Numerical solution of the generalized Kadomtsev-Petviashvili equations with compact finite difference schemes

We propose compact finite difference schemes to solve the KP equations $u\mathrm{\_}t+u\mathrm{\_}xxx+u^{p}u\mathrm{\_}x+$\lambda${\mathrm{\partial }}^{--1}\mathrm{\_}xu\mathrm{\_}yy=0$. 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.