Propagating and annihilating vortex dipoles in the Gross-Pitaevskii equation

TL;DR

This paper investigates vortex dipole dynamics in the Gross-Pitaevskii equation, revealing that initial conditions significantly influence whether vortices annihilate or propagate as solitary waves, and introduces a phase diagram for this transition.

Contribution

It provides more accurate stationary vortex profiles using advanced Pade' approximants and demonstrates the impact of initial conditions on vortex dipole behavior in the GP model.

Findings

Initial vortex profiles strongly affect dipole dynamics.

A phase diagram for annihilation and propagation transition is constructed.

Elliptical phase distributions can tune vortex behavior.

Abstract

Quantum vortex dynamics in Bose-Einstein condensates or superfluid helium can be informatively described by the Gross-Pitaevskii (GP) equation. Various approximate analytical formulae for a single stationary vortex are recalled and their shortcomings demonstrated. Significantly more accurate two-point [2/2] and [3/3] Pade' approximants for stationary vortex profiles are presented. Two straight, singly quantized, antiparallel vortices, located at a distance d 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 (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, 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.