The Whitham Equation with Surface Tension

Abstract

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 the Whitham 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.

19 pages, 5 figures, 1 table, 36 references. Other author's papers can be downloaded at http://www.denys-dutykh.com/. arXiv admin note: text overlap with arXiv:1410.8299

References in corpus (4)

• On the Water Waves Equations with Surface Tension
• The Whitham Equation as a Model for Surface Water Waves
• On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type
• On the nonlinear dynamics of the traveling-wave solutions of the Serre system

Cited by in corpus (6)

• Discontinuous shock solutions of the Whitham modulation equations as dispersionless limits of traveling waves
• A comparative study of bi-directional Whitham systems
• Solitary wave solutions and their interactions for fully nonlinear water waves with surface tension in the generalized Serre equations
• Well-posedness for a Whitham-Boussinesq system with surface tension
• Numerical bifurcation and stability for the capillary-gravity Whitham equation
• Stability of Periodic, Traveling-Wave Solutions to the Capillary-Whitham Equation