arXiv:cond-mat/0306596 · Journal of Physics A Mathematical and General

Padé approximations of solitary wave solutions of the Gross–Pitaevskii equation

Natalia G. Berloff

Published Jan 19, 2004• 102 citations

Abstract

Pade approximants are used to find approximate vortex solutions of any winding number in the context of Gross-Pitaevskii equation for a uniform condensate and condensates with axisymmetric trapping potentials. Rational function and generalised rational function approximations of axisymmetric solitary waves of the Gross-Pitaevskii equation are obtained in two and three dimensions. These approximations are used to establish a new mechanism of vortex nucleation as a result of solitary wave interactions.

DOI: 10.1088/0305-4470/37/5/011

Related papers

Padé approximations of vortex solutions of the Gross-Pitaevskii equation

Natalia G. Berloff

Jan 1, 20030 citations

Padé approximations of quantized-vortex solutions of the Gross–Pitaevskii equation

Weiru Chen, Shan-Quan Lan, Xiyi Liu +3 more

Communications in Theoretical PhysicsMay 6, 20213 citations

Solitary waves on vortex lines in Ginzburg--Landau models

Natalia G. Berloff

arXiv:cond-mat/0407483Jul 19, 20040 citations

Solitary Waves on Vortex Lines in Ginzburg-Landau Models for the Example of Bose-Einstein Condensates

Natalia G. Berloff

Physical Review LettersJan 7, 20059 citations

Solitary Wave Complexes in Two-Component Condensates

Natalia G. Berloff

arXiv:cond-mat/0412099Physical Review LettersMar 28, 200545 citations

High precision solutions to quantized vortices within Gross–Pitaevskii equation

Hao-Hao Peng, Jian Deng, Sen‐Yue Lou +1 more

arXiv:2207.13360Communications in Theoretical PhysicsAug 4, 20220 citations

arXiv:cond-mat/0306596 · Journal of Physics A Mathematical and General

Padé approximations of solitary wave solutions of the Gross–Pitaevskii equation

Natalia G. Berloff

Published Jan 19, 2004• 102 citations

arXiv HTML PDF

Abstract

DOI: 10.1088/0305-4470/37/5/011

Related papers

Padé approximations of vortex solutions of the Gross-Pitaevskii equation

Natalia G. Berloff

Padé approximants are used to find the approximate vortex solutions of any winding number in the context of Gross-Pitaevskii equation for the uniform condensate and condensates with axisymmetric trapping potentials. The approximate solutions are used to elucidate the annihilation of pair of vortices of opposite circulation by finding the approximation to the streamfunction of the flow generated by such a pair. The critical velocity and minimal separation of vortices are calculated. Keywords: Gross-Pitaevskii model, Padé approximations, vortex solutions. 1Padé approximants are used to find the approximate vortex solutions of any winding number in the context of Gross-Pitaevskii equation for the uniform condensate and condensates with axisymmetric trapping potentials. The approximate solutions are used to elucidate the annihilation of pair of vortices of opposite circulation by finding the approximation to the streamfunction of the flow generated by such a pair. The critical velocity and minimal separation of vortices are calculated. Keywords: Gross-Pitaevskii model, Padé approximations, vortex solutions. 1

Jan 1, 20030 citations

Padé approximations of quantized-vortex solutions of the Gross–Pitaevskii equation

Weiru Chen, Shan-Quan Lan, Xiyi Liu +3 more

Abstract Quantized vortices are important topological excitations in Bose–Einstein condensates. The Gross–Pitaevskii equation is a widely accepted theoretical tool. High accuracy quantized-vortex solutions are desirable in many numerical and analytical studies. We successfully derive the Padé approximate solutions for quantized vortices with winding numbers ω = 1, 2, 3, 4, 5, 6 in the context of the Gross–Pitaevskii equation for a uniform condensate. Compared with the numerical solutions, we find that (1) they approximate the entire solutions quite well from the core to infinity; (2) higher-order Padé approximate solutions have higher accuracy; (3) Padé approximate solutions for larger winding numbers have lower accuracy. The healing lengths of the quantized vortices are calculated and found to increase almost linearly with the winding number. Based on experiments performed with 87 Rb cold atoms, the healing lengths of quantized vortices and the number of particles within the healing lengths are calculated, and they may be checked by experiment. Our results show that the Gross–Pitaevskii equation is capable of describing the structure of quantized vortices and physics at length scales smaller than the healing length.Abstract Quantized vortices are important topological excitations in Bose–Einstein condensates. The Gross–Pitaevskii equation is a widely accepted theoretical tool. High accuracy quantized-vortex solutions are desirable in many numerical and analytical studies. We successfully derive the Padé approximate solutions for quantized vortices with winding numbers ω = 1, 2, 3, 4, 5, 6 in the context of the Gross–Pitaevskii equation for a uniform condensate. Compared with the numerical solutions, we find that (1) they approximate the entire solutions quite well from the core to infinity; (2) higher-order Padé approximate solutions have higher accuracy; (3) Padé approximate solutions for larger winding numbers have lower accuracy. The healing lengths of the quantized vortices are calculated and found to increase almost linearly with the winding number. Based on experiments performed with 87 Rb cold atoms, the healing lengths of quantized vortices and the number of particles within the healing lengths are calculated, and they may be checked by experiment. Our results show that the Gross–Pitaevskii equation is capable of describing the structure of quantized vortices and physics at length scales smaller than the healing length.

