EFFECTS OF DISORDER IN PATTERN-FORMATION

The interplay between localization and nonlinearity is investigated fo r a modified Swift-Hohenberg equation. We introduced a spatially stoch astic contribution etaxi(x) in the control parameter that mimics, for instance, the essential effects of irregularities at the top and botto m plate in Rayleigh-Benard-convection experiments. Near the threshold where the trivial solution u = 0 becomes unstable, this randomness lea ds to localized solutions. Furthermore, the threshold value Of the spa tially averaged control parameter is reduced by the disorder etaxi(x). The interaction between localization and nonlinearity leads to a char acteristic 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-Benard convection.