Well-posedness for a higher-order Benjamin–Ono equation

In this paper we prove that the initial value problem associated to the following higher-order Benjamin–Ono equation∂tv−bH∂x2v+a∂x3v=cv∂xv−d∂x(vH∂xv+H(v∂xv)), where x,t∈Rx,t∈R, v   is a real-valued function, HH is the Hilbert transform, a∈Ra∈R, b, c and d are positive constants, is locally well-posed for initial datav(0)=v0∈Hs(R),s⩾2orv0∈Hk(R)∩L2(R;x2dx),k∈Z+,k⩾2.