A MULTISCALE MODEL FOR WEAKLY NONLINEAR SHALLOW WATER WAVES OVER PERIODIC BATHYMETRY

We study the behavior of shallow water waves over periodically varying bathymetry, based on the frst-order hyperbolic Saint-Venant equations. Although solutions of this system are known to generally exhibit wave breaking, numerical experiments suggest a diferent behavior in the presence of periodic bathymetry. Starting from the frst-order variable-coefcient hyperbolic system, we apply a multiple-scale perturbation approach in order to derive a system of constantcoefcient high-order partial diferential equations whose solution approximates that of the original system. The high-order system turns out to be dispersive and exhibits solitary-wave formation, in close agreement with direct numerical simulations of the original system. We show that the constantcoefcient homogenized system can be used to study the properties of solitary waves and to conduct efcient numerical simulations.