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

Kodama, Yuji; Matsutani, Shigeki; Previato, Emma

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.

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