## 2 Full Form of (2+1)-Dimensional Fifth-Order KdV Equation and (2+1)-Dimensional Gardner Equation  
Recall the basic set of [PERSON]'s equations for an ideal fluid in the container with a flat, impenetrable bottom [10; 11; 12]. In dimensional variables, the set of hydrodynamic equations has the following form  
\[\phi_{xx}+\phi_{yy}+\phi_{zz} = 0,\quad\text{in}\quad\text{volume}, \tag{2}\] \[\phi_{z}-(\eta_{x}\phi_{x}+\eta_{y}\phi_{y}+\eta_{t}) = 0,\quad\text{at}\quad\text{the surface},\] (3) \[\phi_{t}+\frac{1}{2}(\phi_{x}^{2}+\phi_{y}^{2}+\phi_{z}^{2})+g \eta+\frac{p_{s}}{\varrho} = 0,\quad\text{at}\quad\text{the surface},\] (4) \[\phi_{z}-h_{x}\phi_{x}-h_{y}\phi_{y} = 0,\quad\text{at}\quad\text{the bottom}. \tag{5}\]  
Here, \(\phi(x,y,z,t)\) denotes the velocity potential, \(\eta(x,y,t)\) denotes the surface profile function, \(g\) is the gravitational acceleration, and \(\varrho\) is the density of the fluid. The indexes here and thereafter denote partial derivatives, i.e., \(\phi_{xx}\equiv\frac{\partial^{2}\phi}{\partial x^{2}}\), \(\eta_{t}=\frac{\partial\eta}{\partial t}\), and so on. In (5), \(p_{s}\) is the additional pressure due to the surface tension [13]  
\[p_{s}=-T\,\
abla\cdot\left(\frac{\
abla\eta}{(1+|\
abla\eta|^{2})^{1/2}}\right) =T\,\frac{\left(1+\eta_{y}^{2}\right)\eta_{xx}+\left(1+\eta_{x}^{2}\right)\eta _{yy}-2\eta_{x}\eta_{y}\eta_{xy}}{\left(1+\eta_{x}^{2}+\eta_{y}^{2}\right)^{3/2 }}, \tag{6}\]  
where \(T\) is the fluid's surface tension coefficient, and \(\
abla=(\partial_{x},\partial_{y})\) is the two-dimensional gradient. In the next step, we perform scaling transformation to dimensionless variables  
\[\bar{x}=x/L_{x},\quad\bar{y}=y/L_{y},\quad\bar{z}=z/H,\quad\bar{t }=t/(L_{x}/\sqrt{gH}),\] \[\bar{\eta}=\eta/A,\quad\bar{\phi}=\phi/(L_{x}\frac{A}{H}\sqrt{gH}), \tag{7}\]where \(A\) is the amplitude of the surface distortions from an equilibrium shape (flat surface), \(H\) is the average fluid depth, \(L_{x}\) is the average wavelength (in \(x\)-direction), and \(L_{y}\) is a wavelength in the \(y\)-direction. In general, \(L_{y}\) should be in the same order as \(L_{x}\), but not necessarily equal. Then, set (2)-(5), in scaled variables, takes the following form (here and thereafter, we omit the tilde signs and work in dimensionless variables) set yields  
\[\beta\phi_{xx}+\gamma\phi_{yy}+\phi_{zz} =0, \tag{8}\] \[\eta_{t}+\alpha(\eta_{x}\phi_{x}+\frac{\gamma}{\beta}\eta_{y}\phi _{y})-\frac{1}{\beta}\phi_{z} =0,\quad\text{for}\quad z=1+\alpha\eta,\] (9) \[\phi_{t}+\frac{1}{2}\alpha\left(\phi_{x}^{2}+\frac{\gamma}{\beta }\phi_{y}^{2}+\frac{1}{\beta}\phi_{z}^{2}\right)+\eta-\tau\left(\beta\eta_{xx }+\gamma\eta_{yy}\right) =0,\quad\text{for}\quad z=1+\alpha\eta,\] (10) \[\phi_{z}-(\eta_{x}\phi_{x}+\eta_{y}\phi_{y}+\eta_{t}) = 0,\quad\text{at}\quad\text{the surface},\]0  
Here, \(\alpha=\frac{A}{H}\), \(\beta=\left(\frac{H}{L_{x}}\right)^{2}\), and \(\gamma=\left(\frac{H}{L_{y}}\right)^{2}\). Equation (10) contains surface tension effects (see, e.g., [12]), which are important for waves on the surface of thin liquid layers (depths of the order of millimetres). For ordinary shallow water waves (depths of the order of metres, \(\tau\in(10^{-8}-10^{-6})\)), surface tension effects can be safely neglected by setting \(\tau=0\). Here, \(\tau=\frac{T}{\alpha\beta H^{2}}\) is the Bond number. \(T\) is the surface tension coefficient, \(\varrho\) is the density of the fluid, \(g\) is the gravitational acceleration, and \(H\) is the depth of the fluid. In (10), we retained only first- and second-order terms originating from the surface tension, neglecting other terms of the order \(O(\beta^{4})\), see the details in Appendix A of [12]. Dimensionless variables allow meaningful approximations to be introduced when solving the complex system of Equations (8)-(11). If we restrict ourselves to the cases where \(\alpha,\beta,\gamma\ll 1\), it is possible to obtain the corresponding wave equations in any order with respect to these parameters.