Abstract:
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 $so(4)$.
This work is partially supported by the grants of RFBR No. 14–01–00119, 16–01–00170, 16–01–00809, and 16–01–00378, common grant of RFBR and Volgograd Region Authorities No. 15–41–02049, and the grant of the President of the Russian Federation for State Support of Leading Scientific Schools No. 7962.2016.1.
Citation:
Pavel E. Ryabov, Andrej A. Oshemkov, Sergei V. Sokolov, “The Integrable Case of Adler – van Moerbeke. Discriminant Set and Bifurcation Diagram”, Regul. Chaotic Dyn., 21:5 (2016), 581–592
\Bibitem{RyaOshSok16}
\by Pavel E. Ryabov, Andrej A. Oshemkov, Sergei V. Sokolov
\paper The Integrable Case of Adler – van Moerbeke. Discriminant Set and Bifurcation Diagram
\jour Regul. Chaotic Dyn.
\yr 2016
\vol 21
\issue 5
\pages 581--592
\mathnet{http://mi.mathnet.ru/rcd211}
\crossref{https://doi.org/10.1134/S1560354716050087}
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Linking options:
https://www.mathnet.ru/eng/rcd211
https://www.mathnet.ru/eng/rcd/v21/i5/p581
This publication is cited in the following 8 articles:
Sergei V. Sokolov, Pavel E. Ryabov, Sergei M. Ramodanov, INTERNATIONAL CONFERENCE OF COMPUTATIONAL METHODS IN SCIENCES AND ENGINEERING ICCMSE 2022, 3030, INTERNATIONAL CONFERENCE OF COMPUTATIONAL METHODS IN SCIENCES AND ENGINEERING ICCMSE 2022, 2024, 080001
Alexey A. Kireenkov, INTERNATIONAL CONFERENCE OF COMPUTATIONAL METHODS IN SCIENCES AND ENGINEERING ICCMSE 2021, 2611, INTERNATIONAL CONFERENCE OF COMPUTATIONAL METHODS IN SCIENCES AND ENGINEERING ICCMSE 2021, 2022, 100001
Alexey A. Kireenkov, INTERNATIONAL CONFERENCE OF COMPUTATIONAL METHODS IN SCIENCES AND ENGINEERING ICCMSE 2021, 2611, INTERNATIONAL CONFERENCE OF COMPUTATIONAL METHODS IN SCIENCES AND ENGINEERING ICCMSE 2021, 2022, 100002
Sergei V. Sokolov, Sergei M. Ramodanov, INTERNATIONAL CONFERENCE OF COMPUTATIONAL METHODS IN SCIENCES AND ENGINEERING ICCMSE 2021, 2611, INTERNATIONAL CONFERENCE OF COMPUTATIONAL METHODS IN SCIENCES AND ENGINEERING ICCMSE 2021, 2022, 100007
A. V. Borisov, P. E. Ryabov, S. V. Sokolov, “On the existence of focus singularities in one model of a Lagrange top with a vibrating suspension point”, Dokl. Math., 102:3 (2020), 468–471
S. V. Sokolov, P. E. Ryabov, “Bifurcation diagram of the two vortices in a Bose–Einstein condensate with intensities of the same signs”, Dokl. Math., 97:3 (2018), 286–290
A. A. Oshemkov, P. E. Ryabov, S. V. Sokolov, “Explicit determination of certain periodic motions of a generalized two-field gyrostat”, Russ. J. Math. Phys., 24:4 (2017), 517–525
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