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.