The Whitham equation with surface tension

Evgueni Dinvay (1) , Daulet Moldabayev (1) , Denys Dutykh (2, 3, 4) , Henrik Kalisch (1)

1 Department of Mathematics
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

The viability of the Whitham equation as a nonlocal model for capillary-gravity waves at the surface of an inviscid incompressible fluid is under study. A nonlocal Hamiltonian system of model equations is derived using the Hamiltonian structure of the free-surface water-wave problem and the Dirichlet-Neumann operator. The system features gravitational and capillary effects, and when restricted to one-way propagation, the system reduces to the capillary Whitham equation. It is shown numerically that in various scaling regimes theWhitham equation gives a more accurate approximation of the free-surface problem for the Euler system than other models like the KdV and Kawahara equation. In the case of relatively strong capillarity considered here, the KdV and Kawahara equations outperform the Whitham equation with surface tension only for very long waves with negative polarity.

Mots clés: Hamiltonian systems; Capillarity; Fully dispersive equations; Surface waves

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

Evgueni Dinvay, Daulet Moldabayev, Denys Dutykh, Henrik Kalisch. The Whitham equation with surface tension. Nonlinear Dynamics, 2017, 88 (2), pp.1125-1138. ⟨10.1007/s11071-016-3299-7⟩. ⟨hal-01504047⟩