Formation and propagation of fundamental and vortex soliton families in 1D and 2D dual-Lévy-index fractional nonlinear Schrödinger equations with cubic–quintic nonlinearity

We address effects of interplay of two different fractional-diffraction terms, corresponding to different Lé vy indices (LIs), in the framework of nonlinear Schrödinger equation (NLSE) with cubic–quintic nonlinearity (dual-LI fractional NLSE), in one- and two-dimensional (1D and 2D) settings. The critical (in 1D) and supercritical (in 2D) wave collapses are suppressed in the presence of a defocusing quintic term, making it possible to produce stable localized modes, including fundamental and vortical ones. In 2D, families of fundamental and vortex solitons, with topological charges S=0,1,2,3, are produced in a numerical form and by dint of the variational approximation (VA). In particular, the threshold power necessary for the existence of 2D fundamental solitons is predicted by the VA, being close to the numerical results. In the case of fractional-diffraction terms with unequal LIs acting in two transverse directions, anisotropic 2D fundamental and vortex solitons are constructed. Stability of the solitons is investigated for small perturbations governed by the linearized equations, and results are corroborated by direct simulations of the perturbed evolution. Those vortex solitons which are unstable are split by azimuthal perturbations into fragments, whose number is determined by the shape of the perturbation eigenmode with the largest growth rate. These results extend the concept of the 2D fractional diffraction and solitons to media with the anisotropic fractionality.