It was recently found that the Lee-Huang-Yang (LHY) correction to the mean-field Hamiltonian of binary atomic boson condensates suppresses the collapse and creates stable localized modes (twocomponent "quantum droplets", QDs) in two and three dimensions (2D and 3D). In particular, the LHY effect modifies the effective Gross-Pitaevskii equation (GPE) in 2D by adding a logarithmic factor to the usual cubic term. In the framework of the accordingly modified two-component GPE system, we construct 2D self-trapped modes in the form of QDs with vorticity S embedded into each component. Due to the effect of the logarithmic factor, the QDs feature a flat-top shape, which expands with the increase of S and norm N . An essential finding, produced by a systematic numerical investigation and analytical estimates, is that the vortical QDs are stable (which is a critical issue for vortex solitons in nonlinear models) up to S = 5, for N exceeding a certain threshold value, which is predicted to scale as N th ∼ S 4 for large S (for three-dimensional QDs, the scaling is N th ∼ S 6 ). The prediction is corroborated by numerical findings. Pivots of QDs with S ≥ 2 are subject to structural instability, as specially selected perturbations can split the single pivot in a set of S or S + 2 pivots corresponding to unitary vortices; however, the structural instability remains virtually invisible, as it occurs in a broad central "hole" of the vortex soliton, where values of fields are very small, and it does not cause any dynamical instability. In the condensate of 39 K atoms, in which QDs with S = 0 and a quasi-2D shape were created recently, the vortical droplets may have radial size 30 µm, with the number of atoms in the range of 10 4 − 10 5 . The role of three-body losses is considered too, demonstrating that they do not prevent the creation of the vortex droplets, but may produce a noteworthy effect, leading to sudden splitting of "light" droplets. In addition, hidden-vorticity states in QDs, with topological charges S+ = −S− = 1 in their components, which are prone to strong instability in other settings, have their stability region too. Unstable HV states tend to spontaneously merge into zero-vorticity solitons. Collisions of QDs, which may lead to their merger, and dynamics of elliptically deformed QDs (which form rotating elongated patterns or ones with oscillations of the eccentricity) are briefly considered too.

We study two-dimensional (2D) matter-wave solitons in spinor Bose-Einstein condensates under the action of the spin-orbit coupling and opposite signs of the self-and cross-interactions. Stable 2D twocomponent solitons of the mixed-mode type are found if the cross-interaction between the components is attractive, while the self-interaction is repulsive in each component. Stable solitons of the semi-vortex type are formed in the opposite case, under the action of competing self-attraction and cross-repulsion. The solitons exist with the total norm taking values below a collapse threshold. Further, in the case of the repulsive self-interaction and inter-component attraction, stable 2D selftrapped modes, which may be considered as quantum droplets (QDs), are created if the beyondmean-field Lee-Huang-Yang terms are added to the self-repulsion in the underlying system of coupled Gross-Pitaevskii equations. Stable QDs of the mixed-mode type, of a large size with an anisotropic density profile, exist with arbitrarily large values of the norm, as the Lee-Huang-Yang terms eliminate the collapse. The effect of the spin-orbit coupling term on characteristics of the QDs is systematically studied. We also address the existence and stability of QDs in the case of SOC with mixed Rashba and Dresselhaus terms, which makes the density profile of the QD more isotropic. Thus, QDs in the spin-orbit-coupled binary Bose-Einstein condensate are for the first time studied in the present work.

We present gap solitons (GSs) that can be created in free nearly two-dimensional (2D) space in dipolar spinor Bose-Einstein condensates with the spin-orbit coupling (SOC), subject to tight confinement, with size a ⊥ , in the third direction. For quasi-2D patterns, with lateral sizes l ≫ a ⊥ , the kinetic-energy terms in the respective spinor Gross-Pitaevskii equations may be neglected in comparison with SOC. This gives rise to a bandgap in the system's spectrum, in the presence of the Zeeman splitting between the spinor components. While the present system with contact interactions does not produce 2D solitons, stable gap solitons (GSs), with vorticities 0 and 1 in the two components, are found, in quasi-analytical and numerical forms, under the action of dipoledipole interaction (DDI). Namely, isotropic and anisotropic 2D GSs are obtained when the dipoles are polarized, respectively, perpendicular or parallel to the 2D plane. The GS families extend, as embedded solitons (ESs), into spectral bands, a part of the ES branch being stable for isotropic solitons. The GSs remain stable if the competing contact interaction, with the sign opposite to that of the DDI, is included, while the addition of the contact term with the same sign destabilizes the GSs, at first replacing them by breathers, and eventually leading to destruction of the solitons. Mobility and collision of the GSs are studied too, revealing negative and positive effective masses of the isotropic and anisotropic solitons, respectively.

