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arXiv:2509.15994

Attractive Multidimensional Solitons in Trapping Potentials

F. Kh. Abdullaev, Mario Salerno

Published Sep 19, 2025• 0 citations
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Abstract

This paper reviews theoretical advances on the formation and stabilization of multidimensional solitons in nonlinear Schrödinger systems with attractive interactions, focusing on atomic Bose-Einstein condensates and nonlinear optics. While 1D solitons are generally stable, their 2D and 3D counterparts are prone to collapse. Several mechanisms have been proposed to mitigate this, including optical lattices, modulation of the nonlinearity via Feshbach resonance management, and Rabi coupling between hyperfine states. Other approaches involve competing nonlinearities and quantum corrections, such as Lee-Huang-Yang effects. Emphasis is placed on conditions enabling long-lived or fully stable solitons. Despite experimental feasibility, achieving robust stabilization remains challenging due to the intricate interplay of nonlinearities and external controls. The paper surveys collapse dynamics, stabilization strategies, and soliton existence based on key theoretical contributions.

DOI
10.48550/arxiv.2509.15994

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This paper reviews theoretical advances on the formation and stabilization of multidimensional solitons in nonlinear Schrödinger systems with attractive interactions, focusing on atomic Bose-Einstein condensates and nonlinear optics. While 1D solitons are generally stable, their 2D and 3D counterparts are prone to collapse. Several mechanisms have been proposed to mitigate this, including optical lattices, modulation of the nonlinearity via Feshbach resonance management, and Rabi coupling between hyperfine states. Other approaches involve competing nonlinearities and quantum corrections, such as Lee-Huang-Yang effects. Emphasis is placed on conditions enabling long-lived or fully stable solitons. Despite experimental feasibility, achieving robust stabilization remains challenging due to the intricate interplay of nonlinearities and external controls. The paper surveys collapse dynamics, stabilization strategies, and soliton existence based on key theoretical contributions.
This concise review aims to provide a summary of the most relevant recent experimental and theoretical results for solitons, i.e., self-trapped bound states of nonlinear waves, in two- and three-dimensional (2D and 3D) media. In comparison with commonly known one-dimensional solitons, which are, normally, stable modes, a challenging problem is the propensity of 2D and 3D solitons to instability, caused by the occurrence of the critical or supercritical wave collapse (catastrophic self-compression) in the same spatial dimension. A remarkable feature of multidimensional solitons is their ability to carry vorticity; however, 2D vortex rings and 3D vortex tori are subject to strong splitting instability. Therefore, it is natural to categorize the basic results according to physically relevant settings which make it possible to maintain stability of fundamental (non-topological) and vortex solitons against the collapse and splitting, respectively. The present review is focused on schemes that were recently elaborated in terms of Bose-Einstein condensates and similar photonic setups. These are two-component systems with spin-orbit coupling, and ones stabilized by the beyond-mean-field Lee-Huang-Yang effect. The latter setting has been implemented experimentally, giving rise to stable self-trapped quasi-2D and 3D "quantum droplets".
Experiments on attractive Bose-Einstein condensates (BECs) have unlocked many intriguing out-of-equilibrium dynamics through the interplay between matter-wave dispersion and nonlinear attractive interaction. Competition between these effects leads to fascinating phenomena such as wave collapse, modulational instability, and formation of multidimensional bright solitons. This chapter reviews experimental studies on attractive condensates, with a primary focus on alkali atoms featuring two-body contact interactions. We review recent experimental advances in optical trapping and interaction control techniques, which have enabled new studies on attractive condensates in three and also in lower dimensions. Specifically, we discuss pioneering and recent experimental observations on the dynamics and stability of attractive BECs, including the formation of bright solitons, their collisions, and excitations in quasi-one-dimensional traps. Recent observations of the elusive two-dimensional Townes solitons and vortex solitons are also discussed in this Chapter. We then highlight an experimental technique revealing the nonclassical signatures of modulational instability in an attractive condensate.
We review the state of the art and recently obtained theoretical and experimental results for two- and three-dimensional (2D and 3D) solitons and related states, such as quantum droplets, in optical systems, atomic Bose-Einstein condensates (BECs), and other fields - in particular, liquid crystals. The central challenge is avoiding the trend of 2D and 3D solitary states supported by the ubiquitous cubic nonlinearity to be strongly unstable - a property far less present in one-dimensional systems. Many possibilities for the stabilization of multi-dimensional states have been theoretically proposed over the years. Most strategies involve non-cubic nonlinearities or using different sorts of potentials, including periodic ones. New important avenues have arisen recently in systems based on two-component BEC with spin-orbit coupling, which have been predicted to support stable 2D and metastable 3D solitons. An important recent breakthrough is the creation of 3D quantum droplets. These are self-sustained states existing in two-component BECs, which are stabilized by the presence of Lee-Hung-Yang quantum fluctuations around the underlying mean-field states. By and large, multi-dimensional geometries afford unique opportunities to explore the existence of complex self-sustained states, including topologically rich ones, which by their very nature are not possible in one-dimensional geometries. In addition to vortex solitons, these are hopfions, skyrmions, and hybrid vortex-antivortex complexes, which have been predicted in different models. Here we review recent landmark findings in this field and outline outstanding open challenges.
A briefly formulated update of the recently published review [1] on the topic of multidimensional solitons (chiefly, in nonlinear optics and atomic Bose-Einstein condensates (BECs)) is presented. The update briefly summarizes some notable results on this topic that have been reported very recently, and offers a compact outline of the directions of the current experiment and theoretical work in this and related fields. In particular, as concerns newest experimental findings, included are the creation of multiple quantum droplets in prolate BEC, and the observation of expanding toroidal light structures in linear optics. The update of theoretical results includes the analysis of 2D solitons in the optical system with the quadratic nonlinearity and fractional diffraction, and the prediction of stable 3D vortex solitons in various BEC schemes with long-range interactions.
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A briefly formulated update of the recently published review [1] on the topic of multidimensional solitons (chiefly, in nonlinear optics and atomic Bose-Einstein condensates (BECs)) is presented. The update briefly summarizes some notable results on this topic that have been reported very recently, and offers a compact outline of the directions of the current experiment and theoretical work in this and related fields. In particular, as concerns newest experimental findings, included are the creation of multiple _quantum droplets_ in prolate BEC, and the observation of expanding toroidal light structures in linear optics. The update of theoretical results includes the analysis of 2D solitons in the optical system with the quadratic nonlinearity and _fractional diffraction_, and the prediction of stable 3D _vortex solitons_ in various BEC schemes with long-range interactions.