Numerical comparisons of two long-wave limit models

The Benney-Luke equation (BL) is a model for the evolution of three-dimensional weakly nonlinear, long water waves of small amplitude. In this paper we propose a nearly conservative scheme for the numerical resolution of (BL). Moreover, it is known ([PQ99] and [Pau03]) that (BL) is linked to the Kadomtsev-Petviashvili equation for almost one-dimensional waves propagating in one direction. We study here numerically the link between (KP) and (BL) and we point out the coupling effects emerging by considering two solitary waves propagating in two opposite directions. Introduction The difficulties met to work with the full water-wave problem lead to derive simplified model in the special case of long waves. The model equation we will consider describe the evolution of three-dimensional weakly nonlinear, long water waves of small amplitude, (1) Φtt −∆Φ+ μ(a∆Φ− b∆Φtt) + ε(Φt∆Φ+ (∇Φ)t ) = 0, where Φ(x, y, t) is the velocity potential on the bottom of the domain after rescaling the variables and ∇ and ∆ are respectively the two-dimensional gradient and Laplacian. The constants a and b are positive and such that a−b = σ− 13 6= 0 where σ is the Bond number. ε is the amplitude parameter (nonlinearity coefficient) and μ = (h0/L) 2 is the long wave parameter (dispersion coefficient), where h0 is the equilibrium depth and L is the length scale. This equation was first derived by Benney and Luke (BL) (see [BL64]) when a = 1/6 and b = 1/2 with no surface tension. Let us remark that the model (1) is not valid for a = b (σ = 1/3) and in this case, it corresponds to a nonlinear wave equation with no dispersion. Precisely, in this special case, (BL) is not linearly well-posed. Then, we can derive (see [Pau02]) an equation still valid when we suppose that σ is equal or close to 1/3, (2) Φtt−∆Φ+ √ ε [ a∆Φ− b∆Φtt ]