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

Abstract

AbstractWe study theoretically the properties of a soliton solution of the fractional Schrödinger equation with quintic nonlinearity. Under “fractional” we understand the Schrödinger equation, where ordinary Laplacian (second spatial derivative in 1D) is substituted by its fractional counterpart with Lévy index $$\alpha$$ α . 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 Schrödinger equation with the ordinary Laplacian (corresponding to Lévy index $$\alpha =2$$ α = 2 ), the soliton is unstable, even infinitesimal difference $$\alpha$$ α from 2 immediately stabilizes the soliton texture. Our analytical and numerical investigations of $$\omega (N)$$ ω ( N ) dependence ($$\omega$$ ω is soliton frequency and N its mass) show (within the famous Vakhitov–Kolokolov criterion) the stability of our soliton texture in the fractional $$\alpha <2$$ α < 2 case. Direct numerical analysis of the linear stability problem of soliton texture also confirms this point. We show analytically and numerically that fractional Schrödinger equation with quintic nonlinearity admits the existence of (stable) soliton textures at $$2/3<\alpha <2$$ 2 / 3 < α < 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.