On the Cauchy problem for higher-order nonlinear dispersive equations

We study the higher-order nonlinear dispersive equation 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≠0 for some k>j, in the sense that this equation cannot have its flow map C2 at the origin in , for any . 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