Diffusive mixing of periodic wave trains in reaction-diffusion systems

Björn Sandstede, Arnd Scheel, Guido Schneider, Hannes Uecker

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20 Scopus citations


We consider reaction-diffusion systems on the infinite line that exhibit a family of spectrally stable spatially periodic wave trains u 0(kx-ωt;k) that are parameterized by the wave number k. We prove stable diffusive mixing of the asymptotic states u 0(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.

Original languageEnglish (US)
Pages (from-to)3541-3574
Number of pages34
JournalJournal of Differential Equations
Issue number5
StatePublished - Mar 1 2012

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