Creation of two-dimensional composite solitons in spin-orbit-coupled self-attractive Bose-Einstein condensates in free space

Stable three-dimensional vortex solitons of high topological charge in a Rydberg-dressed Bose-Einstein condensate with spin-orbit coupling

Stable vortex solitons (VSs) are objects of great interest for fundamental studies and various applications, including particle trapping, microscopy, data encoding, and matter-wave gyroscopes. However, three-dimensional (3D) VSs with high topological charges, supported by self-attractive nonlinearities, are unstable against fragmentation, which eventually leads to internal blowup (supercritical collapse) of the fragments. Here, we propose a scheme for realizing stable 3D VSs with topological charges up to 5 and 6 in the two components of a binary, Rydberg-dressed Bose-Einstein condensate with spin-orbit coupling (SOC). We show that, if the SOC strength exceeds a critical value, the rotational symmetry of the VSs in the transverse plane gets broken, resulting in a separation of the two components. Nevertheless, the VSs with the broken symmetry remain stable. The VS stability domains are identified in the system's parameter space. Moreover, application of torque to the stable VSs sets them in the state of robust gyroscopic precession.

Semivortex solitons and their excited states in spin-orbit-coupled binary bosonic condensates

It is known that two-dimensional two-component fundamental solitons of the semivortex (SV) type, with vorticities (s_{+},s_{-})=(0,1) in their components, are stable ground states (GSs) in the spin-orbit-coupled (SOC) binary Bose-Einstein condensate with the contact self-attraction acting in both components, in spite of the possibility of the critical collapse in the system. However, excited states (ESs) of the SV solitons, with the vorticity set (s_{+},s_{-})=(S_{+},S_{+}+1) and S_{+}=1,2,3,..., are unstable in the same system. We construct ESs of SV solitons in the SOC system with opposite signs of the self-interaction in the two components. The main finding is stability of the ES-SV solitons, with the extra vorticity (at least) up to S_{+}=6. The threshold value of the norm for the onset of the critical collapse, N_{thr}, in these excited states is higher than the commonly known critical value, N_{c}≈5.85, associated with the single-component Townes solitons, N_{thr} increasing with the growth of S_{+}. A velocity interval for stable motion of the GS-SV solitons is found too. The results suggest a solution for the challenging problem of the creation of stable vortex solitons with high topological charges.

Quasi-one- and quasi-two-dimensional symbiotic solitons bound by dipolar interaction

We study the formation of quasi-one- (quasi-1D) and quasi-two-dimensional (quasi-2D) symbiotic solitons bound by an interspecies dipolar interaction in a binary dipolar Bose-Einstein condensate. These binary solitons have a repulsive intraspecies contact interaction stronger than the intraspecies dipolar interaction, so that they can not be bound in isolation in the absence of an interspecies dipolar interaction. These symbiotic solitons are bound in the presence of an interspecies dipolar interaction and zero interspecies contact interaction. The quasi-1D solitons are free to move along the polarization z direction of the dipolar atoms, whereas the quasi-2D solitons move in the x-z plane. To illustrate these, we consider a ^{164}Er-^{166}Er mixture with scattering lengths a(^{164}Er)=81a_{0} and a(^{166}Er)=68a_{0} and with dipolar lengths a_{dd}(^{164}Er)≈a_{dd}(^{166}Er)≈65a_{0}, where a_{0} is the Bohr radius. In each of the two components a>a_{dd}, which stops the binding of solitons in each component in isolation, whereas a binary quasi-1D or a quasi-2D ^{164}Er-^{166}Er soliton is bound in the presence of an interspecies dipolar interaction. The stationary states were obtained by imaginary-time propagation of the underlying mean-field model; dynamical stability of the solitons was established by real-time propagation over a long period of time.

