Derivation of a viscous KP equation including surface tension, and related equations
arXiv:1506.00456
Abstract
The aim of this article is to derive surface wave models in the presence of surface tension and viscosity. Using the Navier-Stokes equations with a free surface, flat bottom and surface tension, we derive the viscous 2D Boussinesq system with a weak transverse variation. The assumed transverse variation is on a larger scale than along the main propagation direction. This Boussinesq system is proved to be consistent with the Navier-Stokes equations. This system is only an intermediate result that enables us to derive the Kadomtsev-Petviashvili (KP) equation which is a 2D generalization of the KdV equation. In addition, we get the 1D KdV equation, and lastly the Boussinesq equation. All these equations are derived for non-vanishing initial conditions.