|
On a theorem of Picard
Author(s):
F.
Gesztesy;
W.
Sticka
Journal:
Proc. Amer. Math. Soc.
126
(1998),
1089-1099.
MSC (1991):
Primary 33E05, 34C25;
Secondary 58F07
Retrieve article in:
PDF
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
Abstract:
We extend Picard's theorem on the existence of elliptic solutions of the second kind of linear homogeneous  -order scalar ordinary differential equations with coefficients being elliptic functions (associated with a common period lattice) to linear homogeneous first-order  systems. In particular, the qualitative Floquet-type structure of fundamental systems of solutions in terms of elliptic and exponential functions, polynomials, and Weierstrass zeta functions of the independent variable is determined. Connections with completely integrable systems are mentioned.
References:
- 1.
- M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972. MR 94b:00012 (reprint of 1972 edition)
- 2.
- N. I. Akhiezer, Elements of the Theory of Elliptic Functions, Amer. Math. Soc., Providence, RI, 1990. MR 91k:33016
- 3.
- E. D. Belokolos, A. I. Bobenko, V. Z. Enol'skii, A. R. Its, and V. B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer, Berlin, 1994.
- 4.
- H. Burkhardt, Elliptische Funktionen, 2nd ed., Verlag von Veit, Leipzig, 1906.
- 5.
- E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Krieger, Malabar, 1985.
- 6.
- L. A. Dickey, Soliton Equations and Hamiltonian Systems, World Scientific, Singapore, 1991. MR 93d:58067
- 7.
- M. V. Fedoryuk, Lamé wave functions in the Jacobi form, J. Diff. Eqs., 23, 1170-1177 (1987). MR 89c:33014
- 8.
- G. Floquet, Sur les équations différentielles linéaires à coefficients doublement périodiques, C. R. Acad. Sci. Paris, 98, 38-39, 82-85 (1884).
- 9.
- G. Floquet, Sur les équations différentielles linéaires à coefficients doublement périodiques, Ann. Sci. Ecole Norm. Sup., 1, 181-238 (1884).
- 10.
- G. Floquet, Addition a un mémorie sur les équations différentielles linéaires, Ann. Sci. Ecole Norm. Sup., 1, 405-408 (1884).
- 11.
- F. R. Gantmacher, Theory of Matrices, Vol. 1, Chelsea, New York, 1960. MR 21:6372c
- 12.
- F. Gesztesy and R. Weikard, Lamé potentials and the stationary (m)KdV hierarchy, Math. Nachr., 176, 73-91 (1995). CMP 96:04
- 13.
- F. Gesztesy and R. Weikard, On Picard potentials, Diff. Int. Eqs., 8, 1453-1476 (1995). MR 96e:34141
- 14.
- F. Gesztesy and R. Weikard, Treibich-Verdier potentials and the stationary (m)KdV hierarchy, Math. Z. 219, 451-476 (1995). MR 96e:14030
- 15.
- F. Gesztesy and R. Weikard, Picard potentials and Hill's equation on a torus, Acta Math., 176, 73-107 (1996). MR 97f:14046
- 16.
- F. Gesztesy and R. Weikard, Toward a characterization of elliptic solutions of hierarchies of soliton equations, Contemp. Math., to appear.
- 17.
- F. Gesztesy and R. Weikard, A characterization of all elliptic solutions of the AKNS hierarchy, Acta Math., to appear.
- 18.
- J. Gray, Linear Differential Equations and Group Theory from Riemann to Poincaré, Birkhäuser, Boston, 1986. MR 89d:01041
- 19.
- G.-H. Halphen, Traité des Fonctions Elliptiques, tome 2, Gauthier-Villars, Paris, 1888.
- 20.
- P. Hartman, Ordinary Differential Equations, 2nd, ed., Birkhäuser, Boston, 1982. MR 83e:34002
- 21.
- C. Hermite, Oeuvres, tome 3, Gauthier-Villars, Paris, 1912.
- 22.
- E. L. Ince, Ordinary Differential Equations, Dover, New York, 1956. MR 6:65f
- 23.
- M. Krause, Theorie der doppeltperiodischen Funktionen einer veränderlichen Grösse, Vol. 2, Teubner, Leipzig, 1897.
- 24.
- G. Mittag-Leffler, Sur les équations différentielles linéaires à coefficients doublement périodiques, C. R. Acad. Sci. Paris, 90, 299-300 (1880).
- 25.
- S. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons, Consultants Bureau, New York, 1984. MR 86k:35142
- 26.
- E. Picard, Sur une généralisation des fonctions périodiques et sur certaines équations différentielles linéaires, C. R. Acad. Sci. Paris, 89, 140-144 (1879).
- 27.
- E. Picard, Sur une classe d'équations différentielles linéaires, C. R. Acad. Sci. Paris, 90, 128-131 (1880).
- 28.
- E. Picard, Sur les équations différentielles linéaires à coefficients doublement périodiques, J. Reine Angew. Math., 90, 281-302 (1881).
- 29.
- E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press, Cambridge, 1996. MR 97k:01072
- 30.
- V. A. Yakubovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients, Vol. 1, Wiley, New York, 1975. MR 51:994
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical Society
with MSC
(1991):
33E05, 34C25,
58F07
Retrieve articles in all Journals with MSC
(1991):
33E05, 34C25,
58F07
Additional Information:
F.
Gesztesy
Affiliation:
Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email:
fritz@math.missouri.edu
W.
Sticka
Affiliation:
Department of Mathematics, University of Missouri, Columbia, Missouri 65211
DOI:
10.1090/S0002-9939-98-04668-1
PII:
S 0002-9939(98)04668-1
Received by editor(s):
September 23, 1996
Additional Notes:
The research was based upon work supported by the National Science Foundation under Grant No. DMS-9623121.
Communicated by:
Hal L. Smith
Copyright of article:
Copyright
1998,
American Mathematical Society
|
Comments: webmaster@ams.org