From the conserved Kuramoto-Sivashinsky equation to a coalescing particles model

TL;DR

This paper maps the conserved Kuramoto-Sivashinsky equation to a coalescing particles model, simplifying simulations, and establishes a  scaling law for interparticle distance growth, providing both numerical and analytical insights.

Contribution

It introduces a particle-based model for the CKS equation, enabling easier simulation and analysis of coarsening dynamics and scaling behavior.

Findings

Interparticle distance scales as .

Numerical probability distribution g() characterized and analytically overestimated.

A new coalescence wave method proposed for better analytical understanding.

Abstract

The conserved Kuramoto-Sivashinsky (CKS) equation, u_t = -(u+u_xx+u_x^2)_xx, has recently been derived in the context of crystal growth, and it is also strictly related to a similar equation appearing, e.g., in sand-ripple dynamics. We show that this equation can be mapped into the motion of a system of particles with attractive interactions, decaying as the inverse of their distance. Particles represent vanishing regions of diverging curvature, joined by arcs of a single parabola, and coalesce upon encounter. The coalescing particles model is easier to simulate than the original CKS equation. The growing interparticle distance \ell represents coarsening of the system, and we are able to establish firmly the scaling \ell(t) \sim \sqrt{t}. We obtain its probability distribution function, g(\ell), numerically, and study it analytically within the hypothesis of uncorrelated intervals, finding an overestimate at large distances. Finally, we introduce a method based on coalescence waves which might be useful to gain better analytical insights into the model.