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

TL;DR

This paper proves the existence of small amplitude solitary wave solutions in the Isobe-Kakinuma water wave model and provides numerical evidence for large amplitude solitary waves, including extreme forms with sharp crests.

Contribution

It is the first to theoretically establish small amplitude solitary waves in the Isobe-Kakinuma model and numerically explore large amplitude solutions.

Findings

Existence of small amplitude solitary waves proven mathematically.

Numerical analysis indicates large amplitude solitary waves with sharp crests.

Potential for extreme solitary wave solutions with sharp features.

Abstract

We consider the Isobe-Kakinuma model for two-dimensional water waves in the case of the 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.