  A discrete analogue of the dressing method is presented and used to derive
integrable nonlinear evolution equations, including two infinite families of
novel continuous and discrete coupled integrable systems of equations of
nonlinear Schrodinger type. First, a demonstration is given of how discrete
nonlinear integrable equations can be derived starting from their linear
counterparts. Then, starting from two uncoupled, discrete one-directional
linear wave equations, an appropriate matrix Riemann-Hilbert problem is
constructed, and a discrete matrix nonlinear Schrodinger system of equations is
derived, together with its Lax pair. The corresponding compatible vector
reductions admitted by these systems are also discussed, as well as their
continuum limits. Finally, by increasing the size of the problem,
three-component discrete and continuous integrable discrete systems are
derived, as well as their generalizations to systems with an arbitrary number
of components.