Discrete and Semi-Discrete Multidimensional Solitons and Vortices: Established Results and Novel Findings

This article presents a concise survey of basic discrete and semi-discrete nonlinear models, which produce two- and three-dimensional (2D and 3D) solitons, and a summary of the main theoretical and experimental results obtained for such solitons. The models are based on the discrete nonlinear Schrödinger (DNLS) equations and their generalizations, such as a system of discrete Gross-Pitaevskii (GP) equations with the Lee-Huang-Yang corrections, the 2D Salerno model (SM), DNLS equations with long-range dipole-dipole and quadrupole-quadrupole interactions, a system of coupled discrete equations for the second-harmonic generation with the quadratic (χ(2)) nonlinearity, a 2D DNLS equation with a superlattice modulation opening mini-gaps, a discretized NLS equation with rotation, a DNLS coupler and its PT-symmetric version, a system of DNLS equations for the spin-orbit-coupled (SOC) binary Bose-Einstein condensate, and others. The article presents a review of the basic species of multidimensional discrete modes, including fundamental (zero-vorticity) and vortex solitons, their bound states, gap solitons populating mini-gaps, symmetric and asymmetric solitons in the conservative and PT-symmetric couplers, cuspons in the 2D SM, discrete SOC solitons of the semi-vortex and mixed-mode types, 3D discrete skyrmions, and some others.

Basic fractional nonlinear-wave models and solitons

This review article provides a concise summary of one- and two-dimensional models for the propagation of linear and nonlinear waves in fractional media. The basic models, which originate from Laskin's fractional quantum mechanics and more experimentally relevant setups emulating fractional diffraction in optics, are based on the Riesz definition of fractional derivatives, which are characterized by the respective Lévy indices. Basic species of one-dimensional solitons, produced by the fractional models which include cubic or quadratic nonlinear terms, are outlined too. In particular, it is demonstrated that the variational approximation is relevant in many cases. A summary of the recently demonstrated experimental realization of the fractional group-velocity dispersion in fiber lasers is also presented.

Integrator for general spin-s Gross-Pitaevskii systems

We provide an algorithm, i-SPin 2, for evolving general spin-s Gross-Pitaevskii or nonlinear Schrödinger systems carrying a variety of interactions, where the 2s+1 components of the "spinor" field represent the different spin-multiplicity states. We consider many nonrelativistic interactions up to quartic order in the Schrödinger field (both short and long range, and spin-dependent and spin-independent interactions), including explicit spin-orbit couplings. The algorithm allows for spatially varying external and/or self-generated vector potentials that couple to the spin density of the field. Our work can be used for scenarios ranging from laboratory systems such as spinor Bose-Einstein condensates (BECs), to cosmological or astrophysical systems such as self-interacting bosonic dark matter. As examples, we provide results for two different setups of spin-1 BECs that employ a varying magnetic field and spin-orbit coupling, respectively, and also collisions of spin-1 solitons in dark matter. Our symplectic algorithm is second-order accurate in time, and is extensible to the known higher-order-accurate methods.

Three-dimensional solitons in Rydberg-dressed cold atomic gases with spin-orbit coupling

We present numerical results for three-dimensional (3D) solitons with symmetries of the semi-vortex (SV) and mixed-mode (MM) types, which can be created in spinor Bose-Einstein condensates of Rydberg atoms under the action of the spin-orbit coupling (SOC). By means of systematic numerical computations, we demonstrate that the interplay of SOC and long-range spherically symmetric Rydberg interactions stabilize the 3D solitons, improving their resistance to collapse. We find how the stability range depends on the strengths of the SOC and Rydberg interactions and the soft-core atomic radius.

Observation of π solitons in oscillating waveguide arrays

Floquet systems with periodically varying in time parameters enable realization of unconventional topological phases that do not exist in static systems with constant parameters and that are frequently accompanied by appearance of novel types of the topological states. Among such Floquet systems are the Su-Schrieffer-Heeger lattices with periodically-modulated couplings that can support at their edges anomalous π modes of topological origin despite the fact that the lattice spends only half of the evolution period in topologically nontrivial phase, while during other half-period it is topologically trivial. Here, using Su-Schrieffer-Heeger arrays composed from periodically oscillating waveguides inscribed in transparent nonlinear optical medium, we report experimental observation of photonic anomalous π modes residing at the edge or in the corner of the one- or two-dimensional arrays, respectively, and demonstrate a new class of topological π solitons bifurcating from such modes in the topological gap of the Floquet spectrum at high powers. π solitons reported here are strongly oscillating nonlinear Floquet states exactly reproducing their profiles after each longitudinal period of the structure. They can be dynamically stable in both one- and two-dimensional oscillating waveguide arrays, the latter ones representing the first realization of the Floquet photonic higher-order topological insulator, while localization properties of such π solitons are determined by their power.

