Dispersionless limit of Hirota equations in some problems of complex analysis

Abstract

The integrable structure, recently revealed in some classical problems of the theory of functions in one complex variable, is discussed. Given a simply connected domain in the complex plane, bounded by a simple analytic curve, we consider the conformal mapping problem, the Dirichlet boundary problem, and to 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 from the family gives a formal solution to the problems listed above. These functions are shown to obey an infinite set of dispersionless Hirota equations. This means that they are $τ$-functions of an integrable hierarchy. The hierarchy is identified with the dispersionless limit of the 2D Toda lattice. In addition to our previous studies, we show that, with a more general definition of the moments, this connection is not specific for any particular solution to the Hirota equations but reflects the structure of the hierarchy itself.