Destabilization of fronts in a class of bi-stable systems

Abstract

In this article, we consider a class of bi-stable reaction-diffusion equations in two components on the real line. We assume that the system is singularly perturbed, i.e. that the ratio of the diffusion coefficients is (asymptotically) small. This class admits front solutions that are asymptotically close to the (stable) front solution of the `trivial' scalar bi-stable limit system $u_t = u_{xx} + u(1-u^2)$. However, in the system these fronts can become unstable by varying parameters. This destabilization is either caused by the essential spectrum associated to the linearized stability problem, or by an eigenvalue that exists near the essential spectrum. We use the Evans function to study the various bifurcation mechanisms and establish an explicit connection between the character of the destabilization and the possible appearance of saddle-node bifurcations of heteroclinic orbits in the existence problem.