Algebro-geometric constructions of semi-discrete Chen-Lee-Liu equations

Resorting to the finite-order expansion of the Lax matrix, the relation between elliptic coordinates and potentials is established, from which the semi-discrete Chen-Lee-Liu equations are decomposed into solvable ordinary differential equations. Based on the theory of algebraic curves, Abel-Jacobi coordinates are introduced to straighten out the continuous flow and discrete flow, by which explicit solutions of the semi-discrete Chen-Lee-Liu equations are obtained in the Abel-Jacobi coordinates.