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]

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.

Here φ (x, y, z,t) denotes the velocity potential, η(x, y,t) denotes the surface function and g is the gravitational acceleration. Indexes denote partial derivatives, i.e. φ xx ≡ ∂ 2 φ ∂ x 2 , and so on. The authors take into account surface tension terms, as well. In this note, we neglect these terms since their presence or absence does not change the source of errors made in [1].
Next, the authors introduce a standard scaling to dimensionless variables (different in x-and y-direction) x =x/L,ỹ = y/y 0 ,z = z/h 0 ,t = t/t 0 , where A is the amplitude of surface distortions from equilibrium shape (flat surface), h 0 is average fluid depth, L is the average wavelength (in x-direction), and y 0 is a wavelength in y-direction. In general, y 0 should be of the same order as L, but not necessarily equal. Notation t 0 = L/ √ gh 0 is not explained in the paper.
Then the set (1)-(4) takes in scaled variables the following form (tildas are now dropped): As usual, small parameters are defined as follows: 2 Details of calculations in [1] limited to first order Next, the authors assume γ = β and write erroneous formula [1, Eq. (11)] for the velocity potential In the following, the authors limit their considerations to the Boussinesq equations up to second order in small parameters. Then it is enough to use the explicit form of the potential (11) up to third order (due to terms 1 β φ z in (8) and 1 β φ 2 z in (9) the velocity potential should be valid up to one order higher than the Boussinesq equations) The fact that formulas (11)-(12) are wrong is easy to check by a direct substitution to the Laplace equation (7). With the above form of the velocity potential, the authors obtained the set of Boussinesq's equations in the form (here surface tension is neglected and only terms of first order are retained) Next, the authors insert the velocity potential into equations (8) and (9) retaining terms up to second order in α, ε = β . They introduce the following notation In the following, the authors apply the perturbative approach described in detail by Burde and Sergyeyev [2] and next extended in [3] to more complicated cases. In this method, one begins from zeroth-order solutions, then uses their properties in the calculation of corrections of the first order, and so on. In zeroth order eqs. [ Looking for the solution in first-order approximation, one introduces corrections of the first order to the equations (18) and requires that the Boussinesq's equations become compatible in this order (what means that these two equations become equivalent). The authors look for first-order solutions in the form Inserting first order corrections (19)-(20) into (16)-(17), retaining only terms up to first order, and using properties B t = −B x , C t = −C x the authors obtained Here, we cite the formulas from [1]. More detailed derivation, with the correct velocity potential formula, is presented in section 3. These equations, after integration give the formulas [1, Eq. (23)]-[1, Eq. (24)]. These formulas read as So, in first order approximation u becomes (in [1, Eq. (25)] terms at α and ε are incorrectly positioned) Insertion of u, given by the above expression into [ In other words, equation

Details of correct calculations in first order perturbation approach
The correct formula for the velocity potential fulfilling (1) has the following form For γ = β , the explicit form of this velocity potential up to third order in small parameters reads as The correct formula for the velocity potential (23) and (24) differs from that used by the authors ( (11) and (12)) by values of coefficients in front of mixed xy-derivatives.
With the correct formula (24), one obtains the following Boussinesq's equations. From (8), limiting to first order, one gets The correct result from (9) in first order (after differentiation over x) is In variables u, v (15), equations (25)-(26) become Equations (25) and consequently (27) differ from equations (13) and (14) obtained from incorrect velocity potential (11) by the factor in front of the term f xxyy . Let us follow the authors' approach with the correct equations (27)-(28). In zeroth order we have (18). In first order assume the correction functions in the same form (19)-(20).
Substitution to (27) and limitation to first order terms gives Substitution to (28) and retention of terms up to first order gives Substraction of (30) from (29) gives In (31), we use the properties B t = −B x , C t = −C x , η 2yt = −η 2yx , η 2xt = −η 3x valid in zeroth order. Since expressions in (31) are already in first order it is sufficient. Recall that the general form, e.g., B t = −B x + αBa + εBe, and so on, do not change the further results since after insertion into (31) second order terms have to be rejected. These properties and freedom of α, ε allow us to obtain simple differential equations for B and C in the form Integrating above equations one obtains Then, the function u becomes With u in the form (33), both Boussinesq's equations should reduce to the same wave equation. Indeed, this is the case, and the final first order wave equation receives the following form η t + η x + α 3 2 ηη x + 1 2 a η y + ε 1 6 η xxx + 1 12 η xyy = 0.

Critics
The authors treat u, v as independent functions, ignoring that, in fact, u, v are partial derivatives of the same function f . Since u = f x and v = f y , and these functions and their partial derivatives should be continuous, then the fundamental condition has to be fulfilled Inserting (36)-(37) into (27)-(28) and neglecting terms of second order one obtains Substraction of (39) from (38) and use the properties B t = −B x , C t = −C x allows us to obtain two equations 2B x + η η x − a a x + a η y + R y = 0, and Integration over x gives (2+1)-dimensional shallow water problem is highly complicated. There is little hope to find either a solution to (49) or a (2+1)-dimensional wave equation for wave profile η(x, y,t) without further significant simplifications.

Conclusions
We have proved that the (2+1)-dimensional KdV-type equation [1, Eq. (26)] has been inconsistently obtained by the authors and therefore cannot describe (2+1)-dimensional surface waves. Moreover, we have shown that when assumptions for first order functions [1, Eqs. (14)-(15)] are used consistently with the properties of the velocity potential, then the solution reduces to the usual KdV equation. Additionally, we have demonstrated that even a consistent extension of the authors' method [1] gives no hope for obtaining appropriate first order (2+1)-dimensional evolution equation for shallow water problem.