Three-dimensional solitons in fractional nonlinear Schrödinger equation with exponential saturating nonlinearity

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.