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Uspekhi Mat. Nauk, 2001, Volume 56, Issue 6(342), Pages 89–136 (Mi umn454)  

This article is cited in 20 scientific papers (total in 20 papers)

Hyperelliptic tangential covers and finite-gap potentials

A. Treibich

Université d'Artois

Abstract: This paper is devoted to the study of finite-gap potentials that can be expressed in terms of elliptic functions. New elliptic finite-gap potentials and new elliptic solitons of the KdV equation are found. A step is taken towards the study of relationships connecting elliptic KdV solitons, elliptic finite-gap potentials, and the Jacobians of the associated spectral curves.


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English version:
Russian Mathematical Surveys, 2001, 56:6, 1107–1151

Bibliographic databases:

UDC: 517.95
MSC: Primary 14H52, 35Q53, 14E20; Secondary 14H40, 14H45, 35Q51, 37K10
Received: 21.05.2001

Citation: A. Treibich, “Hyperelliptic tangential covers and finite-gap potentials”, Uspekhi Mat. Nauk, 56:6(342) (2001), 89–136; Russian Math. Surveys, 56:6 (2001), 1107–1151

Citation in format AMSBIB
\by A.~Treibich
\paper Hyperelliptic tangential covers and finite-gap potentials
\jour Uspekhi Mat. Nauk
\yr 2001
\vol 56
\issue 6(342)
\pages 89--136
\jour Russian Math. Surveys
\yr 2001
\vol 56
\issue 6
\pages 1107--1151

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    This publication is cited in the following articles:
    1. Treibich A., “Difference analogs of elliptic KdV solitons and Schrödinger operators”, Int. Math. Res. Not., 2003, no. 6, 313–360  crossref  mathscinet  zmath  isi  elib
    2. Brezhnev Yu.V., “Elliptic solitons and Gröbner bases”, J. Math. Phys., 45:2 (2004), 696–712  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    3. Federov Yu.N., “Algebraic closed geodesics on a triaxial ellipsoid”, Regul. Chaotic Dyn., 10:4 (2005), 463–485  crossref  mathscinet  adsnasa  isi  elib  scopus  scopus
    4. Khare A., Sukhatme U., “Complex periodic potentials with a finite number of band gaps”, J. Math. Phys., 47:6 (2006), 062103, 22 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    5. Flédrich P., Treibich A., “Hyperelliptic osculating covers and KdV solutions periodic in $t$”, Int. Math. Res. Not., 2006, 73476, 17 pp.  crossref  mathscinet  zmath  isi  scopus  scopus
    6. Flédrich P., “Polynômes 3-tangentiels et solutions $t$-périodiques de KdV”, Lett. Math. Phys., 76:2-3 (2006), 231–247  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    7. Takemura K., “The Heun equation and the Calogero–Moser-Sutherland system. V. Generalized Darboux transformations”, J. Nonlinear Math. Phys., 13:4 (2006), 584–611  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    8. Abenda S., Fedorov Yu., “Closed geodesics and billiards on quadrics related to elliptic KdV solutions”, Lett. Math. Phys., 76:2-3 (2006), 111–134  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    9. Accola R.D.M., Previato E., “Covers of tori: genus two”, Lett. Math. Phys., 76:2-3 (2006), 135–161  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    10. Smirnov A.O., “Finite-gap solutions of the Fuchsian equations”, Lett. Math. Phys., 76:2-3 (2006), 297–316  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    11. Gesztesy F., Unterkofler K., Weikard R., “An explicit characterization of Calogero–Moser systems”, Trans. Amer. Math. Soc., 358:2 (2006), 603–656  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    12. Pawellek M., “Quasi-doubly periodic solutions to a generalized Lamé equation”, J. Phys. A, 40:27 (2007), 7673–7686  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    13. Chalykh O., “Algebro-geometric Schrödinger operators in many dimensions”, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 366:1867 (2008), 947–971  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    14. Maier R.S., “Lame polynomials, hyperelliptic reductions and Lame band structure”, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 366:1867 (2008), 1115–1153  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    15. Takemura K., “The Hermite–Krichever ansatz for Fuchsian equations with applications to the sixth Painlevé equation and to finite-gap potentials”, Math. Z., 263:1 (2009), 149–194  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    16. Abenda S., Grinevich P.G., “Periodic billiard orbits on n-dimensional ellipsoids with impacts on confocal quadrics and isoperiodic deformations”, Journal of Geometry and Physics, 60:10 (2010), 1617–1633  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    17. Pavel Etingof, Eric Rains, “On Algebraically Integrable Differential Operators on an Elliptic Curve”, SIGMA, 7 (2011), 062, 19 pp.  mathnet  crossref  mathscinet
    18. A. Treibich, “Systems of Polynomial Equations Defining Hyperelliptic $d$-Osculating Covers”, Funct. Anal. Appl., 49:1 (2015), 40–49  mathnet  crossref  crossref  zmath  isi  elib
    19. Chen Zh., Kuo T.-J., Lin Ch.-Sh., “The Geometry of Generalized Lame Equation, i”, J. Math. Pures Appl., 127 (2019), 89–120  crossref  isi
    20. Chen Zh., Lin Ch.-Sh., “on Algebro-Geometric Simply-Periodic Solutions of the Kdv Hierarchy”, Commun. Math. Phys., 374:1 (2020), 111–144  crossref  isi
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