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.
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
\Bibitem{Zab01}
\by A.~V.~Zabrodin
\paper Dispersionless Limit of Hirota Equations in Some Problems of Complex Analysis
\jour TMF
\yr 2001
\vol 129
\issue 2
\pages 239--257
\mathnet{http://mi.mathnet.ru/tmf534}
\crossref{https://doi.org/10.4213/tmf534}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1904798}
\zmath{https://zbmath.org/?q=an:1029.37048}
\transl
\jour Theoret. and Math. Phys.
\yr 2001
\vol 129
\issue 2
\pages 1511--1525
\crossref{https://doi.org/10.1023/A:1012883123413}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000173055900007}
Linking options:
https://www.mathnet.ru/eng/tmf534
https://doi.org/10.4213/tmf534
https://www.mathnet.ru/eng/tmf/v129/i2/p239
This publication is cited in the following 51 articles:
Takashi Takebe, Anton Zabrodin, “Dispersionless version of the constrained Toda hierarchy and symmetric radial Löwner equation”, Lett. Math. Phys., 112 (2022), 105–25
Ferapontov V E., Kruglikov B., Novikov V., “Integrability of Dispersionless Hirota-Type Equations and the Symplectic Monge-Ampere Property”, Int. Math. Res. Notices, 2021:18 (2021), 14220–14251
Akhmedova V. Takebe T. Zabrodin A., “Lowner Equations and Reductions of Dispersionless Hierarchies”, J. Geom. Phys., 162 (2021), 104100
Clery F. Ferapontov V E., “Dispersionless Hirota Equations and the Genus 3 Hyperelliptic Divisor”, Commun. Math. Phys., 376:2 (2020), 1397–1412
Natanzon S.M. Zabrodin A.V., “Formal Solutions To the KP Hierarchy”, J. Phys. A-Math. Theor., 49:14 (2016), 145206
Zabrodin A., “Laplacian Growth in a Channel and Hurwitz Numbers”, J. Phys. A-Math. Theor., 46:18 (2013), 185203
Takasaki K., Nakatsu T., “Thermodynamic limit of random partitions and dispersionless Toda hierarchy”, J. Phys. A: Math. Theor., 45:2 (2012), 025403
Takasaki K., “Generalized String Equations for Double Hurwitz Numbers”, J. Geom. Phys., 62:5 (2012), 1135–1156
Carlet G., Dubrovin B., Mertens L.Ph., “Infinite-dimensional Frobenius manifolds for 2+1 integrable systems”, Math Ann, 349:1 (2011), 75–115
Takasaki K., “Differential Fay Identities and Auxiliary Linear Problem of Integrable Hierarchies”, Exploring New Structures and Natural Constructions in Mathematical Physics, Advanced Studies in Pure Mathematics, 61, ed. Hasegawa K. Hayashi T. Hosono S. Yamada Y., Math Soc Japan, 2011, 387–441
Ferapontov, EV, “Integrable Equations of the Dispersionless Hirota type and Hypersurfaces in the Lagrangian Grassmannian”, International Mathematics Research Notices, 2010, no. 3, 496
Hsin-Fu Shen, Niann-Chern Lee, Ming-Hsien Tu, “Kernel formula approach to the universal Whitham hierarchy”, Theoret. and Math. Phys., 165:2 (2010), 1456–1469
Zabrodin A., “Canonical and Grand Canonical Partition Functions of Dyson Gases as Tau-Functions of Integrable Hierarchies and Their Fermionic Realization”, Complex Anal Oper Theory, 4:3 (2010), 497–514
Teo L.-P., “Conformal Mappings and Dispersionless Toda Hierarchy II: General String Equations”, Comm Math Phys, 297:2 (2010), 447–474
Takasaki K., Takebe T., Teo L.P., “Non-degenerate solutions of the universal Whitham hierarchy”, J. Phys. A: Math. Theor., 43:32 (2010), 325205
Lee N.-Ch., Shen H.-F., Tu M.-H., “A note on reductions of the dispersionless Toda hierarchy”, J Math Phys, 51:12 (2010), 122704
Teo, LP, “Conformal Mappings and Dispersionless Toda Hierarchy”, Communications in Mathematical Physics, 292:2 (2009), 391
Kodama, Y, “Combinatorics of Dispersionless Integrable Systems and Universality in Random Matrix Theory”, Communications in Mathematical Physics, 292:2 (2009), 529
Matsutani, S, “A class of solutions of the dispersionless KP equation”, Physics Letters A, 373:34 (2009), 3001