In this work we present a systematic study of the three-dimensional extension of the ring dark soliton, examining its existence, stability, and dynamics in isotropic harmonically trapped Bose-Einstein condensates. Detuning the chemical potential from the linear limit, the ring dark soliton becomes unstable immediately but can be fully stabilized by an external cylindrical potential. The ring has a large number of unstable modes which are analyzed through spectral stability analysis. Furthermore, a few typical destabilization dynamical scenarios are revealed with a number of interesting vortical structures emerging, such as the two or four coaxial parallel vortex rings. In the process of considering the stability of the structure, we also develop a modified version of the degenerate perturbation theory method for characterizing the spectra of the coherent structure. This semianalytical method can be reliably applied to any soliton with a linear limit to explore its spectral properties near this limit. The good agreement of the resulting spectrum is illustrated via a comparison with the full numerical Bogolyubov--de Gennes spectrum. The application of the method to the two-component ring dark-bright soliton is also discussed.
In this work we present a systematic study of the three-dimensional extension of the ring dark soliton, examining its existence, stability, and dynamics in isotropic harmonically trapped Bose-Einstein condensates. Detuning the chemical potential from the linear limit, the ring dark soliton becomes unstable immediately but can be fully stabilized by an external cylindrical potential. The ring has a large number of unstable modes which are analyzed through spectral stability analysis. Furthermore, a few typical destabilization dynamical scenarios are revealed with a number of interesting vortical structures emerging, such as the two or four coaxial parallel vortex rings. In the process of considering the stability of the structure, we also develop a modified version of the degenerate perturbation theory method for characterizing the spectra of the coherent structure. This semianalytical method can be reliably applied to any soliton with a linear limit to explore its spectral properties near this limit. The good agreement of the resulting spectrum is illustrated via a comparison with the full numerical Bogolyubov--de Gennes spectrum. The application of the method to the two-component ring dark-bright soliton is also discussed.
In the present work, we investigate how single- and multi-vortex-ring states can emerge from a planar dark soliton in three-dimensional (3D) Bose-Einstein condensates (confined in isotropic or anisotropic traps) through bifurcations. We characterize such bifurcations quantitatively using a Galerkin-type approach and find good qualitative and quantitative agreement with our Bogoliubov--de Gennes (BdG) analysis. We also systematically characterize the BdG spectrum of the dark solitons, using perturbation theory, and obtain a quantitative match with our 3D BdG numerical calculations. We then turn our attention to the emergence of single- and multi-vortex-ring states. We systematically capture these as stationary states of the system and quantify their BdG spectra numerically. We find that although the vortex ring may be unstable when bifurcating, its instabilities weaken and may even eventually disappear for sufficiently large chemical potentials and suitable trap settings. For instance, we demonstrate the stability of the vortex ring for an isotropic trap in the large-chemical-potential regime.
In the present work, we revisit two-component Bose-Einstein condensates in their fully three-dimensional (3D) form. Motivated by earlier studies of dark-bright solitons in the 1D case, we explore the stability of these structures in their fully 3D form in two variants. In one the dark soliton is planar and trapping a planar bright (disk) soliton. In the other case, a dark spherical shell soliton creates an effective potential in which a bright spherical shell of atoms is trapped in the second component. We identify these solutions as numerically exact states (up to a prescribed accuracy) and perform a Bogolyubov-de Gennes linearization analysis that illustrates that both structures can be dynamically stable in suitable intervals of sufficiently low chemical potentials. We corroborate this finding theoretically by analyzing the stability via degenerate perturbation theory near the linear limit of the system. When the solitary waves are found to be unstable, we explore their dynamical evolution via direct numerical simulations which, in turn, reveal wave forms that are more robust. Finally, using the SO(2) symmetry of the model, we produce multi-dark-bright planar or shell solitons involved in pairwise oscillatory motion.
