Equations of Motion of the Vortices in Bose–Einstein Condensates: Influence of Rotation and The Inhomogeneity of Density

Abstract

We deduce equations of motion of quantized vortices in rotating Bose–Einstein condensates in two cases, namely, for a homogeneous condensate in a rigid rotating cylinder and for an inhomogeneous condensate in a rotating magnetic trap in the Thomas–Fermi approximation. The Schrödinger equations for both media are reduced to a convenient dimensionless form. By the method of expansion in a small parameter, we obtain two asymptotic solutions in different space scales. The comparison of the principal terms of these solutions yields the required equations. The equations of motion of the vortices in the homogeneous condensate are reduced to the well-known equations of vortices in the ideal liquid. The inhomogeneity of the medium results in the appearance of additional terms. We deduce the equations of motion of the vortices in the most general case: for any number of vortices, for vessels of different shapes, and both in the presence and in the absence of rotation. It is shown that, in the partial case of motion of a single vortex, the corresponding equations are reduced to the well-known equations of precession of the vortex. We present the plots of motion of several vortices for various initial data.