The Cauchy problem for the Novikov equation

This work studies the initial value problem for a Camassa-Holm type equation with cubic nonlinearities that has been recently discovered by Vladimir Novikov to be integrable. For s > 3/2, using a Galerkin-type approximation method, it is shown that this equation is well-posed in Sobolev spaces H-s on both the line and the circle with continuous dependence on initial data. Furthermore, it is proved that this dependence is optimal by showing that the data-to-solution map is not uniformly continuous. The nonuniform dependence is proved using the method of approximate solutions in conjunction with well-posedness estimates.