Three-dimensional solitons in fractional nonlinear Schr\"{o}dinger equation with exponential saturating nonlinearity

TL;DR

This paper investigates 3D solitons in a fractional nonlinear Schrödinger equation with exponential saturation, analyzing their stability, modulation instability, and noise self-cleaning properties relevant to plasma turbulence modeling.

Contribution

It introduces the study of 3D solitons in a fractional Schrödinger equation with exponential nonlinearity, including stability analysis and noise resilience.

Findings

Identified regions of modulation instability depending on Lévy index

Numerically obtained stable 3D solitons under noise

Demonstrated self-cleaning of solitons from initial disturbances

Abstract

We study the fractional three-dimensional (3D) nonlinear Schr\"{o}dinger equation with exponential saturating nonlinearity. In the case of the L\'{e}vy index $\alpha=1.9$, this equation can be considered as a model equation to describe strong Langmuir plasma turbulence. The modulation instability of a plane wave is studied, the regions of instability depending on the L\'{e}vy index, and the corresponding instability growth rates are determined. Numerical solutions in the form of 3D fundamental soliton (ground state) are obtained for different values of the L\'{e}vy index. It was shown that in a certain range of soliton parameters it is stable even in the presence of a sufficiently strong initial random disturbance, and the self-cleaning of the soliton from such initial noise was demonstrated.