Solitary wave solutions to the Isobe‐Kakinuma model for water waves

Mathieu Colin,Tatsuo Iguchi

doi:10.1111/sapm.12310

Abstract

AbstractWe consider the Isobe‐Kakinuma model for two‐dimensional water waves in the case of a flat bottom. The Isobe‐Kakinuma model is a system of Euler‐Lagrange equations for a Lagrangian approximating Luke's Lagrangian for water waves. We show theoretically the existence of a family of small amplitude solitary wave solutions to the Isobe‐Kakinuma model in the long wave regime. Numerical analysis for large amplitude solitary wave solutions is also provided and suggests the existence of a solitary wave of extreme form with a sharp crest.

Highlights

In this paper we consider the motion of two-dimensional water waves in the case of the flat bottom
The Green–Naghdi equations are known as higher order shallow water approximate equations for water waves in the strongly nonlinear regime and have the same solitary wave solutions as those of the Korteweg–de Vries equation, but again does not have any solitary wave of extreme form
The Isobe–Kakinuma model even in the simplest case catches the property on the existence of the solitary wave of extreme form, the included angle of the crest is not 120◦

Summary

Introduction

In this paper we consider the motion of two-dimensional water waves in the case of the flat bottom. The Green–Naghdi equations are known as higher order shallow water approximate equations for water waves in the strongly nonlinear regime and have the same solitary wave solutions as those of the Korteweg–de Vries equation, but again does not have any solitary wave of extreme form Compared to these models, the Isobe–Kakinuma model even in the simplest case catches the property on the existence of the solitary wave of extreme form, the included angle of the crest is not 120◦. Marche [12], where it was shown that an extended Green– Naghdi equations for the water waves with a constant vorticity has a solitary wave solution of extreme form.

Conservation laws

Approximate solutions

Reduction of the problem

Green’s functions

Existence of small amplitude solitary waves

Numerical analysis for large amplitude solitary waves