S. Abenda

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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)
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 n. Namely, generic complex invariant manifolds are not Abelian varieties, and the billiard map is no more algebraic. A(More)
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)
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)
Algebraically closed real geodesics on n-dimensional ellipsoids are dense in the parameter space and related to hyperelliptic tangential coverings * Simonetta Abenda Abstract 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(More)