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

Abstract

The geodesic equations of a class of right invariantmetrics on the semi-direct product $Diff(\mathbb{S}^1)$Ⓢ$Diff(\mathbb{S}^1)$ arestudied. The equations are explicitly described, they have theform of a system of coupled equations of Camassa-Holm type andpossess singular (peakon) solutions. Their integrability isfurther investigated, however no compatible bi-Hamiltonianstructures on the corresponding dual Lie algebra$(Vect(\mathbb{S}^1)$Ⓢ$Vect(\mathbb{S}^1))^{*}$ are found.