Dispersive shallow water wave modelling. Part I: Model derivation on a globally flat space

Gayaz Khakimzyanov (1) , Denys Dutykh (2, 3, 4) , Zinaida Fedotova (1) , Dimitrios Mitsotakis (5)

1 Institute of Computational Technologies
2 LAMA - Laboratoire de Mathématiques
3 USMB [Université de Savoie] [Université de Chambéry] - Université Savoie Mont Blanc
4 INSMI-CNRS - Institut National des Sciences Mathématiques et de leurs Interactions - CNRS Mathématiques
5 Victoria University of Wellington

In this paper, we review the history and current state-of-the-art in the modelling of long nonlinear dispersive waves. For the sake of conciseness of this review, we omit the unidirectional models and focus especially on some classical and improved Boussinesq-type and Serre-Green-Naghdi equations. Finally, we propose also a unified modelling framework which incorporates several well-known and some less known dispersive wave models. The present manuscript is the first part of a series of two papers. The second part will be devoted to the numerical discretization of a practically important model on moving adaptive grids.

Mots clés: solitary waves; shallow water equations; long wave approximation; nonlinear dispersive waves

Domaines: Mécanique des fluides [physics.class-ph]; Physique Atmosphérique et Océanique [physics.ao-ph]; Dynamique des Fluides [physics.flu-dyn]; Equations aux dérivées partielles [math.AP]; Systèmes Solubles et Intégrables [nlin.SI]; Formation de Structures et Solitons [nlin.PS]

Gayaz Khakimzyanov, Denys Dutykh, Zinaida Fedotova, Dimitrios Mitsotakis. Dispersive shallow water wave modelling. Part I: Model derivation on a globally flat space. Communications in Computational Physics, 2018, 23 (1), pp.1-29. ⟨10.4208/cicp.OA-2016-0179a⟩. ⟨hal-01547833v3⟩