Fluctuations and Pattern Selection Near an Eckhaus Instability

TL;DR

This paper investigates how fluctuations influence pattern stability and selection near an Eckhaus instability, revealing a fluctuation-dependent stability boundary and new asymptotic states through numerical simulations.

Contribution

It introduces a fluctuation-dependent stability boundary and demonstrates altered pattern selection near an Eckhaus instability using stochastic Swift-Hohenberg simulations.

Findings

Fluctuations smear out the classical stability limit.

The width of the fluctuation-dominated region scales as D^{1/2}.

The selected pattern differs from previous predictions.

Abstract

We study the effect of fluctuations in the vicinity of an Eckhaus instability. The classical stability limit, which is defined in the absence of fluctuations, is smeared out into a region in which fluctuations and nonlinearities dominate the decay of unstable states. The width of this region is shown to grow as $ D^{1/2} $, where $D$ is the intensity of the fluctuations. We find an effective stability boundary that depends on $D$. A numerical solution of the stochastic Swift-Hohenberg equation in one dimension is used to test this prediction and to study pattern selection when the initial unstable state lies within the fluctuation dominated region. The asymptotically selected state differs from the predictions of previous analyses. Finally, the nonlinear relaxation for $D > 0$ is shown to exhibit a scaling form.