## 5 Summary and Conclusions  
The (2+1)-dimensional fifth-order KdV and Gardner equations found in previous work [10] were not complete, i.e., they did not contain all of the possible second-order terms. Equations (17) and (21) in their present form gained soliton solutions analogous to the solutions of the one-dimensional fifth-order KdV and Gardner equations.  
It is worth summarizing our results in [10,11] and the current work. We derived (2+1)-dimensional extensions of the famous equations: Korteweg-de Vries Equation (32) of [10], extended KdV Equation (32) of [11], and the full form of 5 th-order KdV and Gardner Equations ((17) and (21)). These equations were rigorously derived for the irrotational motion of an ideal fluid. Then, we showed that all the above-mentioned (2+1)-dimensional equations have families of solutions analogous to the corresponding one-dimensional equations. Moreover, though the newly derived equations are explicitly (2+1)-dimensional, the found traveling wave solutions were, in fact, one-dimensional. So, a fundamental question can be posed. Are there other truly two-dimensional solutions to these new equations?  
Central to the derivations of these equations is the assumption that the parameter \(\gamma=\left(H/L_{y}\right)^{2}\) is essentially smaller than \(\beta=\left(H/L_{x}\right)^{2}\). In other words, this assumption means that the average wavelength in the \(y\)-direction is much larger than that in the \(x\)-direction. Only this assumption \(\gamma\ll\beta\) allows the system of [PERSON]'s equations to be  
Figure 3: Top row: Three snapshots of the solitary wave solution (52) for the (2+1)-dimensional Gardner equation in two dimensions at \(t=0,1,2\). Bottom row: Three snapshots of the solitary wave solution (53) at \(t=0,2,4\). Left: 3D plots of \(\eta(x,y,t)\). Right: Plots of \(\eta(x^{\prime},t)\) in rotated coordinates.  
reduced to a single nonlinear equation for the wave profile. It is possible as \(y\)-derivatives do not appear in zeroth order Boussinesq equations.  
On the other hand, the assumption \(\gamma\ll\beta\) (or \(L_{y}\gg L_{x}\)) is difficult to maintain. It seems that the wave propagation on the water surface should be the same in all directions. However, when \(\gamma=\beta\) (or both are of the same order), the Boussinesq equations contain \(y\)-derivatives already in the zeroth order [12]. This does not allow us to remove the function \(f(x,y,t)\) from the system of Boussinesq equations, which determines the velocity potential, to obtain the wave equation on the surface of the liquid. In principle, approximate solutions \(\eta(x,y,t)\) can be obtained indirectly. It is possible to eliminate the surface profile function \(\eta(x,y,t)\) from the Boussinesq equations, obtaining the wave equation for the function \(f(x,y,t)\). If the solution of this equation was known, then the function \(\eta(x,y,t)\) could be obtained from it [12]. However, this problem is much more complicated than those considered so far, and has not yet been solved.  
All authors [PERSON] and P.R. contributed equally in all steps of the current work. All authors have read and agreed to the published version of the manuscript. This research received no external funding.  
Not applicable  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The authors declare no conflicts of interest.