We demonstrate that rotating quasi-one-dimensional potentials, periodic or parabolic, support solitons in settings where they are otherwise impossible. Ground-state and vortex solitons are found in defocusing media, if the rotation frequency $\ensuremath{\alpha}$ exceeds a critical value. The revolving periodic potentials exhibit the strongest stabilization capacity at a finite optimum value of their strength, while the rotating parabolic trap features a very sharp transition to stability with the increase of $\ensuremath{\alpha}$.
We demonstrate that rotating quasi-one-dimensional potentials, periodic or parabolic, support solitons in settings where they are otherwise impossible. Ground-state and vortex solitons are found in defocusing media, if the rotation frequency \ensuremathα exceeds a critical value. The revolving periodic potentials exhibit the strongest stabilization capacity at a finite optimum value of their strength, while the rotating parabolic trap features a very sharp transition to stability with the increase of \ensuremathα.
We construct soliton solutions for a self-attractive Bose-Einstein condensate trapped in a rotating optical lattice. The rotation pivot is set at a local maximum of the lattice potential. We demonstrate that fully localized stable solitons, containing $N\ensuremath{\sim}10,000$ atoms, may be supported by the lattice, rotating along with it, if the angular velocity $\ensuremath{\omega}$ is taken below a critical value ${\ensuremath{\omega}}_{c}$ (which is $\ensuremath{\lesssim}10\phantom{\rule{0.3em}{0ex}}\mathrm{kHz}$). A monotonously increasing dependence of ${\ensuremath{\omega}}_{c}$ on N is found. In the regime of rapid rotation, the lattice potential is nearly tantamount to the axisymmetric Bessel potential, with a maximum at the center. The latter potential supports fundamental ring-shaped solitons, and also dipole states, which have a bipolar form of two adjacent ring solitons with opposite signs. Stability regions are found for both species (which is the first example of stable azimuthally uniform solitons in a self-focusing model with a radial potential). Unstable solitons of these types evolve into strongly localized nonrotating states. At smaller $\ensuremath{\omega}$, they start to drift slowly, following the rotating lattice. The second critical value, $\ensuremath{\omega}={\ensuremath{\omega}}_{c}^{(2)}$, is found, below which the drifting solitons are destroyed. The rotating lattice supports no stable states in the interval of ${\ensuremath{\omega}}_{c}<\ensuremath{\omega}<{\ensuremath{\omega}}_{c}^{(2)}$.
We analyze the shape and stability of localized states in nonlinear cubic media with space-dependent potentials modeling an inhomogeneity. By means of a static variational approach, we describe the ground states and vortexlike stationary solutions, either in dilute atom gases or in optical cavities, with an emphasis on parabolic-type potentials. First, we determine the existence conditions for soliton and vortex structures for both focusing and defocusing nonlinearity. It is shown that, even for a defocusing medium, soliton modes can exist with a confining potential. Second, step potentials and boundedness effects in hollow capillaries are investigated, which both proceed from a similar analysis. Finally, we discuss applications of this procedure to charged vortices in dilute quantum gases and to Bose-Einstein condensates trapped in the presence of a light-induced Gaussian barrier.
We report on results of a systematic analysis of two-dimensional solitons and localized vortices in models including a rotating periodic potential and the cubic nonlinearity, with the latter being both self-attractive and self-repulsive. The models apply to Bose-Einstein condensates stirred by rotating optical lattices and to twisted photonic-crystal fibers, or bundled arrays of waveguides, in nonlinear optics. In the case of the attractive nonlinearity, we construct compound states in the form of vortices, quadrupoles, and supervortices, all trapped in the slowly rotating lattice, and identify their stability limits (fundamental solitons in this setting were studied previously). In rapidly rotating potentials, vortices decouple from the lattice in the azimuthal direction and assume an annular shape. In the model with the repulsive nonlinearity, which was not previously explored in this setting, gap solitons and vortices are found in both cases of the slow and rapid rotations. It is again concluded that the increase in the rotation frequency leads to the transition from fully trapped corotating vortices to ring-shaped ones. We also study ``crater-shaped'' vortices in the attraction model, which, unlike their compound counterparts, are trapped, essentially, in one cell of the lattice. Previously, only unstable vortices of this type were reported. We demonstrate that they have a certain stability region. Solitons and vortices are found here in the numerical form, and, in parallel, by means of the variational approximation.
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".
Stabilizing vortex solitons with high values of the topological charge S is a challenging issue in optics, studies of Bose-Einstein condensates (BECs), and other fields. To develop an approach to the solution of this problem, we consider a two-dimensional dipolar BEC under the action of an axisymmetric radially periodic lattice potential, $V(r)\ensuremath{\sim}cos(2r+\ensuremath{\delta})$, with dipole moments polarized perpendicular to the system's plane, which gives rise to isotropic repulsive dipole-dipole interactions. Two radial lattices are considered, with $\ensuremath{\delta}=0$ and $\ensuremath{\pi}$, i.e., a potential maximum or minimum at r=0, respectively. Families of vortex gap soliton (GSs) with S=1 and $S\ensuremath{\ge}2$, the latter ones often being unstable in other settings, are completely stable in the present system (at least up to S=11), being trapped in different annular troughs of the radial potential. The vortex solitons with different S may stably coexist in sufficiently far separated troughs. Fundamental GSs, with S=0, are found too. In the case of $\ensuremath{\delta}=0$, the fundamental solitons are ring-shaped modes, with a local minimum at r=0. At $\ensuremath{\delta}=\ensuremath{\pi}$, they place a density peak at the center.
We study the dynamics of one-dimensional solitons in the attractive and repulsive Bose-Einstein condensates (BECs) loaded into an optical lattice (OL), which is combined with an external parabolic potential. First, we demonstrate analytically that, in the repulsive BEC, where the soliton is of the gap type, its effective mass is negative. This gives rise to a prediction for the experiment: such a soliton cannot be not held by the usual parabolic trap, but it can be captured (performing harmonic oscillations) by an anti-trapping inverted parabolic potential. We also study the motion of the soliton a in long system, concluding that, in the cases of both the positive and negative mass, it moves freely, provided that its amplitude is below a certain critical value; above it, the soliton’s velocity decreases due to the interaction with the OL. At a late stage, the damped motion becomes chaotic. We also investigate the evolution of a two-soliton pulse in the attractive model. The pulse generates a persistent breather, if its amplitude is not too large; otherwise, fusion into a single fundamental soliton takes place. Collisions between two solitons captured in the parabolic trap or anti-trap are considered too. Depending on their amplitudes and phase difference, the solitons either perform stable oscillations, colliding indefinitely many times, or merge into a single soliton. Effects reported in this work for BECs can also be formulated for optical solitons in nonlinear photonic crystals. In particular, the capture of the negative-mass soliton in the anti-trap implies that a bright optical soliton in a self-defocusing medium with a periodic structure of the refractive index may be stable in an anti-waveguide. 1 1