  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.
