(2+1)-dimensional KdV, fifth-order KdV, and Gardner equations derived from the ideal fluid model. Soliton, cnoidal and superposition solutions

Abstract

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 α, the long-wavelength parameter β, the transverse wavelength parameter γ, and the bottom variation parameter δ. Such an approach, known in (1+1)-dimensional theory, is extended for the first time for (2+1)-dimensional shallow water waves. Using this procedure, 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 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. Thus, the Kadomtsev–Petviashvili equation gained a derivation from the fundamental laws of hydrodynamics. 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.