Kink motion for the one-dimensional stochastic Allen-Cahn equation

TL;DR

This paper derives an explicit stochastic differential equation describing the motion of interfaces (kinks) in the one-dimensional stochastic Allen-Cahn equation and its mass-conserving variant, using a slow manifold approach in the sharp interface limit.

Contribution

It introduces a novel explicit SDE for kink motion in the stochastic Allen-Cahn equation, accounting for mass conservation and interface interactions.

Findings

Kinks behave approximately like a projected Wiener process.

In the mass-conserving case, kink dynamics are coupled via the mass constraint.

The derived SDE is valid when solutions stay close to the slow manifold.

Abstract

We study the kink motion for the one-dimensional stochastic Allen-Cahn equation and its mass conserving counterpart. Using a deterministic slow manifold, in the sharp interface limit for sufficiently small noise strength we derive an explicit stochastic differential equation for the motion of the interfaces, which is valid as long as the solution stays close to the manifold. On a relevant time-scale, where interfaces move at most by the minimal allowed distance between interfaces, we show that the kinks behave approximately like the driving Wiener-process projected onto the slow manifold, while in the mass-conserving case they are additionally coupled via the mass constraint.