The Algebro-Geometric Solutions for the Ruijsenaars-Toda Hierarchy
Peng Zhao, Engui Fan, Yu Hou

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 associated with the Burchnall-Chaundy polynomial, Dubrovin-type equations for auxiliary divisors and associated trace formulas. With the help of a foundamental meromorphic function , Baker-Akhiezer vector on , the complex-valued algebro-geometric solutions of RT hierarchy are derived.
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