# A comparative study of bi-directional Whitham systems

## TL;DR

This paper reviews existing bi-directional Whitham systems, introduces a new Hamiltonian semi-linear system, and uses advanced numerical methods to analyze solution existence and stability, enhancing understanding of wave modeling accuracy.

## Contribution

It presents a new Hamiltonian semi-linear bi-directional Whitham system and evaluates its performance and mathematical properties compared to existing models.

## Key findings

- The new system approximates the full Euler equations well.
- It exhibits favorable well-posedness properties.
- Numerical analysis reveals insights into solution stability.

## Abstract

In 1967, Whitham proposed a simplified surface water-wave model which combined the full linear dispersion relation of the full Euler equations with a weakly linear approximation. The equation he postulated which is now called the Whitham equation has recently been extended to a system of equations allowing for bi-directional propagation of surface waves. A number of different two-way systems have been put forward, and even though they are similar from a modeling point of view, these systems have very different mathematical properties. In the current work, we review some of the existing fully dispersive systems. We use state-of-the-art numerical tools to try to understand existence and stability of solutions to the initial-value problem associated to these systems. We also put forward a new system which is Hamiltonian and semi-linear. The new system is shown to perform well both with regard to approximating the full Euler system, and with regard to well posedness properties.