When does coarsening occur in the dynamics of one-dimensional fronts ?

TL;DR

This paper investigates the conditions under which coarsening occurs in one-dimensional front dynamics, establishing a criterion based on the relationship between steady state period and amplitude, supported by rigorous proofs and numerical evidence.

Contribution

It provides a rigorous criterion for coarsening in 1D front dynamics and validates it through analytical proofs and numerical simulations.

Findings

Coarsening occurs if the steady state period increases with amplitude.

The criterion is proven for conserved and nonconserved models.

Numerical evidence supports the criterion in crystal surface growth.

Abstract

Dynamics of a one-dimensional growing front with an unstable straight profile are analyzed. We argue that a coarsening process occurs if and only if the period \lambda of the steady state solution is an increasing function of its amplitude A. This statement is rigorously proved for two important classes of conserved and nonconserved models by investigating the phase diffusion equation of the steady pattern. We further provide clear numerical evidences for the growth equation of a stepped crystal surface.