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Teoreticheskaya i Matematicheskaya Fizika, 2001, Volume 129, Number 2, Pages 239–257
DOI: https://doi.org/10.4213/tmf534
(Mi tmf534)
 

This article is cited in 51 scientific papers (total in 51 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
References:
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.
English version:
Theoretical and Mathematical Physics, 2001, Volume 129, Issue 2, Pages 1511–1525
DOI: https://doi.org/10.1023/A:1012883123413
Bibliographic databases:
Language: Russian
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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\paper Dispersionless Limit of Hirota Equations in Some Problems of Complex Analysis
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\jour Theoret. and Math. Phys.
\yr 2001
\vol 129
\issue 2
\pages 1511--1525
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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:
    1. Takashi Takebe, Anton Zabrodin, “Dispersionless version of the constrained Toda hierarchy and symmetric radial Löwner equation”, Lett. Math. Phys., 112 (2022), 105–25  mathnet  crossref
    2. 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  crossref  mathscinet  isi
    3. Akhmedova V. Takebe T. Zabrodin A., “Lowner Equations and Reductions of Dispersionless Hierarchies”, J. Geom. Phys., 162 (2021), 104100  crossref  mathscinet  isi
    4. Clery F. Ferapontov V E., “Dispersionless Hirota Equations and the Genus 3 Hyperelliptic Divisor”, Commun. Math. Phys., 376:2 (2020), 1397–1412  crossref  mathscinet  isi  scopus
    5. Natanzon S.M. Zabrodin A.V., “Formal Solutions To the KP Hierarchy”, J. Phys. A-Math. Theor., 49:14 (2016), 145206  crossref  mathscinet  zmath  isi  scopus  scopus
    6. Zabrodin A., “Laplacian Growth in a Channel and Hurwitz Numbers”, J. Phys. A-Math. Theor., 46:18 (2013), 185203  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    7. Takasaki K., Nakatsu T., “Thermodynamic limit of random partitions and dispersionless Toda hierarchy”, J. Phys. A: Math. Theor., 45:2 (2012), 025403  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    8. Takasaki K., “Generalized String Equations for Double Hurwitz Numbers”, J. Geom. Phys., 62:5 (2012), 1135–1156  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    9. Carlet G., Dubrovin B., Mertens L.Ph., “Infinite-dimensional Frobenius manifolds for 2+1 integrable systems”, Math Ann, 349:1 (2011), 75–115  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    10. 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  crossref  mathscinet  zmath  isi
    11. Ferapontov, EV, “Integrable Equations of the Dispersionless Hirota type and Hypersurfaces in the Lagrangian Grassmannian”, International Mathematics Research Notices, 2010, no. 3, 496  crossref  mathscinet  zmath  isi  elib  scopus
    12. 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  mathnet  crossref  crossref  isi
    13. 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  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    14. Teo L.-P., “Conformal Mappings and Dispersionless Toda Hierarchy II: General String Equations”, Comm Math Phys, 297:2 (2010), 447–474  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    15. Takasaki K., Takebe T., Teo L.P., “Non-degenerate solutions of the universal Whitham hierarchy”, J. Phys. A: Math. Theor., 43:32 (2010), 325205  crossref  mathscinet  zmath  isi
    16. 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  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    17. Teo, LP, “Conformal Mappings and Dispersionless Toda Hierarchy”, Communications in Mathematical Physics, 292:2 (2009), 391  crossref  mathscinet  zmath  adsnasa  isi  scopus
    18. Kodama, Y, “Combinatorics of Dispersionless Integrable Systems and Universality in Random Matrix Theory”, Communications in Mathematical Physics, 292:2 (2009), 529  crossref  mathscinet  zmath  adsnasa  isi  scopus
    19. Matsutani, S, “A class of solutions of the dispersionless KP equation”, Physics Letters A, 373:34 (2009), 3001  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    20. Khavinson, D, “Planar Elliptic Growth”, Complex Analysis and Operator Theory, 3:2 (2009), 425  crossref  mathscinet  zmath  isi  scopus
    Citing articles in Google Scholar: Russian citations, English citations
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    Теоретическая и математическая физика Theoretical and Mathematical Physics
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