An Alternative Approach to Integrable Discrete Nonlinear Schrodinger Equations

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) .