Superexponential droplet fractalization as a hierarchical formation of dissipative compactons

We study the dynamics of a thin film over a substrate heated from below in a framework of a strongly nonlinear one-dimensional Cahn-Hilliard 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, which means vanishing dimension of the fractal.

Figure 1

(Color online) The shape of the base DC, $\tilde{H}\left(x\right)$.

Figure 2

(Color online) Evolution of the field $h\left(x,t\right)$ illustrating hierarchical formation of droplets, $d=10$. (a) Fragment of the evolution. Notice logarithmic scales of $t$ and $h$. (b) Snapshots of $h$. Panels (c) and (d) are zoomed in fragments of panel (b). Lines represent numerical results for Eq. (1), circles show the profiles of corresponding DCs as in Eq. (6).

Figure 3

(Color online) (a) The distance $L_n$ between two neighboring ${\text{DC}}^{\left(n\right)}$ and ${\text{DC}}^{\left(n-1\right)}$ versus the base $2l_n$ of ${\text{DC}}^{\left(n\right)}$. Squares and circles are numerical results for $d=8$ and $d=10$. Dotted line is a fit, Eq. (10). (b) Mapping $\text{log}{G}_{k+1}\left(\text{log}{G}_k\right)$ calculated in the framework of Eq. (14) (circles); dotted line corresponds to the asymptotic law, Eq. (15).

Figure 4

(Color online) Numerical solution $G\left(\eta \right)$ of Eq. (13). Initial conditions at small $\eta$ are $G\approx 0.5{\eta }^2$ (solid line) and $G\approx 1-0.1{\eta }^2$ (dashed line). Squares and triangles show the local maxima $G_k$ of $G\left(\eta \right)$; for large $k$, the maxima $G_k$ approach the law $G={\eta }^2/\pi$ (dotted line). Inset: a comparison of a piece of $G\left(\eta \right)$ with a single DC (circles), Eq. (6).