Effects of disorder in pattern formation

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Abstract
The interplay between localization and nonlinearity is investigated for a modified Swift-Hohenberg equation. We introduced a spatially stochastic contribution ηξ(x) in the control parameter that mimics, for instance, the essential effects of irregularities at the top and bottom plate in Rayleigh-Bénard-convection experiments. Near the threshold where the trivial solution u≡0 becomes unstable, this randomness leads to localized solutions. Furthermore, the threshold value of the spatially averaged control parameter is reduced by the disorder ηξ(x). The interaction between localization and nonlinearity leads to a characteristic change of the nonlinear bifurcation behavior. When ramping the control parameter in time, the disorder leads to an earlier onset and to a less steep temporal evolution of the pattern. This static and dynamic nonlinear behavior has similarities with recent measurements on Rayleigh-Bénard convection.