# Higher-order dispersive systems

One can study 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.

A one-dimensional special case of these systems are the higher order water wave models

$\partial_t u + \partial_x^{2j+1} u = P(u, u_x , ..., \partial_x^{2j} u)$

where u is real-valued and P is a polynomial with no constant or linear terms; thus KdV and gKdV correspond to j=1, and the higher order equations in the KdV hierarchy correspond to j=2,3,etc. LWP for these equations in high regularity Sobolev spaces is in KnPoVe1994, and independently by Cai (ref?); see also CrKpSr1992.The case j=2 was studied by Choi</span> (ref?).The non-scalar diagonal case was treated in KnSt1997; the periodic case was studied in Bo-p3. Note in the periodic case it is possible to have ill-posedness for every regularity, for instance $\partial_t u + u_{xxx} = u^2 u_x^2$ is ill-posed in every Hs Bo-p3.