Dynamics of straight vortex filaments in a Bose–Einstein condensate with the Gaussian density profile

Abstract

The dynamics of interacting quantized vortex filaments in a rotating trapped Bose-Einstein condensate, which is in the Thomas-Fermi regime at zero temperature and described by the Gross-Pitaevskii equation, is considered in the hydrodynamic "anelastic" approximation. In the presence of a smoothly inhomogeneous array of filaments (vortex lattice), a non-canonical Hamiltonian equation of motion is derived for the macroscopically averaged vorticity, with taking into account the spatial non-uniformity of the equilibrium condensate density determined by the trap potential. A minimum of the corresponding Hamiltonian describes a static configuration of deformed vortex lattice against a given density background. The minimum condition is reduced to a vector nonlinear partial differential equation of the second order, for which some approximate and exact solutions are found. It is shown that if the condensate density has an anisotropic Gaussian profile then equation of motion for the averaged vorticity admits solutions in the form of a spatially uniform vector with a nontrivial time dependence. An integral representation is obtained for the matrix Green function determining the non-local Hamiltonian of a system of arbitrary shaped vortex filaments in a condensate with Gaussian density. ... A simple approximate expression for the two-dimensional Green function is suggested at rather arbitrary density profile, and its successful comparison to the exact result in the Gaussian case is done. Approximate equations of motion are derived which describe a long-wave dynamics of interacting vortex filaments in condensates with the density depending on the transverse coordinates only.