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arXiv · 1902.07317

A comparative study of bi-directional Whitham systems

Evgueni Dinvay·Denys Dutykh·Henrik Kalisch
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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.

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Evgueni Dinvay, Denys Dutykh, Henrik Kalisch. 2019-02-18. A comparative study of bi-directional Whitham systems. https://doi.org/10.1016/j.apnum.2018.09.016

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