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

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.