Pattern formation in 2d stochastic anisotropic Swift-Hohenberg equation

TL;DR

This paper investigates how noise influences pattern formation in a 2D anisotropic Swift-Hohenberg model relevant to electroconvection, proving solution existence and linking solutions to a stochastic Ginzburg-Landau equation.

Contribution

It establishes the existence of global solutions for the stochastic anisotropic Swift-Hohenberg equation and connects its large-scale behavior to a stochastic Ginzburg-Landau equation.

Findings

Solutions can be approximated by a periodic wave under scaling.

Noise affects the pattern formation process.

The model provides insights into electroconvection patterns.

Abstract

In this paper, we study a phenomenological model for pattern formation in electroconvection, and the effect of noise on the pattern. As such model we consider an anisotropic Swift-Hohenberg equation adding an additive noise. We prove the existence of a global solution of that equation on the two dimensional torus. In addition, inserting a scaling parameter, we consider the equation on a large domain near its change of stability. We observe numerically that, under the appropriate scaling, its solutions can be approximated by a periodic wave, which is modulated by the solutions to a stochastic Ginzburg-Landau equation.