Model Equations and Traveling Wave Solutions for Shallow-Water Waves with the Coriolis Effect

• DOI: 10.1007/s00332-018-9510-x
• Publication Date: 2018-11-10T04:12:09Z
• Authors: Guilong Gui; Yue Liu; Ting Luo
• Keywords: Kelvin wave; Shallow water equations; Wave shoaling

In the present study, we start by formally deriving the simplified phenomenological models of long-crested shallow-water waves propagating in the equatorial ocean regions with the Coriolis effect due to the Earth’s rotation. These new model equations are analogous to the Green–Naghdi equations, the first-order approximations of the KdV-, or BBM type, respectively. We then justify rigorously that in the long-wave limit, unidirectional solutions of a class of KdV- or BBM type are well approximated by the solutions of the Camassa–Holm equation in a rotating setting. The modeling and analysis of those mathematical models then illustrate that the Coriolis forcing in the propagation of shallow-water waves can not be neglected. Indeed, the CH-approximation with the Coriolis effect captures stronger nonlinear effects than the nonlinear dispersive rotational KdV type. Furthermore, we demonstrate nonexistence of the Camassa–Holm-type peaked solution and classify various localized traveling wave solutions to the Camassa–Holm equation with the Coriolis effect depending on the range of the rotation parameter.