The Algebro-Geometric Solutions for the Ruijsenaars-Toda Hierarchy

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 $ϕ$, Baker-Akhiezer vector $Ψ$ on $\mathcal{K}_p$, the complex-valued algebro-geometric solutions of RT hierarchy are derived.

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