Amplitude Equations for SPDEs

 

Abstract

We consider a stochastic partial differential equation (Swift-Hohenberg equation) on the real axis with periodic boundary conditions that arises in pattern formation. If the trivial solution is near criticality, and if the stochastic forcing and the deterministic (in)stability are of a comparable magnitude, a complex-valued stochastic ordinary equation can be derived in order to describe the dynamics of the bifurcating solutions. Our result can be used to explain pattern formation below thresholds predicted by deterministic theory. This is an effect frequently observed in experiments (e.g. Benard's problem).