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

I. Krichever·A. Marshakov·A. Zabrodin
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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.

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I. Krichever, A. Marshakov, A. Zabrodin. 2004-02-04. Integrable Structure of the Dirichlet Boundary Problem in Multiply-Connected Domains. https://doi.org/10.1007/s00220-005-1387-5

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