Stochastic amplitude equation for the stochastic generalized Swift–Hohenberg equation

In this paper we derive rigorously the amplitude equation, using the natural separation of time-scales near a change of stability, for the stochastic generalized Swift–Hohenberg equation with quadratic and cubic nonlinearity in this formdu=-(1+∂x2)2u+νεu+γu2-u3dt+σεdW,where W(t)W(t) is a Wiener process. For deterministic PDE it is known that the quadratic term generates an additional cubic term, which is unstable. We consider two cases depending on γ2γ2. If γ2<2738, then we have amplitude equation with cubic nonlinearities. In the other case γ2=2738 the cubic term in the amplitude equation vanishes. Therefore we consider larger solutions to obtain an amplitude equation with quintic nonlinearities.