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

We study the fractional three-dimensional (3D) nonlinear Schrödinger equation with exponential saturating nonlinearity. In the case of the Lévy 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 Lévy 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 Lévy 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.