We introduce spatiotemporal spinning solitons (vortex tori) of the three-dimensional nonlinear Schrödinger equation with focusing cubic and defocusing quintic nonlinearities. The first ever found completely stable spatiotemporal vortex solitons are demonstrated. A general conclusion is that stable spinning solitons are possible as a result of competition between focusing and defocusing nonlinearities.

We find one-, two-, and three-component solitons of the polar and ferromagnetic (FM) types in the general (nonintegrable) model of a spinor (three-component) model of the Bose-Einstein condensate, based on a system of three nonlinearly coupled Gross-Pitaevskii equations. The stability of the solitons is studied by means of direct simulations and, in a part, analytically, using linearized equations for small perturbations. Global stability of the solitons is considered by means of an energy comparison. As a result, ground-state and metastable soliton states of the FM and polar types are identified. For the special integrable version of the model, we develop the Darboux transformation (DT). As an application of the DT, analytical solutions are obtained that display full nonlinear evolution of the modulational instability of a continuous-wave state seeded by a small spatially periodic perturbation. Additionally, by dint of direct simulations, we demonstrate that solitons of both the polar and FM types, found in the integrable system, are structurally stable; i.e., they are robust under random changes of the relevant nonlinear coefficient in time.

This review is dedicated to recent progress in the active field of rogue waves, with an emphasis on the analytical prediction of versatile rogue wave structures in scalar, vector, and multidimensional integrable nonlinear systems. We first give a brief outline of the historical background of the rogue wave research, including referring to relevant up-to-date experimental results. Then we present an in-depth discussion of the scalar rogue waves within two different integrable frameworks-the infinite nonlinear Schrödinger (NLS) hierarchy and the general cubic-quintic NLS equation, considering both the self-focusing and self-defocusing Kerr nonlinearities. We highlight the concept of chirped Peregrine solitons, the baseband modulation instability as an origin of rogue waves, and the relation between integrable turbulence and rogue waves, each with illuminating examples confirmed by numerical simulations. Later, we recur to the vector rogue waves in diverse coupled multicomponent systems such as the long-wave shortwave equations, the three-wave resonant interaction equations, and the vector NLS equations (alias Topical Review

We demonstrate the existence of stable toroidal dissipative solitons with the inner phase field in the form of rotating spirals, corresponding to vorticity $S=0$, 1, and 2, in the complex Ginzburg-Landau equation with the cubic-quintic nonlinearity. The stable solitons easily self-trap from pulses with embedded vorticity. The stability is corroborated by accurate computation of growth rates for perturbation eigenmodes. The results provide the first example of stable vortex tori in a 3D dissipative medium, as well as the first example of higher-order tori (with $S=2$) in any nonlinear medium. It is found that all stable vortical solitons coexist in a large domain of the parameter space; in smaller regions, there coexist stable solitons with either $S=0$ and $S=1$, or $S=1$ and $S=2$.