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arXiv:1010.2100

Development of instability of dark solitons generated by a flow of Bose-Einstein condensate past a concave corner

A. M. Kamchatnov, Korneev, S. V.

Published Oct 11, 2010• 0 citations
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Abstract

The stability of dark solitons generated by a supersonic flow of a Bose-Einstein condensate past a concave corner (or a wedge) is studied. It is shown that solitons in the dispersive shock wave generated at the initial moment of time demonstrate a snake instability during their evolution to stationary curved solitons. Time of decay of soliton to vortices agrees very well with analytical estimates of the instability growth rate.

DOI
10.48550/arxiv.1010.2100

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The stability of dark solitons generated by a supersonic flow of a Bose-Einstein condensate past a concave corner (or a wedge) is studied. It is shown that solitons in the dispersive shock wave generated at the initial moment of time demonstrate a snake instability during their evolution to stationary curved solitons. Time of decay of soliton to vortices agrees very well with analytical estimates of the instability growth rate.
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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.
The stability of nonlinear waves on curved surfaces is a problem of widespread interest across physics. Here, we establish the stability criteria for dark solitons on a spherical Bose-Einstein condensate. We demonstrate a sharp instability threshold in the nonlinear parameter, beyond which solitons decay into vortex dipoles via snake instabilities. Analytically and numerically, we prove this decay is dictated by a single unstable mode for each angular momentum m≥2m \geq 2m≥2, which is a universal mechanism that controls the resulting vortex state. Unlike in the full three-dimensional case, where snake instabilities lead to vortex rings, a dark soliton confined to the surface of a bubble can only decay into vortex pairs.
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 examine the stability and dynamics of a family of crossed dark solitons in a harmonically confined Bose-Einstein condensate in two dimensions. Working in a regime where the fundamental snake instability is suppressed, we show the existence of an instability which leads to an interesting collapse and revival of the initial state for the fundamental case of two crossed solitons. The instability originates from the singular point where the solitons cross, and we characterize it by examining the Bogoliubov spectrum. Finally, we extend the treatment to systems of higher symmetry.
We present numerical calculations of the snake instability in a Fermi superfluid within the Bogoliubov-de Gennes theory of the Bose-Einstein condensate (BEC) to BCS crossover using the random-phase approximation complemented by time-dependent simulations. We examine the snaking behavior across the crossover and quantify the time scale and length scale of the instability. While the dynamics shows extensive snaking before eventually producing vortices and sound on the BEC side of the crossover, the snaking dynamics is preempted by decay into sound due to pair breaking in the deep BCS regime. At the unitarity limit, hydrodynamic arguments allow us to link the rate of snaking to the experimentally observable ratio of inertial to physical mass of the soliton. In this limit we witness an unresolved discrepancy between our numerical estimates for the critical wave number of suppression of the snake instability and recent experimental observations with an ultracold Fermi gas.