Fractional Integrable Nonlinear Soliton Equations

TL;DR

This paper introduces a new class of integrable fractional nonlinear evolution equations, extending classical soliton equations to fractional media, and demonstrates their ability to model super-dispersive soliton transport.

Contribution

It presents a novel method to construct integrable fractional nonlinear equations from classical ones using advanced mathematical techniques.

Findings

Derived fractional extensions of KdV and NLS equations.

Predicted super-dispersive transport of solitons in fractional media.

Established a framework connecting integrable systems with fractional calculus.

Abstract

Nonlinear integrable equations serve as a foundation for nonlinear dynamics, and fractional equations are well known in anomalous diffusion. We connect these two fields by presenting the discovery of a new class of integrable fractional nonlinear evolution equations describing dispersive transport in fractional media. These equations can be constructed from nonlinear integrable equations using a widely generalizable mathematical process utilizing completeness relations, dispersion relations, and inverse scattering transform techniques. As examples, this general method is used to characterize fractional extensions to two physically relevant, pervasive integrable nonlinear equations: the Korteweg-de Vries and nonlinear Schr\"odinger equations. These equations are shown to predict super-dispersive transport of non-dissipative solitons in fractional media.