Well-posedness of a higher order modified Camassa–Holm equation in spaces of low regularity

Abstract

In this paper we consider the Cauchy problem for a higher order modified Camassa–Holm equation. By using the Fourier restriction norm method introduced by Bourgain, we establish the local well-posedness for the initial data in the Hs(R) with \({s > -n+\frac{5}{4},\,n\in {\bf N}^{+}.}\) As a consequence of the conservation of the energy \({{||u||_{H^{1}(R)},}}\) we have the global well-posedness for the initial data in H1(R).