2011/06/21 by Björn Sandstede, Sandstede, Björn, Arnd Scheel +5 · 2 citations
Computer Science · Medicine · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Nonlinear Dynamics and Pattern Formation #Nonlinear Photonic Systems
paper · pdf · doi:10.48550/arxiv.1106.4342
openalex publication_date 2011/06/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider reaction-diffusion systems on the infinite line that exhibit a family of spectrally stable spatially periodic wave trains u0(kx-\om t;k) that are parameterized by the wave number k. We prove stable diffusive mixing of the asymptotic states u0(k x+ϕ±;k) as x\ra ±∞ with different phases ϕ-≠ϕ+ at infinity for solutions that initially converge to these states as x\ra ±∞. 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.