01078nas a2200109   4500008004300000245007500043210006900118520070500187100002200892700001800914856003600932       2005                        en_Ud  00aModulation of the Camassa-Holm equation and reciprocal transformations0 aModulation of the CamassaHolm equation and reciprocal transforma3 aWe 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 transformation is generated by the Casimir of the second Poisson bracket of the KdV averaged flow. We show that the geometry of the bi-Hamiltonian structure of the KdV and CH modulation equations is quite different: indeed the KdV averaged bi-Hamiltonian structure can always be related to a semisimple Frobenius manifold while the CH one cannot.1 aAbenda, Simonetta1 aGrava, Tamara  uhttp://hdl.handle.net/1963/2305