Diffusive mixing of periodic wave trains in reaction–diffusion systems

Abstract: We consider reaction–diffusion systems on the infinite line that exhibit a family of spectrally stable spatially periodic wave trains that are parameterized by the wave number k. We prove stable diffusive mixing of the asymptotic states as with different phases at infinity for solutions that initially converge to these states as . 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. [Copyright &y& Elsevier]