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Solitons are typically stable objects in 1D models, but their straightforwardextensions to 2D and 3D settings tend to be unstable. In particular, theubiquitous nonlinear Schroedinger (NLS) equation with the cubic self-focusing,creates only unstable 2D and 3D solitons, because the same equation gives riseto the critical and supercritical collapse in the 2D and 3D cases,respectively. This article offers, first, a review of relevant settings which,nevertheless, make it possible to create stable 2D and 3D solitons, includingones with embedded vorticity. The main stabilization schemes considered hereare: (i) competing (e.g., cubic-quintic) and saturable nonlinearities; (2)linear and nonlinear trapping potentials; (3) the Lee-Huang-Yang correction tothe BEC dynamics, leading to the formation of robust quantum droplets; (4)spin-orbit-coupling (SOC) effects in binary BEC; (5) emulation of SOC innonlinear optical waveguides, including PT-symmetric ones. Further, a detailedsummary is presented for results which demonstrate the creation of stable 2Dand 3D solitons by schemes based on the usual linear trapping potentials oreffective nonlinear ones, which may be induced by means of spatial modulationof the local nonlinearity strength. The latter setting is especially promising,making it possible to use self-defocusing media, with the local nonlinearitystrength growing fast enough from the center to periphery, for the creation ofa great variety of stable multidimensional modes. In addition to fundamentalones and vortex rings, the 3D modes may be hopfions, i.e., twisted vortex ringswith two independent topological charges. Many results for the multidimensionalsolitons are obtained, in such settings, not only in a numerical form, but alsoby means of analytical methods, such as the variational and Thomas-Fermiapproximations.

Multidimensional self-trapping in linear and nonlinear potentials