Diffusive mixing of periodic wave trains in reaction-diffusion systems

TL;DR

This paper proves that in reaction-diffusion systems with periodic wave trains, different asymptotic states with distinct phases mix diffusively over time, with the analysis revealing Gaussian or Burgers-type profiles depending on the dispersion relation.

Contribution

It provides a rigorous proof of diffusive mixing of periodic wave trains in reaction-diffusion systems using Bloch wave analysis and renormalization techniques, including a detailed perturbation decomposition.

Findings

Stable diffusive mixing occurs with Gaussian or Burgers profiles.

The analysis applies to systems with different phase states at infinity.

The proof combines Bloch wave analysis, renormalization, and phase decomposition.

Abstract

We consider reaction-diffusion systems on the infinite line that exhibit a family of spectrally stable spatially periodic wave trains $u_0(kx-\om t;k)$ that are parameterized by the wave number $k$. We prove stable diffusive mixing of the asymptotic states $u_0(k x+\phi_{\pm};k)$ as $x\ra \pm\infty$ with different phases $\phi_-\neq\phi_+$ at infinity for solutions that initially converge to these states as $x\ra \pm\infty$. The proof is based on Bloch wave analysis, renormalization theory, and a rigorous decomposition of the perturbations of these wave solutions into a phase mode, which shows diffusive behavior, and an exponentially damped remainder. Depending on the dispersion relation, the asymptotic states mix linearly with a Gaussian profile at lowest order or with a nonsymmetric non-Gaussian profile given by Burgers equation, which is the amplitude equation of the diffusive modes in the case of a nontrivial dispersion relation.