SIGMA 2 (2006), 064, 4 pages      nlin.SI/0408027      https://doi.org/10.3842/SIGMA.2006.064

On a 'Mysterious' Case of a Quadratic Hamiltonian

Sergei Sakovich
Institute of Physics, National Academy of Sciences, 220072 Minsk, Belarus

Received June 02, 2006, in final form July 18, 2006; Published online July 28, 2006

Abstract
We show that one of the five cases of a quadratic Hamiltonian, which were recently selected by Sokolov and Wolf who used the Kovalevskaya-Lyapunov test, fails to pass the Painlevé test for integrability.

Key words: Hamiltonian system; nonintegrability; singularity analysis.

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References

1. Sokolov V.V., Wolf T., Integrable quadratic classical Hamiltonians on so(4) and so(3,1), J. Phys. A: Math. Gen., 2006, V.39, 1915-1926, nlin.SI/0405066.
2. Ablowitz M.J., Ramani A., Segur H., A connection between nonlinear evolution equations and ordinary dif ferential equations of P-type. I, J. Math. Phys., 1980, V.21, 715-721.
3. Ramani A., Grammaticos B., Bountis T., The Painlevé property and singularity analysis of integrable and non-integrable systems, Phys. Rep., 1989, V.180, 159-245.
4. Tsiganov A.V., Goremykin O.V., Integrable systems on so(4) related with XXX spin chains with boundaries, J. Phys. A: Math. Gen., 2004, V.37, 4843-4849, nlin.SI/0310049.
5. Sokolov V.V., On a class of quadratic Hamiltonians on so(4), Dokl. Akad. Nauk, 2004, V.394, 602-605 (in Russian).
6. Ramani A., Dorizzi B., Grammaticos B., Painlevé conjecture revisited, Phys. Rev. Lett., 1982, V.49, 1539-1541.
7. Grammaticos B., Dorizzi B., Ramani A., Integrability of Hamiltonians with third- and fourth-degree polynomial potentials, J. Math. Phys., 1983, V.24, 2289-2295.
8. Ablowitz M.J., Clarkson P.A., Solitons, nonlinear evolution equations and inverse scattering, Cambridge, Cambridge University Press, 1991.
9. Sakovich S.Yu., Tsuchida T., Symmetrically coupled higher-order nonlinear Schrödinger equations: singularity analysis and integrability, J. Phys. A: Math. Gen., 2000, V.33, 7217-7226, nlin.SI/0006004.
10. Sakovich S.Yu., Tsuchida T., Coupled higher-order nonlinear Schrödinger equations: a new integrable case via the singularity analysis, nlin.SI/0002023.