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

TL;DR

This paper derives (2+1)-dimensional extensions of the KdV, fifth-order KdV, and Gardner equations from an ideal fluid model, and finds soliton, cnoidal, and superposition solutions for these equations.

Contribution

It systematically derives the only possible (2+1)-dimensional non-local extensions of these classical equations for specific parameter relations.

Findings

Derived (2+1)-dimensional KdV, fifth-order KdV, and Gardner equations.

Found soliton, cnoidal, and superposition solutions for these equations.

Connected the (2+1)-dimensional KdV to the KP equation in flat-bottom cases.

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.