Optical Solitons and Vortices in Fractional Media: A Mini-Review of Recent Results

Abstract

The article produces a brief review of some recent results which predict stable propagation of solitons and solitary vortices in models based on the nonlinear Schrödinger equation (NLSE) including fractional one-dimensional or two-dimensional diffraction and cubic or cubic-quintic nonlinear terms, as well as linear potentials. The fractional diffraction is represented by fractional-order spatial derivatives of the Riesz type, defined in terms of the direct and inverse Fourier transform. In this form, it can be realized by spatial-domain light propagation in optical setups with a specially devised combination of mirrors, lenses, and phase masks. The results presented in the article were chiefly obtained in a numerical form. Some analytical findings are included too, in particular, for fast moving solitons and the results produced by the variational approximation. Moreover, dissipative solitons are briefly considered, which are governed by the fractional complex Ginzburg–Landau equation.

Localized Modes in Nonlinear Fractional Systems with Deep Lattices

“…In the 1D case, the corresponding operator $$(-\partial ^{2}/\partial x^{2}) ^{\alpha /2}$$ is defined by the straightforward counterpart of Equation (3). In 1D systems with the attractive cubic nonlinearity, which corresponds to $$a_{s}<0$$, interval $$0<\alpha \le 1$$ is not considered, as the collapse occurs in it, and in the same 2D setting all values $$\alpha \le 2$$ lead to the collapse for $$a_{s}<0$$ [ 44 ] . However, we here consider Equation (1) with the self‐defocusing sign of the nonlinearity, $$a_{s}>0$$.…”

“…In the framework of these studies, the fractional Schrödinger equation and its extensions in the form of nonlinear fractional Schrödinger equations (NLFSEs) have drawn a great deal of interest, revealing a variety of linear and nonlinear wave patterns in diverse physical media, [ 16–40,42,43 ] see also a recent brief review. [ 44 ] These include gap solitons (GSs) trapped by shallow or moderately deep lattice potentials acting in the combination with the self‐repulsive nonlinearity, [ 23–26,31–33,41,42 ] while the deep‐lattice limit was not studied in detail. It is relevant to mention that optical and matter‐wave GSs, supported by the interplay of periodic potentials, self‐repulsion, and normal (non‐fractional) diffraction, [ 45–56 ] have been experimentally created in fiber Bragg gratings and photonic crystals, [ 57–60 ] in atomic Bose–Einstein condensates loaded into optical lattices (OLs), [ 61 ] and in polariton condensates trapped in semiconductor microcavities.…”

Fractional Integrable Nonlinear Soliton Equations

“…Different versions of the fNLS equation have been studied in, e.g., [20,[27][28][29], and soliton type solutions have been found, but unlike the fNLS and fKdV equations that we introduce, none of these are integrable. The fractional operators in the fNLS and fKdV equations are nonlinear generalizations of the Riesz fractional derivative.…”

“…This form of transport has been observed extensively in biology [5][6][7][8], amorphous materials [9][10][11], porous media [12][13][14], and climate science [15] amongst others. Equations in multiscale media can express fractional derivatives in any governing term [16,17], including dispersion, such as found in the 1D nonlinear Schrödinger equation (NLS) in optics [18][19][20][21][22][23][24] and the Korteweg-de Vries equation (KdV) in water waves [25]. In the case of integer derivatives, NLS and KdV are famously integrable equations, leading to solitonic solutions and an infinite set of conservation laws [26].…”

“…The fractional soliton solutions of fKdV and fNLS are given in equations (20)(21). These correspond to reflectionless bound states of the Schrödinger and AKNS scattering problems with one complex eigenvalue k K = iκ and k S = ξ + iη respectively:…”