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

This article is cited in 46 scientific papers (total in 46 papers)

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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    This publication is cited in the following articles:
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