Fractional dynamics of coupled oscillators with long-range interaction

Abstract

We consider a one-dimensional chain of coupled linear and nonlinear oscillators with long-range powerwise interaction. The corresponding term in dynamical equations is proportional to 1 ∕ ∣ n − m ∣ α + 1 . It is shown that the equation of motion in the infrared limit can be transformed into the medium equation with the Riesz fractional derivative of order α , when 0 < α < 2 . We consider a few models of coupled oscillators and show how their synchronization can appear as a result of bifurcation, and how the corresponding solutions depend on α . The presence of a fractional derivative also leads to the occurrence of localized structures. Particular solutions for fractional time-dependent complex Ginzburg-Landau (or nonlinear Schrödinger) equation are derived. These solutions are interpreted as synchronized states and localized structures of the oscillatory medium.