The main results of this paper are concerned with the "bad" behavior of the KP-I equation with respect to a Picard iteration scheme applied to the associated integral equation, for data in usual or anisotropic Sobolev spaces. This leads to some kind of ill-posedness of the corresponding Cauchy problem: the flow map cannot be of class $C\sp 2$ in any Sobolev space.
"Well-posedness and ill-posedness results for the Kadomtsev-Petviashvili-I equation." Duke Math. J. 115 (2) 353 - 384, 1 November 2002. https://doi.org/10.1215/S0012-7094-02-11525-7