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