Well-posedness of modified Camassa–Holm equations

The Camassa–Holm equation can be viewed as the geodesic equation on some diffeomorphism group with respect to the invariant H1 metric. We derive the geodesic equations on that group with respect to the invariant Hk metric, which we call the modified Camassa–Holm equation, and then study the well-posedness and dynamics of a modified Camassa–Holm equation on the unit circle S, which has some significant difference from that of Camassa–Holm equation, e.g., it does not admit finite time blowup solutions.