Vortex solitons in fractional nonlinear Schrödinger equation with the cubic-quintic nonlinearity

• Vortex-soliton solutions of the fractional nonlinear Schrdinger equation, with vorticities s = 1, 2, and 3, are constructed in free space. • The Lévy index determines the structure and stability of the vortex-solitons. • The stability of the vortex-solitons is identified by means of the linearized equations for small perturbations and corroborated by direct simulations. • Stable subfamilies are found for all values of s . Unstable vortices split into sets of fragments. We address the existence and stability of vortex-soliton (VS) solutions of the fractional nonlinear Schrödinger equation (NLSE) with competing cubic-quintic nonlinearities and the Lévy index (fractionality) taking values 1 ≤ α ≤ 2. Families of ring-shaped VSs with vorticities s = 1 , 2 , and 3 are constructed in a numerical form. Unlike the usual two-dimensional NLSE (which corresponds to α = 2 ), in the fractional model VSs exist above a finite threshold value of the total power, P . Stability of the VS solutions is investigated for small perturbations governed by the linearized equation, and corroborated by direct simulations. Unstable VSs are broken up by azimuthal perturbations into several fragments, whose number is determined by the fastest growing eigenmode of small perturbations. The stability region, defined in terms of P , expands with the increase of α from 1 up to 2 for all s = 1 , 2, and 3, except for steep shrinkage for s = 2 in the interval of 1 ≤ α ≤ 1.3.