Singular Limit Analysis of Stability of Traveling Wave Solutions in Bistable Reaction-Diffusion Systems

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Abstract
The stability properties of the traveling front solutions to bistable reaction-diffusion systems in which there are big differences in both the diffusion rates and the reaction rates between two species are studied. In contrast to the scalar case, this bistable system has multiple existence of traveling waves in the appropriate region of parameters. Each wave can be constructed by using a singular perturbation method, and its stability or instability is determined by a simple algebraic quantity appearing in its construction: namely, the sign of the Jacobian of inner and outer matching conditions. The singular limit approach (which is quite different from formal limiting arguments) adopted in this paper is rigorous and very useful in the study of stability problems of singularly perturbed solutions.