Integrable Structure of the Dirichlet Boundary Problem in Multiply-Connected Domains

TL;DR

This paper explores the integrable structure of the Dirichlet boundary problem in multiply-connected planar domains, introducing a tau-function framework that generalizes known hierarchies and connects to matrix models.

Contribution

It extends the integrable approach to the Dirichlet problem to multiply-connected domains using a generalized tau-function and Hirota-like equations.

Findings

The solution is expressed via a quasiclassical tau-function.

The tau-function satisfies an infinite hierarchy of Hirota-like equations.

Connections to multi-support matrix model solutions are discussed.

Abstract

We study the integrable structure of the Dirichlet boundary problem in two dimensions and extend the approach to the case of planar multiply-connected domains. The solution to the Dirichlet boundary problem in multiply-connected case is given through a quasiclassical tau-function, which generalizes the tau-function of the dispersionless Toda hierarchy. It is shown to obey an infinite hierarchy of Hirota-like equations which directly follow from properties of the Dirichlet Green function and from the Fay identities. The relation to multi-support solutions of matrix models is briefly discussed.