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

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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 the 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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Authors and Affiliations

  1. Department of Mathematics, Columbia University, New York, USA

    I. Krichever

  2. Landau Institute, Moscow, Russia

    I. Krichever

  3. ITEP, Moscow, Russia

    I. Krichever, A. Marshakov & A Zabrodin

  4. Max Planck Institute of Mathematics, Bonn, Germany

    A. Marshakov

  5. Lebedev Physics Institute, Moscow, Russia

    A. Marshakov

  6. Institute of Biochemical Physics, Moscow, Russia

    A Zabrodin

Authors
  1. I. Krichever
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  2. A. Marshakov
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  3. A Zabrodin
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Correspondence to I. Krichever.

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Communicated by L. Takhtajan

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Krichever, I., Marshakov, A. & Zabrodin, A. Integrable Structure of the Dirichlet Boundary Problem in Multiply-Connected Domains. Commun. Math. Phys. 259, 1–44 (2005). https://doi.org/10.1007/s00220-005-1387-5

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  • DOI: https://doi.org/10.1007/s00220-005-1387-5

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