A rigorous link between KP and a Benney-Luke equation

Abstract

Kadomtsev-Petviashvili (KP) equations model weakly nonlinear dispersive waves, which are essentially unidimensional, when weak transverse effects are taken into account. These equations can be formally obtained by asymptotic methods from the Euler equations, but there is no rigorous justification yet. In this paper we consider an intermediate equation (BL) derived first by Benney and Luke. (BL) reduces formally to (KP) if we seek waves propagating essentially in one direction, with a weak variations in time and in the transverse direction measured by a small parameter ${\varepsilon}$. We show rigorously that the $L^2(\mathbb R^2)$-norm of the difference between the amplitude of the wave given by (KP) and the one given by (BL) is of order $\mathcal{O}({\varepsilon}^{3/4})$ during a growing with ${\varepsilon}$ time.