Fractional dynamics of coupled oscillators with long-range interaction

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We consider one-dimensional chain of coupled linear and nonlinear oscillators with long-range power-wise 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 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 fractional derivative leads also to the occurrence of localized structures. Particular solutions for fractional time-dependent complex Ginzburg-Landau (or nonlinear Schrodinger) equation are derived. These solutions are interpreted as synchronized states and localized structures of the oscillatory medium.