Phase sinks and sources around two-dimensional periodic-wave solutions of reaction-diffusion-advection systems

TL;DR

This paper establishes a comprehensive stability framework for two-dimensional periodic traveling waves in reaction-diffusion systems, linking spectral stability to nonlinear stability and describing long-term dynamics influenced by diffusive and dispersive effects.

Contribution

It introduces a diffusive spectral stability assumption that guarantees nonlinear stability and provides detailed asymptotic descriptions for perturbations, extending large-time estimates to complex systems.

Findings

Long-time behavior governed by Whitham modulation system

Enhanced decay rates due to diffusive and dispersive effects

Extended stability analysis to systems with cross-diffusion and anisotropy

Abstract

We develop a complete stability theory for two-dimensional periodic traveling waves of reaction-diffusion systems. More precisely, we identify a diffusive spectral stability assumption, prove that it implies nonlinear stability and provide a sharp asymptotic description of the dynamics resulting from both localized and critically nonlocalized perturbations. In particular, we show that the long-time behavior is governed at leading order by a second-order Whitham modulation system and elucidate how the intertwining of diffusive and dispersive effects may enhance decay rates. The latter requires a non trivial extension of the large-time estimates for constant-coefficient hyperbolic-parabolic operators to some classes of systems with no particular structure, including on one hand systems with a scalar-like - but not scalar - hyperbolic part and a cross-diffusion, and on the other hand anisotropic systems with dispersion.