We numerically demonstrate two-dimensional (2D) matter-wave solitons in the disk-shaped dipolar Bose-Einstein condensates (BECs) trapped in strongly anisotropic optical lattices (OLs) in a disk's plane. The considered OLs are square lattices which can be formed by interfering two pairs of plane waves with different intensities. The hopping rates of the condensates between two adjacent lattices in the orthogonal directions are different, which gives rise to a linearly anisotropic system. We find that when the polarized orientation of the dipoles is parallel to disk's plane with the same direction, the combined effects of the linearly anisotropy and the nonlocal nonlinear anisotropy strongly influence the formations, as well as the dynamics of the lattice solitons. Particularly, the isotropy-pattern solitons (IPSs) are found when these combined effects reach a balance. Motion, collision, and rotation of the IPSs are also studied in detail by means of systematic simulations. We further find that these IPSs can move freely in the 2D anisotropic discrete system, hence giving rise to an anisotropic effective mass. Four types of collisions between the IPSs are identified. By rotating an external magnetic field up to a critical angular velocity, the IPSs can still remain localized and play as a breather. Finally, the influences from the combined effects between the linear and the nonlocal nonlinear anisotropy with consideration of the contact and/or local nonlinearity are discussed too.
We numerically demonstrate two-dimensional (2D) matter-wave solitons in the disk-shaped dipolar Bose-Einstein condensates (BECs) trapped in strongly anisotropic optical lattices (OLs) in a disk's plane. The considered OLs are square lattices which can be formed by interfering two pairs of plane waves with different intensities. The hopping rates of the condensates between two adjacent lattices in the orthogonal directions are different, which gives rise to a linearly anisotropic system. We find that when the polarized orientation of the dipoles is parallel to disk's plane with the same direction, the combined effects of the linearly anisotropy and the nonlocal nonlinear anisotropy strongly influence the formations, as well as the dynamics of the lattice solitons. Particularly, the isotropy-pattern solitons (IPSs) are found when these combined effects reach a balance. Motion, collision, and rotation of the IPSs are also studied in detail by means of systematic simulations. We further find that these IPSs can move freely in the 2D anisotropic discrete system, hence giving rise to an anisotropic effective mass. Four types of collisions between the IPSs are identified. By rotating an external magnetic field up to a critical angular velocity, the IPSs can still remain localized and play as a breather. Finally, the influences from the combined effects between the linear and the nonlocal nonlinear anisotropy with consideration of the contact and/or local nonlinearity are discussed too.
We numerically and analytically investigate the formations and features of two-dimensional discrete Bose–Einstein condensate solitons, which are constructed by quadrupole–quadrupole interactional particles trapped in the tunable anisotropic discrete optical lattices. The square optical lattices in the model can be formed by two pairs of interfering plane waves with different intensities. Two hopping rates of the particles in the orthogonal directions are different, which gives rise to a linear anisotropic system. We find that if all of the pairs of dipole and anti-dipole are perpendicular to the lattice panel and the line connecting the dipole and anti-dipole which compose the quadrupole is parallel to horizontal direction, both the linear anisotropy and the nonlocal nonlinear one can strongly influence the formations of the solitons. There exist three patterns of stable solitons, namely horizontal elongation quasi-one-dimensional discrete solitons, disk-shape isotropic pattern solitons and vertical elongation quasi-continuous solitons. We systematically demonstrate the relationships of chemical potential, size and shape of the soliton with its total norm and vertical hopping rate and analytically reveal the linear dispersion relation for quasi-one-dimensional discrete solitons.
We analyze the formation and dynamics of bright unstaggered solitons in the disk-shaped dipolar Bose-Einstein condensate, which features the interplay of contact (collisional) and long-range dipole-dipole (DD) interactions between atoms. The condensate is assumed to be trapped in a strong optical-lattice potential in the disk's plane, hence it may be approximated by a two-dimensional (2D) discrete model, which includes the on-site nonlinearity and cubic long-range (DD) interactions between sites of the lattice. We consider two such models, which differ by the form of the on-site nonlinearity, represented by the usual cubic term, or more accurate nonpolynomial one, derived from the underlying three-dimensional Gross-Pitaevskii equation. Similar results are obtained for both models. The analysis is focused on the effects of the DD interaction on fundamental localized modes in the lattice (2D discrete solitons). The repulsive isotropic DD nonlinearity extends the existence and stability regions of the fundamental solitons. New families of on-site, inter-site, and hybrid solitons, built on top of a finite background, are found as a result of the interplay of the isotropic repulsive DD interaction and attractive contact nonlinearity. By themselves, these solutions are unstable, but they evolve into robust breathers which exist on an oscillating background. In the presence of the repulsive contact interactions, fundamental localized modes exist if the DD interaction (attractive isotropic or anisotropic) is strong enough. They are stable in narrow regions close to the anticontinuum limit, while unstable solitons evolve into breathers. In the latter case, the presence of the background is immaterial.
