Nonlinear Sciences – Chaotic Dynamics
Scientific paper
2011-07-27
Communications in Nonlinear Science and Numerical Simulation 11 (2006) 885-898
Nonlinear Sciences
Chaotic Dynamics
arXiv admin note: substantial overlap with arXiv:nlin/0512013
Scientific paper
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.
Tarasov Vasily E.
Zaslavsky George M.
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