Dynamic scaling and quasi-ordered states in the two dimensional Swift-Hohenberg equation

TL;DR

This paper investigates pattern formation in the two-dimensional Swift-Hohenberg equation, revealing how noise influences stationary states and demonstrating a simple scaling law for slow ordering dynamics with a specific dynamic exponent.

Contribution

It introduces new dynamic scaling relationships for convective roll ordering and analyzes the impact of noise on stationary states in the Swift-Hohenberg equation.

Findings

Stationary states vary from quasi-ordered to disordered with noise strength.

Ordering dynamics are slow or fast depending on initial conditions and noise.

A scaling law with a dynamic exponent of 1/4 characterizes slow dynamics.

Abstract

The process of pattern formation in the two dimensional Swift-Hohenberg equation is examined through numerical and analytic methods. Dynamic scaling relationships are developed for the collective ordering of convective rolls in the limit of infinite aspect ratio. The stationary solutions are shown to be strongly influenced by the strength of noise. Stationary states for small and large noise strengths appear to be quasi-ordered and disordered respectively. The dynamics of ordering from an initially inhomogeneous state is very slow in the former case and fast in the latter. Both numerical and analytic calculations indicate that the slow dynamics can be characterized by a simple scaling relationship, with a characteristic dynamic exponent of $1/4$ in the intermediate time regime.