Breather solutions in the diffraction managed NLS equation

We show the existence of localized breather solutions in an averaged version of the discrete nonlinear Schrِdinger equation (NLS) with diffraction management, a system that models coupled waveguide arrays with periodic diffraction management geometries. The breather solutions are constrained extrema of the Hamiltonian of the averaged system and their existence is shown by a discrete version of the concentration-compactness principle. The main assumptions are that the averaged diffraction is sufficiently small (compared to strength of the nonlinearity) and that the sign of the nonlinearity corresponds to the focusing case. An interesting feature of the problem is that the nonlinear interaction between neighboring lattice sites can be large and is of infinite range. On the other hand, the interaction decays rapidly at sufficiently large distances, and this plays an important role in the proof. The results also apply to higher dimensional lattices, and to the discrete NLS equation.