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We derive the modulation equations (Whitham equations) for the Camassa-Holm (CH)
equation. We show that the modulation equations are hyperbolic and admit a 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...
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