We consider a quite general class of stochastic partial differential equations with quadratic and cubic nonlinearities and derive rigorously amplitude equations, using the natural separation of time-scales near a change of stability. We show that degenerate additive noise has the potential to stabilize or destabilize the dynamics of the dominant modes, due to additional deterministic terms arising in averaging. We focus on equations with quadratic and cubic nonlinearities and give applications to the Burgers’ equation, the Ginzburg–Landau equation, and generalized Swift–Hohenberg equation.
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Mohammed, W.W., Blömker, D. & Klepel, K. Multi-scale analysis of SPDEs with degenerate additive noise. J. Evol. Equ. 14, 273–298 (2014). https://doi.org/10.1007/s00028-013-0213-3
Mathematics Subject Classification (2010)
- Amplitude equation
- multi-scale analysis
- stochastic partial differential equations
- degenerate noise