Stochastic amplitude equation for the stochastic generalized Swift–Hohenberg equation
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...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2015-10-01
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| Series: | Journal of the Egyptian Mathematical Society |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S1110256X14001278 |
| _version_ | 1818924832340836352 |
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| author | Wael W. Mohammed |
| author_facet | Wael W. Mohammed |
| author_sort | Wael W. Mohammed |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-20T02:31:36Z |
| format | Article |
| id | doaj.art-b75833c162f748c1a196427b965a655b |
| institution | Directory Open Access Journal |
| issn | 1110-256X |
| language | English |
| last_indexed | 2024-12-20T02:31:36Z |
| publishDate | 2015-10-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of the Egyptian Mathematical Society |
| spelling | doaj.art-b75833c162f748c1a196427b965a655b2022-12-21T19:56:33ZengSpringerOpenJournal of the Egyptian Mathematical Society1110-256X2015-10-0123348248910.1016/j.joems.2014.10.005Stochastic amplitude equation for the stochastic generalized Swift–Hohenberg equationWael W. MohammedIn 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.http://www.sciencedirect.com/science/article/pii/S1110256X14001278Multi-scale analysisSPDEsSwift–Hohenberg equationAmplitude equation |
| spellingShingle | Wael W. Mohammed Stochastic amplitude equation for the stochastic generalized Swift–Hohenberg equation Journal of the Egyptian Mathematical Society Multi-scale analysis SPDEs Swift–Hohenberg equation Amplitude equation |
| title | Stochastic amplitude equation for the stochastic generalized Swift–Hohenberg equation |
| title_full | Stochastic amplitude equation for the stochastic generalized Swift–Hohenberg equation |
| title_fullStr | Stochastic amplitude equation for the stochastic generalized Swift–Hohenberg equation |
| title_full_unstemmed | Stochastic amplitude equation for the stochastic generalized Swift–Hohenberg equation |
| title_short | Stochastic amplitude equation for the stochastic generalized Swift–Hohenberg equation |
| title_sort | stochastic amplitude equation for the stochastic generalized swift hohenberg equation |
| topic | Multi-scale analysis SPDEs Swift–Hohenberg equation Amplitude equation |
| url | http://www.sciencedirect.com/science/article/pii/S1110256X14001278 |
| work_keys_str_mv | AT waelwmohammed stochasticamplitudeequationforthestochasticgeneralizedswifthohenbergequation |