Amplitude equation with quintic nonlinearities for the generalized Swift-Hohenberg equation with additive degenerate noise

Advances in Difference Equations

Wael W Mohammed¹ 

[1] Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt

关键词: generalized Swift-Hohenberg equation;    multi-scale analysis;    amplitude equation;    additive noise;    60H10;    60H15;   

DOI  :  10.1186/s13662-016-0814-6

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

In this paper, we are interested in the approximation of a stochastic generalized Swift-Hohenberg equation with quadratic and cubic nonlinearity by using the natural separation of time-scales near a change of stability. The main results show that the behavior of the SPDE is well approximated by a stochastic ordinary differential equation describing the amplitude of the dominant mode. The cubic and the quadratic nonlinearities lead to cubic nonlinearities of opposite sign. Here we study the interesting case, where both contributions cancel and in the right scaling a quintic nonlinearity emerges in the amplitude equation. Also, we give a brief indication of how the effect of additive degenerate noise (i.e. noise that does not act directly to the dominant mode) might lead to the stabilization of the trivial solution.

【 授权许可】

CC BY