Euler equations on a semi-direct product of the diffeomorphisms group by itself
Résumé
The geodesic equations of a class of right invariant metrics on the semi-direct product of the diffeomorphisms group of the circle by itself (acting by conjugacy) 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 are found.