An Alternative Approach to Integrable Discrete Nonlinear Schrödinger Equations

In this article we develop the direct and inverse scattering theory of a discrete matrix Zakharov-Shabat system with solutions Un and W n. Contrary to the discretization scheme enacted by Ablowitz and Ladik, a central difference scheme is applied to the positional derivative term in the matrix Zakharov-Shabat system to arrive at a different discrete linear system. The major effect of the new discretization is that we no longer need the following two conditions in theories based on the Ablowitz-Ladik discretization: (a) invertibility of IN −UnW n and IM −W nUn, and (b) IN −UnW n and IM −W nUn being nonzero multiples of the respective identity matrices IN and IM .