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].

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 (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 $H^{s}$ [Bo-p3]