no. 2
Quasi-periodic and periodic solutions of the Toda lattice via the hyperelliptic sigma function
[Intégration du réseau de Toda quasi-périodique et périodique par la fonction sigma hyperelliptique]
Kodama, Yuji1 ; Matsutani, Shigeki2 ; Previato, Emma3
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, Tome 63 (2013) no. 2, pp. 655-688.

M. Toda a donné la définition et l’intégration au moyen les fonctions elliptiques de Jacobi d’un réseau dont les noeuds réagissent réciproquement exponentiellement. La hiérarchie de Toda des équations (différentielles-différences) ont été beaucoup étudiées via les fonctions thêta hyperelliptiques ; une matrice de Lax donne l’intégration dans le cas périodique. Dans ce travail, utilisant la méthode de Toda et les formules d’addition qu’on vienne d’établir pour les fonctions (“sigma”) de Klein hyperelliptiques de n’importe quel genre, nous donnons la solution du réseau quasi-périodique qui est donc aussi une solution de la fermeture de Poncelet. Les coefficients de la matrice de Lax peuvent être écrits comme fonctions rationnelles des coordonnées affines de la courbe hyperelliptique que nous utilisons pour la solution.

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.

DOI : 10.5802/aif.2772
Classification : 14H70, 37K20, 14H51, 37K60
Keywords: Toda lattice equation, hyperelliptic sigma function
Mot clés : Réseau de Toda, sigma-fonction hyperelliptique

Kodama, Yuji 1 ; Matsutani, Shigeki 2 ; Previato, Emma 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. 
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Kodama, Yuji; Matsutani, Shigeki; Previato, Emma. Quasi-periodic and periodic solutions of the Toda lattice via the hyperelliptic sigma function. Annales de l'Institut Fourier, Tome 63 (2013) no. 2, pp. 655-688. doi : 10.5802/aif.2772. https://aif.centre-mersenne.org/articles/10.5802/aif.2772/

[1] Adler, M.; Haine, L.; van Moerbeke, P. Limit matrices for the Toda flow and periodic flags for loop groups, Math. Ann., Volume 296 (1993), pp. 1-33 | DOI | MR | Zbl

[2] Adler, M.; van Moerbeke, P. The Toda Lattice, Dynkin diagrams, singularities and Abelian varieties, Invent. Math., Volume 103 (1991), pp. 223-278 | DOI | MR | Zbl

[3] Adler, M.; van Moerbeke, P.; Vanhaecke, P. Algebraic integrability, Painlevé geometry and Lie algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 47, Springer-Verlag, Berlin, 2004 | MR | Zbl

[4] Baker, H. F. Abelian Functions, Abel’s theorem and the allied theory of theta functions, Cambridge University Press, Cambridge, 1897 | MR | Zbl

[5] Baker, H. F. On the hyperelliptic sigma functions, Amer. J. Math, Volume 20 (1898), pp. 301-384 | DOI | MR

[6] Baker, H. F. On a system of differential equations leading to periodic functions, Acta Math., Volume 27 (1903), pp. 135-156 | DOI | MR

[7] Belokolos, E. D.; Enolskii, V. Z.; Salerno, M. Wannier functions for quasi-periodic finite-gap potentials, Theor. Math. Phys., Volume 144 (2005), pp. 1081-1099 | DOI | MR | Zbl

[8] Brioschi, F. Sur quelques formules pour la multiplication des fonctions elliptiques, C. R. Acad. Sci. Paris, Volume 59 (1864), pp. 769-775

[9] Buchstaber, V. M.; Enolskii, V. Z.; Leykin, D. V.; Novikov, S.P.; Krichever, I.M. Kleinian Functions, Hyperelliptic Jacobians and Applications, Reviews in Mathematics and Mathematical Physics (London), Gordon and Breach, India, 1997, pp. 1-125 | Zbl

[10] Burskii, V. P.; Zhedanov, A. S. On Dirichlet, Poncelet and Abel problems (arXiv:0903.253)

[11] Cantor, D.G. On the analogue of the division polynomials for hyperelliptic curves, J. reine angew. Math., Volume 447 (1994), pp. 91-145 | MR | Zbl

[12] Casian, L.; Kodama, Y. Compactification of the isospectral varieties of nilpotent Toda lattices, RIMS Proceedings (Kyoto University). Surikaisekiken Kokyuroku, Volume 1400 (2004), pp. 39-87 (mathAG/0404345)