We study dipolar Bose–Einstein condensate (BEC) solitons in a one-dimensional optical lattice with the combined effect of local and nonlocal nonlinearities. The local nonlinearity is imposed by a magnetic or optical field via the Feshbach (FB) resonance. In contrast, the nonlocal nonlinearity is created by the long-range dipole–dipole interaction among the condensates. The orientations of the dipoles are directed by a rotatable uniform external field, which gives rise to a controllable nonlocal nonlinearity resulting from the angle θ between the direction of the dipole and the elongation of the lattice. If the lattice is sufficiently deep, this model can be described by the Gross–Pitaevskii equation (GPE) with tunable local and nonlocal nonlinear strengths (g and θ respectively). The formation, motion, and collision of the solitons in this system are studied by numerical simulations. The combined effect of the local and nonlocal nonlinearities gives a controllable scheme for all these characteristics via the relation between g and θ.
We report some investigations on the existence of matter-wave solitons when considering cross-combined optical lattices (OL) in three dimensions (3D), where we have a nonlinear OL in one of the directions, which is perpendicular to a two-dimensional (2D) plane with linear OLs in either one or both directions. This study can be useful to manage 3D solitons through spatial modulations of the scattering length in one of the OL directions. Another independent study is reported, by considering bright solitons manifested in a bosonic condensate gas carrying collinear dipole moments, which induces an external polarizing field with strength periodically modulated along one of the coordinates. This leads to an effective nonlocal nonlinear lattice, with solitonic solutions. Their dynamics and mobility can be investigated by an effective one-dimensional (1D) model. Interactions between solitons are also reported within this 1D model. In all the cases, we consider full numerical and variational approaches.
We study the spontaneous symmetry breaking of dipolar Bose–Einstein condensates trapped in stacks of two-well systems, which may be effectively built as one-dimensional trapping lattices sliced by a repelling laser sheet. If the potential wells are sufficiently deep, the system is modeled by coupled discrete Gross–Pitaevskii equations with nonlocal self- and cross-interaction terms representing dipole–dipole interactions. When the dipoles are not polarized perpendicular or parallel to the lattice, the crossinteraction is asymmetric, replacing the familiar symmetric two-component solitons with a new species of cross-symmetric or -asymmetric ones. The orientation of the dipole moments and the interwell hopping rate strongly affect the shapes of the discrete two-component solitons as well as the characteristics of the cross-symmetry breaking and the associated phase transition. The sub- and super-critical types of cross-symmetry breaking can be controlled by either the hopping rate between the components or the total norm of the solitons. The effect of the interplay between the contact nonlinearity and the dipole angle on the cross-symmetry breaking is also discussed.
This is a review addressing soliton-like states in systems with nonlocal nonlinearity. The work on this topic has long history in optics and related areas. Some results produced by the work (such as solitons supported by thermal nonlinearity in optical glasses, and orientational nonlinearity, which affects light propagation in liquid crystals) are well known, and have been properly reviewed in the literature, therefore the respective models are outlined in the present review in a brief form. Some other studies, such as those addressing models with fractional diffraction, which is represented by a linear nonlocal operator, have started more recently, therefore it will be relevant to review them in detail when more results will be accumulated; for this reason, the present article provides a short outline of the latter topic. The main part of the article is a summary of results obtained for two-dimensional solitons in specific nonlocal nonlinear models originating in studies of Bose–Einstein condensates (BECs), which are sufficiently mature but have not yet been reviewed previously (some results for three-dimensional solitons are briefly mentioned too). These are, in particular, anisotropic quasi-2D solitons supported by long-range dipole-dipole interactions in a condensate of magnetic atoms and giant vortex solitons (which are stable for high values of the winding number), as well as 2D vortex solitons of the latter type moving with self-acceleration. The vortex solitons are states of a hybrid type, which include matter-wave and electromagnetic-wave components. They are supported, in a binary BEC composed of two different atomic states, by the resonant interaction of the two-component matter waves with a microwave field that couples the two atomic states. The shape, stability, and dynamics of the solitons in such systems are strongly affected by their symmetry. Some other topics are included in the review in a brief form. This review uses the “Harvard style” of referring to the bibliography.