Mathematics – Complex Variables
Scientific paper
2001-04-17
Theor.Math.Phys. 129 (2001) 1511-1525; Teor.Mat.Fiz. 129 (2001) 239-257
Mathematics
Complex Variables
17 pages, LaTeX
Scientific paper
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 $\tau$-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.
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