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arXiv · 1107.5436

Fractional dynamics of systems with long-range interaction

Vasily E. Tarasov·George M. Zaslavsky
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Abstract

We consider one-dimensional chain of coupled linear and nonlinear oscillators with long-range power wise interaction defined by a term proportional to 1/|n-m|^{\alpha+1}. Continuous medium equation for this system can be obtained in the so-called infrared limit when the wave number tends to zero. We construct a transform operator that maps the system of large number of ordinary differential equations of motion of the particles into a partial differential equation with the Riesz fractional derivative of order \alpha, when 0<\alpha<2. Few models of coupled oscillators are considered and their synchronized states and localized structures are discussed in details. Particularly, we discuss some solutions of time-dependent fractional Ginzburg-Landau (or nonlinear Schrodinger) equation.

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Vasily E. Tarasov, George M. Zaslavsky. 2011-07-27. Fractional dynamics of systems with long-range interaction. https://doi.org/10.1016/j.cnsns.2006.03.005

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