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

Abstract

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 ( S 1 ) ⋉ C ∞ ( S 1 ) ˆ , where Diff ( S 1 ) ˆ 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 L 2 metric on the semidirect product space Diff ( S 1 ) ⋉ C ∞ ( S 1 ) ˆ .