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 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.

Nonlinear Stability at the Eckhaus Boundary

“…This idea has been generalized to very general systems where 2 has been replaced by operators which possess a curve of eigenvalues with a parabolic profile for → 0 in Fourier or Bloch space. In many such systems the nonlinear terms turn out be irrelevant [Sch98,DSSS09,SSSU12,JNRZ14].…”

Diffusive Stability of Spatially Periodic Solutions of the Brusselator Model

“…Indeed, the passage from spectral to nonlinear stability has by now been established for small-and large-amplitude patterns alike [S1,S2,JZ,JNRZ1,JNRZ2,SSSU], with in addition considerable information on modulational behavior. However, up to now the rigorous characterization of spectral stability has been carried out in all details only for the particular case of the (scalar) SwiftHohenberg equation [M1, M2, S1] (1.1) ∂ t u = −(1 + ∂ 2 x ) 2 u + ε 2 u − u 3 , u ∈ R 1 , where ε ∈ R 1 is a bifurcation parameter.…”

Diffusive stability against nonlocalized perturbations of planar wave trains in reaction–diffusion systems

“…Throughout this paper we require the spectral stability assumptions (D1)-(D3) to hold true. Similar spectral assumptions are made in one spatial dimension -see [12,13,22]. We emphasize that spectrally-stable planar wave trains to (1.1) can be generated from spectrally-stable wave trains to (1.2) -see §3.…”

“…Notice that the linearization of the reaction-diffusion system (1.2) in one spatial dimension about the wave train solution u(x, t) = u ∞ (kx − ωt) is given by the operator L 0 -see (2.3). Spectral stability of u(x, t) = u ∞ (kx − ωt) as a solution to (1.2), in the sense of [12,13,22], entails that 0 is a simple eigenvalue of L 0 and that the spectrum of L 0 lies to the left of the imaginary axis, except for a quadratic touching at the origin. This leads to the following result.…”