Matter-wave solitons in an array of spin-orbit-coupled Bose-Einstein condensates

We investigate matter-wave solitons in a binary Bose-Einstein condensate (BEC) with spin-orbit (SO) coupling, loaded in a one-dimensional (1D) deep optical lattice and a three-dimensional anisotropic magnetic trap, which creates an array of elongated sub-BECs with transverse tunneling. We show that the system supports 1D continuous and discrete solitons localized in the longitudinal (along the array) and the transverse (across the array) directions, respectively. In addition, such solitons are always unpolarized in the zero-momentum state but polarized in finite-momentum states. We also show that the system supports stable two-dimensional semidiscrete solitons, including single- and multiple-peaked ones, localized in both the longitudinal and transverse directions. Stability diagrams for single-peaked semidiscrete solitons in different parameter spaces are identified. The results reported here are beneficial not only for understanding the physical property of SO-coupled BECs but also for generating new types of matter-wave solitons.

Stability limits for modes held in alternating trapping-expulsive potentials

We elaborate a scheme of trapping-expulsion management (TEM), in the form of the quadratic potential periodically switching between confinement and expulsion, as a means of stabilization of two-dimensional dynamical states against the backdrop of the critical collapse driven by the cubic self-attraction with strength g. The TEM scheme may be implemented, as spatially or temporally periodic modulations, in optics or BEC, respectively. The consideration is carried out by dint of numerical simulations and variational approximation (VA). In terms of the VA, the dynamics amounts to a nonlinear Ermakov equation, which, in turn, is tantamount to a linear Mathieu equation. Stability boundaries are found as functions of g and parameters of the periodic modulation of the trapping potential. Below the usual collapse threshold, which is known, in the numerical form, as g<g_{c}^{(num)}≈5.85 (in the standard notation), the stability is limited by the onset of the parametric resonance. This stability limit, including the setup with the self-repulsive sign of the cubic term (g<0), is accurately predicted by the VA. At g>g_{c}^{(num)}, the collapse threshold is found with the help of full numerical simulations. The relative increase of g_{c} above g_{c}^{(num)} is ≈1.5%. It is a meaningful result, even if its size is small, because the collapse threshold is a universal constant which is difficult to change.

Two-Dimensional Solitons in Bose–Einstein Condensates with Spin–Orbit Coupling and Rydberg–Rydberg Interaction

Applying an imaginary time evolution method (AITEM) to the system of Gross–Pitaevskii equations, we find two-dimensional stable solitons in binary atomic Bose–Einstein condensates with spin–orbit coupling (SOC) and the Rydberg–Rydberg interaction (RRI). The stability of 2D solitons by utilizing their norm and energy is discussed in detail. Depending on the SOC and Rydberg–Rydberg interaction, we find stable zero-vorticity and vortical solitons. Furthermore, we show that the solitons can be effectively tuned by the local and nonlocal nonlinearities of this system.

Symbiotic solitons in quasi-one- and quasi-two-dimensional spin-1 condensates

We study the formation of spin-1 symbiotic spinor solitons in a quasi-one- (quasi-1D) and quasi-two-dimensional (quasi-2D) hyperfine spin F=1 ferromagnetic Bose-Einstein condensate (BEC). The symbiotic solitons necessarily have a repulsive intraspecies interaction and are bound due to an attractive interspecies interaction. Due to a collapse instability in higher dimensions, an additional spin-orbit coupling is necessary to stabilize a quasi-2D symbiotic spinor soliton. Although a quasi-1D symbiotic soliton has a simple Gaussian-type density distribution, novel spatial periodic structure in density is found in quasi-2D symbiotic SO-coupled spinor solitons. For a weak SO coupling, the quasi-2D solitons are of the (-1,0,+1) or (+1,0,-1) type with intrinsic vorticity and multiring structure, for Rashba or Dresselhaus SO coupling, respectively, where the numbers in the parentheses are angular momenta projections in spin components F_{z}=+1,0,-1, respectively. For a strong SO coupling, stripe and superlattice solitons, respectively, with a stripe and square-lattice modulation in density, are found in addition to the multiring solitons. The stationary states were obtained by imaginary-time propagation of a mean-field model; dynamical stability of the solitons was established by real-time propagation over a long period of time. The possibility of the creation of such a soliton by removing the trap of a confined spin-1 BEC in a laboratory is also demonstrated.

