×

Quasi-periodic solutions of the integrable dynamical systems related to Hill’s equation. (English) Zbl 0691.58022

The author considers the relationship between Hill’s equation and the Garnier-type systems. Finite-gap solutions to the Garnier system and to g-dimensional anisotropic harmonic oscillator in a radial quartic potential are presented. The interconnection between these solutions and solutions to Neumann-type systems is discussed.
Reviewer: Yu.E.Gliklikh

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Dubrovin, B. A., Matveev, V. B., and Novikov, S. P., Russian Math. Surveys 31, 59 (1976). · Zbl 0346.35025
[2] McKean, H. P. and Van Moerbeke, P., Invent. Math. 30, 217 (1975). · Zbl 0319.34024
[3] Marchenko, V. A. and Ostrovski, I. V., Math. Sb. 97, 493 (1975). · Zbl 0343.34016
[4] McKean, H. P. and Trubowitz, E., Comm. Pure Appl. Math 29, 14 (1976).
[5] Moser, J., Various Aspects of Integrable Hamiltonian Systems, in Progress in Mathematics, Vol. 8, Birkhäuser, Boston, 1980, p. 233. · Zbl 0468.58011
[6] Moser, J., Geometry of quadrics and spectral theory in The Chern Symposium, Springer, New York, 1980, p. 147. · Zbl 0455.58018
[7] Veselov, V. P., Funct. Anal. Appl. 14, 48 (1980).
[8] Flaschka, H., Relations between infinite-dimensional and finite-dimensional isospectral equations, in Non-linear Integrable Systems-Classical Theory and Quantum Theory, World Scientific, Singapore, 1983, p. 221. · Zbl 0551.58017
[9] Flaschka, H., Tohoku Math. J. 36, 407 (1984). · Zbl 0582.35102
[10] Schilling, R. J., Bull. Amer. Math. Soc. 14, 287 1986); Generalizations of the Neumann system ? A curve theoretical approach, Comm. Pure Appl. Math. 40, 455 (1987). · Zbl 0591.58016
[11] Dubrovin, B. A., Russian Math. Surveys 36, 11 (1981). · Zbl 0549.58038
[12] Krichever, I. M., Funct. Anal. Appl. 11, 12 (1977). · Zbl 0368.35022
[13] Cherednik, I. V., Funct. Anal. Appl. 12, 195 (1978). · Zbl 0404.35030
[14] Garnier, R., Circolo Mat. Palermo 43, 155 (1919). · JFM 47.0404.01
[15] Choodnovsky, D. V., and Choodnovsky, G. V., Lett. Nuovo Cim. 22, 47 (1978).
[16] Wojciechowski, S., Phys. Scripta 31, 433 (1985). · Zbl 1063.70521
[17] Mumford, D., Tata Lectures on Theta II, Progress in Mathematics, Vol. 43, Birkhäuser, Boston, 1984. · Zbl 0549.14014
[18] Horozov, E. I., Dokl. Bulg. Acad. Sci 37, 145 (1984).
[19] McKean, H. P., Comm. Pure Appl. Math. 38, 669 (1985). · Zbl 0591.34025
[20] Levitan, B. M., Inverse Sturm-Liouville Problems, Nauka, Moscow, 1984 (in Russian).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.