# Modulation of the Camassa-Holm equation and reciprocal transformations

Simonetta Abenda^{[1]}; Tamara Grava

- [1] Università degli Studi di Bologna, Dipartimento di Matematica e CIRAM, (Italie), SISSA, Via Beirut 9, Trieste (Italie)

Annales de l’institut Fourier (2005)

- Volume: 55, Issue: 6, page 1803-1834
- ISSN: 0373-0956

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topAbenda, Simonetta, and Grava, Tamara. "Modulation of the Camassa-Holm equation and reciprocal transformations." Annales de l’institut Fourier 55.6 (2005): 1803-1834. <http://eudml.org/doc/116234>.

@article{Abenda2005,

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.},

affiliation = {Università degli Studi di Bologna, Dipartimento di Matematica e CIRAM, (Italie), SISSA, Via Beirut 9, Trieste (Italie)},

author = {Abenda, Simonetta, Grava, Tamara},

journal = {Annales de l’institut Fourier},

keywords = {Camassa-Holm equation; Korteweg de Vries hierarchy; modulation equations; Whitham equations; reciprocal transformations; Hamiltonian structures},

language = {eng},

number = {6},

pages = {1803-1834},

publisher = {Association des Annales de l'Institut Fourier},

title = {Modulation of the Camassa-Holm equation and reciprocal transformations},

url = {http://eudml.org/doc/116234},

volume = {55},

year = {2005},

}

TY - JOUR

AU - Abenda, Simonetta

AU - Grava, Tamara

TI - Modulation of the Camassa-Holm equation and reciprocal transformations

JO - Annales de l’institut Fourier

PY - 2005

PB - Association des Annales de l'Institut Fourier

VL - 55

IS - 6

SP - 1803

EP - 1834

AB - 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.

LA - eng

KW - Camassa-Holm equation; Korteweg de Vries hierarchy; modulation equations; Whitham equations; reciprocal transformations; Hamiltonian structures

UR - http://eudml.org/doc/116234

ER -

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