Analytic doubly periodic wave patterns for the integrable discrete nonlinear Schrödinger (Ablowitz–Ladik) model

We derive two new solutions in terms of elliptic functions, one for the dark and one for the bright soliton regime, for the semi-discrete cubic nonlinear Schrödinger equation of Ablowitz and Ladik. When considered in the complex plane, these two solutions are identical. In the continuum limit, they reduce to known elliptic function solutions. In the long wave limit, the dark one reduces to the collision of two discrete dark solitons, and the bright one to a discrete breather.