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 u0(kx−ωt;k) that are parameterized by the wave number k. We prove stable diffusive mixing of the asymptotic states u0(kx+ϕ±;k) as x→±∞ with different phases ϕ−≠ϕ+ at infinity for solutions that initially converge to these states as x→±∞. 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.