Steady states and coarsening in one-dimensional driven Allen-Cahn system

TL;DR

This paper investigates steady states and coarsening behavior in a one-dimensional driven Allen-Cahn system, deriving phase boundary dynamics, analyzing interactions, and performing bifurcation and stability analyses.

Contribution

It introduces a phase boundary approach for driven Allen-Cahn systems, deriving equations of motion and analyzing coarsening modes and bifurcations.

Findings

Derived equations of motion for phase boundaries.

Analyzed kink interactions and domain size scaling.

Identified coarsening modes and bifurcation structures.

Abstract

We study the steady states and the coarsening dynamics in a one dimensional driven non-conserved system modelled by the so called driven Allen-Cahn equation, which is the standard Allen-Cahn equation with an additional driving force. In particular, we derive equations of motion for the phase boundaries in a phase ordering system obeying this equation using a nearest neighbour interaction approach. Using the equations of motion we explore kink binary and ternary interactions and analyze how the average domain size scale with respect to time. Further, we employ numerical techniques to perform a bifurcation analysis of the one-period stationary solutions of the equation. We then investigate the linear stability of the two-period solutions and thereby identify and study various coarsening modes.