Integrable Structure of the Dirichlet Boundary Problem in Multiply-Connected Domains
High Energy Physics - Theory
2009-11-10 v3 Mathematical Physics
math.MP
Exactly Solvable and Integrable Systems
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.
Cite
@article{arxiv.hep-th/0309010,
title = {Integrable Structure of the Dirichlet Boundary Problem in Multiply-Connected Domains},
author = {I. Krichever and A. Marshakov and A. Zabrodin},
journal= {arXiv preprint arXiv:hep-th/0309010},
year = {2009}
}
Comments
41 pages, 5 figures, LaTeX; some revision of exposition, misprints corrected, the version to appear in Commun. Math. Phys