Approximate Model Equations for Water Waves

Abstract

We present two new model equations for the unidirectional propagation of long waves in dispersive media for the specific purpose of modeling water waves. The derivation of the new equations uses a Pade(2,2) approximation of the phase velocity that arises in the linear water wave theory. Unlike the Korteweg-deVries (KdV) equation and similarly to the Benjamin-Bona-Mahony (BBM) equation, our models have a bounded dispersion relation. At the same time, the equations we propose provide the best approximation of the phase velocity for small wave numbers that can be obtained with third-order equations. We note that the new model equations can be transformed into previously studied models, such as the BBM and the Burgers-Poisson equations. It is therefore straightforward to establish the existence and uniqueness of solutions to the new equations. We also show that the distance between the solutions of one of the new equations, the KdV equation, and the BBM equation, is of the small order that is formally neglected by all models.