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

Abstract

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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 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.

How to cite

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Abenda, 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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