KP description of unidirectional long waves. The model case

The Kadomtsev-Petviashvili (KP) equation can be formally derived as an enve lope equation for three-dimensional unidirectional water waves in the limit of long waves. As a first step towards a mathematical justification, we co nsider here a two-dimensional Boussinesq equation, which is a realistic mod el for three-dimensional water waves, Using rigorous estimates, we show tha t part of the dynamics of the XP equation can be found approximately in the two-dimensional Boussinesq equation. On the other hand, there exist initia l data for the KP equation such that the corresponding solutions of the two -dimensional Boussinesq equation behave in no way according to the KP predi ction. We expect that similar results hold for the three-dimensional water wave problem too.