Universitetskaya 1, Izhevsk, 426034 Russia
Institute of Computer Science, Udmurt State University
Sokolov S. V., Ryabov P. E.
Bifurcation Analysis of the Dynamics of Two Vortices in a Bose – Einstein Condensate. The Case of Intensities of Opposite Signs
2017, vol. 22, no. 8, pp. 976–995
This paper is concerned with a system two point vortices in a Bose – Einstein condensate enclosed in a trap. The Hamiltonian form of equations of motion is presented and its Liouville integrability is shown. A bifurcation diagram is constructed, analysis of bifurcations of Liouville tori is carried out for the case of opposite-signed vortices, and the types of critical motions are identified.
Ryabov P. E., Oshemkov A. A., Sokolov S. V.
The Integrable Case of Adler – van Moerbeke. Discriminant Set and Bifurcation Diagram
2016, vol. 21, no. 5, pp. 581-592
The Adler – van Moerbeke integrable case of the Euler equations on the Lie algebra $so(4)$ is investigated. For the $L-A$ pair found by Reyman and Semenov-Tian-Shansky for this system, we explicitly present a spectral curve and construct the corresponding discriminant set. The singularities of the Adler – van Moerbeke integrable case and its bifurcation diagram are discussed. We explicitly describe singular points of rank 0, determine their types, and show that the momentum mapping takes them to self-intersection points of the real part of the discriminant set. In particular, the described structure of singularities of the Adler – van Moerbeke integrable case shows that it is topologically different from the other known integrable cases on $so(4)$.
Sokolov S. V., Ramodanov S. M.
Falling Motion of a Circular Cylinder Interacting Dynamically with a Point Vortex
2013, vol. 18, no. 1-2, pp. 184-193
The dynamical behavior of a heavy circular cylinder and a point vortex in an unbounded volume of ideal liquid is considered. The liquid is assumed to be irrotational and at rest at infinity. The circulation about the cylinder is different from zero. The governing equations are Hamiltonian and admit an evident autonomous integral of motion — the horizontal component of the linear momentum. Using the integral we reduce the order and thereby obtain a system with two degrees of freedom. The stability of equilibrium solutions is investigated and some remarkable types of partial solutions of the system are presented.