Fractional dynamics of systems with long-range interaction

TL;DR

This paper derives a fractional differential equation model for long-range interacting oscillators, revealing new insights into their synchronized states and localized structures, including solutions to fractional Ginzburg-Landau equations.

Contribution

It introduces a transform operator linking discrete long-range oscillator systems to fractional PDEs, advancing understanding of their dynamics.

Findings

Derivation of a fractional PDE with Riesz derivative from oscillator chains

Analysis of synchronized states and localized structures in fractional models

Solutions to fractional Ginzburg-Landau and nonlinear Schrödinger equations

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.