Spatial order in a two-dimensional spin-orbit-coupled spin-1/2 condensate: superlattice, multi-ring and stripe formation

We demonstrate the formation of stable spatially-ordered states in auniformand alsotrappedquasi-two-dimensional (quasi-2D) Rashba or Dresselhaus spin-orbit (SO) coupled pseudo spin-1/2 Bose-Einstein condensate using the mean-field Gross-Pitaevskii equation. For weak SO coupling, one can have a circularly-symmetric (0, +1)- or (0, -1)-type multi-ring state with intrinsic vorticity, for Rashba or Dresselhaus SO coupling, respectively, where the numbers in the parentheses denote the net angular momentum projection in the two components, in addition to a circularly-asymmetric degenerate state with zero net angular momentum projection. For intermediate SO couplings, in addition to the above two types, one can also have states with stripe pattern in component densities with no periodic modulation in total density. The stripe state continues to exist for large SO coupling. In addition, a new spatially-periodic state appears in the uniform system: asuperlatticestate, possessing some properties of asupersolid, with a square-lattice pattern in component densities and also in total density. In a trapped system the superlattice state is slightly different with multi-ring pattern in component density and a square-lattice pattern in total density. For an equal mixture of Rashba and Dresselhaus SO couplings, in both uniform and trapped systems, only stripe states are found for all strengths of SO couplings. In a uniform system all these states are quasi-2D solitonic states.

Modulation instability in helicoidal spin-orbit coupled open Bose-Einstein condensates

We introduce a vector form of the cubic complex Ginzburg-Landau equation describing the dynamics of dissipative solitons in the two-component helicoidal spin-orbit coupled open Bose-Einstein condensates (BECs), where the addition of dissipative interactions is done through coupled rate equations. Furthermore, the standard linear stability analysis is used to investigate theoretically the stability of continuous-wave (cw) solutions and to obtain an expression for the modulational instability gain spectrum. Using direct simulations of the Fourier space, we numerically investigate the dynamics of the modulational instability in the presence of helicoidal spin-orbit coupling. Our numerical simulations confirm the theoretical predictions of the linear theory as well as the threshold for amplitude perturbations.

Stability and quantum escape dynamics of spin-orbit-coupled Bose-Einstein condensates in the shallow trap

The Bose-Einstein condensates in a finite depth potential well provide an ideal platform to study the quantum escape dynamics. In this paper, the ground state, tunneling, and diffusion dynamics of the spin-orbit coupling (SOC) of Bose-Einstein condensates with two pseudospin components in a shallow trap are studied analytically and numerically. The phase transition between the plane-wave phase and zero-momentum phase of the ground state is obtained. Furthermore, the stability of the ground state is discussed, and the stability diagram in the parameter space is provided. The bound state (in which condensates are stably trapped in the potential well), the quasibound state (in which condensates tunnel through the well), and the unstable state (in which diffusion occurs) are revealed. We find that the finite depth potential well has an important effect on the phase transition of the ground state, and, interestingly, SOC can stabilize the system against the diffusion and manipulate the tunneling and diffusion dynamics. In particular, spatial anisotropic tunneling and diffusion dynamics of the two pseudospin components induced by SOC in quasibound and unstable states are observed. We provide an effective model and method to study and control the quantum tunneling and diffusion dynamics.

Stable Nonlinear Modes Sustained by Gauge Fields

We reveal the universal effect of gauge fields on the existence, evolution, and stability of solitons in the spinor multidimensional nonlinear Schrödinger equation. Focusing on the two-dimensional case, we show that when gauge field can be split in a pure gauge and a nonpure gauge generating effective potential, the roles of these components in soliton dynamics are different: the localization characteristics of emerging states are determined by the curvature, while pure gauge affects the stability of the modes. Respectively the solutions can be exactly represented as the envelopes which may depend on the pure gauge implicitly through the effective potential, and modulating stationary carrier-mode states, which are independent of the curvature. Our central finding is that nonzero curvature can lead to the existence of unusual modes, in particular, enabling stable localized self-trapped fundamental and vortex-carrying states in media with constant repulsive interactions without additional external confining potentials and even in the expulsive external traps.

