Modeling water waves beyond perturbations

Didier Clamond (1) , Denys Dutykh (2, 3, 4)

1 LJAD - Laboratoire Jean Alexandre Dieudonné
2 LAMA - Laboratoire de Mathématiques
3 INSMI-CNRS - Institut National des Sciences Mathématiques et de leurs Interactions - CNRS Mathématiques
4 USMB [Université de Savoie] [Université de Chambéry] - Université Savoie Mont Blanc

In this chapter, we illustrate the advantage of variational principles for modeling water waves from an elementary practical viewpoint. The method is based on a `relaxed' variational principle, i.e., on a Lagrangian involving as many variables as possible, and imposing some suitable subordinate constraints. This approach allows the construction of approximations without necessarily relying on a small parameter. This is illustrated via simple examples, namely the Serre equations in shallow water, a generalization of the Klein-Gordon equation in deep water and how to unify these equations in arbitrary depth. The chapter ends with a discussion and caution on how this approach should be used in practice.

Mots clés: approximate models; variational methods; water waves

Domaines: Mécanique des fluides [physics.class-ph]; Dynamique des Fluides [physics.flu-dyn]; Physique Atmosphérique et Océanique [physics.ao-ph]

Didier Clamond, Denys Dutykh. Modeling water waves beyond perturbations. Springer. New approaches to nonlinear waves, 908, Springer International Publishing, pp.197-210, 2015, New Approaches to Nonlinear Waves, 978-3-319-20690-5. ⟨10.1007/978-3-319-20690-5_7⟩. ⟨hal-01102120⟩