Quasiperiodic solutions of the discrete Chen–Lee–Liu hierarchy

Using the Lax matrix and elliptic variables, we decompose the discrete Chen–Lee–Liu hierarchy into solvable ordinary differential equations. Based on the theory of the algebraic curve, we straighten the continuous and discrete flows related to the discrete Chen–Lee–Liu hierarchy in Abel–Jacobi coordinates. We introduce the meromorphic function $\phi$, Baker–Akhiezer vector $\bar\psi$, and hyperelliptic curve $\mathcal{K}_N$ according to whose asymptotic properties and the algebro-geometric characters we construct quasiperiodic solutions of the discrete Chen–Lee–Liu hierarchy.