t 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 ¼ ð1 þ @2
xÞ
2
u þ meu þ cu2  u3
h idt þ redW;
where WðtÞ 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
c2. If c2 < 27
38, then we have amplitude equation with cubic nonlinearities. In the other case
c2 ¼ 27
38 the cubic term in the amplitude equation vanishes. Therefore we consider larger solutions to obtain an amplitude equation with quintic nonlinearities.