Open Access
Article
Normally generated line bundle and laurent series solutions of nonlinear differential equations
*A. LesfariCorresponding authorlesfariahmed@yahoo.frDepartment of Mathematics, Faculty of Sciences, University of Chouaib Doukkali, B.P. 20, El Jadida,B.P. 20, El-Jadida, MoroccoView full profile →
* Corresponding author · click or hover a name for details
- Received:
- 08 Feb 2017
- Accepted:
- 24 Sep 2017
- Published Online:
- 03 Jul 2018
- Article type:
- Article
- Language:
- EN
- Article no.:
- 1726037X.2018.1437941
- Pages:
- 151–171
Abstract
The aim of this paper is to demonstrate the rich interaction between the properties of dynamical systems, the geometry of its asymptotic solutions, and the theory of Abelian varieties. We are going to illustrate as well that the nature of many methods for finding solutions of some dynamical systems is determined by the Laurent series for solutions of nonlinear differential equations. We apply the methods to the Kowalewski’top a solid body rotating about a fixed point, the Kirchhoff’s equations of motion of a solid in an ideal fluid and the Ramani-Dorizzi-Grammaticos (RDG) series of integrable potentials.
Keywords
Subject Classifications
34M4534M0511G1014H4014K0214H70
References
M.Adler and P.van Moerbeke, The complex geometry of the Kowalewski-Painlevé analysis , Invent. Math ., 97 , 3 - 51 ( 1989 ). doi: 10.1007/BF01850654 M.Adler, P.van Moerbeke and P., Vanhaecke, Algebraic integrability, Painlev geometry and Lie algebras, A series of modern surveys in mathematics , Volume 47 . Springer-Verlag , 2004 . V.I.Arnold, Mathematical methods in classical mechanics , Springer-Verlag , Berlin-Heidelberg-New York, 1978 . E.Arbarello, M.Cornalba, P.A., Griffiths and J.Harris, Geometry of algebraic curves I , Springer-Verlag , 1994 . CH.Birkenhake and H.Lange, Complex abelian varieties . Springer-Verlag , 1992 . A.Clebsch, Der Bewegung eines starren Körpers in einen Flüssigkeit , Math. Ann ., 3 , 238 - 268 ( 1871 ). doi: 10.1007/BF01443985 B.A.Dubrovin, Theta functions and non-linear equations, Russian Math . Surveys , 36 : 2 , 11 - 92 ( 1981 ). P.A.Griffiths and J.Harris, Principles of algebraic geometry , Wiley-Interscience , New York, 1978 . L.Haine, Geodesic flow on SO(4) and Abelian surfaces , Math. Ann ., 263 : 435 - 472 ( 1983 ). doi: 10.1007/BF01457053 R.Hartshorne, Algebraic geometry . Springer-Verlag , 1977 . doi: 10.1007/978-1-4757-3849-0 J.Hietarinta, Direct methods for the search of the second invariant , Phys. Rep ., 147 , 87154 ( 1987 ). S.Koizumi, Theta relations and projective normality of abelian varieties , Am. J. of Math ., 98 , 865 - 889 (1976) . doi: 10.2307/2374034 S.Kowalewski, Sur le problème de la rotation d’un corps solide autour d’un point fixe , Acta Math ., 12 : 177 - 232 ( 1989 ). doi: 10.1007/BF02592182 R.Lazarsfeld, Projectivité normale des surfaces abéliennes. Rédigé par Debarre , Prépublication , 14 , Europroj-C.I.M.P.A., Nice ( 1990 ). A.Lesfari, Abelian surfaces and Kowalewski’s top , Ann. Scient. Éc. Norm. Sup., Paris, 4 e série , t.21 , 193 - 223 ( 1988 ). A.Lesfari, Etude des solutions méromorphes d’équations différentielles , Ren. Semin. Mat. Univ. Politec. Torino , Vol. 65 , 4 , 451 - 464 ( 2007 ). A.Lesfari, Algebraic integrability : the Adler-van Moerbeke approach , Regul. Chaotic Dyn ., Vol. 16 , Nos. 3-4 , pp. 187 - 209 ( 2011 ). doi: 10.1134/S1560354711030026 A.Lesfari, The Hénon-Heiles system as part of an integrable system , Journal of Advanced Research in Dynamical and Control Systems , Vol. 6 , Issue 3 , 24 - 31 ( 2014 ). A.Lesfari, Introduction à la géométrie algébrique complexe . Hermann , Paris, 2015 . B.G.Moishezon, On n-dimensional compact varieties with n algebraically independent mero- morphic functions, I, II and III , Izv. Akad. Nauk SSSR Ser. Mat ., 30 : 133 - 174 , 345-386, 621-656. English translation. AMS Translation Ser. 2, 63 51-177 ( 1966 ). D.Mumford, On the equations defining abelian varieties I, II , III. Invent. Math. 1 , 287 - 354 ( 1966 ), Invent. Math., 3, 75-135 (1967). Invent. Math., 3, 215-244 (1967) . doi: 10.1007/BF01389737 D.Mumford, Varieties defined by quadratic equations, in: “Questions on Algebraic Varieties” CIME , Roma, 29 - 94 ( 1970 ). D.Mumford, Algebraic geometry I: complex projective varieties . Springer-Verlag , 1975 . A. B.Ramani, B.Dorizzi and B.Grammaticos, Painlevé conjecture revisited , Phys. Rev. Lett. , 49 , 1539 – 1541 ( 1982 ). doi: 10.1103/PhysRevLett.49.1539 P.van Moerbeke, Introduction to algebraic integrable systems and their Painlev analysis. Theta functions-Bowdoin 1987, Part 1 (Brunswick, ME, 1987), 107-131, Proc. Sympos. Pure Math. 49, Part 1, Amer. Math. Soc., Providence, RI, 1989 . P.Vanhaecke, Integrable systems in the realm of algebraic geometry , Lecture Notes in Math., 1638 , Springer-Verlag , Second edition 2001 .
Views: 14Downloads: 4Citations: 2
