Laplacian growth in a channel and Hurwitz numbers

TL;DR

This paper explores the integrable structure of zero-tension Laplacian growth in a channel, linking it to the 2D Toda hierarchy and Hurwitz numbers, revealing new mathematical connections.

Contribution

It identifies a novel solution to the 2D Toda hierarchy for Laplacian growth and connects the dispersionless tau-function to double Hurwitz numbers.

Findings

The dispersionless tau-function matches the genus-zero generating function for double Hurwitz numbers.

The solution differs from radial geometry cases but shares similar integrable structures.

The work bridges Laplacian growth dynamics with algebraic geometry via Hurwitz numbers.

Abstract

We study the integrable structure of the 2D Laplacian growth problem with zero surface tension in an infinite channel with periodic boundary conditions in the transverse direction. Similar to the Laplacian growth in radial geometry, this problem can be embedded into the 2D Toda lattice hierarchy in the zero dispersion limit. However, the relevant solution to the hierarchy is different. We characterize this solution by the string equations and construct the corresponding dispersionless tau-function. This tau-function is shown to coincide with the genus-zero part of the generating function for double Hurwitz numbers.