Modulation of Camassa-Holm equation and reciprocal transformations

The authors derive the modulation equations for the one-phase periodic solution of the Camassa-Holm (CH) equation by using a Lagrangian formalism. Following the method of Haynes and Whitham, the modulation equations are shown to be Hamiltonian with a local Poisson bracket of Dubrovin-Novikov type. Subsequently, the modulation equations are rewritten in Riemann-invariant form and are shown to be hyperbolic. The authors then investigate the bi-Hamiltonian structure and the integration of the one-phase Whitham equations in Riemann-invariant form. Finally, in the last section, the authors show that the modulation equations of the CH equation are transformed to the modulation equations of the first negative KdV flow by the average of the reciprocal transformation which links the two equations.