A Mel'nikov approach to soliton-like solutions of systems of discretized Nonlinear Schrödinger Equations

We investigate a class of N coupled discretized nonlinear Schrödinger equations of interacting chains in a nonlinear lattice, which, in the limit of zero coupling, become integrable Ablowitz=Ladik differential-difference equations. We study the existence of stationary localized excitations, in the form of soliton-like time-periodic states, by reducing the system to a perturbed 2N-dimensional symplectic map, whose homoclinic orbits are obtained by a recently developed Mel'nikov analysis. We find that, depending on the perturbation, homoclinic orbits can be accurately located from the simple zeros of a Mel'nikov vector and illustrate our results in the cases N = 2 and 3.