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Verhoeven, Caroline; Musette, Micheline; Conte, Robert

doi:10.1023/a:1021880025669

Cited by 7 publications

(2 citation statements)

References 18 publications

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“…Note that, in the particular case βγ = 0, i.e. κ 2 1 = κ 2 2 , these coefficients become rational, see [29]. Remark.…”

Section: Equivalent Fourth Order Odesmentioning

confidence: 95%

“…In the 1 : 6 : 1, 1 : 6 : 8 case, the hyperelliptic curve y 2 = P (s) of F-VI (see (28)) reduces in the separated cases βγ = 0 to the hyperelliptic curve of the separating variables. Therefore, F-VI is a good ODE to consider, and the only missing item is to find a Hamiltonian structure of F-VI that is necessarily distinct from (30) and admits a canonical transformation to 1 : 6 : 1, 1 : 6 : 8.…”

Section: Conclusion and Unresolved Problemsmentioning

confidence: 99%

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Completeness of the Cubic and Quartic Henon-Heiles Hamiltonians

2005

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The quartic Hénon-Heiles Hamiltonian passes the Painlevé test for only four sets of values of the constants. Only one of these, identical to the traveling-wave reduction of the Manakov system, has been explicitly integrated (Wojciechowski, 1985), while the other three have not yet been integrated in the general case (α, β, γ) = (0, 0, 0). We integrate them by building a birational transformation to two fourth-order firstdegree equations in the Cosgrove classification of polynomial equations that have the Painlevé property. This transformation involves the stationary reduction of various partial differential equations. The result is the same as for the three cubic Hénon-Heiles Hamiltonians, namely, a general solution that is meromorphic and hyperelliptic with genus two in all four quartic cases. As a consequence, no additional autonomous term can be added to either the cubic or the quartic Hamiltonians without destroying the Painlevé integrability (the completeness property).

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“…Note that, in the particular case βγ = 0, i.e. κ 2 1 = κ 2 2 , these coefficients become rational, see [29]. Remark.…”

Section: Equivalent Fourth Order Odesmentioning

confidence: 95%

Section: Conclusion and Unresolved Problemsmentioning

confidence: 99%