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.

💡 Research Summary

The paper investigates the zero‑surface‑tension Laplacian growth problem in a two‑dimensional infinite channel with periodic boundary conditions across the transverse direction. While the radial version of Laplacian growth is known to be embedded in the dispersionless limit of the 2‑D Toda lattice hierarchy, the channel geometry imposes a different set of constraints, requiring a distinct solution of the hierarchy.

The authors begin by formulating the growth dynamics in complex coordinates (z=x+iy), where the channel is periodic in the (y)‑direction with period (L). The harmonic potential (\phi) satisfies (\Delta\phi=0) together with a kinematic condition that the normal velocity of the moving interface equals the normal derivative of (\phi). By introducing a conformal map to a single‑valued “spectral” variable (\lambda), the evolution of the interface can be expressed as a flow in the space of Lax functions of the Toda hierarchy.

In the dispersionless (zero‑dispersion) limit the hierarchy is described by a pair of Lax functions (L(p)) and (\bar L(p)) together with Orlov–Schulman operators (M,\bar M). Because of the channel’s symmetry the two sets of times coincide ((\bar t_n=t_n)), and the hierarchy is reduced to a single set of commuting flows. The crucial ingredient is a pair of string equations
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