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]

TL;DR

This paper critically examines a recent derivation of two-dimensional nonlinear evolution equations for shallow water waves, finding that the original derivation is inconsistent and thus the claimed results are invalid.

Contribution

It provides a detailed critique showing the derivation's inconsistency, clarifying the correct properties of the velocity potential in such models.

Findings

The original derivation violates fundamental properties of the velocity potential.

The claimed equations and solutions are not valid due to derivation errors.

The critique clarifies the correct approach for deriving such equations.

Abstract

The authors of the paper "Two-dimensional third- and fifth-order nonlinear evolution equations for shallow water waves with surface tension" \cite{Fok} claim that they derived the equation which generalizes the KdV equation to two space dimensions both in first and second order in small parameters. Moreover, they claim to obtain soliton solution to the derived first order (2+1)-dimension equation. The equation has been obtained by applying the perturbation method \cite{burde} for small parameters of the same order. The results, if correct, would be significant. In this comment, it is shown that the derivation presented in \cite{Fok} is inconsistent because it violates fundamental properties of the velocity potential. Therefore, the results, particularly the new evolution equation and the dynamics that it describes, bear no relation to the problem under consideration.