We report families of two-dimensional (2D) composite solitons in spinor dipolar Bose-Einstein condensates, with two localized components linearly mixed by the spin-orbit coupling (SOC), and the intrinsic nonlinearity represented by the dipole-dipole interaction (DDI) between atomic magnetic moments polarized in-plane by an external magnetic field. Recently, stable solitons were predicted in the form of semi-vortices (composites built of coupled fundamental and vortical components) in the 2D system combining the SOC and contact attractive interactions. Replacing the latter by the anisotropic long-range DDI, we demonstrate that, for a fixed norm of the soliton, the system supports a continuous family of stable spatially asymmetric vortex solitons (AVSs), parameterized by an offset of the pivot of the vortical component relative to its fundamental counterpart. The offset is limited by a certain maximum value, while the energy of the AVS practically does not depend on the offset. At small values of the norm, the vortex solitons are subject to a weak oscillatory instability. In the present system, with the Galilean invariance broken by the SOC, the composite solitons are set in motion by a kick whose strength exceeds a certain depinning value. The kicked solitons feature a negative effective mass, drifting along a spiral trajectory opposite to the direction of the kick. A critical angular velocity, up to which the semi-vortices may follow rotation of the polarizing magnetic field, is found too.

We introduce a 2D network built of PT-symmetric dimers with on-site cubic nonlinearity, the gain and loss elements of the dimers being linked by parallel square-shaped lattices. The system may be realized as a set of PT-symmetric dual-core waveguides embedded into a photonic crystal. The system supports PT-symmetric and antisymmetric fundamental solitons (FSs) and on-site-centered solitary vortices (OnVs). Stability of these discrete solitons is the central topic of the consideration. Their stability regions in the underlying parameter space are identified through the computation of stability eigenvalues, and verified by direct simulations. Symmetric FSs represent the system's ground state, being stable at lowest values of the power, while anti-symmetric FSs and OnVs are stable at higher powers. Symmetric OnVs, which are also stable at lower powers, are remarkably robust modes: on the contrary to other PT-symmetric states, unstable OnVs do not blow up, but spontaneously rebuild themselves into stable FSs.

We study two-dimensional (2D) solitons in the mean-field models of ultracold gases with long-range quadrupole-quadrupole interaction (QQI) between particles. The condensate is loaded into a deep optical-lattice (OL) potential, therefore the model is based on the 2D discrete nonlinear Schr\"{o}dinger equation with contact onsite and long-range intersite interactions, which represent the QQI. The quadrupoles are built as pairs of electric dipoles and anti-dipoles orientated perpendicular to the 2D plane to which the gas is confined. Because the quadrupoles interact with the local gradient of the external field, they are polarized by inhomogeneous dc electric field that may be supplied by a tapered capacitor. Shapes, stability, mobility, and collisions of fundamental discrete solitons are studied by means of systematic simulations. In particular, threshold values of the norm, necessary for the existence of the solitons, are found, and anisotropy of their static and dynamical properties is explored. As concerns the mobility and collisions, it is the first analysis of such properties for discrete solitons on 2D lattices with long-range intersite interactions of any type. Estimates demonstrate that the setting can be realized under experimentally available conditions, predicting solitons built of $\sim$ 10,000 particles.Comment: 13 pages, 11 figures, 97 references, Physical Review A, in pres

a These two authors contributed equally to this paper.We consider a binary bosonic condensate with weak mean-field (MF) residual repulsion, loaded in an array of nearly one-dimensional traps coupled by transverse hopping. With the MF force balanced by the effectively one-dimensional attraction, induced in each trap by the Lee-Hung-Yang correction (produced by quantum fluctuations around the MF state), stable onsite-centered and intersite-centered semi-discrete quantum droplets (QDs) emerge in the array, as fundamental ones and self-trapped vortices, with winding numbers, at least, up to 5, in both tightly-bound and quasi-continuum forms. The application of a relatively strong trapping potential leads to squeezing transitions, which increase the number of sites in fundamental QDs, and eventually replace vortex modes by fundamental or dipole ones. The results provide the first realization of stable semi-discrete vortex QDs, including ones with multiple vorticity.

We proposed an innovative method to achieve dynamic control of particle separation by employing viscoelastic fluids in deterministic lateral displacement (DLD) arrays. The effects of shear-thinning and elasticity of working fluids on the critical separation size in DLD arrays are investigated. It is observed that each effect can lead to the variation of the critical separation size by approximately 40%. Since the elasticity strength of the fluid is related to the shear rate, the dynamic control can for the first time be easily realized through tuning the flow rate in microchannels.