Quasi-periodic and periodic solutions of the Toda lattice via the hyperelliptic sigma function

Yuji Kodama[1]; Shigeki Matsutani[2]; Emma Previato[3]

  • [1] Department of Mathematics The Ohio State University Columbus, OH 43210, U.S.A.
  • [2] 8-21-1 Higashi-Linkan, Minami-ku Sagamihara 252-0311, JAPAN
  • [3] Department of Mathematics and Statistics, Boston University, Boston, MA 02215-2411, U.S.A.

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 2, page 655-688
  • ISSN: 0373-0956

Abstract

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A lattice model with exponential interaction, was proposed and integrated by M. Toda in the 1960s; it was then extensively studied as one of the completely integrable (differential-difference) equations by algebro-geometric methods, which produced both quasi-periodic solutions in terms of theta functions of hyperelliptic curves and periodic solutions defined on suitable Jacobians by the Lax-pair method. In this work, we revisit Toda’s original approach to give solutions of the Toda lattice in terms of hyperelliptic Kleinian (“sigma”) functions for arbitrary genus. We then show that periodic solutions of the Toda lattice correspond to the zeros of Kiepert-Brioschi’s division polynomials, and note these are related to solutions of Poncelet’s closure problem. The hyperelliptic curve of our approach is related in a non-trivial way to the one given by the Lax pair.

How to cite

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Kodama, Yuji, Matsutani, Shigeki, and Previato, Emma. "Quasi-periodic and periodic solutions of the Toda lattice via the hyperelliptic sigma function." Annales de l’institut Fourier 63.2 (2013): 655-688. <http://eudml.org/doc/275599>.

@article{Kodama2013,
abstract = {A lattice model with exponential interaction, was proposed and integrated by M. Toda in the 1960s; it was then extensively studied as one of the completely integrable (differential-difference) equations by algebro-geometric methods, which produced both quasi-periodic solutions in terms of theta functions of hyperelliptic curves and periodic solutions defined on suitable Jacobians by the Lax-pair method. In this work, we revisit Toda’s original approach to give solutions of the Toda lattice in terms of hyperelliptic Kleinian (“sigma”) functions for arbitrary genus. We then show that periodic solutions of the Toda lattice correspond to the zeros of Kiepert-Brioschi’s division polynomials, and note these are related to solutions of Poncelet’s closure problem. The hyperelliptic curve of our approach is related in a non-trivial way to the one given by the Lax pair.},
affiliation = {Department of Mathematics The Ohio State University Columbus, OH 43210, U.S.A.; 8-21-1 Higashi-Linkan, Minami-ku Sagamihara 252-0311, JAPAN; Department of Mathematics and Statistics, Boston University, Boston, MA 02215-2411, U.S.A.},
author = {Kodama, Yuji, Matsutani, Shigeki, Previato, Emma},
journal = {Annales de l’institut Fourier},
keywords = {Toda lattice equation; hyperelliptic sigma function},
language = {eng},
number = {2},
pages = {655-688},
publisher = {Association des Annales de l’institut Fourier},
title = {Quasi-periodic and periodic solutions of the Toda lattice via the hyperelliptic sigma function},
url = {http://eudml.org/doc/275599},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Kodama, Yuji
AU - Matsutani, Shigeki
AU - Previato, Emma
TI - Quasi-periodic and periodic solutions of the Toda lattice via the hyperelliptic sigma function
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 2
SP - 655
EP - 688
AB - A lattice model with exponential interaction, was proposed and integrated by M. Toda in the 1960s; it was then extensively studied as one of the completely integrable (differential-difference) equations by algebro-geometric methods, which produced both quasi-periodic solutions in terms of theta functions of hyperelliptic curves and periodic solutions defined on suitable Jacobians by the Lax-pair method. In this work, we revisit Toda’s original approach to give solutions of the Toda lattice in terms of hyperelliptic Kleinian (“sigma”) functions for arbitrary genus. We then show that periodic solutions of the Toda lattice correspond to the zeros of Kiepert-Brioschi’s division polynomials, and note these are related to solutions of Poncelet’s closure problem. The hyperelliptic curve of our approach is related in a non-trivial way to the one given by the Lax pair.
LA - eng
KW - Toda lattice equation; hyperelliptic sigma function
UR - http://eudml.org/doc/275599
ER -

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