On a model system for the oblique interaction of internal gravity waves

We give local and global well-posedness results for a system of two Kadomtsev-Petviashvili (KP) equations derived by R. Grimshaw and Y. Zhu to model the oblique interaction of weakly nonlinear, two dimensional, long internal waves in shallow fluids. We also prove a smoothing effect for the amplitudes of the interacting waves. We use the Fourier transform restriction norms introduced by J. Bourgain and the Stricharta estimates for the linear KP group. Finally we extend the result of [3] for lower order perturbation of the system in the absence of transverse effects.