these
PDAEs generates moving coordinate systems in which the
above-mentioned patterns become stationary.
