Long wave approximations for water waves

In this paper, we obtain new nonlinear systems describing the interaction of long water waves in both two and three dimensions. These systems are symmetric and conservative. Rigorous convergence results are provided showing that solutions of the complete free-surface Euler equations tend to associated solutions of these systems as the amplitude becomes small and the wavelength large. Using this result as a tool, a rigorous justification of all the two-dimensional, approximate systems recently put forward and analysed by Bona, Chen and Saut is obtained. In the two-dimensional context, our methods also allows a significant improvement of the convergence estimate obtained by Schneider and Wayne in their justification of the decoupled Korteweg-de Vries approximation of the two-dimensional Euler equations. It also follows from our theory that coupled models provide a better description than the decoupled ones over short time scales. Results are obtained both on an unbounded domain for solutions that evanesce at infinity as well as for solutions that are spatially periodic.