Geodesic
Dispersionless Limit of Hirota Equations in Some Problems of Complex Analysis
Teoretičeskaâ i matematičeskaâ fizika, Tome 129 (2001) no. 2, pp. 239-257 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study the integrable structure recently revealed in some classical problems in the theory of functions in one complex variable. Given a simply connected domain bounded by a simple analytic curve in the complex plane, we consider the conformal mapping problem, the Dirichlet boundary problem, and the 2D inverse potential problem associated with the domain. A remarkable family of real-valued functionals on the space of such domains is constructed. Regarded as a function of infinitely many variables, which are properly defined moments of the domain, any functional in the family gives a formal solution of the above problems. These functions satisfy an infinite set of dispersionless Hirota equations and are therefore tau-functions of an integrable hierarchy. The hierarchy is identified with the dispersionless limit of the 2D Toda chain. In addition to our previous studies, we show that within a more general definition of the moments, this connection pertains not to a particular solution of the Hirota equations but to the hierarchy itself. 
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A. V. Zabrodin. Dispersionless Limit of Hirota Equations in Some Problems of Complex Analysis. Teoretičeskaâ i matematičeskaâ fizika, Tome 129 (2001) no. 2, pp. 239-257. http://geodesic.mathdoc.fr/item/TMF_2001_129_2_a6/

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