A mathematical justification of the Isobe-Kakinuma model for water waves with and without bottom topography

TL;DR

This paper rigorously justifies the Isobe-Kakinuma water wave model as a higher order shallow water approximation, providing error estimates for both flat and variable bottom topographies in nonlinear regimes.

Contribution

It offers the first rigorous error bounds for the Isobe-Kakinuma model compared to full water wave equations, including cases with variable bottom topography.

Findings

Error is of order O(δ^{4N+2}) for flat bottoms.

Error is of order O(δ^{4[ N/2]+2}) for variable bottoms.

Model is validated as a higher order approximation in nonlinear water wave regimes.

Abstract

We consider the Isobe-Kakinuma model for water waves in both cases of the flat and the variable bottoms. The Isobe-Kakinuma model is a system of Euler-Lagrange equations for an approximate Lagrangian which is derived from Luke's Lagrangian for water waves by approximating the velocity potential in the Lagrangian appropriately. The Isobe-Kakinuma model consists of $(N+1)$ second order and a first order partial differential equations, where $N$ is a nonnegative integer. We justify rigorously the Isobe-Kakinuma model as a higher order shallow water approximation in the strongly nonlinear regime by giving an error estimate between the solutions of the Isobe-Kakinuma model and of the full water wave problem in terms of the small nondimensional parameter $\delta$, which is the ratio of the mean depth to the typical wavelength. It turns out that the error is of order $O(\delta^{4N+2})$ in the case of the flat bottom and of order $O(\delta^{4[N/2]+2})$ in the case of variable bottoms.