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arXiv:1004.3060 · Physical Review A

Controlling the transverse instability of dark solitons and nucleation of vortices by a potential barrier

Manjun Ma, R. Carretero-González, P. G. Kevrekidis, D. J. Frantzeskakis, Boris A. Malomed

Published Aug 30, 2010• 101 citations
arXivHTMLPDF

Abstract

We study possibilities to suppress the transverse modulational instability (MI) of dark-soliton stripes in two-dimensional Bose-Einstein condensates (BEC's) and self-defocusing bulk optical waveguides by means of quasi-one-dimensional structures. Adding an external repulsive barrier potential (which can be induced in BEC by a laser sheet, or by an embedded plate in optics), we demonstrate that it is possible to reduce the MI wave number band, and even render the dark-soliton stripe completely stable. Using this method, we demonstrate the control of the number of vortex pairs nucleated by each spatial period of the modulational perturbation. By means of the perturbation theory, we predict the number of the nucleated vortices per unit length. The analytical results are corroborated by the numerical computation of eigenmodes of small perturbations, as well as by direct simulations of the underlying Gross-Pitaevskii/nonlinear Schr\"odinger equation.

DOI
10.1103/physreva.82.023621

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We study possibilities to suppress the transverse modulational instability (MI) of dark-soliton stripes in two-dimensional Bose-Einstein condensates (BEC's) and self-defocusing bulk optical waveguides by means of quasi-one-dimensional structures. Adding an external repulsive barrier potential (which can be induced in BEC by a laser sheet, or by an embedded plate in optics), we demonstrate that it is possible to reduce the MI wave number band, and even render the dark-soliton stripe completely stable. Using this method, we demonstrate the control of the number of vortex pairs nucleated by each spatial period of the modulational perturbation. By means of the perturbation theory, we predict the number of the nucleated vortices per unit length. The analytical results are corroborated by the numerical computation of eigenmodes of small perturbations, as well as by direct simulations of the underlying Gross-Pitaevskii/nonlinear Schr"odinger equation.
In the present work we examine the statics and dynamics of multiple parallel dark soliton stripes in a two-dimensional Bose-Einstein condensate. Our principal goal is to study the effect of the interaction between the stripes on the transverse instability of the individual stripes. The cases of two-, three-, and four-stripe states are studied in detail. We use a recently developed adiabatic invariant formulation to derive a quasianalytical prediction for the stripe equilibrium position and for the Bogoliubov--de Gennes spectrum of excitations of stationary stripes. We subsequently test our predictions against numerical simulations of the full two-dimensional Gross-Pitaevskii equation. We find that the number of unstable eigenmodes increases as the number of stripes increases due to (unstable) relative motions between the stripes. Their corresponding growth rates do not significantly change, although for large chemical potentials, the larger the stripe number, the larger the maximal instability growth rate. The instability induced dynamics of multiple stripe states and their decay into vortices are also investigated.
In the present work, we develop an adiabatic invariant approach for the evolution of quasi-one-dimensional (stripe) solitons embedded in a two-dimensional Bose-Einstein condensate. The results of the theory are obtained both for the one-component case of dark soliton stripes, as well as for the considerably more involved case of the two-component dark-bright (alias ``filled dark'') soliton stripes. In both cases, analytical predictions regarding the stability and dynamics of these structures are obtained. One of our main findings is the determination of the instability modes of the waves as a function of the parameters of the system (such as the trap strength and the chemical potential). Our analytical predictions are favorably compared with results of direct numerical simulations.
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.
Straight dark solitons in two-dimensional (2D) quantum fluids usually decay by transverse modulational instability, with no intrinsic suppression in contact-interacting Bose--Einstein condensates (BECs). We theoretically show that anisotropic long-range interactions in a quasi-2D dipolar BEC stabilize an embedded straight soliton, with spontaneous stripe order providing stronger pinning. The excitation spectra show that the lowest transverse solitonic branch remains gapped, while stripe-supersolid density modulation further hardens this branch and increases the soliton bending stiffness, penalizing transverse deformation. Accessible in current 166^{166}166Er and 164^{164}164Dy platforms, these results establish interaction-driven protection for straight dark solitons in structured quantum fluids.
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.
A self-consistent theory of a cylindrically shaped Bose-Einstein condensate (BEC) periodically modulated by a laser beam is presented. We show, both analytically and numerically, that modulational instability/stability is the mechanism by which wave functions of soliton type can be generated in a cylindrically shaped BEC subject to a one-dimensional optical lattice. The theory explains why bright solitons can exist in a BEC with positive scattering length and why condensates with negative scattering length can be stable and give rise to dark solitary pulses.