Coupled oscillators with power-law interaction and their fractional dynamics analogues

TL;DR

This paper investigates a one-dimensional chain of coupled oscillators with long-range power-law interactions, mapping their dynamics to fractional differential equations, and compares numerical solutions of discrete and continuous models for soliton-like structures.

Contribution

It introduces a novel mapping of long-range oscillator chains to fractional differential equations and compares discrete and continuous models for nonlinear wave structures.

Findings

Numerical solutions show good agreement between discrete and continuous models.

Soliton-like and breather-like structures are effectively described by fractional dynamics.

The fractional derivative order $ extalpha$ influences the evolution of nonlinear structures.

Abstract

The one-dimensional chain of coupled oscillators with long-range power-law interaction is considered. The equation of motion in the infrared limit are mapped onto the continuum equation with the Riesz fractional derivative of order $\alpha$, when $0<\alpha<2$. The evolution of soliton-like and breather-like structures are obtained numerically and compared for both types of simulations: using the chain of oscillators and using the continuous medium equation with the fractional derivative.