Diffusive mixing of periodic wave trains in reaction-diffusion systems

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.