Boussinesq's equations for (2+1)-dimensional surface gravity waves in an ideal fluid model

We study the problem of gravity surface waves for the ideal fluid model in the (2+1)-dimensional case. We apply a systematic procedure for deriving the Boussinesq equations for a prescribed relationship between the orders of four expansion parameters, the amplitude parameter α, the long-wavelength parameter β, the transverse wavelength parameter γ, and the bottom variation parameter δ. We also take into account surface tension effects when relevant. For all considered cases, the (2+1)-dimensional Boussinesq equations cannot be reduced to a single nonlinear wave equation for surface elevation function. On the other hand, they can be reduced to a single, highly nonlinear partial differential equation for an auxiliary function f(x, y, t) which determines the velocity potential but is not directly observed quantity. The solution f of this equation, if known, determines the surface elevation function. We also show that limiting the obtained the Boussinesq equations to (1+1)-dimensions one recovers well-known cases of the KdV, extended KdV, fifth-order KdV, and Gardner equations.