Fractional dynamics of systems with long-range interaction

✍ Scribed by Vasily E. Tarasov; George M. Zaslavsky

• Publisher: Elsevier Science
• Year: 2006
• Tongue: English
• Weight: 623 KB
• Volume: 11
• Category: Article
• ISSN: 1007-5704
• DOI: 10.1016/j.cnsns.2006.03.005

✦ Synopsis

We consider one-dimensional chain of coupled linear and nonlinear oscillators with long-range powerwise interaction defined by a term proportional to 1/jn À mj a+1 . Continuous medium equation for this system can be obtained in the socalled 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 a, when 0 < a < 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.