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Dispersionless limit of Hirota equations in some problems of complex analysis. (English. Russian original) Zbl 1029.37048

Theor. Math. Phys. 129, No. 2, 1511-1525 (2001); translation from Teor. Mat. Fiz. 129, No. 2, 239-257 (2001).
Summary: 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.

MSC:

37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
30E25 Boundary value problems in the complex plane
35Q53 KdV equations (Korteweg-de Vries equations)
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