Transition to Chaos in Discrete Nonlinear Schrödinger Equation with Long-Range Interaction

Discrete nonlinear Schrödinger equation (DNLS) 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+α with fractional α < 2 and l as a distance between oscillators. This model is called αDNLS. It exhibits competition between the nonlinearity and a level of correlation between interacting far-distanced oscillators, that is defined by the value of α. We consider transition to chaos in this system as a function of α and nonlinearity. It is shown that decreasing of α with respect to nonlinearity stabilize the system. Connection of the model to the fractional genezalization of the NLS (called FNLS) in the long-wave approximation is also discussed and some of the results obtained for αDNLS can be correspondingly extended to the FNLS. PACS: 45.05.+x; 45.50.-j; 45.10.Hj