Diffusive stability of spatial periodic solutions of the Swift-Hohenberg equation

Abstract

We are interested in the nonlinear stability of the Eckhaus-stable equilibria of the Swift-Hohenberg equation on the infinite line with respect to small integrable perturbations. The difficulty in showing stability for these stationary solutions comes from the fact that their linearizations possess continuous spectrum up to zero. The nonlinear stability problem is treated with renormalization theory which allows us to show that the nonlinear terms are irrelevant, i.e. that the fully nonlinear problem behaves asymptotically as the linearized one which is under a diffusive regime.

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 of oscillations in reaction-diffusion systems

“…For this reason, diffusive stability is often easier to establish in high space dimensions, when diffusion is strong enough to control all possible nonlinear terms, whereas serious problems can occur in low dimensions. This is the case in particular in the stability analysis of one-dimensional spatially periodic patterns [28,10], where a key step of the proof is to show that relevant "self-coupling" terms actually do not occur in the evolution equation for the neutral translational mode.…”

“…Estimate (37) reduces to (34) when t − s = mT , and follows from (35) when t − s ≤ T . In the case where t − s > T , we decompose t − s = τ 1 + mT + τ 2 with τ 1 , τ 2 ∈ [0, T ) and m ∈ N, and we factorize M (t, s; k) as in (28). Using (34), (35), we find…”

Stable transport of information near essentially unstable localized structures

“…R X s ' # " , 9 R X ' I 9 s Q " , and we always assume that Ö c d X is sufficiently small (in particular, X w Ö ) Ç Ö ¡ ). Although the linearization around the bifurcating equilibria i { ae È F ç has continuous spectrum all the way to the imaginary axis, the nonlinear stability of these solutions with respect to spatially localized perturbations can be shown using the techniques developed in [Sch96,Sch98a,Sch98b]. Theorem 2.8 shows that small, spatially localized perturbations of the modulated front do not destroy the form of i © , but lead to a finite shift % z þ .…”

“…[CE90] for more details. The stability proof is based on previous results by the second author [Sch96,Sch98a,Sch98b].…”