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arXiv · 2408.14869

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

Benjamin Melinand·L. Miguel Rodrigues
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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.

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Benjamin Melinand, L. Miguel Rodrigues. 2024-08-27. Phase sinks and sources around two-dimensional periodic-wave solutions of reaction-diffusion-advection systems. https://arxiv.org/abs/2408.14869

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