Abstract

Surface water waves of significant interest such as tsunamis, solitary waves and undular bores are nonlinear and dispersive waves. Unluckily, the equations describing the propagation of surface water waves known as Euler’s equations are immensely hard to solve. For this reason, several simplified systems of PDEs have been proposed as alternative approximations to Euler’s equations. In this presentation we review the theoretical properties of such systems. We show that only some of the asymptotically derived systems obey to the laws of mathematics and physics while there is only one with complete mathematical theory for physically sound initial-boundary value problems. We also discuss the numerical modeling of such problems. In particular, we focus on Galerkin / Finite element methods, which is a class of high-order methods that has been proved convergent to certain initial-boundary value problems of physical interest (perhaps the only one). We close this presentation with conditions for the existence of special solutions and validation with laboratory data.