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    Phase Topology of Two Vortices of Identical Intensities in a Bose – Einstein Condensate

    2019, Vol. 15, no. 1, pp.  59-66

    Author(s): 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

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    [1] Fetter, A. L., “Rotating Trapped Bose – Einstein Condensates”, Rev. Mod. Phys., 81:2 (2009), 647–691  crossref  adsnasa  elib
    [2] Torres, P. J., Kevrekidis, P. G., Frantzeskakis, D. J., Carretero-González, R., Schmelcher, P., and Hall, D. S., “Dynamics of Vortex Dipoles in Confined Bose – Einstein Condensates”, Phys. Lett. A, 375:33 (2011), 3044–3050  crossref  adsnasa
    [3] Navarro, R., Carretero-González, R., Torres, P. J., Kevrekidis, P. G., Frantzeskakis, D. J., Ray, M. W., Altuntaş, E., and Hall, D. S., “Dynamics of a Few Corotating Vortices in Bose – Einstein Condensates”, Phys. Rev. Lett., 110:22 (2013), 225301, 6 pp.  crossref  adsnasa
    [4] Koukouloyannis, V., Voyatzis, G., and Kevrekidis, P. G., “Dynamics of Three Noncorotating Vortices in Bose – Einstein Condensates”, Phys. Rev. E, 89:4 (2014), 042905, 14 pp.  crossref  adsnasa  elib
    [5] Borisov, A. V. and Kilin, A. A., “Stability of Thomson's Configurations of Vortices on a Sphere”, Regul. Chaotic Dyn., 5:2 (2000), 189–200  crossref  mathscinet  zmath
    [6] Kilin, A. A., Borisov, A. V., and Mamaev, I. S., “The Dynamics of Point Vortices inside and outside a Circular Domain”, Basic and Applied Problems of the Theory of Vortices, eds. A. V. Borisov, I. S. Mamaev, M. A. Sokolovskiy, R&C Dynamics, Institute of Computer Science, Izhevsk, 2003, 414–440 (Russian)  mathscinet
    [7] Borisov, A. V., Mamaev, I. S., and Kilin, A. A., “Absolute and Relative Choreographies in the Problem of Point Vortices Moving on a Plane”, Regul. Chaotic Dyn., 9:2 (2004), 101–111  crossref  mathscinet  zmath  adsnasa
    [8] Borisov, A. V., Kilin, A. A., and Mamaev, I. S., “The Dynamics of Vortex Rings: Leapfrogging, Choreographies and the Stability Problem”, Regul. Chaotic Dyn., 18:1–2 (2013), 33–62  mathnet  crossref  mathscinet  zmath  adsnasa  elib
    [9] Borisov, A. V., Ryabov, P. E., and Sokolov, S. V., “Bifurcation Analysis of a Problem on the Motion of a Cylinder and a Point Vortex in an Ideal Fluid”, Math. Notes, 99:5–6 (2016), 834–839  mathnet  crossref  mathscinet  zmath  elib; Mat. Zametki, 99:6 (2016), 848–854 (Russian)  crossref  zmath
    [10] Sokolov, S. V. and Ryabov, P. E., “Bifurcation Analysis of the Dynamics of Two Vortices in a Bose – Einstein Condensate. The Case of Intensities of Opposite Signs”, Regul. Chaotic Dyn., 22:8 (2017), 979–998  mathnet  crossref  mathscinet  adsnasa
    [11] Sokolov, S. V. and Ryabov, P. E., “Bifurcation Diagram of the Two Vortices in a Bose – Einstein Condensate with Intensities of the Same Signs”, Dokl. Math., 97:3 (2018), 286–290  crossref  zmath; Dokl. Akad. Nauk, 480:6 (2018), 652–656 (Russian)  zmath
    [12] Ryabov, P. E., Bifurcation Diagram of One Perturbed Vortex Dynamics Problem, 2018, arXiv: 1811.10512 [nlin.SI]
    [13] Ryabov, P. E., On One Unstable Bifurcation in the Dynamics of Vortex Structure, 2018, arXiv: 1812.03563 [nlin.SI]
    [14] Kharlamov, M. P., Topological Analysis of Integrable Problems of Rigid Body Dynamics, Leningr. Gos. Univ., Leningrad, 1988, 197 pp. (Russian)  mathscinet
    [15] Bolsinov, A. V., Matveev, S. V., and Fomenko, A. T., “Topological Classification of Integrable Hamiltonian Systems with Two Degrees of Freedom. List of Systems of Small Complexity”, Russian Math. Surveys, 45:2 (1990), 59–94  mathnet  crossref  mathscinet  zmath  adsnasa; Uspekhi Mat. Nauk, 45:2(272) (1990), 49–77, 240 (Russian)  mathscinet
    [16] Oshemkov, A. A. and Tuzhilin, M. A., “Integrable Perturbations of Saddle Singularities of Rank $0$ of Integrable Hamiltonian Systems”, Sb. Math., 209:9 (2018), 1351–1375  mathnet  crossref  mathscinet  zmath; Mat. Sb., 209:9 (2018), 102–127 (Russian)  crossref  mathscinet  zmath
    [17] Kharlamov, M. P., “Extensions of the Appelrot Classes for the Generalized Gyrostat in a Double Force Field”, Regul. Chaotic Dyn., 19:2 (2014), 226–244  mathnet  crossref  mathscinet  zmath  adsnasa  elib
    [18] Bolsinov, A. V., Borisov, A. V., and Mamaev, I. S., “Topology and Stability of Integrable Systems”, Russian Math. Surveys, 65:2 (2010), 259–318  mathnet  crossref  mathscinet  adsnasa  elib; Uspekhi Mat. Nauk, 65:2 (2010), 71–132 (Russian)  crossref  mathscinet  zmath
    [19] Ivanov, A. P., “On Singular Points of Equations of Mechanics”, Dokl. Math., 97:2 (2018), 167–169  crossref  mathscinet  zmath; Dokl. Akad. Nauk, 479:5 (2018), 493–496 (Russian)  zmath

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