This talk study the long wave asymptotic behavior of Green-Naghdi equations, which could be used to describe the propagation of long-crested shallow-water waves in the equatorial ocean regions with the Coriolis effect due to Earth's rotation. This model equation is called the rotation-Green-Naghdi (R-GN) equations modeling the propagation of wave allowing large amplitude in shallow water. 1. We demonstrate the lift span of the solution to the R-GN model equations in a Sobolev space by the refined energy estimates. 2.We provide a rigorous justification from the solutions of the R-GN equations to the associated solution of the right-left R-BBM or KdV equation in the KdV regime with the small amplitude and the large wavelength. This is the first result on the issue that the solution of R-GN equations are well approximated by the bi-directional R-BBM in the general regular initial data.It is a jointed work with Prof. Yue Liu.
