An Alternative Approach to Integrable Discrete Nonlinear Schrödinger Equations

Abstract

In this article we develop the direct and inverse scattering theory of a discrete matrix Zakharov-Shabat system with solutions U n 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 I N − U n W n and I M − W n U n , and (b) I N − U n W n and I M − W n U n being nonzero multiples of the respective identity matrices I N and I M .