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