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

Presented by: 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.

Abstract
Résumé

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.

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.

Received: 2017-05-12
Accepted: 2017-05-30
Published online: 2017-06-12

DOI: 10.1016/j.crma.2017.05.009

Author's affiliations:

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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