  The Kadomtsev-Petviashvili (KP) equation possesses a four-parameter family of
one-dimensional periodic traveling waves. We study the spectral stability of
the waves with small amplitude with respect to two-dimensional perturbations
which are either periodic in the direction of propagation, with the same period
as the one-dimensional traveling wave, or non-periodic (localized or bounded).
We focus on the so-called KP-I equation (positive dispersion case), for which
we show that these periodic waves are unstable with respect to both types of
perturbations. Finally, we briefly discuss the KP-II equation, for which we
show that these periodic waves are spectrally stable with respect to
perturbations which are periodic in the direction of propagation, and have long
wavelengths in the transverse direction.
