The Algebro-Geometric Initial Value Problem for the Relativistic Lotka-Volterra Hierarchy and Quasi-Periodic Solutions

We provide a detailed treatment of relativistic Lotka-Volterra hierarchy and a kind of initial value problem with special emphasis on its the theta function representation of all algebro-geometric solutions. The basic tools involve hyperelliptic curve ${\mathcal{K}}_n$ associated with the Burchnall-Chaundy polynomial, Dubrovin-type equations for auxiliary divisors and associated trace formulas. With the help of a foundamental meromorphic function $\tilde{\varphi }$ on ${\mathcal{K}}_p$ and trace formulas, the complex-valued algebro-geometric solutions of of RLV hierarchy are derived.