TY - JOUR
T1 - Modulation of the Camassa-Holm equation and reciprocal transformations
JF - Ann. Inst. Fourier (Grenoble) 55 (2005) 1803-1834
Y1 - 2005
A1 - Simonetta Abenda
A1 - Tamara Grava
AB - 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 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.
UR - http://hdl.handle.net/1963/2305
U1 - 1711
U2 - Mathematics
U3 - Mathematical Physics
ER -