Growth of fat slits and dispersionless KP hierarchy

TL;DR

This paper links the growth of fat slits in the upper half plane to the dispersionless KP hierarchy, extending known conformal map solutions and introducing a new family of solutions through harmonic moments.

Contribution

It introduces a novel connection between fat slit growth and the dispersionless KP hierarchy, expanding the class of conformal map solutions.

Findings

Fat slit conformal maps obey Lax equations of the dispersionless KP hierarchy.

Deformation of fat slits models a growth process similar to Laplacian growth.

Provides a new large family of solutions to the hierarchy.

Abstract

A "fat slit" is a compact domain in the upper half plane bounded by a curve with endpoints on the real axis and a segment of the real axis between them. We consider conformal maps of the upper half plane to the exterior of a fat slit parameterized by harmonic moments of the latter and show that they obey an infinite set of Lax equations for the dispersionless KP hierarchy. Deformation of a fat slit under changing a particular harmonic moment can be treated as a growth process similar to the Laplacian growth of domains in the whole plane. This construction extends the well known link between solutions to the dispersionless KP hierarchy and conformal maps of slit domains in the upper half plane and provides a new, large family of solutions.