Subharmonic Dynamics of Wave Trains in Reaction Diffusion Systems

TL;DR

This paper studies the stability of wave trains in reaction-diffusion systems under subharmonic perturbations, introducing a new method to achieve uniform stability results across different perturbation scales.

Contribution

The authors develop a methodology that ensures uniform stability of wave trains against subharmonic perturbations in reaction-diffusion systems.

Findings

Stability of wave trains is non-uniform for large N without new methods.

New methodology achieves uniform stability results for all N.

Results are relevant for localized perturbations in reaction-diffusion systems.

Abstract

We investigate the stability and nonlinear local dynamics of spectrally stable wave trains in reaction-diffusion systems. For each $N\in\mathbb{N}$, such $T$-periodic traveling waves are easily seen to be nonlinearly asymptotically stable (with asymptotic phase) with exponential rates of decay when subject to $NT$-periodic, i.e., subharmonic, perturbations. However, both the allowable size of perturbations and the exponential rates of decay depend on $N$, and, in particular, they tend to zero as $N\to\infty$, leading to a lack of uniformity in such subharmonic stability results. In this work, we build on recent work by the authors and introduce a methodology that allows us to achieve a stability result for subharmonic perturbations which is uniform in $N$. Our work is motivated by the dynamics of such waves when subject to perturbations which are localized (i.e. integrable on the line), which has recently received considerable attention by many authors.