Communications in Mathematical Physics

, Volume 259, Issue 1, pp 1–44 | Cite as

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

  • I. Krichever
  • A. Marshakov
  • A Zabrodin


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.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Green Function 
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© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Department of MathematicsColumbia UniversityNew YorkUSA
  2. 2.Landau InstituteMoscowRussia
  3. 3.ITEPMoscowRussia
  4. 4.Max Planck Institute of MathematicsBonnGermany
  5. 5.Lebedev Physics InstituteMoscowRussia
  6. 6.Institute of Biochemical PhysicsMoscowRussia

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