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

Discrete nonlinear Schrodinger 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 $1/l^{1+\alpha}$ with fractional $\alpha < 2$ and $l$ as a distance between oscillators. This model is called $\alpha$DNLS. It exhibits competition between the nonlinearity and a level of correlation between interacting far-distanced oscillators, that is defined by the value of $\alpha$. We consider transition to chaos in this system as a function of $\alpha$ and nonlinearity. It is shown that decreasing of $\alpha$ 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 $\alpha$DNLS can be correspondingly extended to the FNLS.