Singular Laplacian Growth

TL;DR

This paper derives equations for two-dimensional Laplacian growth with singularities on the unit circle, revealing that solution non-uniqueness and singularity creation lead to complex growth behaviors independent of initial conditions.

Contribution

It introduces a novel formulation of Laplacian growth equations for singular conformal maps and highlights the non-uniqueness and complexity arising from singularity creation mechanisms.

Findings

Equations describe singularity motions on the unit circle.

Solutions are not sensitive to initial conditions.

Singularity creation leads to increased complexity without noise.

Abstract

The general equations of motion for two dimensional Laplacian growth are derived using the conformal mapping method. In the singular case, all singularities of the conformal map are on the unit circle, and the map is a degenerate Schwarz-Christoffel map. The equations of motion describe the motions of these singularities. Despite the typical fractal-like outcomes of Laplacian growth processes, the equations of motion are shown to be not particularly sensitive to initial conditions. It is argued that the sensitivity of this system derives from a novel cause, the non-uniqueness of solutions to the differential system. By a mechanism of singularity creation, every solution can become more complex, even in the absence of noise, without violating the growth law. These processes are permitted, but are not required, meaning the equation of motion does not determine the motion, even in the small.