Abstract : The Kadomtsev-Petviashvili equations (KP) describe the small amplitude long wave moving mainly in the x-direction in shallow water. As for the Benjamin-Ono equation (BO), it describes such waves moving inside water. We are interested in these equations seen as equations of Benjamin-Bona-Mahony type (BBM).
Our work is subdivided in three parts. In the first one, we recall the modelling of the different equations. More particularly, we show that the BBM models are obtained from the fundamental principle of dynamics via an asymptotic analysis. We compare then the solutions of the KP equations, respectively of the BO one, with the solutions of the equations of BBM type.
In the second part, we are interested in some qualitative properties of the generalized equations of BBM type. Some results of continuation in time of bounds on Sobolev norms, decay in time and unique continuation of the solutions, are established.
Finally, we conclude with a numerical study of the solutions of the generalized KP equations in space dimension 3. In this last part, in collaboration
with F. Hamidouche and S. Mefire, we inspect numerically the phenomena of dispersion, blow-up in finite time, solitonic behaviour and transverse