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

TL;DR

This paper develops a comprehensive algebro-geometric framework for the relativistic Lotka-Volterra hierarchy, deriving quasi-periodic solutions using hyperelliptic curves, trace formulas, and theta functions.

Contribution

It introduces a detailed theta function representation of solutions for the relativistic Lotka-Volterra hierarchy, linking integrable systems with algebraic geometry techniques.

Findings

Explicit algebro-geometric solutions derived

Connection between hyperelliptic curves and RLV hierarchy established

Framework for quasi-periodic solutions developed

Abstract

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 $\tilde{\phi}$ on $\mathcal{K}_p$ and trace formulas, the complex-valued algebro-geometric solutions of of RLV hierarchy are derived.