Spectra and stability of spatially periodic pulse patterns: Evans function factorization via Riccati transformation

TL;DR

This paper introduces an analytic method using Riccati transformation and exponential dichotomies to factorize the Evans function, enabling spectral stability analysis of spatially periodic pulse patterns in singularly perturbed reaction-diffusion systems.

Contribution

It develops a novel analytic Evans function factorization technique that generalizes previous geometric methods for stability analysis of periodic pulses.

Findings

Provides explicit formulas for slow and fast Evans functions.

Extends stability analysis to a broad class of reaction-diffusion systems.

Derives explicit instability criteria for localized periodic patterns.

Abstract

In the spectral stability analysis of localized patterns to singular perturbed evolution problems, one often encounters that the Evans function respects the scale separation. In such cases the Evans function of the full linear stability problem can be approximated by a product of a slow and a fast reduced Evans function, which correspond to properly scaled slow and fast singular limit problems. This feature has been used in several spectral stability analyses in order to reduce the complexity of the linear stability problem. In these studies the factorization of the Evans function was established via geometric arguments that need to be customized for the specific equations and solutions under consideration. In this paper we develop an alternative factorization method. In this analytic method we use the Riccati transformation and exponential dichotomies to separate slow from fast dynamics. We employ our factorization procedure to study the spectra associated with spatially periodic pulse solutions to a general class of multi-component, singularly perturbed reaction-diffusion equations. Eventually, we obtain expressions of the slow and fast reduced Evans functions, which describe the spectrum in the singular limit. The spectral stability of localized periodic patterns has so far only been investigated in specific models such as the Gierer-Meinhardt equations. Our spectral analysis significantly extends and formalizes these existing results. Moreover, it leads to explicit instability criteria.