Integrable Structure of the Dirichlet Boundary Problem in Multiply-Connected Domains
TLDR
In this article, the integrable structure of the Dirichlet boundary problem in two dimensions was studied and a quasiclassical tau-function was proposed for the multiply-connected case.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.read more
Citations
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Laplacian growth and Whitham equations of soliton theory
TL;DR: The Laplacian growth of multiply-connected domains in the case of zero surface tension is proven to be equivalent to an integrable system of Whitham equations known in soliton theory as mentioned in this paper.
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The characterization of two-component (2+1)-dimensional integrable systems of hydrodynamic type
TL;DR: In this article, the necessary and sufficient conditions for a two-component (2 + 1)-dimensional system of hydrodynamic type to possess infinitely many hydrodynamic reductions were obtained.
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Normal random matrix ensemble as a growth problem
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References
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TL;DR: Riemann's theta function as discussed by the authors is the prime-form function of Riemann surfaces, and it can be expressed in terms of cyclic unramified coverings and Ramified double coverings.
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Kanehisa Takasaki,Takashi Takebe +1 more
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TL;DR: The Schwarz Function of the analytic arc as mentioned in this paper is an analytic function of a complex variable that can be expressed as an antianalytic mapping of the complex conjugate of the function.