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Journal of Physics A Mathematical and General

Motions in a Bose condensate. IV. Axisymmetric solitary waves

CHRIS A. JONES, Paul Harry Roberts

Published Aug 1, 1982• 297 citations
Publisher

Abstract

For pt.III see ibid., vol.7, p.260 (1974). Axisymmetric disturbances that preserve their form as they move through a Bose condensate are obtained numerically by the solution of the appropriate nonlinear Schrodinger equation. A continuous family is obtained that, in the momentum (p)-energy (E) plane, consists of two branches meeting at a cusp of minimum momentum around 0.140 rho kappa 3 /c 2 and minimum energy about 0.145 rho kappa 3/c , where rho is density, c is the speed of sound and kappa is the quantum of circulation. For all larger p, there are two possible energy states. One (the lower branch) is (for large enough p) a vortex ring of circulation kappa ; as p to infinity its radius omega approximately (p/ pi kappa ) 1/2 becomes infinite and its forward velocity tends to zero. The other (the upper branch) lacks vorticity and is a rarefaction sound pulse that becomes increasingly one dimensional as p to infinity ; its velocity approaches c for large p. The velocity of any member of the family is shown, both numerically and analytically, to be delta E/ delta p, the derivative being taken along the family.

DOI
10.1088/0305-4470/15/8/036

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For pt.III see ibid., vol.7, p.260 (1974). Axisymmetric disturbances that preserve their form as they move through a Bose condensate are obtained numerically by the solution of the appropriate nonlinear Schrodinger equation. A continuous family is obtained that, in the momentum (p)-energy (E) plane, consists of two branches meeting at a cusp of minimum momentum around 0.140 rho kappa 3 /c 2 and minimum energy about 0.145 rho kappa 3/c , where rho is density, c is the speed of sound and kappa is the quantum of circulation. For all larger p, there are two possible energy states. One (the lower branch) is (for large enough p) a vortex ring of circulation kappa ; as p to infinity its radius omega approximately (p/ pi kappa ) 1/2 becomes infinite and its forward velocity tends to zero. The other (the upper branch) lacks vorticity and is a rarefaction sound pulse that becomes increasingly one dimensional as p to infinity ; its velocity approaches c for large p. The velocity of any member of the family is shown, both numerically and analytically, to be delta E/ delta p, the derivative being taken along the family.
Axisymmetric disturbances that preserve their form as they move along the vortex lines in uniform Bose-Einstein condensates are obtained numerically by the solution of the Gross-Pitaevskii equation. A continuous family of such solitary waves is shown in the momentum (ppp) -- substitution energy (E^\hat{\cal E}E^) plane with p→0.09ρκ3/c2,E^→0.091ρκ3/cp\to 0.09 ρκ^3/c^2, \hat{\cal E}\to 0.091 ρκ^3/cp→0.09ρκ3/c2,E^→0.091ρκ3/c as U→cU \to cU→c, where ρρρ is the density, ccc is the speed of sound, κκκ is the quantum of circulation and UUU is the solitary wave velocity. It is shown that collapse of a bubble captured by a vortex line leads to the generation of such solitary waves in condensates. The various stages of collapse are elucidated. In particularly, it is shown that during collapse the vortex core becomes significantly compressed and after collapse two solitary wave trains moving in opposite directions are formed on the vortex line.
Axisymmetric disturbances that preserve their form as they move along the vortex lines in uniform Bose-Einstein condensates are obtained numerically by the solution of the Gross-Pitaevskii equation. A continuous family of such solitary waves is shown in the momentum (ppp)-substitution energy ($\stackrel{^}{\mathcal{E}}$) plane with $p\ensuremath{\rightarrow}0.09\ensuremath{\rho}{\ensuremath{\kappa}}^{3}/{c}^{2}$, $\stackrel{^}{\mathcal{E}}\ensuremath{\rightarrow}0.091\ensuremath{\rho}{\ensuremath{\kappa}}^{3}/c$ as $U\ensuremath{\rightarrow}c$, where $\ensuremath{\rho}$ is the density, ccc is the speed of sound, $\ensuremath{\kappa}$ is the quantum of circulation, and UUU is the solitary wave velocity. It is shown that collapse of a bubble captured by a vortex line leads to the generation of such solitary waves in condensates. The various stages of collapse are elucidated. In particular, it is shown that during collapse the vortex core becomes significantly compressed, and after collapse two solitary wave trains moving in opposite directions are formed on the vortex line.
A procedure is given which in principle permits the exact evaluation of any term in the asymptotic expansion in a/c of the velocity U o and energy E of a circular vortex line of radius c in a Bose condensate (where a is the healing length). The procedure is used to obtain the two leading terms (previously only inaccurately determined by a variational calculation), namely U 0 = kappa /4 pi c(In(8c/a)-0.615); E= 1 / 2 rho kappa 2 c(1n(8c/a)-1.615); where kappa denotes the quantum of circulation, and rho denotes the fluid density at infinity.
Axisymmetric three-dimensional solitary waves in uniform two-component mixture Bose-Einstein condensates are obtained as solutions of the coupled Gross-Pitaevskii equations with equal intracomponent but varying intercomponent interaction strengths. Several families of solitary wave complexes are found: (1) vortex rings of various radii in each of the components; (2) a vortex ring in one component coupled to a rarefaction solitary wave of the other component; (3) two coupled rarefaction waves; (4) either a vortex ring or a rarefaction pulse coupled to a localized disturbance of a very low momentum. The continuous families of such waves are shown in the momentum-energy plane for various values of the interaction strengths and the relative differences between the chemical potentials of two components. Solitary wave formation, their stability, and solitary wave complexes in two dimensions are discussed.
Nonlocal nonlinear Schrödinger equations are considered as models of liquid helium II. The models contain a nonlocal interaction potential that leads to a phonon-roton-like dispersion relation. Also, a higher-order term in the local density approximation for the correlation energy is introduced into the model to overcome nonphysical mass concentrations. These equations are solved for straight-line vortices. It is demonstrated that the parameters of the equation can be chosen to bring into agreement the vortex core parameter and the healing length. The structure of vortex rings of large radius is studied. The family of the vortex rings of different radii propagating with different velocities is found numerically. As the velocity of the vortex ring reaches the Landau critical velocity the sequence of rings terminates.
Quasi-one-dimensional solitons that may occur in an elongated Bose-Einstein condensate become unstable at high particle density. We study two basic modes of instability and the corresponding bifurcations to genuinely three-dimensional solitary waves such as axisymmetric vortex rings and nonaxisymmetric solitonic vortices. We calculate the profiles of the above structures and examine their dependence on the velocity of propagation along a cylindrical trap. At sufficiently high velocity, both the vortex ring and the solitonic vortex transform into an axisymmetric soliton. We also calculate the energy-momentum dispersions and show that a Lieb-type mode appears in the excitation spectrum for all particle densities.