Modulation of the Camassa-Holm equation and reciprocal transformations
Annales de l'Institut Fourier, Volume 55 (2005) no. 6, p. 1803-1834

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.

Nous construisons les équations modulées (équations de Whitham) pour l\rq équation de Camassa-Holm (CH). Nous démontrons que ces équations modulées sont hyperboliques et bi- hamiltoniennes. En particulier, il existe une transformation réciproque telle qu'aux équations modulées du premier flot négatif de l\rq équation de Korteweg-de Vries (KdV) correspondent aux équations modulées de CH. Cette transformation réciproque est engendrée par le Casimir du deuxième crochet de Poisson associé au flot moyenné de KdV. Nous démontrons que la géométrie des structures bi-hamiltoniennes des équations modulées de KdV et CH sont très différentes : en effet, la structure de Poisson moyennée de KdV est liée à une variété semi-simple de Frobenius et non celle de CH.

DOI : https://doi.org/10.5802/aif.2142
Classification:  37K05,  35L60,  35Q53,  37K20
Keywords: Camassa-Holm equation, Korteweg de Vries hierarchy, modulation equations, Whitham equations, reciprocal transformations, Hamiltonian structures
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     author = {Abenda, Simonetta and Grava, Tamara},
     title = {Modulation of the Camassa-Holm equation and reciprocal transformations},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {55},
     number = {6},
     year = {2005},
     pages = {1803-1834},
     doi = {10.5802/aif.2142},
     zbl = {02230058},
     mrnumber = {2187936},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2005__55_6_1803_0}
}
Abenda, Simonetta; Grava, Tamara. Modulation of the Camassa-Holm equation and reciprocal transformations. Annales de l'Institut Fourier, Volume 55 (2005) no. 6, pp. 1803-1834. doi : 10.5802/aif.2142. http://www.numdam.org/item/AIF_2005__55_6_1803_0/

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