The Algebro-Geometric Solutions for the Ruijsenaars-Toda Hierarchy

TL;DR

This paper develops a comprehensive algebro-geometric framework for the Ruijsenaars-Toda hierarchy, utilizing hyperelliptic curves and theta functions to explicitly construct solutions.

Contribution

It introduces a detailed theta function representation of all solutions, employing hyperelliptic curves, Dubrovin equations, and trace formulas for the RT hierarchy.

Findings

Explicit algebro-geometric solutions via theta functions

Use of hyperelliptic curves and trace formulas

Derivation of solutions using Baker-Akhiezer functions

Abstract

We provide a detailed treatment of Ruijsenaars-Toda (RT) hierarchy with special emphasis on its the theta function representation of all algebro-geometric solutions. The basic tools involve hyperelliptic curve $\mathcal{K}_p$ associated with the Burchnall-Chaundy polynomial, Dubrovin-type equations for auxiliary divisors and associated trace formulas. With the help of a foundamental meromorphic function $\phi$, Baker-Akhiezer vector $\Psi$ on $\mathcal{K}_p$, the complex-valued algebro-geometric solutions of RT hierarchy are derived.