Nonlocalized modulation of periodic reaction diffusion waves: The Whitham equation

Abstract

Abstract. In a companion paper, we established nonlinear stability with detailed diffusive rates of decay of spectrally stable periodic traveling-wave solutions of reaction diffusion systems under small perturbations consisting of a nonlocalized modulation plus a localized (L 1 ) perturbation. Here, we determine time-asymptotic behavior under such perturbations, showing that solutions consist to leading order of a modulation whose parameter evolution is governed by an associated Whitham averaged equation.

Pointwise nonlinear stability of nonlocalized modulated periodic reaction–diffusion waves

“…Thus, if we could find out how to handle the Bloch solution operator for |x| >> Ct, this would be a nice improvement both in Theorem 1.1, and in the analysis of the previous work [J1, J2]. Another, very interesting, open problem is to determine not only pointwise derivative decay of the modulation ψ, but also its pointwise behavior to lowest order, similarly as done in the L p context in [JNRZ2]. Indeed, this might be a route also to the elimination of hypotheses on decay of h − h ±∞ , since subtracting off this principal behavior would leave only localized terms more amenable to pointwise estimates.…”

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

“…where ( ) is the solution operator of and the formula of ( ) is given by (13). The goal of this paper is to estimate (23) in 1 ( ; R) for an appropriate nonlocalized modulation ( , ).…”

“…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]. For other related works on modulated periodic traveling waves, we refer readers to [11][12][13].…”