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Abenda, Simonetta; Fedorov, Yu. N.

doi:10.1023/a:1006425609939

Cited by 41 publications

(18 citation statements)

References 26 publications

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“…So this perturbed Hénon-Heiles system is another example of an integrable system with deficiency one, and instead of the standard Jacobi inversion involving the threefold symmetric product of the curve, one has the twofold product corresponding to pairs of points (λ 1 , µ 1 ), (λ 2 , µ 2 ). The image (z 1 , z 2 , z 3 ) of a pair of points under the Abel map lies on a stratum in the Jacobian of the curve, and this leads to an explanation for the algebraic branching in the solutions [1].…”

Section: Travelling Waves and Perturbed Hénon-heilesmentioning

confidence: 99%

“…Note that, because the inversion problem (4.7) involves a single integral of one holomorphic differential on a curve of genus two, the simple ODE (4.3) (which can be rewritten in Hamiltonian form) has deficiency one in the terminology of Abenda and Fedorov [1,2]. This can be seen as the origin of the weak Kowalevski-Painlevé property for the solutions.…”

Section: Travelling Waves and Perturbed Hénon-heilesmentioning

confidence: 99%

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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%

Section: Travelling Waves and Perturbed Hénon-heilesmentioning

confidence: 99%

An integrable hierarchy with a perturbed Hénon–Heiles system

2006

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“…Composing the successive transformations (20), (25), and (24), we obtain a meromorphic general solution for q 1 and q 2 2 ,…”

Section: General Solution Of the Quartic 1 : 6 : 1 And 1 : 6 : 8 Casesmentioning

confidence: 99%

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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“…In order to avoid excluding such basic integrable systems from singularity classification, Ramani et al [86] proposed an extension of the Painlevé property. There are many examples of finite-dimensional many-body Hamiltonian systems which are Liouville integrable and yet have algebraic branching in their solutions [1,2]. Among these examples [2] is the geodesic flow on an ellipsoid, which was solved classically by Jacobi [58].…”

Section: Weak Painlevé Testsmentioning

confidence: 99%

“…dt (1) In the above, ℓ and ω are respectively the angular momentum and angular velocity of the body, g is the gravity vector with respect to a moving frame, and the centre of mass vector c and inertia tensor I are both constant. The remarkable insight of Kowalevski was that the system of equations (1) could be solved explicitly whenever the dependent variables ℓ and g are meromorphic functions of time t extended to the complex plane, t ∈ C. By requiring that the solutions should admit Laurent expansions around singular points, she found constraints on the constants c and I = diag(I 1 , I 2 , I 3 ) (diagonalized in a suitable frame).…”

Section: Introductionmentioning

confidence: 99%

Painlevé Tests, Singularity Structure and Integrability

Hone

Integrability

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Summary. After a brief introduction to the Painlevé property for ordinary differential equations, we present a concise review of the various methods of singularity analysis which are commonly referred to as Painlevé tests. The tests are applied to several different examples, and the connection between singularity structure and integrability for ordinary and partial differential equation is discussed.

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Cited by 41 publications

References 26 publications

An integrable hierarchy with a perturbed Hénon–Heiles system

An integrable hierarchy with a perturbed Hénon–Heiles system

Completeness of the Cubic and Quartic Henon-Heiles Hamiltonians

Painlevé Tests, Singularity Structure and Integrability

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