The impact of multiplicative noise in SPDEs close to bifurcation via amplitude equations. (5th June 2020)

• Record Type: Journal Article
• Authors: Blömker, Dirk; Fu, Hongbo
• Is Part Of: Nonlinearity. Volume 33:Number 8(2020)
• Page Start: 3905
• Page End: 3927
• Publication Date: 2020-06-05
• Subjects: amplitude equation -- multiplicative noise -- multi-scale analysis -- bifurcation -- slow–fast system -- stochastic partial differential equations; 60H10; 60H15; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ab801e
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: This article deals with the approximation of a stochastic partial differential equation (SPDE) via amplitude equations. We consider an SPDE with a cubic nonlinearity perturbed by a general multiplicative noise that preserves the constant trivial solution and we study the dynamics around it for the deterministic equation being close to a bifurcation. Based on the separation of time-scales close to a change of stability, we rigorously derive an amplitude equation describing the dynamics of the bifurcating pattern.This allows us to approximate the original infinite dimensional dynamics by a simpler effective dynamics associated with the solution of the amplitude equation. To illustrate the abstract result we apply it to a simple one-dimensional stochastic Ginzburg–Landau equation.