Equilibrium and oscillations in a turbulent Bose-Einstein condensate

A general approach to equilibrium and oscillations in a turbulent Bose-Einstein condensate, valid in the mean-field approximation, is proposed. Two different wave equations, describing the quasi-static equilibrium and the elementary excitations of the medium are derived. In particular, it is shown that a modified Thomas-Fermi equilibrium can be defined in the presence of an arbitrary spectrum of oscillations. The oscillating modes are also determined using Bogoliubov-de Gennes (BdG) equations, valid in different geometrical configurations, such as infinite, cylindrical and toroidal. The case of a non-homogenous density is considered, which includes the modified equilibrium profiles. A general form of dispersion relation for the elementary excitations is derived. Twisted phonons, carrying a finite amount of angular momentum, are shown to be a particular class of BdG modes. They were previously derived from fluid equations. Nonlinear analysis also shows that envelope dark solitons can exist, and correspond to a rarefaction of twisted phonons.