Solitons in spin-orbit-coupled spin-2 spinor Bose-Einstein condensates

We investigate the different types of matter-wave solitons in spin-orbit-coupled spin-2 spinor Bose-Einstein condensates. Using mean-field theory and adopting the multiscale perturbation method, the original five-component Gross-Pitaevskii spin-orbit-coupled spin-2 spinor Bose-Einstein condensate model can be reduced to a single effective nonlinear Schrödinger equation, which allows us to find analytical soliton solutions of this system. In this way, for different regimes of the spin-orbit coupling, Raman coupling, and interatomic interactions, we find approximate bright and dark soliton solutions. Particularly, the type of solitons depends on the dispersion properties of the system. When the lowest-energy band has a single-well structure, we find there only exist positive mass bright or dark solitons due to the dispersion coefficient of effective nonlinear Shrödinger equation always positive. However, when the lowest-energy band has a double-well structure, there will appear positive (negative) mass bright or dark solitons because the sign of the dispersion coefficient can be positive (negative) under different momentum. We employ direct numerical simulation of the original five-component Gross-Pitaevskii equations to confirm the analytical results.

Tunable spinful matter wave valve

We investigate the transport problem that a spinful matter wave is incident on a strong localized spin-orbit-coupled Bose-Einstein condensate in optical lattices, where the localization is admitted by atom interaction only existing at one particular site, and the spin-orbit coupling arouse spatial rotation of the spin texture. We find that tuning the spin orientation of the localized Bose-Einstein condensate can lead to spin-nonreciprocal/spin-reciprocal transport, meaning the transport properties are dependent on/independent of the spin orientation of incident waves. In the former case, we obtain the conditions to achieve transparency, beam-splitting, and blockade of the incident wave with a given spin orientation, and furthermore the ones to perfectly isolate incident waves of different spin orientation, while in the latter, we obtain the condition to maximize the conversion of different spin states. The result may be useful to develop a novel spinful matter wave valve that integrates spin switcher, beam-splitter, isolator, and converter. The method can also be applied to other real systems, e.g., realizing perfect isolation of spin states in magnetism, which is otherwise rather difficult.

Stable Multiring and Rotating Solitons in Two-Dimensional Spin-Orbit-Coupled Bose-Einstein Condensates with a Radially Periodic Potential

We consider two-dimensional spin-orbit-coupled atomic Bose-Einstein condensate in a radially periodic potential. The system supports different types of stable self-sustained states including radially symmetric vorticity-carrying modes with different topological charges in two spinor components that may have multiring profiles and at the same time remain remarkably stable for repulsive interactions. Solitons of the second type show persistent rotation with constant angular frequency. They can be stable for both repulsive and attractive interatomic interactions. Because of the inequivalence between clockwise and counterclockwise rotation directions introduced by spin-orbit coupling, the properties of such solitons strongly differ for positive and negative rotation frequencies. The collision of solitons located in the same or different rings is accompanied by a change of the rotation frequency that depends on the phase difference between colliding solitons.

Nonlinear Management of Topological Solitons in a Spin-Orbit-Coupled System

We consider possibilities to control dynamics of solitons of two types, maintained by the combination of cubic attraction and spin-orbit coupling (SOC) in a two-component system, namely, semi-dipoles (SDs) and mixed modes (MMs), by making the relative strength of the cross-attraction, γ , a function of time periodically oscillating around the critical value, γ = 1 , which is an SD/MM stability boundary in the static system. The structure of SDs is represented by the combination of a fundamental soliton in one component and localized dipole mode in the other, while MMs combine fundamental and dipole terms in each component. Systematic numerical analysis reveals a finite bistability region for the SDs and MMs around γ = 1 , which does not exist in the absence of the periodic temporal modulation (“management”), as well as emergence of specific instability troughs and stability tongues for the solitons of both types, which may be explained as manifestations of resonances between the time-periodic modulation and intrinsic modes of the solitons. The system can be implemented in Bose-Einstein condensates (BECs), and emulated in nonlinear optical waveguides.

