Dynamics of a few vortices in an ideal fluid and in a Bose-Einstein condensate

Completely Liouville integrable Hamiltonian system with two degrees of freedom is considered. This Hamiltonian system describes the dynamics of two vortex filaments in a Bose-Einstein condensate enclosed in a cylindrical trap and dynamics of two vortices in the ideal fluid contained in a domain with the form of a circular cylinder, generatrix of which is parallel to vortices. An explicit reduction to a system with one degree of freedom for the case of arbitrary signs of the vortex intensities is derived from the original system under consideration. Along with the fact that in the case of two vortices with intensities of opposite signs, a bifurcation diagram was obtained for a generalized two-parameter integrable family, similar to that obtained earlier for the case of a Bose-Einstein condensate, a new bifurcation diagram was found that had not been encountered before. Various types of vortex dynamics corresponding to points inside chambers of bifurcation diagrams are presented.