Modulation of Camassa--Holm equation and reciprocal transformations

Details Modulation of Camassa--Holm equation and reciprocal transformations Modulation of Camassa--Holm equation and reciprocal transformations

2005-06-15

arxiv.org/abs/math-ph/0506042v2

Annales de L' Institut Fourier vol. 55, 2005

Physics

Mathematical Physics

Scientific paper

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.

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