The patterns of traveling wave on shallow water modeled by Green-Naghdi equation

Haitong Wei,J Zhu

doi:10.1051/matecconf/202235502005

Abstract

The Green-Naghdi equations are a shallow water waves model which play important roles in nonlinear wave fields. By using the trial equation method and the Complete discrimination system for the polynomial we obtained the classification of travelling wave patterns. Among those patterns, new singular patterns and double periodic patterns are obtained in the first time. And we draw the graphs which help us to understand the dynamics behaviors of the Green-Naghdi model intuitionally.

Highlights

Green-Naghdi equations is a shallow water waves model which can be written as following[1]vt vvx ux [u3 (vvxx vxt vx2 )]x (1) utx 0, (2)where u(x, t) is the free upper surface and v(x, t) is the horizontal velocity of the fluid.Green-Naghdi model describes a kind weekly dispersive nonlinear shallow 1water waves motion which can simulate the propagation of solitons in dispersive media
Compared weekly dispersive nonlinear equations named Boussinesq equations basing small amplitude assumption, Green-Naghdi equations apply to big amplitude question, Green-Naghdi equations have more extensive application
The paper is organized as follows: In section two, we introduce reduction of Green-Naghdi equations

Summary

Introduction

Green-Naghdi equations is a shallow water waves model which can be written as following[1]. Green-Naghdi model describes a kind weekly dispersive nonlinear shallow 1water waves motion which can simulate the propagation of solitons in dispersive media. In 1953, Serre[2] deduced one-dimensional weekly dispersive shallow water waves model in even bottom. Gained same shallow water waves equations by different method. In 1976, Green and Naghdi[4] deduced two-dimensional weekly dispersive shallow water waves equations system (1) and (2) namely Green-Naghdi equations now. In 1987, Santos[5] et al obtained one-dimensional Green-Naghdi model in uneven bottom. JHQHUDOL]HG WKH HTXDWLRQV WR JHQHUDO IRUP 'XH WR WKH LP SRUWDQFH RI WKH HTXDWLRQV IRU shallow water waves in physics, the Green-Naghdi equations have been studied by various

Reduction of Green-Naghdi equations

Expression of solutions

Conclusion