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Prolongation algebras and Hamiltonian operators for peakon equations

Hone, Andrew N.W., Wang, Jing Ping (2003) Prolongation algebras and Hamiltonian operators for peakon equations. Inverse Problems, 19 (1). pp. 129-145. ISSN 0266-5611. (doi:10.1088/0266-5611/19/1/307) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided)

The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided. (Contact us about this Publication)
Official URL
http://dx.doi.org/10.1088/0266-5611/19/1/307

Abstract

We consider a family of non-evolutionary partial differential equations, labelled by a single parameter b, all of which admit multi-peakon solutions. For the two special integrable cases, namely the Camassa-Holm and Degasperis-Procesi equations (b = 2 and 3), we explain how their spectral problems have reciprocal links to Lax pairs for negative flows, in the Korteweg-de Vries and Kaup-Kupershmidt hierarchies respectively. An analogous construction is presented in the case of the Sawada-Kotera hierarchy, leading to a new zero-curvature representation for the integrable Vakhnenko equation. We show how the two special peakon equations are isolated via the Wahlquist-Estabrook prolongation algebra method. Using the trivector technique of Olver, we provide a proof of the Jacobi identity for the non-local Hamiltonian structures of the whole peakon family. Within this class of Hamiltonian operators (also labelled by b), we present a uniqueness theorem which picks out the special cases b = 2, 3.

Item Type: Article
DOI/Identification number: 10.1088/0266-5611/19/1/307
Uncontrolled keywords: CAMASSA-HOLM EQUATION; BACKLUND-TRANSFORMATIONS; EVOLUTION-EQUATIONS; SYMMETRY APPROACH; KDV EQUATION; SOLITONS
Subjects: Q Science > QA Mathematics (inc Computing science)
Divisions: Faculties > Sciences > School of Mathematics Statistics and Actuarial Science
Depositing User: Andrew N W Hone
Date Deposited: 19 Dec 2007 18:28 UTC
Last Modified: 06 Feb 2020 04:00 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/759 (The current URI for this page, for reference purposes)
Hone, Andrew N.W.: https://orcid.org/0000-0001-9780-7369
Wang, Jing Ping: https://orcid.org/0000-0002-6874-5629
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