On the Cauchy problem for higher-order nonlinear dispersive equations

We study the higher-order nonlinear dispersive equation∂tu+∂x2j+1u=∑0⩽j1+j2⩽2jaj1,j2∂xj1u∂xj2u,x,t∈R, where u   is a real- (or complex-) valued function. We show that the associated initial value problem is well-posed in weighted Besov and Sobolev spaces for small initial data. We also prove ill-posedness results when a0,k≠0a0,k≠0 for some k>jk>j, in the sense that this equation cannot have its flow map C2C2 at the origin in Hs(R)Hs(R), for any s∈Rs∈R. The same technique leads to similar ill-posedness results for other higher-order nonlinear dispersive equation as higher-order Benjamin–Ono and intermediate long wave equations.