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We consider algebraic geometrical properties of the integrable billiard on a quadric Q with elastic impacts along another quadric confocal to Q. These properties are in sharp contrast with those of the ellipsoidal Birkhoff billiards in R. Namely, generic complex invariant manifolds are not Abelian varieties, and the billiard map is no more algebraic. A… (More)

We derive the modulation equations or Whitham equations for the Camassa– Holm (CH) equation. We show that the modulation equations are hyperbolic and admit bi-Hamiltonian structure. Furthermore they are connected by a reciprocal transformation to the modulation equations of the first negative flow of the Korteweg de Vries (KdV) equation. The reciprocal… (More)

- Simonetta Abenda
- 2009

We start from a hyperbolic DN hydrodynamic type system of dimension n which possesses Riemann invariants and we settle the necessary conditions on the conservation laws in the reciprocal transformation so that, after such a transformation of the independent variables, one of the metrics associated to the initial system be flat. We prove the following… (More)

We consider hydrodynamic systems which possess a local Hamiltonian structure of Dubrovin-Novikov type. To such a system there are also associated an infinite number of nonlocal Hamiltonian strctures of Mokhov-Ferapontov type. We give necessary and sufficient conditions so that, after a nonlinear transformation of the independent variables, the reciprocal… (More)

- Simonetta Abenda
- 2008

We prove that any real doubly periodic geodesic on an n dimensional ellipsoid with distinct semiaxes and caustic parameters is uniquely associated to a real hyperelliptic tangential cover and that the following density property holds: given a real closed geodesic on the ellipsoid Q = {X2 1/a1 + · · ·+X2 n+1/an+1 = 1} with caustic parameters cj, j = 1, . . .… (More)

- S ABENDA, Yu FEDOROV, Yu Fedorov
- 2001

We propose Dirac formalism for constraint Hamiltonian systems as an useful tool for the algebro-geometrical and dynamical characterizations of a class of integrable systems, the so called hyperelliptically separable systems. As a model example, we apply it to the classical geodesic flow on an ellipsoid. Consider an n-dimensional Hamiltonian system on the… (More)

- S. Abenda, Tamara Grava, Christian Klein
- SIAM Journal of Applied Mathematics
- 2010

The small dispersion limit of solutions to the Camassa-Holm (CH) equation is characterized by the appearance of a zone of rapid modulated oscillations. An asymptotic description of these oscillations is given, for short times, by the one-phase solution to the CH equation, where the branch points of the corresponding elliptic curve depend on the physical… (More)

- SIMONETTA ABENDA, PETR G. GRINEVICH
- 2009

One of the classical integrable problems is the geodesic motion on the ellipsoid. Complete integrability of the 3-dimensional problem was proven by Jacobi [25]. Explicit formulas in terms of genus 2 θ-functions were found by Weierstrass [44]. Integrability of the multidimensional problem was established by Moser [32]. Connections with the Neumann problem… (More)

- Simonetta Abenda
- 2008

The closedness condition for real geodesics on n–dimensional ellipsoids is in general transcendental in the parameters (semiaxes of the ellipsoid and constants of motion). We show that it is algebraic in the parameters if and only if both the real and the imaginary geodesics are closed and we characterize such double–periodicity condition via real… (More)

- Jianting Zhou, S. Abenda
- Intelligent Automation & Soft Computing
- 2010

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