[13] Dragović, V.; Radnović, M. Cayley-type conditions for billiards within k quadrics in ℝ d , J. Phys. A, Volume 37 (2004) no. 4, pp. 1269-1276 | DOI | MR | Zbl

[14] Eilbeck, J. C.; Enolski, V. Z.; Gibbons, J. Sigma, tau and Abelian functions of algebraic curves, J. Phys. A, Volume 43 (2010) no. 45 (455216, 20 pp) | DOI | MR | Zbl

[15] Eilbeck, J. C.; Enolskii, D. V. V. Z.and Leykin On the Kleinian construction of Abelian functions of canonical algebraic curves, Proceedings of the 1998 SIDE III conference, 2000 | Zbl

[16] Eilbeck, J. C.; Enol’skii, V. Z.; Matsutani, S.; Ônishi, Y.; Previato, E. Abelian functions for trigonal curves of genus three, Int. Math. Res. Notices (2008) no. 1 (Art. ID rnm 140, 38 pp) | Zbl

[17] Eilbeck, J. C.; Enol’skii, V. Z.; Matsutani, S.; Ônishi, Y.; Previato, E. Addition formulae over the Jacobian pre-image of hyperelliptic Wirtinger varieties, J. Reine Angew. Math., Volume 619 (2008), pp. 37-48 | MR | Zbl

[18] Enolski, V. Z.; Gibbons, J. Addition theorems on the strata of the theta divisor of genus three hyperelliptic curves, draft, 2002

[19] Fay, J. D. Theta functions on Riemann Surfaces, Lecture Notes in Mathematics, 352, Springer-Verlag, Berlin-New York, 1973 | MR | Zbl

[20] Flaschka, H.; Haine, L. Variétés de drapeaux et réseaux de Toda, C. R. Acad. Sci. Paris Sér. I Math., Volume 312 (1991) no. 3, pp. 255-258 | MR | Zbl

[21] Gesztesy, F.; Holden, H.; Michor, J.; Teschl, G. Soliton equations and their algebro-geometric solutions. Vol. II. (1 + 1)-dimensional discrete models, Cambridge Studies in Advanced Mathematics, 114, Cambridge University Press, Cambridge, 2008 | MR | Zbl

[22] Griffiths, P.; Harris, J. On Cayley’s explicit solution to Poncelet’s porism, Enseign. Math., Volume 24 (1978), pp. 31-40 | MR | Zbl

[23] Griffiths, P. A. Variations on a theorem of Abel, Invent. Math., Volume 35 (1976), pp. 321-390 | DOI | MR | Zbl

[24] Hashimoto, K.; Sakai, Y. General form of Humbert’s modular equation for curves with real multiplication of Δ=5, Proc. Japan Acad., Volume 85 (2009), pp. 171-176 | MR | Zbl

[25] Hirota, R. Soliton no suuri: Mathematics in soliton (Japanese) Iwanami Tokyo 1992. The direct method in soliton theory, translated from the 1992 Japanese original and edited by Atsushi Nagai, Jon Nimmo and Claire Gilson. With a foreword by Jarmo Hietarinta and Nimmo. Cambridge Tracts in Mathematics, vol. 155, Cambridge University Press, Cambridge, 2004 | MR | Zbl

[26] Humbert, G. Sur les fonctions abéliennes singulières, Oeuvres de G. Humbert 2 (pub. par les soins de Pierre Humbert et de Gaston Julia), Gauthier-Villars, Paris, 1936, pp. 297-401

[27] Kac, M.; van Moerbeke, P. On some periodic Toda lattices, Proc. Nat. Acad. Sci. U.S.A., Volume 72 (1975) no. 4, pp. 1627-1629 A complete solution of the periodic Toda problem, ibid., 72 (1975), no. 8, 2879–2880 | DOI | MR | Zbl

[28] Kiepert, L. Wirkliche Ausführung der ganzzahligen Multiplication der elliptischen Functionen, J. reine angew. Math., Volume 76 (1873), pp. 21-33 | DOI

[29] Klein, F. Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade, Teubner, Leipzig, 1884

[30] Kodama, Y. Topology of the real part of hyperelliptic Jacobian associated with the periodic Toda lattice, Teoret. Mat. Fiz., Volume 133 (2002) no. 3, pp. 439-462 (Translation in Theoret. and Math. Phys. 133, 2002 no. 3, P. 1692–1711) | MR | Zbl

