Amplitude equation for the generalized Swift–Hohenberg equation with noise

Abstract

Abstract. We derive an amplitude equation for a stochastic partial differential equation (SPDE) of Swift-Hohenberg type with a nonlinearity that is composed of a stable cubic and an unstable quadratic term, under the assumption that the noise acts only on the constant mode. Due to the natural separation of timescales, solutions are approximated well by the slow modes. Nevertheless, via the nonlinearity, the noise gets transmitted to those modes too, such that multiplicative noise appears in the amplitude equation.

Multi-scale analysis of SPDEs with degenerate additive noise

“…Here we revisit the case σ ε = ε and generalize the previously obtained results in [2,4,16] in a unified framework. The new interesting case of noise strength σ ε = ε 3/2 with noise not acting directly on the dominant modes leads to deterministic constant forcing in the amplitude equations.…”

“…To illustrate our approximation result of Theorem 13 we consider here the setting of [16], which is a stochastic Swift-Hohenberg equation with respect to periodic boundary conditions on [0, 2π] and forced by spatially constant noise:…”

“…Section 4 give bounds for high non-dominant modes. In Section 5 we give the proof of the approximation Theorem I and we give some applications like the Burgers' equation, treated in [2], the Ginzburg-Landau Equation treated in [4] and the generalized Swift-Hohenberg equation treated in [16]. Finally, we prove the the approximation Theorem II and apply this result on the generalized SwiftHohenberg equation.…”

“…Due to averaging additional linear deterministic terms appear that have the potential to stabilize or destabilize the dominant behavior. In [16] a generalized SwiftHohenberg equation was studied with polynomial nonlinearity containing cubic and quadratic terms was studied.…”

Dynamical Bifurcation of the Generalized Swift–Hohenberg Equation

“…This attractor determines the final patterns of the system. See [Choi & Han, 2015;Gao & Xiao, 2010;Han & Hsia, 2012;Han & Yari, 2012;Klepel et al, 2013;Ma & Wang, 2009;Peletier & Rottschäfer, 2004;Peletier & Williams, 2007;Yari, 2007] for recent developments in fourth order model equations including SHE.…”