On the Dirichlet Boundary Problem and Hirota Equations

TL;DR

This paper explores the integrable structure of the 2D Dirichlet boundary problem, linking solutions to Hirota equations and proposing a generalization to multiply-connected domains related to matrix models.

Contribution

It reviews the connection between the Dirichlet boundary problem and integrable hierarchies, introducing a tau-function satisfying Hirota equations and suggesting a generalization to complex domains.

Findings

Solution via quasiclassical tau-function for simply-connected domains

Hirota equations of dispersionless Toda hierarchy derived from Green function properties

Proposed extension to multiply-connected domains related to matrix models

Abstract

We review the integrable structure of the Dirichlet boundary problem in two dimensions. The solution to the Dirichlet boundary problem for simply-connected case is given through a quasiclassical tau-function, which satisfies the Hirota equations of the dispersionless Toda hierarchy, following from properties of the Dirichlet Green function. We also outline a possible generalization to the case of multiply-connected domains related to the multi-support solutions of matrix models.