On the fully-nonlinear shallow-water generalized Serre equations

A fully-nonlinear weakly dispersive system for the shallow water wave regime is presented. In the simplest case the model was first derived by Serre in 1953 and rederived various times since then. Two additions to this system are considered: the effect of surface tension, and that of using a different reference fluid level to describe the velocity field. It is shown how the system can be further expanded by consistent exchanges of spatial and time derivatives. Properties of the solitary waves of the resulting system as well as a symmetric splitting of the equations based on the Riemann invariants of the hyperbolic shallow water system are presented. The latter leads to a fully-nonlinear one-way model and, upon further approximations, existing weakly nonlinear models. Our study also helps clarify the differences or similarities between existing models.