Euler equations on a semi-direct product of the diffeomorphisms group by itself

Joachim Escher; Rossen Ivanov; Boris Kolev

doi:10.3934/jgm.2011.3.313

Article Contents

2011, Volume 3, Issue 3: 313-322. Doi: 10.3934/jgm.2011.3.313

This issue Previous Article Infinitesimal gauge symmetries of closed forms Next Article Killing's equations for invariant metrics on Lie groups

Euler equations on a semi-direct product of the diffeomorphisms group by itself

1.
Institute for Applied Mathematics, University of Hanover, D-30167 Hanover
2.
School of Mathematical Science, Dublin Institute of Technology, Kevin Street, Dublin 8
3.
LATP, CNRS & University of Provence, 39 Rue F. Joliot-Curie, 13453 Marseille Cedex 13

Received: July 31, 2011

Revised: August 31, 2011

Published: August 2011

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Abstract

The geodesic equations of a class of right invariant metrics on the semi-direct product $Diff(\mathbb{S}^1)$Ⓢ$Diff(\mathbb{S}^1)$ are studied. The equations are explicitly described, they have the form of a system of coupled equations of Camassa-Holm type and possess singular (peakon) solutions. Their integrability is further investigated, however no compatible bi-Hamiltonian structures on the corresponding dual Lie algebra $(Vect(\mathbb{S}^1)$Ⓢ$Vect(\mathbb{S}^1))^{*}$ are found.

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Mathematics Subject Classification: Primary: 58D05, 37K65, 58B20; Secondary: 35Q53.

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