An Explicit Solution with Correctors for the Green-Naghdi Equations

In this paper, the water waves problem for uneven bottoms in a highly nonlinear regime is studied. It is well known that, for such regimes, a generalization of the Boussinesq equations called the Green–Naghdi equations can be derived and justified when the bottom is variable (Lannes and Bonneton in Phys Fluids 21, 2009). Moreover, the Green–Naghdi and Boussinesq equations are fully nonlinear and dispersive systems. We derive here new linear asymptotic models of the Green–Naghdi and Boussinesq equations so that they have the same accuracy as the standard equations. We solve explicitly the new linear models and numerically validate the results.