Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-18T12:18:46.743Z Has data issue: false hasContentIssue false
  • English
  • Français

Elliptic solutions and blow-up in an integrable Hénon–Heiles system

Published online by Cambridge University Press:  14 November 2011

J. C. Eilbeck  and
V. Z. Enol'skii
J. C. Eilbeck
Affiliation:
Department of Mathematics, Heriot–Watt University, Riccarton, Edinburgh EH 14 4AS, U.K.
V. Z. Enol'skii
Affiliation:
Department of Theoretical Physics, Institute of Metal Physics, Vernadsky str. 36, Kiev-680, 252142, Ukraine
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider an integrable case of the Henon-Heiles system and use an isomorphism with the two-gap KdV-flow to construct families of real elliptic trajectories which are associated with two-gap elliptic solitons of the KdV equation. Some of these solutions exhibit blow-up in finite time.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 6 , 1994 , pp. 1151 - 1164
Copyright
Copyright © Royal Society of Edinburgh 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Adler, M., and van Moerbeke, P.. Representing the Kowalewski and Henon-Heiles motion as Manakov geodesic flow on S0(4) and a two-dimensional family of Lax pairs. Comm. Math. Phys. 113(1988), 659–700.CrossRefGoogle Scholar
2Airault, H., McKean, H. P..and Moser, J., Rational and elliptic solutions of the KdV equation and a related many-body problem. Comm. Pure Appl. Math. 30 (1977), 94–148.CrossRefGoogle Scholar
3Antonowicz, M., and Rauch-Wojciechowski, S.. Bihamiltonian formulation of the Henon-Heiles system and its multi-dimensional extensions (Preprint, LiTH-MAT-R-91-43, Linkoping University, 1991).Google Scholar
4Bateman, H., and Erdelyi, A., Higher Transcendental Functions, Vol. 2 (New York: McGraw-Hill, 1955).Google Scholar
5Belokolos, E. D., Bobenko, A. I., Enol'skii, V.Z., Its, A. R. and Matveev, V. B.. Algebraic-geometrical Methods in the Theory of Integrable Equations (Berlin: Springer, 1994).Google Scholar
6Belokolos, E. D. and Enol'skii, V. Z.. Verdier elliptic solitons and the Weierstrass reduction theory. Funktsional. Anal, i Prilozhen. (in Russian) 23 (1989), 70–72.Google Scholar
7Belokolos, E. D. and Enol'skii, V. Z.. Reduction of theta functions and elliptic finite-gap potentials. In Elliptic Solitons, ed. Krichever, I. M. (Berlin: Springer, 1992).Google Scholar
8Bountis, T., Segur, H., and Vivaldi, F., Integrable Hamiltonian systems and the Painleve property. Phys. Rev. A 25 (1982), 1257–1264.CrossRefGoogle Scholar
9Chang, Y. F., Tabor, M., and Weiss, J., Analytic structure of the Henon-Heiles Hamiltonian in integrable and nonintegrable regimes. J. Math, Phys. 23 (1982), 531–538.CrossRefGoogle Scholar
10Christiansen, P. L., Eilbeck, J. C., Enol'skii, V. Z. and Gaididey, Ju. B., On ultrasonic Davydov solitons and the Henon-Heiles system. Phys. Lett. A 166 (1992), 129–134.CrossRefGoogle Scholar
11Eilbeck, J. C. and Enol'skii, V.Z.. Elliptic Baker-Akhiezer functions and an application to an integrable dynamical system. J. Math, Phys. 35 (1994) 1192–1201.CrossRefGoogle Scholar
12Enol'skii, V.Z. and Kostov, N. A.. On the geometry of elliptic solitons. In Elliptic Solitons, ed. Krichever, I. M., (Berlin: Springer, 1992).Google Scholar
13Fordy, A. P.. The Henon-Heiles system revisited. Phys. D 52 (1991), 204–210.CrossRefGoogle Scholar
14Gavrilov, L., Bifurcation of invariant manifolds in the generalized Henon-Heiles system. Phys. D 34 (1989), 223–239.CrossRefGoogle Scholar
15Gavrilov, L., and Gaboz, R., Normal modes of an integrable Henon-Heiles system (Preprint, University of Pau, 1989).Google Scholar
16Grammaticos, G., Dorizzi, B., and Padjen, R., Painleve property and integrals of motion for the Henon-Heiles system. Phys. Lett. A 89 (1982), 111–113.CrossRefGoogle Scholar
17Halphen, G. H.. Memoire sur la reduction des equations differentielles lineaires aux formes integrales. Mem. pres. VAcad de Sci. de France 28 (1884), 1–300.Google Scholar
18Henon, M., and Heiles, C., The applicability of the third integral of motion: some numerical experiments. Astrophys. 63 (1964), 73–78.Google Scholar
19Hermite, C., Oeuvres de Charles Hermite, Vol. III (Paris: Gauthier-Villar, 1912).Google Scholar
20Ince, E. L.. Ordinary Differential Equations (New York: Dover, 1956).Google Scholar
21Krazer, A., Lehrbuch der Thetafunktionen (Leipzig: Teubner, 1903).Google Scholar
22Krichever, I. M. ed. Elliptic Solitons (Heidelberg: Springer, 1992).Google Scholar
23Mumford, D., Tata Lectures on Theta, Vol. 1 (Boston: Birkhauser, 1983).CrossRefGoogle Scholar
24Mumford, D., Tata Lectures on Theta, Vol. 2 (Boston: Birkhauser, 1984).Google Scholar
25Novikov, S. P.. The periodic problem for Korteweg de Vries equation. Funktsional. Anal, i Prilozhen. 8(1974), 54–66.Google Scholar
26Rao, N. N. and Kaup, D. J.. A new class of exact solutions for coupled scalar field equations. J. Phys, A: Math. Gen. 24 (1991), L993–L999.CrossRefGoogle Scholar
27Smirnov, A. O.. Elliptic solutions of integrable equations. In Elliptic Solitons, ed. Krichever, I. M. (Berlin: Springer, 1992).Google Scholar
28Treibich, A., and Verdier, J.-L.. Revetements tangentiels et sommes de 4 nombres triangulaires. CR Acad. Sci. Paris 311 (1990), 51–54.Google Scholar
29Treibich, A., and Verdier, J.-L.. Solitons Elliptiques (Boston: Birkhauser, 1991).Google Scholar
30Udry, S., and Martinet, L., Orbital behaviour of transition from the Henon-Heiles to the threeparticle Toda lattice Hamiltonian. Phys. D 44 (1890), 61–74.CrossRefGoogle Scholar
31Verdier, J. L.. New elliptic solitons. In Algebraic Analysis, Vol II Special volume for 60th anniversary of Prof. M. Sato, eds. Kashiwara, M., & Kawai, T., (New York: Academic Press, 1988).Google Scholar
32Wojciechowski, S., Separability of an integrable case of the Henon-Heiles system. Phys. Lett. A 100 (1984), 277–278.CrossRefGoogle Scholar
33Wolfram, S., Mathematica, 2nd edn (New York: Addison-Wesley, 1991).Google Scholar
34Zakharov, V. E., Manakov, S. V., Novikov, S. P. and Pitaevskii, L. P.. Soliton Theory: Inverse Scattering Method (Moscow: Nauka, 1980).Google Scholar