In this work we study spherical shell dark soliton states in three-dimensional atomic Bose-Einstein condensates. Their symmetry is exploited in order to analyze their existence, as well as that of topologically charged variants of the structures, and, importantly, to identify their linear stability Bogoliubov--de Gennes spectrum. We compare our effective one-dimensional spherical and two-dimensional cylindrical computations with the full three-dimensional numerics. An important conclusion is that such spherical shell solitons can be stable sufficiently close to the linear limit of the isotropic condensates considered herein. We have also identified their instabilities leading to the emergence of vortex line and vortex ring cages. In addition, we generalize effective particle pictures of lower-dimensional dark solitons and ring dark solitons to the spherical shell solitons concerning their equilibrium radius and effective dynamics around it. In this case too, we favorably compare, qualitatively and quantitatively, the resulting predictions such as the shell equilibrium radius with full numerical solutions in three dimensions.
Earlier work has shown that ring dark solitons in two-dimensional Bose-Einstein condensates are generically unstable. In this work, we propose a way of stabilizing the ring dark soliton via a radial Gaussian external potential. We investigate the existence and stability of the ring dark soliton upon variations of the chemical potential and also of the strength of the radial potential. Numerical results show that the ring dark soliton can be stabilized in a suitable interval of external potential strengths and chemical potentials. We also explore different proposed particle pictures considering the ring as a moving particle and find, where appropriate, results in very good qualitative and also reasonable quantitative agreement with the numerical findings.
Soliton is an exotic topological excitation, and it widely exists in various nonlinear systems, such as nonlinear optics, Bose-Einstein condensates, classical and quantum fluids, plasma, magnetic materials, etc. A stable soliton can propagate with constant amplitude and velocity, and maintain its shape. Two-dimensional and three-dimensional solitons are usually hard to stabilize, and how to realize stable two-dimensional or three-dimensional solitons has aroused the great interest of the researchers. Ring dark soliton is a kind of two-dimensional soliton, which was first theoretically predicted and experimentally realized in nonlinear optical systems. Compared with the usual two-dimensional solitons, ring dark solitons have good stability and rich dynamical behaviors. Owing to their highly controllable capability, Bose-Einstein condensates provide a new platform for studying the ring dark solitons. Based on the recent progress in Bose-Einstein condensates and solitons, this paper reviews the research on the analytic solutions, stability, as well as the decay dynamics of ring dark solitons in Bose-Einstein condensates. A transform method is introduced, which generalizes the analytic solutions of ring dark solitons from a homogeneous system with time-independent nonlinearity to a harmonically trapped inhomogeneous system with time-dependent nonlinearity. The stability phase diagram of the ring dark soliton under deformation perturbations is discussed by numerically solving the Gross-Pitaevskii equations in the mean-field theory. A method of enhancing the stability of ring dark solitons by periodically modulating the nonlinear coefficients is introduced. It is also shown that the periodically modulated nonlinear coefficient can be experimentally realized by the Feshbach resonance technology. In addition, we discuss the dynamics of the decay of ring dark solitons. It is found that the ring dark soliton can decay into various vortex clusters composed of vortices and antivortices. This opens a new avenue to the investigation of vortex dynamics and quantum turbulence. It is also found that the ring dark solitons combined with periodic modulated nonlinearity can give rise to the pattern formation, which is an interesting nonlinear phenomenon widely explored in all the fields of nature. Finally, some possible research subjects about ring dark solitons in future research are also discussed.
We report on the existence and stability of multidimensional bound solitonic states in harmonically trapped scalar Bose-Einstein condensates. Their equilibrium separation, as a measure of the strength of the soliton-soliton or the solitonic vortex-vortex interaction, is provided for varying chemical potential $\ensuremath{\mu}$. Static bound dark solitons are shown to be dynamically stable in elongated condensates within a range of intermediate (repulsive) interparticle-interaction strength. Beyond this range the snaking instability manifests during the time evolution of the planar solitons and produces the decay into nonstationary vortex states. A subsequent dynamical recurrence of solitons and vortices can be observed at low $\ensuremath{\mu}$. At equilibrium, the bifurcations of bound dark solitons are bound solitonic vortices. Among them, both two-open and two-ring vortex lines are demonstrated to exist with both counter- and co-rotating steady velocity fields. The latter flow configurations evolve, for high chemical potential, into a stationary three-dimensional (3D)-chain-shaped vortex and a three vortex-antivortex-vortex ring sequence that arrest the otherwise increasing angular or linear momentum respectively. As a feature common to the bifurcated families of vortex states, their excitation spectra present unstable modes with associated oscillatory dynamics. In spite of this, the family of two-open counter-rotating vortices support dynamically stable 3D states.