Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik
Doi
Abstract
In this paper, we study a class of singularly perturbed
reaction-diffusion systems, which exhibit under certain conditions slowly
varying multi-pulse solutions. This class contains among others the
Gray-Scott and several versions of the Gierer-Meinhardt model. We first use a
classical singular perturbation approach for the stationary problem and
determine in this way a manifold of quasi-stationary N-pulse solutions.
Then, in the context of the time-dependent problem, we derive an equation for
the leading order approximation of the slow motion along this manifold. We
apply this technique to study 1-pulse and 2-pulse solutions for classical and
modified Gierer-Meinhardt system. In particular, we are able to treat
different types of boundary conditions, calculate folds of the slow manifold,
leading to slow-fast motion, and to identify symmetry breaking singularities
in the manifold of 2-pulse solutions