On the Galilean invariance of some dispersive wave equations

Angel Duran (1) , Denys Dutykh (2, 3, 4, 5) , Dimitrios Mitsotakis (6)

1 MA - Departamento de Matemática Aplicada [Valladolid]
2 LAMA - Laboratoire de Mathématiques
3 School of Mathematical Sciences [Dublin]
4 INSMI-CNRS - Institut National des Sciences Mathématiques et de leurs Interactions - CNRS Mathématiques
5 USMB [Université de Savoie] [Université de Chambéry] - Université Savoie Mont Blanc
6 School of Natural Sciences

Surface water waves in ideal fluids have been typically modeled by asymptotic approximations of the full Euler equations. Some of these simplified models lose relevant properties of the full water wave problem. One of them is the Galilean symmetry, which is not present in important models such as the BBM equation and the Peregrine (Classical Boussinesq) system. In this paper we propose a mechanism to modify the above mentioned classical models and derive new, Galilean invariant models. We present some properties of the new equations, with special emphasis on the computation and interaction of their solitary-wave solutions. The comparison with full Euler solutions shows the relevance of the preservation of Galilean invariance for the description of water waves.

Mots clés: water waves; Galilean invariance; Boussinesq equations; Peregrine system; BBM equation; dispersive waves; solitary waves

Domaines: Mécanique des fluides [physics.class-ph]; Mécanique des fluides [physics.class-ph]; Dynamique des Fluides [physics.flu-dyn]; Physique Atmosphérique et Océanique [physics.ao-ph]; Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]; Physique Numérique [physics.comp-ph]

Angel Duran, Denys Dutykh, Dimitrios Mitsotakis. On the Galilean invariance of some dispersive wave equations. Studies in Applied Mathematics, 2013, 131 (4), pp.359-388. ⟨10.1111/sapm.12015⟩. ⟨hal-00664143v3⟩