Nonlocalized Modulation of Periodic Reaction Diffusion Waves: Nonlinear Stability

Abstract

By a refinement of the technique used by Johnson and Zumbrun to show stability under localized perturbations, we show that spectral stability implies nonlinear modulational stability of periodic traveling-wave solutions of reaction diffusion systems under small perturbations consisting of a nonlocalized modulation plus a localized perturbation. The main new ingredient is a detailed analysis of linear behavior under modulational dataū ′ (x)h0(x), whereū is the background profile and h0 is the initial modulation.

Diffusive Stability of Spatially Periodic Solutions of the Brusselator Model

“…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.…”

“…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.…”

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

“…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].…”

“…In this paper, by using the results of [1][2][3], we consider 1 (R)-nonlinear modulational stability of spatially periodic traveling waves in a system of reaction-diffusion equations:…”

“…For 2 ≤ ≤ ∞, (R)-estimates of such nonlocalized modulated perturbations have been already established in [1] and [9] for systems of reaction-diffusion equations and of conservation laws, respectively, by using the generalized Hausdorff-Young inequality…”