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

  • J. C. Eilbeck (a1) and V. Z. Enol'skii (a2)


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.



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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), 659700.
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), 94148.
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).
4Bateman, H., and Erdelyi, A., Higher Transcendental Functions, Vol. 2 (New York: McGraw-Hill, 1955).
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).
6Belokolos, E. D. and Enol'skii, V. Z.. Verdier elliptic solitons and the Weierstrass reduction theory. Funktsional. Anal, i Prilozhen. (in Russian) 23 (1989), 7072.
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).
8Bountis, T., Segur, H., and Vivaldi, F., Integrable Hamiltonian systems and the Painleve property. Phys. Rev. A 25 (1982), 12571264.
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), 531538.
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), 129134.
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) 11921201.
12Enol'skii, V.Z. and Kostov, N. A.. On the geometry of elliptic solitons. In Elliptic Solitons, ed. Krichever, I. M., (Berlin: Springer, 1992).
13Fordy, A. P.. The Henon-Heiles system revisited. Phys. D 52 (1991), 204210.
14Gavrilov, L., Bifurcation of invariant manifolds in the generalized Henon-Heiles system. Phys. D 34 (1989), 223239.
15Gavrilov, L., and Gaboz, R., Normal modes of an integrable Henon-Heiles system (Preprint, University of Pau, 1989).
16Grammaticos, G., Dorizzi, B., and Padjen, R., Painleve property and integrals of motion for the Henon-Heiles system. Phys. Lett. A 89 (1982), 111113.
17Halphen, G. H.. Memoire sur la reduction des equations differentielles lineaires aux formes integrales. Mem. pres. VAcad de Sci. de France 28 (1884), 1300.
18Henon, M., and Heiles, C., The applicability of the third integral of motion: some numerical experiments. Astrophys. 63 (1964), 7378.
19Hermite, C., Oeuvres de Charles Hermite, Vol. III (Paris: Gauthier-Villar, 1912).
20Ince, E. L.. Ordinary Differential Equations (New York: Dover, 1956).
21Krazer, A., Lehrbuch der Thetafunktionen (Leipzig: Teubner, 1903).
22Krichever, I. M. ed. Elliptic Solitons (Heidelberg: Springer, 1992).
23Mumford, D., Tata Lectures on Theta, Vol. 1 (Boston: Birkhauser, 1983).
24Mumford, D., Tata Lectures on Theta, Vol. 2 (Boston: Birkhauser, 1984).
25Novikov, S. P.. The periodic problem for Korteweg de Vries equation. Funktsional. Anal, i Prilozhen. 8(1974), 5466.
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.
27Smirnov, A. O.. Elliptic solutions of integrable equations. In Elliptic Solitons, ed. Krichever, I. M. (Berlin: Springer, 1992).
28Treibich, A., and Verdier, J.-L.. Revetements tangentiels et sommes de 4 nombres triangulaires. CR Acad. Sci. Paris 311 (1990), 5154.
29Treibich, A., and Verdier, J.-L.. Solitons Elliptiques (Boston: Birkhauser, 1991).
30Udry, S., and Martinet, L., Orbital behaviour of transition from the Henon-Heiles to the threeparticle Toda lattice Hamiltonian. Phys. D 44 (1890), 6174.
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).
32Wojciechowski, S., Separability of an integrable case of the Henon-Heiles system. Phys. Lett. A 100 (1984), 277278.
33Wolfram, S., Mathematica, 2nd edn (New York: Addison-Wesley, 1991).
34Zakharov, V. E., Manakov, S. V., Novikov, S. P. and Pitaevskii, L. P.. Soliton Theory: Inverse Scattering Method (Moscow: Nauka, 1980).


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