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Uspekhi Mat. Nauk, 2011, Volume 66, Issue 4(400), Pages 137–178 (Mi umn9435)  

This article is cited in 21 scientific papers (total in 21 papers)

Schur function expansions of KP $\tau$-functions associated to algebraic curves

J. Harnadab, V. Z. Enolskic

a Université de Montréal, Centre de recherches mathématiques
b Concordia University
c Institute of Magnetism, National Academy of Sciences of Ukraine, Kiev

Abstract: The Schur function expansion of Sato–Segal–Wilson KP $\tau$-functions is reviewed. The case of $\tau$-functions related to algebraic curves of arbitrary genus is studied in detail. Explicit expressions for the Plücker coordinate coefficients appearing in the expansion are obtained in terms of directional derivatives of the Riemann $\theta$-function or Klein $\sigma$-function along the KP flow directions. By using the fundamental bi-differential it is shown how the coefficients can be expressed as polynomials in terms of Klein's higher-genus generalizations of Weierstrass' $\zeta$- and $\wp$-functions. The cases of genus-two hyperelliptic and genus-three trigonal curves are detailed as illustrations of the approach developed here.
Bibliography: 53 titles.

Keywords: $\tau$-functions, $\sigma$-functions, $\theta$-functions, Schur functions, KP equation, algebro-geometric solutions to soliton equations.

DOI: https://doi.org/10.4213/rm9435

Full text: PDF file (872 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2011, 66:4, 767–807

Bibliographic databases:

UDC: 515.178.2+517.958+514
MSC: Primary 14H42, 35Q53; Secondary 14H70, 14H55
Received: 07.12.2010

Citation: J. Harnad, V. Z. Enolski, “Schur function expansions of KP $\tau$-functions associated to algebraic curves”, Uspekhi Mat. Nauk, 66:4(400) (2011), 137–178; Russian Math. Surveys, 66:4 (2011), 767–807

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. Bernatska J., Enolski V., Nakayashiki A., “Sato Grassmannian and Degenerate SIGMA Function”, Commun. Math. Phys.  crossref  isi
    2. Eilbeck J.C., Enolski V.Z., Gibbons J., “Sigma, tau and Abelian functions of algebraic curves”, J. Phys. A, 43:45 (2010), 455216, 20 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib
    3. Korotkin D., Shramchenko V., “On higher genus Weierstrass sigma-function”, Phys. D, 241:23-24 (2012), 2086–2094  crossref  mathscinet  zmath  isi  elib
    4. A. V. Zabrodin, “The master $T$-operator for vertex models with trigonometric $R$-matrices as a classical $\tau$-function”, Theoret. and Math. Phys., 174:1 (2013), 52–67  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. Takanori Ayano, Atsushi Nakayashiki, “On Addition Formulae for Sigma Functions of Telescopic Curves”, SIGMA, 9 (2013), 046, 14 pp.  mathnet  crossref  mathscinet
    6. Trans. Moscow Math. Soc., 74 (2013), 245–260  mathnet  crossref  mathscinet  zmath  elib
    7. Alexandrov A., Kazakov V., Leurent S., Tsuboi Z., Zabrodin A., “Classical tau-function for quantum spin chains”, J. High Energy Phys., 2013, no. 9, 064  crossref  mathscinet  zmath  isi  elib
    8. Novikov D.P., Romanovskii R.K., Sadovnichuk S.G., Nekotorye novye metody konechnozonnogo integrirovaniya solitonnykh uravnenii, Nauka, Novosibirsk, 2013, 252 pp.  elib
    9. Anton Zabrodin, “The Master $T$-Operator for Inhomogeneous $XXX$ Spin Chain and mKP Hierarchy”, SIGMA, 10 (2014), 006, 18 pp.  mathnet  crossref  mathscinet
    10. A Zabrodin, “Quantum Gaudin model and classical KP hierarchy”, J. Phys.: Conf. Ser, 482 (2014), 012047  crossref  adsnasa  isi
    11. A. Alexandrov, S. Leurent, Z. Tsuboi, A. Zabrodin, “The master $T$-operator for the Gaudin model and the KP hierarchy”, Nuclear Phys. B, 883 (2014), 173–223  crossref  mathscinet  zmath  isi  elib
    12. F. Balogh, T. Fonseca, J. Harnad, “Finite dimensional Kadomtsev-Petviashvili $\tau$-functions. I. Finite Grassmannians”, J. Math. Phys., 55:8 (2014), 083517  crossref  mathscinet  zmath  isi
    13. Tsuboi Z., Zabrodin A., Zotov A., “Supersymmetric Quantum Spin Chains and Classical Integrable Systems”, no. 5, 2015, 086  crossref  mathscinet  isi
    14. Shigyo Y., “On the expansion coefficients of Tau-function of the BKP hierarchy”, J. Phys. A-Math. Theor., 49:29 (2016), 295201  crossref  mathscinet  zmath  isi  elib  scopus
    15. 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  elib  scopus
    16. Balogh F., Yang D., “Geometric Interpretation of Zhou'S Explicit Formula For the Witten-Kontsevich Tau Function”, Lett. Math. Phys., 107:10 (2017), 1837–1857  crossref  mathscinet  zmath  isi
    17. Gatto L., Salehyan P., “On Plucker Equations Characterizing Grassmann Cones”, Schubert Varieties, Equivariant Cohomology and Characteristic Classes, Impanga 15, EMS Ser. Congr. Rep., eds. Buczynski J., Michalek M., Postinghel E., Eur. Math. Soc., 2018, 97–125  mathscinet  zmath  isi
    18. Mattia Cafasso, Ann du Crest de Villeneuve, Di Yang, “Drinfeld–Sokolov Hierarchies, Tau Functions, and Generalized Schur Polynomials”, SIGMA, 14 (2018), 104, 17 pp.  mathnet  crossref
    19. Harnad J., Lee E., “Symmetric Polynomials, Generalized Jacobi-Trudi Identities and Tau-Fuctions”, J. Math. Phys., 59:9, SI (2018), 091411  crossref  mathscinet  zmath  isi  scopus
    20. Cafasso M. Gavrylenko P. Lisovyy O., “Tau Functions as Widom Constants”, Commun. Math. Phys., 365:2 (2019), 741–772  crossref  mathscinet  zmath  isi  scopus
    21. Gerdjikov V.S. Smirnov A.O. Matveev V.B., “From Generalized Fourier Transforms to Spectral Curves For the Manakov Hierarchy. i. Generalized Fourier Transforms”, Eur. Phys. J. Plus, 135:8 (2020), 659  crossref  isi
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