  This paper presents the second-order perturbation theory of the Navier-Stokes
equations for free surface flows, with the wave amplitude considered as the
perturbation parameter. Gravity-capillary surface waves in incompressible
viscous fluids are subjected. The results provide a systematic derivation of a
nonlinear surface wave equation that fully takes into account dispersion, while
nonlinearity is included in the leading order. However, the presence of
infinitely many overdamped modes has been neglected, and only the two least
damped modes are considered. Finally, to describe the elevations and evolution
of the surface wave, we introduce a differential equation.
