Attractive Multidimensional Solitons in Trapping Potentials

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.

Figures (10)

• Figure 1: (a) Numerical solution to Eqs. (\ref{['2Dvar']}) for $\varepsilon =0.4$ and $\varepsilon = \varepsilon_{0}=0.92$ (the latter value is the one at which $N_{\mathrm{thr}}$ vanishes). Inset displays evolution of the amplitude of a directly simulated solution initiated by the variational ansatz (\ref{['ansatz']}) with $A$ and $a$ taken as solutions of Eqs. (\ref{['2Dvar']}), in the case of $\varepsilon =0.92$, $N=2\pi$, $a=1.3$ (the VK-stable branch). (b) Dependence of the amplitude $A$ of established 2D solitons vs. the initial norm $N$, as obtained from direct simulations of Eq. (\ref{['gpe']}) starting with the configuration predicted by VA. The undulations in the dependence is a result of radiation loss in the course of the soliton formation. The dashed curve is the dependence $A(N)$ as given by VA. The figure is reproduced from Ref. Baizakov-2003.
• Figure 2: Left Panel. An established single-cell 2D soliton in a moderately strong lattice potential, found from direct simulations of Eq. (\ref{['gpe']}) with $\varepsilon =0.92$. The initial configuration was taken as per the VA-predicted stable soliton, i.e., Eqs. (\ref{['ansatz']}) and (\ref{['2Dvar']}) were used, with $a=1.3$, the corresponding norm being $N=2\pi$. Right panel. Examples of 3D solitons formed in a strong OL. The $z=0$ cross-section is shown for $\varepsilon =10$ and $N=10$. Figure extracted from Ref. Baizakov-2003.
• Figure 3: Top panels. (a) : Variational parameters $A, a, b$ versus $N$ of stable 2D solitons in the quasi-1D potential with $\epsilon=2$, as found from numerical simulations of the Gross-Pitaevskii (solid lines) and as predicted by the VA for amplitude A (dotted line) and inverse squared widths, a and b, (dot dashed and dashed lines, respectively). (b): The numerically found (connected squares) and VA predicted (dashed lines) existence limits for stable 2D solitons in the quasi-1D potential. Bottom panels . The $\mu(N)$ dependence for: (a): 2D solitons, and (b): 3D solitons, in quasi-1D and quasi-2D potentials, respectively. In each panel, the solid and dashed curves show two different branches of the solution family. Figure extracted from Ref. Baizakov-2004.
• Figure 4: Left panel: Chemical potential spectrum $(E_n \equiv \mu_n)$ obtained from the nonlinear eigenvalue problem (\ref{['reducedradial']}) with $\chi =+1$ (attraction), for $l=10$ and $\varepsilon =-10$ ($\varepsilon <0$ means the presence of a potential maximum at $r=0$). Right panel: The same, but for the repulsive model, $\chi =-1$, with $l=2$ and $\varepsilon =4$. Figure extracted from Ref. Baizakov-2006.
• Figure 5: Left and right top panels: 3D view of the annular gap solitons corresponding to bound states (a) and (b), respectively, of Fig. \ref{['fig4']}. Left and right bottom panels: The same for bound states (c) and (d) from Fig. \ref{['fig4']}. Figure extracted from Ref. Baizakov-2006.