Stabilization of 1D solitons by fractional derivatives in systems with quintic nonlinearity

We study theoretically the properties of a soliton solution of the fractional Schrodinger equation with quintic nonlinearity. Under "fractional" we understand the Schrodinger equation, where ordinary Laplacian (second spatial derivative in 1D) is substituted by its fractional counterpart with Levy index alpha. We speculate that the latter substitution corresponds to phenomenological account for disorder in a system. Using analytical (variational and perturbative) and numerical arguments, we have shown that while in the case of Schrodinger equation with the ordinary Laplacian (corresponding to Levy index alpha = 2), the soliton is unstable, even infinitesimal difference alpha from 2 immediately stabilizes the soliton texture. Our analytical and numerical investigations of omega (N) dependence (omega is soliton frequency and N its mass) show (within the famous Vakhitov-Kolokolov criterion) the stability of our soliton texture in the fractional alpha < 2 case. Direct numerical analysis of the linear stability problem of soliton texture also confirms this point. We show analytically and numerically that fractional Schrodinger equation with quintic nonlinearity admits the existence of (stable) soliton textures at 2/3 < alpha < 2, which is in accord with existing literature data. These results may be relevant to both Bose-Einstein condensates in cold atomic gases and optical solitons in the disordered media.