Phase sinks and sources around two-dimensional periodic-wave solutions of reaction-diffusion-advection systems
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.
Keywords
Cite
@article{arxiv.2408.14869,
title = {Phase sinks and sources around two-dimensional periodic-wave solutions of reaction-diffusion-advection systems},
author = {Benjamin Melinand and L. Miguel Rodrigues},
journal= {arXiv preprint arXiv:2408.14869},
year = {2024}
}