Nonlocalized modulation of periodic reaction diffusion waves: Nonlinear stability

TL;DR

This paper extends stability analysis of reaction-diffusion waves to include nonlocalized modulations, showing spectral stability implies nonlinear stability under combined localized and nonlocalized perturbations.

Contribution

It refines existing techniques to demonstrate that spectral stability ensures nonlinear modulational stability for periodic traveling waves with nonlocalized perturbations.

Findings

Spectral stability implies nonlinear stability for nonlocalized modulations.

Detailed analysis of linear behavior under modulational data.

Extension of stability results to broader perturbation classes.

Abstract

By a refinement of the technique used by Johnson and Zumbrun to show stability under localized perturbations, we show that spectral stability implies nonlinear modulational stability of periodic traveling-wave solutions of reaction diffusion systems under small perturbations consisting of a nonlocalized modulation plus a localized perturbation. The main new ingredient is a detailed analysis of linear behavior under modulational data $\bar u'(x)h_0(x)$, where $\bar u$ is the background profile and $h_0$ is the initial modulation