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.

We study two-dimensional (2D) matter-wave solitons in spinor Bose-Einstein condensates under the action of the spin-orbit coupling and opposite signs of the self-and cross-interactions. Stable 2D twocomponent solitons of the mixed-mode type are found if the cross-interaction between the components is attractive, while the self-interaction is repulsive in each component. Stable solitons of the semi-vortex type are formed in the opposite case, under the action of competing self-attraction and cross-repulsion. The solitons exist with the total norm taking values below a collapse threshold. Further, in the case of the repulsive self-interaction and inter-component attraction, stable 2D selftrapped modes, which may be considered as quantum droplets (QDs), are created if the beyondmean-field Lee-Huang-Yang terms are added to the self-repulsion in the underlying system of coupled Gross-Pitaevskii equations. Stable QDs of the mixed-mode type, of a large size with an anisotropic density profile, exist with arbitrarily large values of the norm, as the Lee-Huang-Yang terms eliminate the collapse. The effect of the spin-orbit coupling term on characteristics of the QDs is systematically studied. We also address the existence and stability of QDs in the case of SOC with mixed Rashba and Dresselhaus terms, which makes the density profile of the QD more isotropic. Thus, QDs in the spin-orbit-coupled binary Bose-Einstein condensate are for the first time studied in the present work.

We study spatiotemporal three-dimensional bright solitons in optical media whose non-linear response includes third- and fifth-order terms. By means of numerical simulations, lower and upper stability and existence borders for the solitons without the internal “spin” are identified. Using the variational method based on two different trial functions and collating the results, we obtain approximate solutions for spinning (vortex) solitons. The presence of the lower stability border for both the zero-spin and spinning solitons is a drastic difference of the three-dimensional solitons from those in one and two dimensions. The results show that the corresponding stability and existence borders are chiefly determined by the spatial dimension, quite weakly depending on the soliton’s “spin.” However, the energy of the spinning soliton is much larger than that of the zero-spin one.

We consider a two-component pancake-shaped, i.e., effectively two-dimensional (2D), Bose-Einstein condensate coupled by the spin-orbit (SO) and Rabi terms. The SO coupling adopted here is of the mixed Rashba-Dresselhaus type. For this configuration, we derive a system of two 2D nonpolynomial Schrödinger equations (NPSEs), for both attractive and repulsive interatomic interactions. In the low- and high-density limits, the system amounts to previously known models, namely, the usual 2D Gross-Pitaevskii equation, or the Schrödinger equation with the nonlinearity of power $7/3$. We present simple approximate localized solutions, obtained by treating the SO and Rabi terms as perturbations. Localized solutions of the full NPSE system are obtained in a numerical form. Remarkably, in the case of the attractive nonlinearity acting in free space (i.e., without any 2D trapping potential), we find parameter regions where the SO and Rabi couplings make 2D fundamental solitons dynamically stable.