Geodesic flows, bi-Hamiltonian structure and coupled KdV type systems

We show that all the Antonowicz–Fordy type coupled KdV equations have the same symmetry group and similar bi-Hamiltonian structures. It turns out that their configuration space is Diff(S1) C∞(S1), where Diff(S1) is the Bott–Virasoro group of orientation preserving diffeomorphisms of the circle, and all these systems can be interpreted as equations of a geodesic flow with respect to L2 metric on the semidirect product space Diff(S1) C∞(S1).  2005 Elsevier Inc. All rights reserved