Serre-type Equations in Deep Water

¹ LAMA, UMR 5127 CNRS, Université Savoie Mont Blanc, Campus Scientifique 73376 Le Bourget-du-Lac Cedex, France
² Université de Nice – Sophia Antipolis, Laboratoire J. A. Dieudonné Parc Valrose, 06108 Nice cedex 2, France
³ LOCIE, UMR 5271 CNRS, Université Savoie Mont Blanc, Campus Scientifique 73376 Le Bourget-du-Lac Cedex, France

^* Corresponding author. E-mail: Denys.Dutykh@univ-smb.fr

Abstract

This manuscript is devoted to the modelling of water waves in the deep water regime with some emphasis on the underlying variational structures. The present article should be considered as a review of some existing models and modelling approaches even if new results are presented as well. Namely, we derive the deep water analogue of the celebrated SERRE–GREEN–NAGHDI equations which have become the standard model in shallow water environments. The relation to existing models is discussed. Moreover, the multi-symplectic structure of these equations is reported as well. The results of this work can be used to develop various types of robust structure-preserving variational integrators in deep water. The methodology of constructing approximate models presented in this study can be naturally extrapolated to other physical flow regimes as well.