Discrete and continuous coupled nonlinear integrable systems via the dressing method

A discrete analog 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 Schrödinger 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 Schrödinger 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.