Matched asymptotic expansion approach to pulse dynamics for a three-component reaction diffusion systems

TL;DR

This paper develops a matched asymptotic expansion method to analyze pulse solutions in a three-component reaction diffusion system, providing insights into their existence, stability, and bifurcations, with advantages over traditional geometric methods.

Contribution

It introduces a novel application of MAE and SLEP methods to higher-dimensional systems, offering more precise stability analysis and bifurcation insights compared to existing approaches.

Findings

Existence of standing pulse solutions established.

Identification of codimension two bifurcations involving drift and Hopf bifurcations.

Numerical confirmation of stable breathers near bifurcation points.

Abstract

We study the existence and stability of standing pulse solutions to a singularly perturbed three-component reaction diffusion system with one-activator and two-inhibitor type. We apply the MAE (matched asymptotic expansion) method to the construction of solutions and the SLEP (Singular Limit Eigenvalue Problem) method to their stability properties. This approach is not just an alternative approach to geometric singular perturbation and the associated Evans function, but gives us two advantages: one is the extendability to higher dimensional case, and the other is to allow us to obtain more precise information on the behaviors of critical eigenvalues. This implies the existence of codimension two singularity of drift and Hopf bifurcations for the standing pulse solution and it is numerically confirmed that stable standing and traveling breathers emerge around the singularity in a physically-acceptable regime.