On quasi-periodic solutions of the discrete Chen-Lee-Liu hierarchy

TL;DR

This paper constructs quasi-periodic solutions for the discrete Chen-Lee-Liu hierarchy using algebraic geometric methods, involving elliptic variables, Lax matrices, and hyperelliptic curves, advancing the understanding of integrable discrete systems.

Contribution

It introduces an algebro-geometric approach to explicitly construct quasi-periodic solutions of the discrete Chen-Lee-Liu hierarchy, linking discrete and continuous flows via algebraic curves.

Findings

Explicit quasi-periodic solutions are constructed.

The discrete and continuous flows are unified under Abel-Jacobi coordinates.

The approach connects Lax pairs, elliptic variables, and hyperelliptic curves.

Abstract

Resorting to the Lax matrix and elliptic variables, the discrete Chen-Lee-Liu hierarchy is decomposed into solvable ordinary differential equations. Based on the theory of algebraic curve, the continuous flow and discrete flow related to the discrete Chen-Lee-Liu hierarchy are straightened under the Abel-Jacobi coordinates. The meromorphic function $\phi$, the Baker-Akhiezer vector $\bar\psi $ and the hyperelliptic curve $\mathcal{K}_N$ are introduced, by which quasi-periodic solutions of the discrete Chen-Lee-Liu hierarchy are constructed according to the asymptotic properties and the algebro-geometric characters of $\phi,\ \bar\psi $ and $\mathcal{K}_N$.