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Within the framework of the Gross–Pitaevskii model, we deduce a system of equations that describes the motion of quantized vortices in Bose–Einstein condensates. We consider a strongly anisotropic magnetic trap whose trapping potential is much higher in the direction z than in the transverse direction. Under the action of this potential, the condensate takes the form of a plane disk. The transition to the two-dimensional case enables us to apply the method of matched asymptotic expansions and obtain the equations of vortex motion in the explicit form. We take into account the rotation of the condensate as a whole and the effect of dissipative processes as a result of which the system of vortices comes to the equilibrium state. Some examples of the vortex motion are presented for different values of the external parameters. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]