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]

Abstract

The authors of the paper “Two-dimensional third-and fifth-order nonlinear evolution equations for shallow water waves with surface tension” Fokou et al. (Nonlinear Dyn 91:1177–1189, 2018) 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)-dimensional equation. The equation has been obtained by applying the perturbation method Burde (J Phys A: Math Theor 46:075501, 2013) 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 Fokou et al. (Nonlinear Dyn 91:1177–1189, 2018) 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.

Boussinesq’s equations for (2+1)-dimensional surface gravity waves in an ideal fluid model

“…In [13], the authors presented the derivation of two new (2+1)-dimensional third-and fifth-order nonlinear evolution equations when α, β, γ are of the same order and the bottom is flat (δ =0 ). However, as we proved in [14], all results shown in [13] are false since the derivation is inconsistent and violates the fundamental property of the velocity potential. When the method used by the authors is applied consistently, the problem is reduced to well known KdV equation.…”

“…The only such attempt known to us is the work [13]. Unfortunately, this particular paper is erroneous, as we demonstrated in [14]. To our best knowledge, there are no correct (2+1)-dimensional studies that take into account the model of an ideal fluid in full detail.…”

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

“…Note that the last three terms in both equations are second order ones. Zeroth-order equations ( 45)-( 46) are the same as those of ( 18)- (19). Therefore in zeroth-order relations (20) hold.…”

“…Zeroth-order equations ( 68)-( 69) are the same as those of ( 18)- (19). Therefore in zeroth-order relations (20) hold.…”

“…The only such attempt known to us is the work [18]. Unfortunately, this particular paper is erroneous, as we have shown in the work [19]. To the best of our knowledge, there are no correct (2+1)-dimensional papers that consider the ideal fluid model in full detail.…”