Higher-order dispersive systems

One can study more general dispersive equations of the general form

$u_{t}+P(\nabla )u=F(u,\nabla u,\dots )$

where P(nabla) is an anti-selfadjoint constant coefficient operator, and F involves fewer derivatives on u than P and contains only quadratic and higher terms, and may possibly contain non-local operations such as Hilbert transforms, Hartree-type potentials, or Riesz transforms. Such equations arise as various approximations to wave equations, see e.g. [Dy1979], [Hog1985]. Smoothing effects for the linear part of the equation were established in [BenKocSau2003], [Hs-p]. Nonlinear local existence in the analytic category was established in [Bd1993]. For smooth but not analytic data some local existence results have been established in [Tar1995], [Tar1997], [Ci-p].