Evolution equation for weakly nonlinear waves in a two-layer fluid with gently sloping bottom and lid

A second-order differential model for three-dimensional perturbations of the interface of two fluids of different density is constructed. An evolution equation for traveling quasistationary waves of arbitrary length and small but finite amplitude is obtained. In the case of the horizontal bottom and lid, there are perturbations of the Stokes-wave type among steady-state periodic solutions. For moderately long perturbations, solutions in the form of solitary waves which are in agreement with the available experimental and analytical results are found. The problem of a smooth transition from the deep-fluid to the shallow-fluid region is studied.