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    Pavel Ryabov

    Leningradsky prospect, 49, Moscow, 125993
    Financial University


    Ryabov P. E., Sokolov S. V.
    A completely Liouville integrable Hamiltonian system with two degrees of freedom describing the dynamics of two vortex filaments in a Bose – Einstein condensate enclosed in a cylindrical trap is considered. For the system of two vortices with identical intensities a bifurcation of three Liouville tori into one is detected. Such a bifurcation is found in the integrable case of Goryachev – Chaplygin – Sretensky in rigid body dynamics.
    Keywords: Vortex dynamics, Bose – Einstein condensate, completely integrable Hamiltonian systems, bifurcation diagram of momentum mapping, bifurcations of Liouville tori
    Citation: Ryabov P. E., Sokolov S. V.,  Phase Topology of Two Vortices of Identical Intensities in a Bose – Einstein Condensate, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 1, pp.  59-66
    Ryabov P. E., Biryucheva E.
    The paper presents explicitly the spectral curve and the discriminant set of the integrable case of M. Adler and P. van Moerbeke. For critical points of rank 0 and 1 of the momentum map we explicitly calculate the characteristic values defining their type. An algorithm is proposed for finding the bifurcation diagram from the real part of the discriminant set with the help of critical points of rank 0 and 1. The algorithm works under the condition that the real part of the discriminant set contains the bifurcation diagram.
    Keywords: integrable Hamiltonian systems, spectral curve, discriminant set, bifurcation diagram
    Citation: Ryabov P. E., Biryucheva E.,  The discriminant set and bifurcation diagram of the integrable case of M. Adler and P. van Moerbeke, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 4, pp.  633–650
    Ryabov P. E., Savushkin A. Y.
    Phase topology of the Kowalevski–Sokolov top
    2015, Vol. 11, No. 2, pp.  287-317
    The phase topology of the integrable Hamiltonian system on $e(3)$ found by V. V. Sokolov (2001) and generalizing the Kowalevski case is investigated. The generalization contains, along with a homogeneous potential force field, gyroscopic forces depending on the configurational variables. Relative equilibria are classified, their type is calculated and the character of stability is defined. The Smale diagrams of the case are found and the classification of iso-energy manifolds of the reduced systems with two degrees of freedom is given. The set of critical points of the complete momentum map is represented as a union of critical subsystems; each critical subsystem is a oneparameter family of almost Hamiltonian systems with one degree of freedom. For all critical points we explicitly calculate the characteristic values defining their type. We obtain the equations of the surfaces bearing the bifurcation diagram of the momentum map. We give examples of the existing iso-energy diagrams with a complete description of the corresponding rough topology (of the regular Liouville tori and their bifurcations).
    Keywords: integrable Hamiltonian systems, relative equilibria, iso-energy surfaces, critical subsystems, bifurcation diagrams, rough topology
    Citation: Ryabov P. E., Savushkin A. Y.,  Phase topology of the Kowalevski–Sokolov top, Rus. J. Nonlin. Dyn., 2015, Vol. 11, No. 2, pp.  287-317

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