The Cauchy Problem for Higher-Order KP Equations

We study the local well-posedness of higher-order KP equations. Our well-posedness results make an essential use of a global smoothing effect for the linearized equation established in Ben-Artzi and Saut (preprint, 1997), injected into the framework of Fourier transform restriction spaces introduced by Bourgain. Our ill-posedness results rely on the existence of solitary wave solutions and on scaling arguments. The method was first applied in the context of the KdV and Schrödinger equations (cf. Birniret al., Ann. Inst. H. Poincaré Anal. Non Linéaire13(1996), 529–535; Birniret al., J. London Math. Soc.53(1996), 551–559).