Explicit quasi-periodic solutions to the discrete derivative nonlinear Schrödinger equation

Quasi-periodic solutions to the discrete derivative nonlinear Schrödinger equation are investigated. A Lax pair of the discrete integrable equation is derived, by which we obtain a nonlinear integrable symplectic map. Resorting to this map, the Riemann theta function expressions for discrete potentials are calculated. Consequently, we get the explicit solutions for the discrete derivative nonlinear Schrödinger equation, whose quasi-periodic properties are illustrated through the cases of genus one and genus two.