### Abstract

Quantum vortex dynamics in Bose-Einstein condensates or superfluid helium can be informatively described by the Gross-Pitaevskii (GP) equation. Various approximate analytical formulas for a single stationary vortex are recalled and their shortcomings demonstrated. Significantly more accurate two-point [2/2] and [3/3] Padé approximants for stationary vortex profiles are presented. Two straight, singly quantized, antiparallel vortices, located at a distance d_{0} apart, form a vortex dipole, which, in the GP model, can either annihilate or propagate indefinitely as a "solitary wave." We show, through calculations performed in a periodic domain, that the details and types of behavior displayed by vortex dipoles depend strongly on the initial conditions rather than only on the separation distance d_{0} (as has been previously claimed). It is found, indeed, that the choice of the initial two-vortex profile (i.e., the modulus of the "effective wave function"), strongly affects the vortex trajectories and the time scale of the process: annihilation proceeds more rapidly when low-energy (or "relaxed") initial profiles are imposed. The initial "circular" phase distribution contours, customarily obtained by multiplying an effective wave function for each individual vortex, can be generalized to explicit elliptical forms specified by two parameters; then by "tuning" the elliptical shape at fixed d_{0}, a sharp transition between solitary-wave propagation and annihilation is captured. Thereby, a "phase diagram" for this "AnSol" transition is constructed in the space of ellipticity and separation and various limiting forms of the boundary are discussed.

Original language | English (US) |
---|---|

Article number | 134522 |

Journal | Physical Review B - Condensed Matter and Materials Physics |

Volume | 88 |

Issue number | 13 |

DOIs | |

State | Published - Oct 29 2013 |

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### ASJC Scopus subject areas

- Condensed Matter Physics
- Electronic, Optical and Magnetic Materials

### Cite this

*Physical Review B - Condensed Matter and Materials Physics*,

*88*(13), [134522]. https://doi.org/10.1103/PhysRevB.88.134522

**Propagating and annihilating vortex dipoles in the Gross-Pitaevskii equation.** / Rorai, Cecilia; Sreenivasan, K. R.; Fisher, Michael E.

Research output: Contribution to journal › Article

*Physical Review B - Condensed Matter and Materials Physics*, vol. 88, no. 13, 134522. https://doi.org/10.1103/PhysRevB.88.134522

}

TY - JOUR

T1 - Propagating and annihilating vortex dipoles in the Gross-Pitaevskii equation

AU - Rorai, Cecilia

AU - Sreenivasan, K. R.

AU - Fisher, Michael E.

PY - 2013/10/29

Y1 - 2013/10/29

N2 - Quantum vortex dynamics in Bose-Einstein condensates or superfluid helium can be informatively described by the Gross-Pitaevskii (GP) equation. Various approximate analytical formulas for a single stationary vortex are recalled and their shortcomings demonstrated. Significantly more accurate two-point [2/2] and [3/3] Padé approximants for stationary vortex profiles are presented. Two straight, singly quantized, antiparallel vortices, located at a distance d0 apart, form a vortex dipole, which, in the GP model, can either annihilate or propagate indefinitely as a "solitary wave." We show, through calculations performed in a periodic domain, that the details and types of behavior displayed by vortex dipoles depend strongly on the initial conditions rather than only on the separation distance d0 (as has been previously claimed). It is found, indeed, that the choice of the initial two-vortex profile (i.e., the modulus of the "effective wave function"), strongly affects the vortex trajectories and the time scale of the process: annihilation proceeds more rapidly when low-energy (or "relaxed") initial profiles are imposed. The initial "circular" phase distribution contours, customarily obtained by multiplying an effective wave function for each individual vortex, can be generalized to explicit elliptical forms specified by two parameters; then by "tuning" the elliptical shape at fixed d0, a sharp transition between solitary-wave propagation and annihilation is captured. Thereby, a "phase diagram" for this "AnSol" transition is constructed in the space of ellipticity and separation and various limiting forms of the boundary are discussed.

AB - Quantum vortex dynamics in Bose-Einstein condensates or superfluid helium can be informatively described by the Gross-Pitaevskii (GP) equation. Various approximate analytical formulas for a single stationary vortex are recalled and their shortcomings demonstrated. Significantly more accurate two-point [2/2] and [3/3] Padé approximants for stationary vortex profiles are presented. Two straight, singly quantized, antiparallel vortices, located at a distance d0 apart, form a vortex dipole, which, in the GP model, can either annihilate or propagate indefinitely as a "solitary wave." We show, through calculations performed in a periodic domain, that the details and types of behavior displayed by vortex dipoles depend strongly on the initial conditions rather than only on the separation distance d0 (as has been previously claimed). It is found, indeed, that the choice of the initial two-vortex profile (i.e., the modulus of the "effective wave function"), strongly affects the vortex trajectories and the time scale of the process: annihilation proceeds more rapidly when low-energy (or "relaxed") initial profiles are imposed. The initial "circular" phase distribution contours, customarily obtained by multiplying an effective wave function for each individual vortex, can be generalized to explicit elliptical forms specified by two parameters; then by "tuning" the elliptical shape at fixed d0, a sharp transition between solitary-wave propagation and annihilation is captured. Thereby, a "phase diagram" for this "AnSol" transition is constructed in the space of ellipticity and separation and various limiting forms of the boundary are discussed.

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U2 - 10.1103/PhysRevB.88.134522

DO - 10.1103/PhysRevB.88.134522

M3 - Article

AN - SCOPUS:84887091233

VL - 88

JO - Physical Review B-Condensed Matter

JF - Physical Review B-Condensed Matter

SN - 1098-0121

IS - 13

M1 - 134522

ER -