Toda Equations and $\sigma$-Functions of Genera One and Two

TL;DR

This paper explores the Toda equations across continuous, discrete, and ultradiscrete levels using elliptic and hyperelliptic sigma functions of genus one and two, revealing ultradiscrete Toda as a valuation of recursion relations.

Contribution

It introduces a unified approach to Toda equations through sigma functions of low genus, connecting ultradiscrete Toda to recursion relations of psi functions.

Findings

Ultradiscrete Toda equation derived as a valuation of recursion relations.

Unified framework for Toda equations using elliptic and hyperelliptic sigma functions.

Connections established between continuous, discrete, and ultradiscrete Toda equations.

Abstract

We study the Toda equations in the continuous level, discrete level and ultradiscrete level in terms of elliptic and hyperelliptic $\sigma$ and $\psi$ functions of genera one and two. The ultradiscrete Toda equation appears as a discrete-valuation of recursion relations of $\psi$ functions.