Transition to chaos in discrete nonlinear Schrödinger equation with long-range interaction

Discrete nonlinear Schrödinger (DNLS) equation describes a chain of oscillators with nearest-neighbor interactions and a specific nonlinear term. We consider its modification with long-range interaction through a potential proportional to 1/l1+α1/l1+α with fractional α<2α<2 and l   as a distance between oscillators. This model is called ααDNLS. It exhibits competition between the nonlinearity and a level of correlation between interacting far-distanced oscillators, that is defined by the value of αα. We consider transition to chaos in this system as a function of αα and nonlinearity. It is shown that decreasing of αα with respect to nonlinearity stabilize the system. Connection of the model to the fractional generalization of the NLS (called FNLS) in the long-wave approximation is also discussed and some of the results obtained for ααDNLS can be correspondingly extended to the FNLS.