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Simonetta Abenda; Tamara Grava
Modulation of the Camassa-Holm equation and reciprocal transformations
(Équations de modulation de Camassa-Holm et transformations réciproques)
Annales de l'institut Fourier, 55 no. 6 (2005), p. 1803-1834, doi: 10.5802/aif.2142
Article PDF | Reviews MR 2187936 | Zbl 02230058
Class. Math.: 37K05, 35L60, 35Q53, 37K20
Keywords: Camassa-Holm equation, Korteweg de Vries hierarchy, modulation equations, Whitham equations, reciprocal transformations, Hamiltonian structures

Résumé - Abstract

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 of the bi-Hamiltonian structure of the KdV and CH modulation equations are 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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