Dynamics of vortex structures and their interaction with each other in many respects determine key aspects of evolution for the ultracold Bose gas cloud with repulsing interaction between atoms. Quantum vortices (topological defects or phase singularities) are rightly associated with the breaking of the superfluidity mode and the transition of the Bose-Einstein condensate (BEC) to the turbulent state. Thus, it is important to make a maximal progress in solving the problem of describing different kinds of motion for interacting vortex formations.
As it was shown in our works [1-3], in the case when the distance between a vortex and an antivortex is substantially smaller than the characteristic scale of the medium inhomogenity the distributions of both the density and the velocities field in vortex pairs are similar to those that take place for the homogeneous flow-free condensate in correspoding solitary solutions of Gross ? Pitayevsky (GP) equation (equivalent to nonlinear Shrodinger equation (NSE) in terms of dimensionless variables). In this case one may introduce a concept of the ?two-dimensional dark quasisolitons?. Such quasisolitons represent the holes of BEC concentration, propagating in the inhomogeneous Bose gas at subsonic velocities with acceleration and, generally speaking, along the curved paths. We have developed the asymptotic theory describing behaviour of two-dimensional quasisolitons structures in a smoothly inhomogeneous resting condensate and taking into account many peculiarities of the Bose gas.
In the present work this theory was essentially modified and generalized to the case of the two-dimensional dark quasisolitons motion in the smoothly inhomogeneous flows of BEC with repulsing interaction between atoms. Using this theory we have succeeded in detailed analizing and explaining peculiarities of scattering for both vortex and vortex-free quasisolitons formations on a single vortex and on condensate flows arising in the process of laminar flowing around barriers which move in the Bose gas at the constant subsonic velocity. The results of the direct numerical simulation performed within the framework of the GP equation are demonstrated a good agreement with the developed theory.

References
[1] L. A. Smirnov and V. A. Mironov, Phys. Rev. A., 2012, 85(5), 053620-1 ? 053620-1.
[2] V. A. Mironov and L. A. Smirnov, JETP Lett., 2012, 95(11), 549 ? 554.
[3] V. A. Mironov and L. A. Smirnov, Physics of Wave Phenomena, 2013, 21(1), 62 ? 67.