Superexponential droplet fractalization as a hierarchical formation of dissipative compactons

TL;DR

This paper investigates the formation of a fractal hierarchy of dissipative compactons in a heated thin film modeled by a nonlinear Cahn-Hilliard equation, revealing superexponential scale reduction and vanishing fractal dimension.

Contribution

It introduces the concept of superexponential droplet fractalization driven by dissipative compactons within a nonlinear thin film model.

Findings

Hierarchical structure composed of dissipative compactons

Superexponential decrease in scale of compactons

Fractal dimension approaches zero at small scales

Abstract

We study the dynamics of a thin film over a substrate heated from below in a framework of a strongly nonlinear one-dimensional Cahn-Hillard equation. The evolution leads to a fractalization into smaller and smaller scales. We demonstrate that a primitive element in the appearing hierarchical structure is a dissipative compacton. Both direct simulations and the analysis of a self-similar solution show that the compactons appear at superexponentially decreasing scales, what means vanishing dimension of the fractal.