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arXiv:2509.13874

Oscillating ring ferrodark solitons with breathing nematic core in a homogeneous spinor superfluid

Xiaoquan Yu

Published Sep 17, 2025• 0 citations
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Abstract

We study the dynamics of ring ferrodark solitons (FDSs) in a homogeneous quasi-two-dimensional (2D) ferromagnetic spin-1 Bose-Einstein condensate (BEC). In contrast to the usual expanding dynamics of ring dark solitons in a homogeneous system, the ring FDS radius exhibits self-sustained oscillations accompanied by the nematic tensor breathing at the magnetization-vanishing ring FDS core. When the ring radius greatly exceeds the FDS width, motion is nearly elastic, and we derive the ring-radius equation of motion (EOM) which admits exact solutions. This equation can be recast into a form analogous to the inviscid Rayleigh-Plesset equation governing spherical bubble dynamics in classical fluids, but with anomalous terms. At the ring FDS core, the nematic tensor motion is parameterized by a single parameter that connects the two types of FDSs monotonically. Beyond the hydrodynamics regime, density and spin wave emissions become significant and cause energy loss, shrinking the ring FDS radius oscillation; below a threshold, collapses occur followed by the ring FDS annihilation. In the zero quadratic Zeeman energy limit, the ring radius and eigenvalues of the nematic tensor become stationary, while oscillations of the nematic tensor components, driven by the ring curvature, persist at the core. Excellent agreements are found between analytical predictions and numerical simulations.

DOI
10.48550/arxiv.2509.13874

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We study the dynamics of ring ferrodark solitons (FDSs) in a homogeneous quasi-two-dimensional (2D) ferromagnetic spin-1 Bose-Einstein condensate (BEC). In contrast to the usual expanding dynamics of ring dark solitons in a homogeneous system, the ring FDS radius exhibits self-sustained oscillations accompanied by the nematic tensor breathing at the magnetization-vanishing ring FDS core. When the ring radius greatly exceeds the FDS width, motion is nearly elastic, and we derive the ring-radius equation of motion (EOM) which admits exact solutions. This equation can be recast into a form analogous to the inviscid Rayleigh-Plesset equation governing spherical bubble dynamics in classical fluids, but with anomalous terms. At the ring FDS core, the nematic tensor motion is parameterized by a single parameter that connects the two types of FDSs monotonically. Beyond the hydrodynamics regime, density and spin wave emissions become significant and cause energy loss, shrinking the ring FDS radius oscillation; below a threshold, collapses occur followed by the ring FDS annihilation. In the zero quadratic Zeeman energy limit, the ring radius and eigenvalues of the nematic tensor become stationary, while oscillations of the nematic tensor components, driven by the ring curvature, persist at the core. Excellent agreements are found between analytical predictions and numerical simulations.
We propose a framework for topological soliton dynamics in trapped spinor superfluids, decomposing the force acting on the soliton by the surrounding fluid into the buoyancy force and spin corrections arising from the density depletion at soliton core and the coupling between the orbital motion and the spin mixing, respectively. Our formulation applies to large-amplitude soliton motion in general superfluids with spin degrees of freedom under arbitrary external potentials. For ferrodark solitons (FDSs) in spin-1 Bose-Einstein condensates, the spin correction could diverge, change the direction of the total force, and enable mapping the FDS motion in a harmonic trap to the atomic-mass particle dynamics in an emergent quartic potential. Initially placing a type-I FDS near the trap center, a single-sided oscillation happens, which maps to the particle moving around a local minimum of the emergent double-well potential. As the initial distance of a type-II FDS from the trap center increases, the motion exhibits three regimes: trap-centered harmonic and anharmonic oscillations followed by single-sided oscillations. Correspondingly the emergent quartic potential undergoes a transition from a single minimum to a double-well shape, where the particle motion shifts from oscillating around the single minimum to crossing between two minima via the local maximum, then the symmetry-breaking motion around one of the two minima. In a hard-wall trap with linear potential, the FDS motion maps to a harmonic oscillator.
We develop an analytical description of static ferrodark solitons, the Z2{\mathbb{Z}}_{2}Z2​ topological defects in the magnetic order, in the easy-plane phase of ferromagnetic spin-1 Bose-Einstein condensates. We find that the type-I ferrodark soliton has a single width while the type-II ferrodark soliton exhibits two characteristic length scales. The proposed Ans\"atze show excellent agreement with numerical results. Spin-singlet amplitudes, nematic tensor densities, and nematic currents of ferrodark solitons are also discussed. The Z2{\mathbb{Z}}_{2}Z2​ topological defects in the mass superfluid order, dark-dark-dark vector solitons, are obtained exactly in the parameter regime where exact ferrodark solitons exist. The dark-dark-dark vector soliton has higher excitation energy than ferrodark solitons.
We study collisions between a ferrodark soliton (FDS) and an antiFDS (Z2\mathbb{Z}_2Z2​ kinks in the spin order) in the easy-plane phase of spin-1 Bose-Einstein condensates (BECs). For a type-I pair (type-I FDS-antiFDS pair) at low incoming velocities, the pair annihilates followed by the formation of an extremely long-lived dissipative breather on a stable background, a spatially localized wave packet with out-of-phase oscillating magnetization and mass superfluid densities. Periodic emissions of spin and density waves cause breather energy dissipation and we find that the breather energy decays logarithmically in time. When the incoming velocity is larger than a critical velocity at which a stationary FDS-antiFDS pair forms, a pair with finite separating velocity is reproduced. When approaching the critical velocity from below, we find that the lifetime of the stationary type-I pair shows a power-law divergence, resembling a critical behavior. In contrast, a type-II pair (type-II FDS-antiFDS pair) never annihilates and only exhibits reflection. For collisions of a mixed type FDS-antiFDS pair, as Z2\mathbb{Z}_2Z2​ kinks in the spin order, reflection occurs in the topological structure of the magnetization while the mass superfluid density profiles pass through each other, manifesting spin-mass separation.
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.
We consider the dynamics and formation of vortices from ring dark solitons in a two-dimensional Bose-Einstein condensate with the Rashba spin-orbit coupling based on the time-dependent coupled Gross-Pitaevskii equation. Compared with previous results, the system exhibits complex dynamical behaviors in the presence of the spin-orbit coupling. With the modulation of the spin-orbit coupling, not only the lifetime of ring dark solitons is greatly prolonged, but also their attenuation kinetics is significantly affected. For two shallow ring dark solitons with the equal strength of the spin-orbit coupling, the radius of ring dark solitons increases to a maximum value over time and then shrinks into a minimum value. Due to the effect of the snake instability, ring dark solitons split into a series of ring-like clusters of vortex pairs, which perform complex oscillations. This indicates that the system is strongly dependent on the presence of the spin-orbit coupling. Furthermore, the effect of different initial modulation depths on the dynamics of ring dark solitons is investigated.
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.