  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.
