Abstract
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.
Highlights
We consider the stochastic generalized Swift-Hohenberg equation (SGSH) in the following form: du = – + ∂x u + νεu + γ u – u dt + με dW, ( )where νε is the control parameter, W is a finite dimensional Wiener process
The quadratic term γ u plays an essential role; it was first introduced into the GSH equation mathematically in [ ] in order to model the threshold character of periodic pattern formation
In this paper we deal with the case γ and the noise does not act directly to the dominant mode, which is not treated in [ – ]
Summary
In this paper we deal with the case and the noise does not act directly to the dominant mode, which is not treated in [ – ] In this case the amplitude equation ( ) loses its cubic nonlinearity term and it becomes a linear equation only. If we consider ( ) with respect to periodic boundary conditions on the interval [ , π], we obtain the stochastic amplitude equation with multiplicative noise and with an additional deterministic linear term, appearing due to noise and nonlinear interaction, in the Stratonovich form:. The main result of this paper is that near a change of stability on a time-scale of order ε– the solution of ( ) is of the type u(t) = εb ε t + error, where b is the solution of the amplitude equation on the slow time-scale db =.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