[31] Leprévost, F.; Markushevich, D. A tower of genus two curves related to the Kowalewski top, J. Reine Angew. Math., Volume 514 (1999), pp. 103-111 | MR | Zbl

[32] Markushevich, D. Kowalevski top and genus-2 curves, Kowalevski Workshop on Mathematical Methods of Regular Dynamics (Leeds, 2000), J. Phys. A, Volume 34 (2001), pp. 2125-2135 | DOI | MR | Zbl

[33] Matsutani, S. Hyperelliptic solution of KdV and KP equations: re-evaluation of Baker’s study on hyperelliptic sigma functions, J. Phys. A: Math. Gen., Volume 34 (2001), pp. 4721-4732 | DOI | MR | Zbl

[34] Matsutani, S. Toda Equation and σ-Function of Genera One and Two, J. Nonlinear Math. Phys., Volume 10 (2003), pp. 555-561 | DOI | MR | Zbl

[35] Matsutani, S. Appendix in [45], Proc. Edinburgh Math. Soc., Volume 48 (2005), pp. 736-742 | MR

[36] Matsutani, S. Neumann system and hyperelliptic al functions, Surv. Math. Appl., Volume 3 (2007), pp. 13-25 | MR | Zbl

[37] Matsutani, S.; Previato, E. Jacobi inversion on strata of the Jacobian of the C rs curve y r =f(x). II (arXiv:1006.1090)

[38] Matsutani, S.; Previato, E. A generalized Kiepert Formula for C ab curves, Israel J. Math., Volume 171 (2009), pp. 305-323 | DOI | MR | Zbl

[39] McKean, H. P.; van Moerbeke, P. Hill and Toda curves, Comm. Pure Appl. Math., Volume 33 (1980) no. 1, pp. 23-42 | DOI | MR | Zbl

[40] Mestre, J.-F. Courbes hyperelliptiques à multiplications réelles, C. R. Acad. Sci. Paris Sér. I Math., Volume 307 (1988), pp. 721-724 | MR | Zbl

[41] van Moerbeke, P. The spectrum of Jacobi matrices, Invent. Math., Volume 37 (1976), pp. 45-81 | DOI | MR | Zbl

[42] van Moerbeke, P.; Mumford, D. The spectrum of difference operators and algebraic curves, Acta Math., Volume 143 (1979) no. 1-2, pp. 93-154 | DOI | MR | Zbl

[43] Mumford, D. Tata Lectures on Theta, vols. 1-2, Birkhäuser, Boston, 1984 | MR | Zbl

[44] Nakayashiki, A. Sigma function as a tau function, Int. Math. Res. Notices, Volume 2010 (2009), pp. 373-394 | MR | Zbl

[45] Ônishi, Y. Determinant expressions for hyperelliptic functions, Proc. Edinburgh Math. Soc., Volume 48 (2005), pp. 705-742 | DOI | MR | Zbl

[46] Schilling, R. J. Generalizations of the Neumann system. A curve-theoretical approach. I, Comm. Pure Appl. Math., Volume 40 (1987) no. 4, pp. 455-522 | DOI | MR | Zbl

[47] Schilling, R. J. Generalizations of the Neumann system. A curve-theoretical approach. II, Comm. Pure Appl. Math., Volume 42 (1989) no. 4, pp. 409-442 | DOI | MR | Zbl

[48] Schilling, R. J. Generalizations of the Neumann system — a curve-theoretical approach. III, Comm. Pure Appl. Math., Volume 45 (1992) no. 7, pp. 775-820 | DOI | MR | Zbl

[49] Tanaka, S.; Date, E. KdV houteishiki, Kinokuniya Press, Tokyo, 1979 (in Japanese)

[50] Toda, M. Vibration of a Chain with Nonlinear Interaction, J. Phys. Soc. Japan, Volume 22 (1967), pp. 431-436 Wave propagation in anharmonic lattices, ibid. 23 (1967), pp. 501-506 | DOI

[51] Vanhaecke, P. Integrable systems in the realm of algebraic geometry, Lecture Notes in Mathematics, 1638, Springer-Verlag, Berlin, 2001 | MR | Zbl

[52] Weierstrass, K. Zur Theorie der Abelschen Funktionen, J. Reine Angew. Math., Volume 47 (1854), pp. 289-306 | DOI | Zbl

[53] Whittaker, E. T.; Watson, G. N. A Course of Modern Analysis, Cambridge University Press, 1927 | MR

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