  A complete classification of compacton solutions is carried out for a
generalization of the Kadomtsev-Petviashvili (KP) equation involving nonlinear
dispersion in two and higher spatial dimensions. In particular, precise
conditions are given on the nonlinearity powers in this equation under which a
travelling wave can be cut off to obtain a compacton. Numerous explicit
examples having various profiles are derived, including a quadratic function,
powers of a cosine, and powers of Jacobi $\cn$ functions, all of which are
symmetric. The cosine and $\cn$ symmetric compactons have an anti-symmetric
counterpart. In comparison, explicit solitary waves of the generalized KP
equation are found to have profiles given by a power of a sech and a reciprocal
quadratic function. Kinematic properties of all of the different types of
compactons and solitary waves are discussed, along with conservation laws of
the generalized KP equation.
