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Pure mathematics, Applied mathematics
The geodesic equations of a class of right invariant metrics on the semi-direct product of two Diff(S) groups 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.
J. Escher, R. Ivanov and B. Kolev, Euler equations on a semi-direct product of the diffeomorphisms group by itself, J. Geom. Mechanics 3 (2011) 313-322 doi: 10.21427/e77w-0z73