Abstract
We 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. 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), 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 Schrödinger 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.
Highlights
We study theoretically the properties of a soliton solution of the fractional Schrödinger equation with quintic nonlinearity
When considering the systems with different kinds of nonlinearity that originate from various physical problems (like nonlinear optics, Bose-Einstein condensation (BEC), theory of e lasticity5–7) the so-called nonlinear Schrödinger equation (NLSE) is usually invoked
The NLSE plays an important role in sciences, where solitons appear[8,11]
Summary
In the spirit of what was said above, here we consider the substitution of the ordinary Laplacian (second derivative in one spatial dimension) in the quintic NLSE of the form. Our aim is to substitute the ordinary second spatial derivative in (2) by the 1D fractional Laplacian. We note that at α = 2 the fractional 1D Laplacian converts into ordinary second spatial derivative. The easiest way to check that is via Fourier transformation, which for second derivative gives −k2. This implies that the Fourier image of the fractional Laplacian (3) gives −|k|α or explicitly. This generates following fractional equation for y(x).
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