Stochastic amplitude equation for the stochastic generalized Swift–Hohenberg equation | ||||
| Journal of the Egyptian Mathematical Society | ||||
| Volume 23, Issue 3, 2015, Page 482-489 PDF (550.59 K) | ||||
| Document Type: Review article | ||||
| DOI: 10.1016/j.joems.2014.10.005 | ||||
 View on SCiNiTO | ||||
| Author | ||||
| Wael W. Mohammed* | ||||
| Department of Mathematics, Faculty of Science, Mansoura University, Egypt | ||||
| Abstract | ||||
| 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. | ||||
| Keywords | ||||
| KEYWORDS Multi-scale analysis; SPDEs; Swift–Hohenberg equation; Amplitude equation | ||||
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