Practical use of variational principles for modeling water waves

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

1 JAD - 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

This paper describes a method for deriving approximate equations for irrotational water waves. The method is based on a 'relaxed' variational principle, i.e., on a Lagrangian involving as many variables as possible. This formulation is particularly suitable for the construction of approximate water wave models, since it allows more freedom while preserving the variational structure. The advantages of this relaxed formulation are illustrated with various examples in shallow and deep waters, as well as arbitrary depths. Using subordinate constraints (e.g., irrotationality or free surface impermeability) in various combinations, several model equations are derived, some being well-known, other being new. The models obtained are studied analytically and exact travelling wave solutions are constructed when possible.

Mots clés: water waves; variational principles; Lagrangian; Hamiltonian; shallow water; deep water

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

Didier Clamond, Denys Dutykh. Practical use of variational principles for modeling water waves. Physica D: Nonlinear Phenomena, 2012, 241 (1), pp.25-36. ⟨10.1016/j.physd.2011.09.015⟩. ⟨hal-00456891v6⟩