Abstract
The generalized Hénon–Heiles system is considered. New special solutions for two nonintegrable cases are obtained using the Painlevé test. The solutions have the form of the Laurent series depending on three parameters. One parameter determines the singularity-point location, and the other two parameters determine the coefficients in the Laurent series. For certain values of these two parameters, the series becomes the Laurent series for the known exact solutions. It is established that such solutions do not exist in other nonintegrable cases.
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REFERENCES
S. Kowalevski, Acta Math., 12, 177 (1889); 14, 81 (1890).
V. V. Golubev, Lecture Notes on Integrating Equations of Motion of a Massive Solid Body Near Fixed Point [in Russian], Gostekhizdat, Moscow (1953); Reprinted by Regul. Chaotic Dyn., Moscow (2002).
A. Goriely, Regul.Chaotic Dyn., 5, 1 (2000).
V. V. Golubev, Lecture Notes on the Analytical Theory of Differential Equations [in Russian], Gostekhizdat, Moscow (1950).
E. Hille, Ordinary Differential Equations in the Complex Domain, Wiley, New York (1976).
P. Painlevé, Le¸cons sur la théorie analytique des Équations Différentelles (Le¸cons de Stockholm, 1895), Hermann, Paris (1896); Reprinted in:, Oeuvres de P.Painlevé, Vol. 1, Editions du CNRS, Paris (1973); Bull Soc.Math.France, 28, 201 (1900); Acta Math., 25, 1 (1902).
T. Bountis, H. Segur, and F. Vivaldi, Phys.Rev.A, 25, 1257 (1982).
A. Ramani, B. Grammaticos, and T. Bountis, Phys.Rep., 180, 159 (1989).
M. Tabor, Chaos and Integrability in Nonlinear Dynamics: An Introduction, Wiley, New York (1989).
N. Ercolani and E. D. Siggia, Phys.Lett.A, 119, 112 (1986); Phys.D, 34, 303 (1989).
H. Yoshida, Celest.Mech., 31, 363, 381 (1983).
H. Yoshida, Phys.D, 128, 53 (1999).
V. V. Kozlov, Symmetry, Topology, and Resonances in Hamiltonian Mechanics [in Russian], Udmurtia State University, Izhevsk (1995); English transl., Springer, Berlin (1995).
M. J. Ablowitz, A. Ramani, and H. Segur, Lett.Nuovo Cimento, 23, 333 (1978); J.Math.Phys., 21, 715, 1006 (1980).
J. Weiss, M. Tabor, and G. Carnevale, J.Math.Phys., 24, 522 (1983).
J. Weiss, J.Math.Phys., 24, 1405 (1983).
A. C. Newell, M. Tabor, and Y. B. Zeng, Phys.D, 29, 1 (1987).
R. Conte, A. P. Fordy, and A. Pickering, Phys.D, 69, 33 (1993).
R. Conte, ed., The Painlevé Property: One Century Later (CRM Ser. Math. Phys.), Springer, Berlin (1999).
R. Conte, “Exact solutions of nonlinear partial differential equations by singularity analysis,” nlin.SI/0009024 (2000).
J. Weiss, Phys.Lett.A, 102, 329 (1984); 105, 387 (1984).
R. Sahadevan, Theor.Math.Phys., 99, 776 (1994).
J. Springael, R. Conte, and M. Musette, Regul.Chaotic Dyn., 3, No. 1, 3 (1998); solv-int/9804008 (1998).
G. Contopoulos, Z.Astrophys., 49, 273 (1960); Astron.J., 68, 1 (1963).
M. Hénon and C. Heiles, Astron.J., 69, 73 (1964).
Y. F. Chang, M. Tabor, J. Weiss, and G. Corliss, Phys.Lett.A, 85, 211 (1981).
Y. F. Chang, M. Tabor, and J. Weiss, J.Math.Phys., 23, 531 (1982).
S. Yu. Vernov, “The Painlevé analysis and special solutions for nonintegrable systems,” math-ph/0203003 (2002).
D. L. Rod, J.Differential Equations, 14, 129 (1973).
J. Podolský and K. Veselý, Phys.Rev.D, 58, 081501 (1998).
F. Kokubun, Phys.Rev.D, 57, 2610 (1998).
Y. Guo and C. Grotta Ragazza, Comm.Pure Appl.Math., 49, 1145 (1996).
G. Tondo, Theor.Math.Phys., 99, 796 (1994).
F. Kokubun, Phys.Lett.A, 245, 358 (1998).
R. Conte, M. Musette, and C. Verhoeven, J.Math.Phys., 43, 1906 (2002); nlin.SI/0112030 (2001).
A. Erdélyi et al., eds., Higher Transcendental Functions (Based on notes left by H. Bateman), Vol. 3, McGraw-Hill, New York (1955).
E. I. Timoshkova, Astron.Rep., 43, 406 (1999).
A. C. Hearn, REDUCE User's Manual Vers.3.6, RAND Publ., Santa Monica, Calif. (1995); http://www.unikoeln. de/REDUCE/3.6/doc/reduce/.
V. F. Edneral, A. P. Kryukov, and A. Ya. Rodionov, REDUCE: Programming Language for Analytical Calculations [in Russian], Moscow State University, Moscow (1989).
V. A. Murav'ev and D. E. Burlankov, MATHEMATICA [in Russian], Regul. Chaotic Dyn., Moscow (2000).
S. Melkonian, J.Nonlinear Math.Phys., 6, 139 (1999); math.DS/9904186 (1999).
S. Melkonian and A. Zypchen, Nonlinearity, 8, 1143 (1995).
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Vernov, S.Y. Constructing Solutions for the Generalized Hénon–Heiles System Through the Painlevé Test. Theoretical and Mathematical Physics 135, 792–801 (2003). https://doi.org/10.1023/A:1024074702960
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DOI: https://doi.org/10.1023/A:1024074702960