Paper detail

Laplacian growth in a channel and Hurwitz numbers

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.