Multidimensional Self-Trapping in Linear and Nonlinear Potentials

Boris A. Malomed*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review


Solitons are typically stable objects in diverse one-dimensional (1D) models, but their straightforward extensions to 2D and 3D settings tend to be unstable. In particular, the ubiquitous nonlinear Schrödinger (NLS) equation with the cubic self-focusing, which is also widely known as the Gross-Pitaevskii (GP) equation in the theory of Bose-Einstein condensates (BECs), creates only unstable 2D and 3D solitons, because the same equation gives rise to destructive effects in the form of the critical and supercritical wave collapse in the 2D and 3D cases, respectively. This chapter offers, first, a review of physically 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: (1) competing (e.g., cubic-quintic) and saturable nonlinearities; (2) linear and nonlinear trapping potentials; (3) the Lee-Huang-Yang correction to the mean-field 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 -symmetric ones. Further, the chapter presents a detailed summary of results which demonstrate the creation of stable 2D and 3D solitons by the 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 states and vortex rings, the respective 3D modes may be hopfions, i.e., twisted vortex rings which carry two independent topological charges. Many results for the multidimensional solitons have been obtained, in such settings, not only in a numerical form, but also by means of analytical methods, such as the variational and Thomas-Fermi approximation.

Original languageEnglish
Title of host publicationLecture Notes Series, Institute for Mathematical Sciences
EditorsWeizhu Bao, Yongyong Cai, Ionut Danaila, Peter A. Markowich
PublisherWorld Scientific
Number of pages95
StatePublished - 1 Feb 2023

Publication series

NameLecture Notes Series, Institute for Mathematical Sciences
ISSN (Print)1793-0758


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