Integration of a generalized Hénon–Heiles Hamiltonian

Verhoeven, Caroline; Musette, Micheline; Conte, Robert

doi:10.1063/1.1456948

Cited by 23 publications

(20 citation statements)

References 17 publications

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“…The separation of variables for the original three integrable cases of the Hénon-Heiles system (with only cubic and harmonic terms in the potential) was obtained in [11] (see also [10]). The generalized integrable cases (i) and (iii), which Fordy had shown to be reductions of the Sawada-Kotera and KaupKupershmidt equations respectively [19], were only separated quite recently [42]. As already mentioned, the Hamiltonian (4.9) that we consider here is a perturbation of the integrable case (ii) of the Hénon-Heiles system, and so the separation of variables is straightforward.…”

Section: Travelling Waves and Perturbed Hénon-heilesmentioning

confidence: 99%

An integrable hierarchy with a perturbed Hénon–Heiles system

2006

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We consider an integrable scalar partial differential equation (PDE) that is second order in time. By rewriting it as a system and applying the Wahlquist-Estabrook prolongation algebra method, we obtain the zero curvature representation of the equation, which leads to a Lax representation in terms of an energy-dependent Schrödinger spectral problem of the type studied by Antonowicz and Fordy. The solutions of this PDE system, and of its associated hierarchy of commuting flows, display weak Painlevé behaviour, i.e. they have algebraic branching. By considering the travelling wave solutions of the next flow in the hierarchy, we find an integrable perturbation of the case (ii) Hénon-Heiles system which has the weak Painlevé property. We perform separation of variables for this generalized Hénon-Heiles system, and describe the corresponding solutions of the PDE.

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Section: Travelling Waves and Perturbed Hénon-heilesmentioning

confidence: 99%

An integrable hierarchy with a perturbed Hénon–Heiles system

2006

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“…The general solutions of the Hénon-Heiles system are known only in integrable cases [37,38,11], in other cases not only four-, but even three-parameter exact solutions have yet to be found. The generalized Hénon-Heiles system has attracted enormous attention over the years and has used as a model in astronomy [30] and in gravitation [25,32].…”

Section: The Hénon-heiles Systemmentioning

confidence: 99%

Construction of Special Solutions for Nonintegrable Systems

Vernov¹

2006

JNMP

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The Painlevé test is very useful to construct not only the Laurent series solutions of systems of nonlinear ordinary differential equations but also the elliptic and trigonometric ones. The standard methods for constructing the elliptic solutions consist of two independent steps: transformation of a nonlinear polynomial differential equation into a nonlinear algebraic system and a search for solutions of the obtained system. It has been demonstrated by the example of the generalized Hénon-Heiles system that the use of the Laurent series solutions of the initial differential equation assists to solve the obtained algebraic system. This procedure has been automatized and generalized on some type of multivalued solutions. To find solutions of the initial differential equation in the form of the Laurent or Puiseux series we use the Painlevé test. This test can also assist to solve the inverse problem: to find the form of a polynomial potential, which corresponds to the required type of solutions. We consider the five-dimensional gravitational model with a scalar field to demonstrate this.

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“…Finally the Hamilton's equations in the variables (Q 1 ,Q 2 ,P 1 ,P 2 ) are identified [46] to a hyperelliptic system of the canonical form (7.8),…”

Section: )mentioning

confidence: 99%

Explicit integration of the Hénon-Heiles Hamiltonians 1

Conte¹,

Musette²,

Verhoeven³

2005

JNMP

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We consider the cubic and quartic Hénon-Heiles Hamiltonians with additional inverse square terms, which pass the Painlevé test for only seven sets of coefficients. For all the not yet integrated cases we prove the singlevaluedness of the general solution. The seven Hamiltonians enjoy two properties: meromorphy of the general solution, which is hyperelliptic with genus two and completeness in the Painlevé sense (impossibility to add any term to the Hamiltonian without destroying the Painlevé property).

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Integration of a generalized Hénon–Heiles Hamiltonian

Cited by 23 publications

References 17 publications

An integrable hierarchy with a perturbed Hénon–Heiles system

An integrable hierarchy with a perturbed Hénon–Heiles system

Construction of Special Solutions for Nonintegrable Systems

Explicit integration of the Hénon-Heiles Hamiltonians 1

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