Pulses in singularly perturbed reaction-diffusion systems with slowly mixed nonlinearity.

Abstract

This article is concerned with the existence and spectral stability of pulses in singularly perturbed two-component reaction-diffusion systems with slowly mixed nonlinearity. In this paper, the slow nonlinearity is referred to be "mixed" in the sense that it is generated by a trigonometric function multiplied by a power function. We demonstrate via geometric singular perturbation theory that this model can support both the single-pulse and the double-hump solutions. The presence of the slowly mixed nonlinearity complicates the stability analysis on pulses, since the conditions that govern their stability can no longer be explicitly computed. We remove this difficulty by introducing the hypergeometric functions followed by a comparison theorem. By doing so, the "slow-fast" eigenvalues can be determined via the nonlocal eigenvalue problem method. We prove that the double-hump solution is always unstable, while the single-pulse solution can be stable under certain parameter conditions.