Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction–diffusion equations

Abstract

Using spatial domain techniques developed by the authors and Myunghyun Oh in the context of parabolic conservation laws, we establish under a natural set of spectral stability conditions nonlinear asymptotic stability with decay at Gaussian rate of spatially periodic traveling-waves of systems of reaction diffusion equations. In the case that wave-speed is identically zero for all periodic solutions, we recover and slightly sharpen a well-known result of Schneider obtained by renormalization/Bloch transform techniques; by the same arguments, we are able to treat the open case of nonzero wave-speeds to which Schneider's renormalization techniques do not appear to apply.

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.…”

“…Going beyond the question of existence, an equally fundamental topic is stability, or "selection," of periodic patterns, and linear and nonlinear behavior under perturbation [E,NW,M1,M2,M3,S1,S2,DSSS,SSSU,JZ,JNRZ1,JNRZ2]. Here, two particular landmarks are the formal "weakly unstable," or small-amplitude, theory of Eckhaus [E] deriving the Ginzburg Landau equation as a canonical model for behavior near the threshold of instability in a variety of processes, and the rigorous linear and nonlinear verification of this theory in [M1, M2, S1] for the Swift-Hohenberg equation, a canonical model for hydrodynamic pattern formation.…”

L1(R)-Nonlinear Stability of Nonlocalized Modulated Periodic Reaction-Diffusion Waves

“…Recalling the linear operator in (7), we review the nonlinear perturbation equation about V established in [1,7].…”

“…Stable diffusive mixing of periodic reaction-diffusion waves has been obtained in [6] based on a nonlinear decomposition of phase and amplitude variables and renormalization techniques. Johnson, Zumbrun, and their collaborators also showed (R) ( ≥ 2) nonlinear modulational stability of periodic traveling waves of systems of reaction-diffusion equations and of conservation under both localized and nonlocalized perturbations in [1,[7][8][9]. By using pointwise linear estimates together with a nonlinear iteration scheme developed by Johnson-Zumbrun, pointwise nonlinear stability of such 2 Advances in Mathematical Physics waves has been also studied in [2,3,10].…”

“…We consider this linear operator on 2 (R) with densely defined domain 2 (R). In this section, we recall Bloch analysis [1,3,7] which is a key idea of spectral analysis of linear operators with periodic coefficients. If we apply Floquet theory [14] to the first-order ODE system obtained by (7), ∈ C lies in 2 (R)-spectrum of if and only if V has the form…”