  The numerical discretization of the Zakharov-Shabat Scattering problem using
integrators based on the implicit Euler method, trapezoidal rule and the
split-Magnus method yield discrete systems that qualify as Ablowitz-Ladik
systems. These discrete systems are important on account of their layer-peeling
property which facilitates the differential approach of inverse scattering. In
this paper, we study the Darboux transformation at the discrete level by
following a recipe that closely resembles the Darboux transformation in the
continuous case. The viability of this transformation for the computation of
multisoliton potentials is investigated and it is found that irrespective of
the order of convergence of the underlying discrete framework, the numerical
scheme thus obtained is of first order with respect to the step size.
