On geodesic exponential maps of the Virasoro group

We study the geodesic exponential maps corresponding to Sobolev type right-invariant (weak) Riemannian metrics μ (k ≥ 0) on the Virasoro group and show that for k ≥ 2, but not for k = 0, 1, each of them defines a smooth Fréchet chart of the identity. For k = 0 and k = 1 the corresponding geodesic flows are related to the Korteweg de Vries and Camassa Holm equations. In particular, the geodesic exponential map corresponding to the KdV equation (k = 0) is not a local diffeomorphism near the origin.