Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
integrable systems; curves; abelian varieties
integrable systems; curves; abelian varieties
Summary:
In this paper, a finite dimensional algebraic completely integrable system is considered. We show that the intersection of levels of integrals completes into an abelian surface (a two dimensional complex algebraic torus) of polarization $\left( 2,8\right) $ and that the flow of the system can be linearized on it.
References:
[1] Adler M., van Moerbeke P.: The algebraic complete integrability of geodesic flow on $SO(4)$. Invent. Math. 67 (1982), 297–331. MR 0665159
[2] Arbarello E., Cornalba M., Griffiths P. A., Harris J.: Geometry of algebraic curves I. Springer-Verlag, 1994.
[3] Arnold V. I.: Mathematical methods in classical mechanics. Springer-Verlag, Berlin-Heidelberg-New York, 1978. MR 0690288
[4] Belokolos E. D., Bobenko A. I., Enolskii V. Z., Its A. R., Matveev V. B.: Algebro-Geometric approach to nonlinear integrable equations. Springer-Verlag, 1994.
[5] Christiansen P. L., Eilbeck J. C., Enolskii V. Z., Kostov N. A.: Quasi-periodic solutions of the coupled nonlinear Schrödinger equations. Proc. Roy. Soc. London Ser. A 451 (1995), 685–700. MR 1369055
[6] Griffiths P. A., Harris J.: Principles of algebraic geometry. Wiley-Interscience, 1978. MR 0507725 | Zbl 0408.14001
[7] Haine L.: Geodesic flow on $SO(4)$ and Abelian surfaces. Math. Ann. 263 (1983), 435–472. MR 0707241 | Zbl 0521.58042
[8] Lesfari A.: Une approche systématique à la résolution du corps solide de Kowalewski. C. R. Acad. Sc. Paris, série I, t. 302 (1986), 347–350. MR 0837502
[9] Lesfari A.: Abelian surfaces and Kowalewski’s top. Ann. Scient. École Norm. Sup. 4, 21 (1988), 193–223. MR 0956766 | Zbl 0667.58019
[10] Lesfari A.: On affine surface that can be completed by a smooth curve. Results Math. 35 (1999), 107–118. MR 1678068 | Zbl 0947.14022
[11] Lesfari A.: Une méthode de compactification de variétés liées aux systèmes dynamiques. Les cahiers de la recherche, Rectorat-Université Hassan II-Aïn Chock, Casablanca, Maroc, Vol. I, No. 1, (1999), 147–157.
[12] Lesfari A.: Geodesic flow on $SO(4)$, Kac-Moody Lie algebra and singularities in the complex t-plane. Publ. Mat. 43 (1999), 261–279. MR 1697525
[13] Lesfari A.: Completely integrable systems: Jacobi’s heritage. J. Geom. Phys. 31 (1999), 265–286. MR 1711527 | Zbl 0937.37046
[14] Lesfari A.: The problem of the motion of a solid in an ideal fluid. Integration of the Clebsch’s case. Nonlinear Differential Equations Appl. 8 (2001), 1–13. MR 1828945 | Zbl 0982.35085
[15] Lesfari A.: The generalized Hénon-Heiles system, Abelian surfaces and algebraic complete integrability. Rep. Math. Phys. 47 (2001), 9–20. MR 1823005 | Zbl 1054.37038
[16] Lesfari A.: A new class of integrable systems. Arch. Math. 77 (2001), 347–353. MR 1853551 | Zbl 0996.70014
[17] Lesfari A.: The Hénon-Heiles system via the Kowalewski-Painlevé analysis. Int. J. Theor. Phys. Group Theory Nonlinear Opt. 9, N${{}^{\circ }}$4 (2003), 305–330. MR 2128205
[18] Lesfari A.: Le théorème d’Arnold-Liouville et ses conséquences. Elem. Math. 58, Issue 1 (2003), 6–20. MR 1961831 | Zbl 1112.37043
[19] Lesfari A.: Le système différentiel de Hénon-Heiles et les variétés Prym. Pacific J. Math. 212, No. 1 (2003), 125–132. MR 2016973
[20] Mumford D.: Tata lectures on theta II. Progr. Math., Birkhaüser, Boston, 1982.
[21] Mumford D.: On the equations defining Abelian varieties. Invent. Math. 1 (1966), 287–354. MR 0204427 | Zbl 0219.14024
[22] Perelomov A. M.: Integrable systems of classical mechanics and Lie algebras. Birkhäuser, 1990. MR 1048350 | Zbl 0717.70003
Search
Partner of
