Nonlocalized modulation of periodic reaction diffusion waves: The Whitham equation

TL;DR

This paper analyzes the long-term behavior of reaction-diffusion waves under combined localized and nonlocalized perturbations, demonstrating that their evolution is governed by an associated Whitham averaged equation.

Contribution

It establishes the time-asymptotic behavior of stable periodic waves under complex perturbations, linking their evolution to a Whitham averaged equation.

Findings

Solutions are dominated by a modulation governed by the Whitham equation.

The paper extends stability analysis to include nonlocalized modulations.

It provides a detailed description of the asymptotic decay rates.

Abstract

In a companion paper, we established nonlinear stability with detailed diffusive rates of decay of spectrally stable periodic traveling-wave solutions of reaction diffusion systems under small perturbations consisting of a nonlocalized modulation plus a localized perturbation. Here, we determine time-asymptotic behavior under such perturbations, showing that solutions consist to leading order of a modulation whose parameter evolution is governed by an associated Whitham averaged equation.