Zabrodin, A. V. 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. Cited in 19 Documents 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) Keywords:conformal mapping problem; Dirichlet boundary problem; 2D inverse potential problem; Hirota equations; tau-functions; integrable hierarchy; dispersionless limit PDFBibTeX XMLCite \textit{A. V. Zabrodin}, Theor. Math. Phys. 129, No. 2, 1511--1525 (2001; Zbl 1029.37048); translation from Teor. Mat. Fiz. 129, No. 2, 239--257 (2001) Full Text: DOI arXiv