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 TMF, 2001, Volume 129, Number 2, Pages 239–257 (Mi tmf534)

Dispersionless Limit of Hirota Equations in Some Problems of Complex Analysis

A. V. Zabrodinab

a Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center)
b Institute of biochemical physics of the Russian Academy of Sciences

Abstract: 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.

DOI: https://doi.org/10.4213/tmf534

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English version:
Theoretical and Mathematical Physics, 2001, 129:2, 1511–1525

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Citation: A. V. Zabrodin, “Dispersionless Limit of Hirota Equations in Some Problems of Complex Analysis”, TMF, 129:2 (2001), 239–257; Theoret. and Math. Phys., 129:2 (2001), 1511–1525

Citation in format AMSBIB
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Related articles on Google Scholar: Russian articles, English articles

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