Evans Functions and Instability of a Standing Pulse Solution of a Nonlinear System of Reaction Diffusion Equations

Abstract

In this paper, we consider a nonlinear system of reaction diffusion equations arising from mathematical neuroscience and two nonlinear scalar reaction diffusion equations under some assumptions on their coefficients.
The main purpose is to couple together linearized stability criterion (the equivalence of the nonlinear stability, the linear stability and the spectral stability of the standing pulse solutions) and Evans functions to accomplish the existence and instability of standing pulse solutions of the nonlinear system of reaction diffusion equations and the nonlinear scalar reaction diffusion equations. The Evans functions for the standing pulse solutions are constructed explicitly.