Integrable Fractional Modified Korteweg-de Vries, Sine-Gordon, and Sinh-Gordon Equations

TL;DR

This paper extends classical integrable equations like mKdV, sine-Gordon, and sinh-Gordon to fractional versions using inverse scattering, revealing new solutions and anomalous dispersion properties.

Contribution

It introduces fractional integrable equations for mKdV, sine-Gordon, and sinh-Gordon, and derives explicit one-soliton solutions demonstrating anomalous dispersion effects.

Findings

Explicit fractional equations for mKdV, sine-Gordon, sinh-Gordon

Derived one-soliton solutions exhibiting anomalous dispersion

Connected fractional equations with spectral expansions

Abstract

The inverse scattering transform allows explicit construction of solutions to many physically significant nonlinear wave equations. Notably, this method can be extended to fractional nonlinear evolution equations characterized by anomalous dispersion using completeness of suitable eigenfunctions of the associated linear scattering problem. In anomalous diffusion, the mean squared displacement is proportional to $t^{\alpha}$, $\alpha>0$, while in anomalous dispersion, the speed of localized waves is proportional to $A^{\alpha}$, where $A$ is the amplitude of the wave. Fractional extensions of the modified Korteweg-deVries (mKdV), sine-Gordon (sineG) and sinh-Gordon (sinhG) and associated hierarchies are obtained. Using symmetries present in the linear scattering problem, these equations can be connected with a scalar family of nonlinear evolution equations of which fractional mKdV (fmKdV), fractional sineG (fsineG), and fractional sinhG (fsinhG) are special cases. Completeness of solutions to the scalar problem is obtained and, from this, the nonlinear evolution equation is characterized in terms of a spectral expansion. In particular, fmKdV, fsineG, and fsinhG are explicitly written. One-soliton solutions are derived for fmKdV and fsineG using the inverse scattering transform and these solitons are shown to exhibit anomalous dispersion.