A comparative study of bi-directional Whitham systems

Evgueni Dinvay, Denys Dutykh, Henrik Kalisch

Research output: Contribution to journalArticlepeer-review

14 Scopus citations


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, such as found in [1,4,9,17,22,23]. 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.

Original languageBritish English
Pages (from-to)248-262
Number of pages15
JournalApplied Numerical Mathematics
StatePublished - Jul 2019


  • Fully dispersive system
  • Split-step scheme
  • Surface water waves
  • Whitham equation


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