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