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 form du = - ( 1 + ∂ x 2 ) 2 u + ν ε u + γ u 2 - u 3 dt + σ ε dW , where 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 γ 2 . If γ 2 < 27 38 , then we have amplitude equation with cubic nonlinearities. In the other case γ 2 = 27 38 the cubic term in the amplitude equation vanishes. Therefore we consider larger solutions to obtain an amplitude equation with quintic nonlinearities.