Rigorous derivation of weakly dispersive shallow-water models with large amplitude topography variations

Abstract

We derive rigorously from the water waves equations new irrotational shallow-water models for the propagation of surface waves in the case of uneven topography in horizontal dimensions one and two. The systems are made to capture the possible change in the waves' propagation, which can occur in the case of large amplitude topography. The main contribution of this work is the construction of new multiscale shallow-water approximations of the Dirichlet–Neumann operator. We prove that the precision of these approximations is given at the order $O\left(\mu \epsilon \right)$, $O(\mu \epsilon +{\mu }^2{\beta }^2)$, and $O({\mu }^2\epsilon +\mu \epsilon \beta +{\mu }^2{\beta }^2)$. Here, $\mu$, $\epsilon$, and $\beta$ denote, respectively, the shallow-water parameter, the nonlinear parameter, and the bathymetry parameter. From these approximations, we derive models with the same precision as the ones above. The model with precision $O\left(\mu \epsilon \right)$ is coupled with an elliptic problem, while the other models do not present this inconvenience.