Two-Dimensional Vortex Solitons in Spin-Orbit-Coupled Dipolar Bose–Einstein Condensates

Solitons are self-trapped modes existing in various nonlinear systems. Creating stable solitons in two- and three-dimensional settings is a challenging goal in various branches of physics. Several methods have been developed theoretically and experimentally to achieve this, but few of them can support stable multi-dimensional solitons in free space. Recently, a new scheme using spin-orbit-coupling (SOC) has been proposed to create stable 2D solitons in Bose–Einstein condensates (BECs). This paper reviews recent theoretical progress on creating stable 2D solitons in spinor dipolar BEC with SOC, combined with long-range dipole-dipole interaction (DDI), Zeeman splitting (ZS) and contact nonlinearity, in free space. The continuous family of stable symmetric vortex solitons (SVS), asymmetric vortex solitons (AVS), as well as gap solitons (GS) is found via different settings. Their existence and stability conditions are summarized and discussed in detail. The mobility properties of these types of solitons are also addressed. For SVS, a potential method to manipulate its shape and mobility is investigated. These results are supposed to enrich our understanding of 2D solitons and help create multi-dimensional solitons in experiments.

Exotic complexes in one-dimensional Bose-Einstein condensates with spin-orbit coupling

By means of the F-expansion method and intensive numerical simulations, the existence of three families of nonlinear matter waves including Jacobi elliptic functions, solitons, and triangular periodic functions, is demonstrated for spin-orbit coupled Bose-Einstein condensates with a linear potential. In addition, several complexes are obtained by taking two distinct solutions of each family or two distinct families. These solutions sustain different types of two-body interactions in the condensate that can be repulsive, attractive, or attractive and repulsive. Whereas the spin-orbit coupling destabilized these nonlinear matter waves, the linear potential leads to a stabilization. The numerical results are in excellent agreement with our analytical findings and it can be expected that the proposed robust solutions should be observable for experimentally relevant conditions.

The phase diagram and stability of trapped D-dimensional spin-orbit coupled Bose-Einstein condensate

By variational analysis and direct numerical simulation, we study the phase transition and stability of a trapped D-dimensional Bose-Einstein condensate with spin-orbit coupling. The complete phase and stability diagrams of the system are presented in full parameter space, while the collapse dynamics induced by the mean-filed attraction and the mechanism for stabilizing the collapse by spin-orbit coupling are illustrated explicitly. Particularly, a full and deep understanding of the dependence of phase transition and stability mechanism on geometric dimensionality and external trap potential is revealed. It is shown that the spin-orbit coupling can modify the dispersion relations, which can balance the mean-filed attractive interaction and result in a spin polarized or overlapped state to stabilize the collapse, then changes the collapsing threshold dependent on the geometric dimensionality and external trap potential. Moreover, from 2D to 3D system, the mean-field attraction for inducing the collapse is reduced and the collapse speed is enhanced, namely, the collapse can be more easily stabilized in 2D system. That is, the collapse can be manipulated by adjusting the spin-orbit coupling, Raman coupling, geometric dimensionality and the external trap potential, which can provide a possible way for elaborating the collapse dynamics experimentally.

Vortex-soliton complexes in coupled nonlinear Schrödinger equations with unequal dispersion coefficients

We consider a two-component, two-dimensional nonlinear Schrödinger system with unequal dispersion coefficients and self-defocusing nonlinearities, chiefly with equal strengths of the self- and cross-interactions. In this setting, a natural waveform with a nonvanishing background in one component is a vortex, which induces an effective potential well in the second component, via the nonlinear coupling of the two components. We show that the potential well may support not only the fundamental bound state, but also multiring excited radial state complexes for suitable ranges of values of the dispersion coefficient of the second component. We systematically explore the existence, stability, and nonlinear dynamics of these states. The complexes involving the excited radial states are weakly unstable, with a growth rate depending on the dispersion of the second component. Their evolution leads to transformation of the multiring complexes into stable vortex-bright solitons ones with the fundamental state in the second component. The excited states may be stabilized by a harmonic-oscillator trapping potential, as well as by unequal strengths of the self- and cross-repulsive nonlinearities.

