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arXiv:2402.05351 · Physical Review A

Absence of the breakdown of ferrodark solitons exhibiting a snake instability

Xiaoquan Yu, P. B. Blakie

Published Dec 4, 2024• 5 citations
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Abstract

The authors consider the dynamics of the recently discovered ferrodark soliton as a line defect in a quasi-two-dimensional ferromagnetic spin-1 Bose-Einstein condensate. This line ferrodark soliton exhibits snake instability however does not suffer the expected breakdown. Instead, it shows a chaotic motion combined with the proliferation of confined mass vortex pairs while the topological structure of the magnetic domain wall is kept.

DOI
10.1103/physreva.110.l061303

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The authors consider the dynamics of the recently discovered ferrodark soliton as a line defect in a quasi-two-dimensional ferromagnetic spin-1 Bose-Einstein condensate. This line ferrodark soliton exhibits snake instability however does not suffer the expected breakdown. Instead, it shows a chaotic motion combined with the proliferation of confined mass vortex pairs while the topological structure of the magnetic domain wall is kept.
We report a stable magnetic domain wall in a uniform ferromagnetic spin-1 condensate, characterized by the magnetization having a dark soliton profile with nonvanishing superfluid density.We find exact stationary solutions for a particular ratio of interaction parameters with and without magnetic fields, and develop an accurate analytic solution applicable to the whole ferromagnetic phase.In the absence of magnetic fields, this domain wall relates various distinct solitary excitations in binary condensates through SO(3) spin rotations, which otherwise are unconnected.Remarkably, studying the dynamics of a quasi-two-dimensional (quasi-2D) system we show that standing wave excitations of the domain wall oscillate without decay, being stable against the snake instability.The domain wall is dynamically unstable to modes that cause the magnetization to grow perpendicularly while leaving the domain wall unchanged.Real-time dynamics in the presence of white noise reveals that this "spin twist" instability does not destroy the topological structure of the magnetic domain wall.
The connection between quantized vortices and dark solitons in a waveguidelike trap geometry is explored in the framework of the nonlinear Schr\"odinger equation. Variation of the transverse confinement leads from the quasi-one-dimensional (1D) regime, where solitons are stable, to 2D (or 3D) confinement, where soliton stripes are subject to a transverse modulational instability known as the ``snake instability.'' We present numerical evidence of a regime of intermediate confinement where solitons decay into single, deformed vortices with solitonic properties rather than vortex pairs as associated with the ``snake'' metaphor. Further relaxing the transverse confinement leads to the production of two and then three vortices, which correlates perfectly with a Bogoliubov stability analysis. The decay of a stationary dark soliton (or, planar node) into a single solitonic vortex is predicted to be experimentally observable in a 3D harmonically confined dilute-gas Bose-Einstein condensate.
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.
Solitons are striking manifestations of nonlinearity, encountered in diverse physical systems such as water waves, nonlinear optics, and Bose-Einstein condensates (BECs). In BECs, dark solitons emerge as exact stationary solutions of the one-dimensional Gross-Pitaevskii equation. While they can be long-lived in elongated traps, their stability is compromised in higher dimensions due to the snake instability, which leads to the decay of the soliton into vortex structures among other excitations. We investigate the dynamics of a dark soliton in a Bose-Einstein condensate confined in an anisotropic harmonic trap. Using a variational ansatz that incorporates both the transverse bending of the soliton plane and the emergence of vortices along the nodal line, we derive equations of motion governing the soliton's evolution. This approach allows us to identify stable oscillation modes as well as the growth rates of the unstable perturbations. In particular, we determine the critical trap anisotropy required to suppress the snake instability. Our analytical predictions are in good agreement with full numerical simulations of the Gross-Pitaevskii equation.
We show by numerically solving the Bogoliubov--de Gennes equations and numerically integrating the Gross-Pitaevskii equation that the onset of the snake instability of ring dark solitons requires a broken symmetry. We elucidate explicitly the connection between imaginary Bogoliubov modes and the snake instability, predicting the number of vortex-antivortex pairs produced. In addition, we propose a simple model to give a physical motivation as to why the snake instability takes place and needs a broken symmetry. Finally, we show that tight confinement in a toroidal potential actually enhances soliton decay due to inhibition of soliton motion, but can lead to a periodic revival of the ring dark soliton after the first snake instability.