Geometric Integrability of the Camassa-Holm Equation. II

Hernández Heredero, Rafael and Reyes, Enrique G. (2011). Geometric Integrability of the Camassa-Holm Equation. II. "International Mathematics Research Notices" ; pp. 1-37. ISSN 1073-7928. https://doi.org/10.1093/imrn/rnr120.

Description

Title: Geometric Integrability of the Camassa-Holm Equation. II
Author/s:
  • Hernández Heredero, Rafael
  • Reyes, Enrique G.
Item Type: Article
Título de Revista/Publicación: International Mathematics Research Notices
Date: July 2011
ISSN: 1073-7928
Subjects:
Faculty: E.U.I.T. Telecomunicación (UPM)
Department: Matemática Aplicada a la Ingeniería Técnica de Telecomunicación [hasta 2014]
Creative Commons Licenses: Recognition - No derivative works - Non commercial

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Abstract

It is known that the Camassa–Holm (CH) equation describes pseudo-spherical surfaces and that therefore its integrability properties can be studied by geometrical means. In particular, the CH equation admits nonlocal symmetries of “pseudo-potential type”: the standard quadratic pseudo-potential associated with the geodesics of the pseudo-spherical surfaces determined by (generic) solutions to CH, allows us to construct a covering π of the equation manifold of CH on which nonlocal symmetries can be explicitly calculated. In this article, we present the Lie algebra of (first-order) nonlocal π-symmetries for the CH equation, and we show that this algebra contains a semidirect sum of the loop algebra over sl(2,R) and the centerless Virasoro algebra. As applications, we compute explicit solutions, we construct a Darboux transformation for the CH equation, and we recover its recursion operator. We also extend our results to the associated Camassa–Holm equation introduced by J. Schiff.

More information

Item ID: 11245
DC Identifier: http://oa.upm.es/11245/
OAI Identifier: oai:oa.upm.es:11245
DOI: 10.1093/imrn/rnr120
Official URL: http://imrn.oxfordjournals.org/content/early/2011/07/10/imrn.rnr120.abstract
Deposited by: Memoria Investigacion
Deposited on: 18 Sep 2012 09:58
Last Modified: 20 Apr 2016 19:23
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