Dynamics of fractional N-soliton solutions with anomalous dispersions of integrable fractional higher-order nonlinear Schr\"odinger equations

TL;DR

This paper derives explicit fractional N-soliton solutions for integrable fractional higher-order nonlinear Schrödinger equations, revealing their wave dynamics and velocities, which are relevant for understanding super-dispersion in nonlinear media.

Contribution

It provides exact fractional N-soliton solutions for a class of integrable fractional nonlinear Schrödinger equations using inverse scattering and Riemann-Hilbert methods, a novel analytical achievement.

Findings

Explicit formulas for fractional N-soliton solutions in reflectionless case

Analysis of wave, group, and phase velocities related to amplitude power laws

Potential applications in explaining super-dispersion transport in nonlinear media

Abstract

In this paper, we explore the anomalous dispersive relations, inverse scattering transform and fractional N-soliton solutions of the integrable fractional higher-order nonlinear Schrodinger (fHONLS) equations, containing the fractional Hirota (fHirota), fractional complex mKdV (fcmKdV), and fractional Lakshmanan-Porsezian-Daniel (fLPD) equations, etc. The inverse scattering problem can be solved exactly by means of the matrix Riemann-Hilbert problem with simple poles. As a consequence, an explicit formula is found for the fractional N-soliton solutions of the fHONLS equations in the reflectionless case. In particular, we analyze the fractional one-, two- and three-soliton solutions with anomalous dispersions of fHirota and fcmKdV equations. The wave, group, and phase velocities of these envelope fractional 1-soliton solutions are related to the power laws of their amplitudes. These obtained fractional N-soliton solutions may be useful to explain the related super-dispersion transports of nonlinear waves in fractional nonlinear media.