Multidimensional self-trapping in linear and nonlinear potentials

Solitons are typically stable objects in 1D models, but their straightforward extensions to 2D and 3D settings tend to be unstable. In particular, the ubiquitous nonlinear Schroedinger (NLS) equation with the cubic self-focusing, creates only unstable 2D and 3D solitons, because the same equation gives rise to 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, including ones with embedded vorticity. The main stabilization schemes considered here are: (i) competing (e.g., cubic-quintic) and saturable nonlinearities; (2) linear and nonlinear trapping potentials; (3) the Lee-Huang-Yang correction to the BEC dynamics, leading to the formation of robust quantum droplets; (4) spin-orbit-coupling (SOC) effects in binary BEC; (5) emulation of SOC in nonlinear optical waveguides, including PT-symmetric ones. Further, a detailed summary is presented for results which demonstrate the creation of stable 2D and 3D solitons by schemes based on the usual linear trapping potentials or effective nonlinear ones, which may be induced by means of spatial modulation of the local nonlinearity strength. The latter setting is especially promising, making it possible to use self-defocusing media, with the local nonlinearity strength growing fast enough from the center to periphery, for the creation of a great variety of stable multidimensional modes. In addition to fundamental ones and vortex rings, the 3D modes may be hopfions, i.e., twisted vortex rings with two independent topological charges. Many results for the multidimensional solitons are obtained, in such settings, not only in a numerical form, but also by means of analytical methods, such as the variational and Thomas-Fermi approximations.