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.
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.
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 investigate two-dimensional (2D) states in Bose-Einstein condensates with self-attraction or self-repulsion, trapped in an axially symmetric optical-lattice potential periodic along the radius. The states trapped both in the central potential well and in remote circular troughs are studied. In the repulsive mode, a new soliton species is found, in the form of radial gap solitons. The latter solitons are completely stable if they carry zero vorticity (l=0) , while with l not equal 0 they develop a weak azimuthal modulation, which makes them rotating patterns, that persist indefinitely long. In addition, annular gap solitons may support stable azimuthal dark-soliton pairs on their crests. In remote troughs of the attractive model, stable localized states may assume a ringlike shape with weak azimuthal modulation, or shrink into solitons strongly localized in the azimuthal direction, which is explained in the framework of an averaged 1D equation with the cyclic azimuthal coordinate. Numerical simulations of the attractive model also reveal stable necklacelike patterns, built of several strongly localized peaks. Dynamics of strongly localized solitons circulating in the troughs is studied too. While the solitons with sufficiently small velocities are completely stable, fast solitons gradually decay, due to the leakage of matter into the adjacent trough, under the action of the centrifugal force. Investigation of head-on collisions between strongly localized solitons traveling in circular troughs shows that collisions between in-phase solitons in a common trough lead to collapse, while pi-out-of-phase solitons bounce many times, but eventually merge into a single one, without collapsing. In-phase solitons colliding in adjacent circular troughs also tend to merge into a single soliton.
We investigate vortex solitons in models representing “pancake-shaped” Bose-Einstein condensates (BECs) with attractive interatomic interactions and an anisotropic harmonic trapping potential, or polarization-preserving waveguides, such as a photonic-crystal or radial-bandgap fiber with an elliptically deformed core. Elliptic-vortex states bifurcate from dipole modes. In spite of strongly asymmetric shapes, the elliptic vortices with topological charge 1 are stable in a wide region in their existence domain. The dependence of the stability region on the eccentricity of the elliptic trap is reported. Input circular wave functions with embedded vorticity slowly relax into stationary elliptic vortices, featuring persistent rotation in the course of the evolution. All higher-order vortices are unstable.
We study two-dimensional (2D) matter-wave gap solitons trapped in an elliptically deformed concentric lattice potential, within the framework of the Gross-Pitaevskii equation (GPE) with self-attraction or self-repulsion. For a fixed eccentricity of the lattice, soliton families are found in both the repulsive and attractive models. In the former case, the analysis reveals two kinds of gap solitons trapped in the first oval trough (the ring-shaped potential minimum closest to the center): elliptic annular solitons (EASs), and double solitons (DSs), which are formed by two tightly localized density peaks located at diametrically opposite points of the trough, with zero phase difference between them. With the decrease of the norm, the density distribution in the EAS along the azimuthal direction changes from nearly uniform to double-peaked and, eventually, to the DS. In the attractive model, there exist only DSs in the oval trough, while EASs are not found. All such solitons without the angular momentum (l=0) are fully stable. For $l\ensuremath{\ne}0$, vortical solitons---both EASs with a sufficiently large norm (in the repulsive model) and DSs (in models with both signs of the nonlinearity)---are quasistable, exhibiting rocking motion in the elliptic trough (we consider the cases of l=1andl=2). At smaller values of the norm, the vortical annular solitons (in the repulsive model) are unstable. Stable fundamental solitons trapped in the central potential well are investigated, too, in both the attractive and repulsive models, by means of the variational approximation and numerical methods.
Bose–Einstein condensate (BEC) exhibits a variety of fascinating and unexpected macroscopic phenomena, and has attracted sustained attention in recent years—particularly in the field of solitons and associated nonlinear phenomena. Meanwhile, optical lattices have emerged as a versatile toolbox for understanding the properties and controlling the dynamics of BEC, among which the realization of bright gap solitons is an iconic result. However, the dark gap solitons are still experimentally unproven, and their properties in more than one dimension remain unknown. In light of this, we describe, numerically and theoretically, the formation and stability properties of gap-type dark localized modes in the context of ultracold atoms trapped in optical lattices. Two kinds of stable dark localized modes—gap solitons and soliton clusters—are predicted in both the one- and two-dimensional geometries. The vortical counterparts of both modes are also constructed in two dimensions. A unique feature is the existence of a nonlinear Bloch-wave background on which all above gap modes are situated. By employing linear-stability analysis and direct simulations, stability regions of the predicted modes are obtained. Our results offer the possibility of observing dark gap localized structures with cutting-edge techniques in ultracold atoms experiments and beyond, including in optics with photonic crystals and lattices.
We introduce a wave-propagation model for a pair of linearly coupled two-dimensional (2D) cores with intrinsic cubic nonlinearity—self-focusing (attractive) or self-defocusing (repulsive). Each core carries a periodic potential corresponding to a 1D lattice, the two lattices being mutually perpendicular. Localized states in the model are classified according to their skew-symmetry or skew-antisymmetry, that combine the interchange of the cores and the rotation of coordinates by 90°. Such a setting would be difficult to implement in ordinary optics, but it has a straightforward realization in terms of atom-wave optics, using a Bose–Einstein condensate confined to a pair of tunnel-coupled parallel 'pancake-shaped' traps, which are equipped with crossed 1D optical lattices (an example of a greater versatility of matter-wave optics, as compared to the optics of photons). For the attractive nonlinearity in both cores, symmetry-breaking bifurcations are found, with the help of the variational approximation and, chiefly, by means of numerical methods, in families of skew-symmetric fundamental solitons, dipoles, and vortices. A new species of topologically organized states, which is specific to the dual-core system with crossed 1D lattices, is reported in the form of crosses. They are built as crossed dipole pairs in the two cores, with a phase shift of π/2. Actually, the crosses represent another species of skew-symmetric vortices, different from the ordinary ones. All topologically structured modes in the system exist in pairs: straight and diagonal dipoles, rhombus- and square-shaped vortices (alias on- and off-site ones), and straight and oblique crosses. A noteworthy dynamical feature is the fact that the instability which breaks the skew-symmetry of square- and rhombus-shaped vortices does not lead to their transformation into stable asymmetric counterparts, even though such states are found as stationary solutions. Actually, unstable squares and rhombuses always transform themselves, respectively, into stable crosses of the oblique and straight types. Fundamental solitons and vortices, which do not feature the symmetry-breaking instability, are also found in the first bandgap of the repulsive system, and in the semi-infinite gap of the mixed attractive–repulsive model.