Stochastic amplitude equation for the stochastic generalized Swift–Hohenberg equation

Journal of the Egyptian Mathematical Society

关键词: Multi-scale analysis;    SPDEs;    Swift–Hohenberg equation;    Amplitude equation;   

DOI  :  10.1016/j.joems.2014.10.005

来源: DOAJ

【 摘 要 】

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+∂x2)2u+νεu+γu2-u3dt+σεdW, 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 γ2. 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.

【 授权许可】

Unknown