Compatible Hamiltonian operators for the Krichever–Novikov equation

Sylvain Carpentier

doi:10.1016/j.crma.2017.05.009

Lie algebras/Partial differential equations

Compatible Hamiltonian operators for the Krichever–Novikov equation
[Opérateurs hamiltoniens compatibles pour l'équation de Krichever–Novikov]

Présenté par : Michèle Vergne

Sylvain Carpentier ¹

¹ Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Comptes Rendus. Mathématique, Volume 355 (2017) no. 7, pp. 744-747.

Résumé
Abstract

Il a été démontré par Sokolov que la hiérarchie de l'équation de Krichever–Novikov est hamiltonienne pour l'opérateur hamiltonien $H_{0} = u_{x} \partial^{- 1} u_{x}$ et possède deux opérateurs de récursion faiblement non locaux de degrés 4 et 6, $L_{4}$ et $L_{6}$ . Nous montrons ici que $H_{0}$ , $L_{4} H_{0}$ et $L_{6} H_{0}$ sont des opérateurs hamiltoniens compatibles pour lesquels la hiérarchie de l'équation de Krichever–Novikov est hamiltonienne.

It has been proved by Sokolov that Krichever–Novikov equation's hierarchy is hamiltonian for the Hamiltonian operator $H_{0} = u_{x} \partial^{- 1} u_{x}$ and possesses two weakly non-local recursion operators of degrees 4 and 6, $L_{4}$ and $L_{6}$ . We show here that $H_{0}$ , $L_{4} H_{0}$ and $L_{6} H_{0}$ are compatible Hamiltonians operators for which the Krichever–Novikov equation's hierarchy is hamiltonian.

Reçu le : 2017-05-12
Accepté le : 2017-05-30
Publié le : 2017-06-12

DOI : 10.1016/j.crma.2017.05.009

Affiliations des auteurs :

Sylvain Carpentier ¹

¹ Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

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     title = {Compatible {Hamiltonian} operators for the {Krichever{\textendash}Novikov} equation},
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Sylvain Carpentier. Compatible Hamiltonian operators for the Krichever–Novikov equation. Comptes Rendus. Mathématique, Volume 355 (2017) no. 7, pp. 744-747. doi : 10.1016/j.crma.2017.05.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.05.009/

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