Towards a rigorous derivation of the fifth order KP equation

The Kadomtsev-Petviashvili equations (KP) are universal models for dispersive, weakly nonlinear, almost one-dimensional long waves of small amplitude. In [L. Paumond, A rigorous link between KP and a Benney-Luke equation, Differential Integral Equations 16 (9) (2003) 1039-1064], we proved a rigorous link between KP and a Benney-Luke equation. Here, we derive a new model still valid when the Bond number is equal to 1/3. Following the work [R.L. Pego, J.R. Quintero, Two-dimensional solitary waves for a Benney-Luke equation, Physica D 132 (4) (1999) 476-496], we show the existence of solitary waves for this equation and their convergence towards the solitary waves of the fifth order KP equation. We also link rigorously the dynamics of the two equations.