On the Galilean Invariance of Some Dispersive Wave Equations

Surface water waves in ideal fluids have been typically modeled by asymptotic approximations of the full Euler equations. Some of these simplified models lose relevant properties of the full water wave problem. One of these properties is the Galilean symmetry, i.e. the invariance under Galilean transformations. In this paper, a mechanism to incorporate Galilean invariance in classical water wave models is proposed. The technique is applied to the Benajmin-Bona-Mahony (BBM) equation and the Peregrine (classical Boussinesq) system, leading to the corresponding Galilean invariant versions of these models. Some properties of the new equations are presented, with special emphasis on the computation and interaction of solitary wave solutions. A comparison with the Euler equations demonstrates the relevance of the Galilean invariance in the description of water waves.