On the Geophysical Green-Naghdi System

In this paper, a modified Green-Naghdi system with the effect of the Coriolis force is derived, which is a model in the equatorial oceanography to describe the propagation of large amplitude surface waves. The effects of the Coriolis force caused by the Earth’s rotation and nonlinearities on local well-posedness and traveling wave solutions are then investigated. Employing Kato’s theory, the local well-posedness in Sobolev space Hs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^s$$\end{document} with s>52\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s>\frac{5}{2}$$\end{document} is established. Based on the qualitative method combined with the bifurcation method of dynamical systems, the classification of all traveling wave solutions, all possible phase portraits of bifurcations and exact traveling wave solutions to this system are obtained under various conditions about the parameters depending on the value of the rotation Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document}