Stabilization of single- and multi-peak solitons in the fractional nonlinear Schroedinger equation with a trapping potential

We address the existence and stability of localized modes in the framework of the fractional nonlinear Schroedinger equation (FNSE) with the focusing cubic or focusing-defocusing cubic-quintic nonlinearity and a confining harmonic-oscillator (HO) potential. Approximate analytical solutions are obtained in the form of Hermite-Gauss modes. The linear stability analysis and direct simulations reveal that, under the action of the cubic self-focusing, the single-peak ground state and dipole mode are stabilized by the HO potential at values of the Levy index (the fractionality degree) alpha = 1 and alpha < 1, which lead, respectively, to the critical or supercritical collapse in free space. In addition to that, the inclusion of the quintic self-defocusing provides stabilization of higher-order modes, with the number of local peaks up to seven, at least.