# Isobe-Kakinuma model for water waves as a higher order shallow water approximation

## TL;DR

This paper rigorously justifies the Isobe-Kakinuma model as a higher order shallow water approximation for water waves, achieving an error of order $O( ext{delta}^6)$, surpassing previous models.

## Contribution

It demonstrates that the Isobe-Kakinuma model provides a significantly more accurate higher order approximation to water wave equations than existing models.

## Key findings

- Isobe-Kakinuma model has an error of order $O( ext{delta}^6)$
- It improves upon Green-Naghdi equations with higher accuracy
- Provides rigorous mathematical justification for the model

## Abstract

We justify rigorously an Isobe-Kakinuma model for water waves as a higher order shallow water approximation in the case of a flat bottom. It is known that the full water wave equations are approximated by the shallow water equations with an error of order $O(\delta^2)$, where $\delta$ is a small nondimensional parameter defined as the ratio of the mean depth to the typical wavelength. The Green-Naghdi equations are known as higher order approximate equations to the water wave equations with an error of order $O(\delta^4)$. In this paper we show that the Isobe-Kakinuma model is a much higher order approximation to the water wave equations with an error of order $O(\delta^6)$.