Vortex solitons in two-dimensional spin-orbit coupled Bose-Einstein condensates: Effects of the Rashba-Dresselhaus coupling and Zeeman splitting

We present an analysis of two-dimensional (2D) matter-wave solitons, governed by the pseudospinor system of Gross-Pitaevskii equations with self- and cross attraction, which includes the spin-orbit coupling (SOC) in the general Rashba-Dresselhaus form, and, separately, the Rashba coupling and the Zeeman splitting. Families of semivortex (SV) and mixed-mode (MM) solitons are constructed, which exist and are stable in free space, as the SOC terms prevent the onset of the critical collapse and create the otherwise missing ground states in the form of the solitons. The Dresselhaus SOC produces a destructive effect on the vortex solitons, while the Zeeman term tends to convert the MM states into the SV ones, which eventually suffer delocalization. Existence domains and stability boundaries are identified for the soliton families. For physically relevant parameters of the SOC system, the number of atoms in the 2D solitons is limited by ∼1.5×10^{4}. The results are obtained by means of combined analytical and numerical methods.

Dynamics of dipoles and vortices in nonlinearly coupled three-dimensional field oscillators

The dynamics of a pair of harmonic oscillators represented by three-dimensional fields coupled with a repulsive cubic nonlinearity is investigated through direct simulations of the respective field equations and with the help of the finite-mode Galerkin approximation (GA), which represents the two interacting fields by a superposition of 3+3 harmonic-oscillator p-wave eigenfunctions with orbital and magnetic quantum numbers l=1 and m=1, 0, -1. The system can be implemented in binary Bose-Einstein condensates, demonstrating the potential of the atomic condensates to emulate various complex modes predicted by classical field theories. First, the GA very accurately predicts a broadly degenerate set of the system's ground states in the p-wave manifold, in the form of complexes built of a dipole coaxial with another dipole or vortex, as well as complexes built of mutually orthogonal dipoles. Next, pairs of noncoaxial vortices and/or dipoles, including pairs of mutually perpendicular vortices, develop remarkably stable dynamical regimes, which feature periodic exchange of the angular momentum and periodic switching between dipoles and vortices. For a moderately strong nonlinearity, simulations of the coupled-field equations agree very well with results produced by the GA, demonstrating that the dynamics is accurately spanned by the set of six modes limited to l=1.

Stable Solitons in Three Dimensional Free Space without the Ground State: Self-Trapped Bose-Einstein Condensates with Spin-Orbit Coupling

By means of variational methods and systematic numerical analysis, we demonstrate the existence of metastable solitons in three dimensional (3D) free space, in the context of binary atomic condensates combining contact self-attraction and spin-orbit coupling, which can be engineered by available experimental techniques. Depending on the relative strength of the intra- and intercomponent attraction, the stable solitons feature a semivortex or mixed-mode structure. In spite of the fact that the local cubic self-attraction gives rise to the supercritical collapse in 3D, and hence the setting produces no true ground state, the solitons are stable against small perturbations, motion, and collisions.

Dark solitons in dual-core waveguides with dispersive coupling

We report on new types of two-component one-dimensional dark solitons (DSs) in a dual-core waveguide model with normal group-velocity dispersion and Kerr nonlinearity in both cores, the coupling between which is also dispersive. In the presence of the dispersive coupling, quiescent DSs supported by the zero-frequency background are always gray, being stable with the out-of-phase background, i.e., for opposite signs of the fields in the cores. In contrast, the background with a nonzero frequency supports quiescent black solitons which may be stable for both out-of-phase and in-phase backgrounds, if the dispersive coupling is sufficiently strong. Only DSs supported by the out-of-phase background admit an extension to the case of nonzero phase mismatch between the cores.

Creation of vortices by torque in multidimensional media with inhomogeneous defocusing nonlinearity

Recently, a new class of nonlinear systems was introduced, in which the self-trapping of fundamental and vortical localized modes in space of dimension D is supported by cubic self-repulsion with a strength growing as a function of the distance from the center, r, at any rate faster that r^D. These systems support robust 2D and 3D modes which either do not exist or are unstable in other nonlinear systems. Here we demonstrate a possibility to create solitary vortices in this setting by applying a phase-imprinting torque to the ground state. Initially, a strong torque completely destroys the ground state. However, contrary to usual systems, where the destruction is irreversible, the present ones demonstrate a rapid restabilization and the creation of one or several shifted vortices orbiting the center. For the sake of comparison, we show analytically that, in the linear system with a 3D trapping potential, the action of a torque on the ground state is inefficient and creates only even-vorticity states with a small probability.

