(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 $\alpha$, 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.

Section 4, with soliton, cnoidal and superposition solutions to (2+1)-dimensional nonlocal KdV equation, added. In section 5 mistakes corrected. In Section 6 mistakes corrected

References in corpus (4)

• Superposition of Elliptic Functions as Solutions For a Large Number of Nonlinear Equations
• Boussinesq's equations for (2+1)-dimensional gravity-surface waves in an ideal fluid model
• Deformed two-dimensional rogue waves in the (2+1)-dimensional Korteweg-de Vries equation
• Comment on "Two-dimensional third- and fifth-order nonlinear evolution equations for shallow water waves with surface tension" [Nonlinear Dyn, doi:10.1007/s11071-017-3938-7]