  We propose compact finite difference schemes to solve the KP equations $u\_t
+ u\_{xxx} + u^p u\_x + $\lambda$ \partial^{--1}\_x u\_{yy} = 0$. When $p = 1$,
this equation describes the propagation of small amplitude long waves in
shallow water with weak transverse effects. We first present the numerical
schemes which are compared to the Fourier spectral method. After establishing
the numerical convergence, the scheme is validated. We then depict the behavior
of solutions in the context of solitons instabilities and the blow-up.
