It is commonly known that the interplay of linear and nonlinear effects
gives rise to solitons, i.e., self-trapped localized structures, in a wide range of physical
settings, including optics, Bose--Einstein condensates (BECs), hydrodynamics,
plasmas, condensed-matter physics, etc. Nowadays, solitons are considered as an
interdisciplinary class of modes, which feature diverse internal structures.
While most experimental realizations and theoretical models of solitons
have been elaborated in one-dimensional (1D) settings, a challenging issue is
the prediction of stable solitons in 2D and 3D media. In particular, multidimensional
solitons may carry an intrinsic topological structure in the form of vorticity. In
addition to the "simple" vortex solitons, fascinating objects featuring complex
structures such as hopfions, i.e., vortex rings with internal twist, have been predicted too.
A fundamental problem is the propensity of multidimensional solitons
to be unstable (naturally, solitons with a more sophisticated structure, such as
vortex solitons, are more vulnerable to instabilities). Recently, novel perspectives for the
creation of stable 2D and 3D solitons were brought to the attention of researchers in
optics and BEC. The present talk aims to provide an overview of the main results
and ongoing developments in this vast field. An essential conclusion is the benefit
offered by the exchange of concepts between different areas, such as optics, BEC, and
hydrodynamics.
A thorough survey of the topic is provided in a new book:
B. A. Malomed, "Multidimensional Solitons" (American Institute of Physics, Melville, NY, 2022).