Stabilization of spatiotemporal solitons in Kerr media by dispersive coupling

We introduce a mechanism to stabilize spatiotemporal solitons in Kerr nonlinear media, based on the dispersion of linear coupling between the field components forming the soliton states. Specifically, we consider solitons in a two-core guiding structure with inter-core coupling dispersion (CD). We show that CD profoundly affects properties of the solitons, causing the complete stabilization of the otherwise highly unstable spatiotemporal solitons in Kerr media with focusing nonlinearity. We also find that the presence of CD stimulates the formation of bound states, which, however, are unstable.

Twisted toroidal vortex solitons in inhomogeneous media with repulsive nonlinearity

Toroidal modes in the form of so-called Hopfions, with two independent winding numbers, a hidden one (twist s), which characterizes a circular vortex thread embedded into a three-dimensional soliton, and the vorticity around the vertical axis (m), appear in many fields, including field theory, ferromagnetics, and semi- and superconductors. Such topological states are normally generated in multicomponent systems, or as trapped quasilinear modes in toroidal potentials. We uncover that stable solitons with this structure can be created, without any linear potential, in the single-component setting with the strength of repulsive nonlinearity growing fast enough from the center to the periphery, for both steep and smooth modulation profiles. Toroidal modes with s=1 and vorticity m=0, 1, 2 are produced. They are stable for m≤1, and do not exist for s>1. An approximate analytical solution is obtained for the twisted ring with s=1, m=0. Under the application of an external torque, it rotates like a solid ring. The setting can be implemented in a Bose-Einstein condensate (BEC) by means of the Feshbach resonance controlled by inhomogeneous magnetic fields.

Discrete and continuum composite solitons in Bose-Einstein condensates with the Rashba spin-orbit coupling in one and two dimensions

We introduce one- and two-dimensional (1D and 2D) continuum and discrete models for the two-component BEC, with the spin-orbit (SO) coupling of the Rashba type between the components, and attractive cubic interactions, assuming that the condensate is fragmented into a quasidiscrete state by a deep optical-lattice potential. In 1D, it is demonstrated, in analytical and numerical forms, that the ground states of both the discrete system and its continuum counterpart switch from striped bright solitons, featuring deep short-wave modulations of its profile, to smooth solitons, as the strength ratio of the inter- and intracomponent attraction, γ, changes from γ<1 to γ>1. At the borderline, γ=1, there is a continuous branch of stable solitons, which share a common value of the energy and interpolate between the striped and smooth ones. Unlike the 2D system, the 1D solitons, which do not represent the ground state at given γ, are nevertheless stable against small perturbations, and they remain stable too in collisions with other solitons. In 2D, a transition between two different types of discrete solitons, which represent the ground state, viz., semivortices and mixed modes, also takes place at γ=1. A specific property of 2D discrete solitons of both types is their discontinuous transition into a delocalized state at a critical value of the SO-coupling strength. We also address the continuum 2D model in the borderline case of γ=1, which was not studied previously, and demonstrate the existence of an energy-degenerate branch of dynamically stable solitons connecting the semivortex and the mixed mode. Last, it is demonstrated that 1D and 2D discrete solitons are mobile, in a limited interval of velocities.

Stabilization of solitons under competing nonlinearities by external potentials

We report results of the analysis for families of one-dimensional (1D) trapped solitons, created by competing self-focusing (SF) quintic and self-defocusing (SDF) cubic nonlinear terms. Two trapping potentials are considered, the harmonic-oscillator (HO) and delta-functional ones. The models apply to optical solitons in colloidal waveguides and other photonic media, and to matter-wave solitons in Bose-Einstein condensates loaded into a quasi-1D trap. For the HO potential, the results are obtained in an approximate form, using the variational and Thomas-Fermi approximations, and in a full numerical form, including the ground state and the first antisymmetric excited one. For the delta-functional attractive potential, the results are produced in a fully analytical form, and verified by means of numerical methods. Both exponentially localized solitons and weakly localized trapped modes are found for the delta-functional potential. The most essential conclusions concern the applicability of competing Vakhitov-Kolokolov (VK) and anti-VK criteria to the identification of the stability of solitons created under the action of the competing SF and SDF terms.