Communications in Theoretical PhysicsMay 6, 20213 citations

Solitary waves on vortex lines in Ginzburg--Landau models

Natalia G. Berloff

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 ( $p$ ) -- substitution energy ( $\hat{\cal E}$ ) plane with $p\to 0.09 ρκ^3/c^2, \hat{\cal E}\to 0.091 ρκ^3/c$ as $U \to c$ , where $ρ$ is the density, $c$ is the speed of sound, $κ$ is the quantum of circulation and $U$ 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 ($p$) -- substitution energy ($\hat{\cal E}$) plane with $p\to 0.09 ρκ^3/c^2, \hat{\cal E}\to 0.091 ρκ^3/c$ as $U \to c$, where $ρ$ is the density, $c$ is the speed of sound, $κ$ is the quantum of circulation and $U$ 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.

arXiv:cond-mat/0407483Jul 19, 20040 citations

Solitary Waves on Vortex Lines in Ginzburg-Landau Models for the Example of Bose-Einstein Condensates

Natalia G. Berloff

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 ( $p$ )-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, $c$ is the speed of sound, $\ensuremath{\kappa}$ is the quantum of circulation, and $U$ 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.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 ($p$)-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, $c$ is the speed of sound, $\ensuremath{\kappa}$ is the quantum of circulation, and $U$ 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.

Physical Review LettersJan 7, 20059 citations

Solitary Wave Complexes in Two-Component Condensates

Natalia G. Berloff

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.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.

arXiv:cond-mat/0412099Physical Review LettersMar 28, 200545 citations

High precision solutions to quantized vortices within Gross–Pitaevskii equation

Hao-Hao Peng, Jian Deng, Sen‐Yue Lou +1 more

Abstract The dynamics of vortices in Bose–Einstein condensates of dilute cold atoms can be well formulated by Gross–Pitaevskii equation. To better understand the properties of vortices, a systematic method to solve the nonlinear differential equation for the vortex to very high precision is proposed. Through two-point Padé approximants, these solutions are presented in terms of simple rational functions, which can be used in the simulation of vortex dynamics. The precision of the solutions is sensitive to the connecting parameter and the truncation orders. It can be improved significantly with a reasonable extension in the order of rational functions. The errors of the solutions and the limitation of two-point Padé approximants are discussed. This investigation may shed light on the exact solution to the nonlinear vortex equation.Abstract The dynamics of vortices in Bose–Einstein condensates of dilute cold atoms can be well formulated by Gross–Pitaevskii equation. To better understand the properties of vortices, a systematic method to solve the nonlinear differential equation for the vortex to very high precision is proposed. Through two-point Padé approximants, these solutions are presented in terms of simple rational functions, which can be used in the simulation of vortex dynamics. The precision of the solutions is sensitive to the connecting parameter and the truncation orders. It can be improved significantly with a reasonable extension in the order of rational functions. The errors of the solutions and the limitation of two-point Padé approximants are discussed. This investigation may shed light on the exact solution to the nonlinear vortex equation.

arXiv:2207.13360Communications in Theoretical PhysicsAug 4, 20220 citations

p

) -- substitution energy (

\hat{\cal E}

) plane with

p\to 0.09 ρκ^3/c^2, \hat{\cal E}\to 0.091 ρκ^3/c

U \to c

, where

ρ

is the density,

c

is the speed of sound,

κ

is the quantum of circulation and

U

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.

p

)-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,

c

is the speed of sound, $\ensuremath{\kappa}$ is the quantum of circulation, and

U

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.