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arXiv:2504.01385 · Physical review. E

Dynamics of ring solitons in an expanding cloud of a Bose-Einstein condensate

A. M. Kamchatnov, Bulat Irekovich Suleimanov, E. N. Tsoy

Published Jun 3, 2025• 1 citation
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Abstract

In this paper, we derive equations for the dynamics of ring dark solitons in an expanding cloud of a two-dimensional Bose-Einstein condensate. Assuming that the soliton's width is much smaller than its radius, we obtain the Hamilton equations for its evolution. Then they are transformed into the Newton equation, which is more convenient for applications. The general theory is illustrated by the solution of the Newton equation for the case of the axially symmetric condensate cloud, which expands after switching off a harmonic trap. The validity of our approximate analytical approach is confirmed by comparison with the results of numerical simulations of the Gross-Pitaevskii equation.

DOI
10.1103/physreve.111.064203

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In this paper, we derive equations for the dynamics of ring dark solitons in an expanding cloud of a two-dimensional Bose-Einstein condensate. Assuming that the soliton's width is much smaller than its radius, we obtain the Hamilton equations for its evolution. Then they are transformed into the Newton equation, which is more convenient for applications. The general theory is illustrated by the solution of the Newton equation for the case of the axially symmetric condensate cloud, which expands after switching off a harmonic trap. The validity of our approximate analytical approach is confirmed by comparison with the results of numerical simulations of the Gross-Pitaevskii equation.
The dynamics of the ring dark soliton in a Bose–Einstein condensate (BEC) with thin disc-shaped potential is investigated analytically and numerically. Analytical investigation shows that the ring dark soliton in the radial non-symmetric cylindrical BEC is governed by a cylindrical Kadomtsev–Petviashvili equation, while the ring dark soliton in the radial symmetric cylindrical BEC is governed by a cylindrical Korteweg–de Vries equation. The reduction to the cylindrical KP or KdV equation may be useful to understand the dynamics of a ring dark soliton. The numerical results show that the evolution properties and the snaking of a ring dark soliton are modified significantly by the trapping.
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 investigate both analytically and numerically the dynamics and modulation of ring dark solitons in two-dimensional Bose-Einstein condensates with tunable interaction. The analytic solutions for the ring dark soliton are derived by using a transformation method. A shallow ring dark soliton is stable when the ring is slightly distorted, while for large deformation of the ring, vortex pairs appear and demonstrate different dynamical behaviors: the vortex pairs transform into dark lumplike solitons and revert to ring dark solitons periodically. Moreover, our results show that the dynamical evolution of the ring dark soliton can be dramatically affected by the Feshbach resonance, and the lifetime of the ring dark soliton can be greatly extended, which offers a useful method for observing ring dark solitons in future experiments.
We investigate the dynamics and pattern formation of two ring dark solitons in a two-dimensional binary Bose-Einstein condensate with tunable intercomponent interaction via numerical simulation of the time-dependent Gross-Pitaevskii equation. Both the black and gray ring dark solitons are considered for cases where the modulation frequency of the intercomponent interaction is resonant or nonresonant with the one of the trapping potential. Our results show that in the presence of periodic modulation of the intercomponent interaction not only are the lifetimes of the ring dark solitons largely extended but also their decaying dynamics are dramatically affected. Before snaking instability sets in, new ring dark solitons are formed, and both the numbers and depths of the ring dark solitons exhibit collective oscillations. With the development of instability, the system exhibits different decaying processes, and a variety of decay profiles, such as vortex necklace, distorted octagon, vortex-antivortex ring, and cross, are formed, showing a strong dependence on the modulation frequency of the intercomponent interaction and the initial depth of the ring dark soliton.
We use the time-dependent Gross-Pitaevskii equation to study the motion of a vortex ring produced by phase imprinting on an elongated cloud of cold atoms. Our approach models the experiments of Yefsah et al. [Nature (London) 499, 426 (2013)] on 6{}^{6}6Li in the Bose-Einstein-condensate regime where the fermions are tightly bound into bosonic dimers. We find ring oscillation periods which are much larger than the period of the axial harmonic trap. Our results lend further strength to Bulgac et al.'s arguments (arXiv:1306.4266) that the ``heavy solitons'' seen in those experiments are actually vortex rings. We numerically calculate the periods of oscillation for the vortex rings as a function of interaction strength, trap aspect ratio, and minimum vortex ring radius. In the presence of axial anisotropies the rings undergo complicated internal dynamics where they break into sets of vortex lines, then later combine into rings. These structures oscillate with a similar frequency to simple axially symmetric rings.
A variational technique is applied to solve the time-dependent nonlinear Schr\"odinger equation (Gross-Pitaevskii equation) with the goal to model the dynamics of dilute ultracold atom clouds in the Bose-Einstein condensed phase. We derive analytical predictions for the collapse, equilibrium widths, and evolution laws of the condensate parameters and find them to be in very good agreement with our numerical simulations of the nonlinear Schr\"odinger equation. It is found that not only the number of particles, but also both the initial width of the condensate and the effect of different perturbations to the condensate may play a crucial role in the collapse dynamics. The results are applicable when the shape of the condensate is sufficiently simple.