  We study the problem of gravity surface waves for an ideal fluid model in the
(2+1)-dimensional case. We apply a systematic procedure to derive the
Boussinesq equations for a given relation between the orders of four expansion
parameters, the amplitude parameter $\alpha$, the long-wavelength parameter
$\beta$, the transverse wavelength parameter $\gamma$, and the bottom variation
parameter $\delta$. We derived the only possible (2+1)-dimensional extensions
of the Korteweg-de Vries equation, the fifth-order KdV equation, and the
Gardner equation in three special cases of the relationship between these
parameters. All these equations are non-local. When the bottom is flat, the
(2+1)-dimensional KdV equation can be transformed to the Kadomtsev-Petviashvili
equation in a fixed reference frame and next to the classical KP equation in a
moving frame. We have found soliton, cnoidal, and superposition solutions
(essentially one-dimensional) to the (2+1)-dimensional Korteweg-de Vries
equation and the Kadomtsev-Petviashvili equation.
