## 4 Soliton Solutions to (2+1)-Dimensional Gardner Equation
For \(k,l\geq 0\), the condition \(A_{2}^{2}>0\) holds when  
\[+\frac{\alpha\gamma}{\beta}\bigg{(}\frac{1}{8}\int(\eta_{y}^{2}+ \eta\eta_{yy})\,dx+\frac{1}{8}\eta\int\eta_{yy}\,dx+\eta_{y}\int\eta_{y}\,dx -\frac{1}{2}\eta_{x}\int(\int\eta_{yy}\,dx)\,dx\bigg{)}=0.\]4The second case, when \(\tau\) is of the order of \(1/3\), corresponds to thin fluid layers. When \(0<\tau<\frac{1}{3}\), then \(A_{2}^{2}>0\) holds when  
\[+\frac{\alpha\gamma}{\beta}\bigg{(}\frac{1}{8}\int(\eta_{y}^{2}+ \eta\eta_{yy})\,dx+\frac{1}{8}\eta\int\eta_{yy}\,dx+\eta_{y}\int\eta_{y}\,dx -\frac{1}{2}\eta_{x}\int(\int\eta_{yy}\,dx)\,dx\bigg{)}=0.\]5  
\[+\frac{\alpha\gamma}{\beta}\bigg{(}\frac{1}{8}\int(\eta_{y}^{2}+ \eta\eta_{yy})\,dx+\frac{1}{8}\eta\int\eta_{yy}\,dx+\eta_{y}\int\eta_{y}\,dx -\frac{1}{2}\eta_{x}\int(\int\eta_{yy}\,dx)\,dx\bigg{)}=0.\]6  
When \(\tau\geq\frac{1}{3}\), then \(A_{2}^{2}>0\) holds for arbitrary \(k,l\). Therefore, for an arbitrary value of the Bond number \(\tau\), a family of solutions (37) exists, i.e., there always exist some ranges of admissible values of parameters \(k,l\) of solution. It is worth noting that by changing \(k\to-k\) and \(l\to-l\), one obtains solutions moving in the opposite directions. To illustrate possible solutions, consider a particular example. Set, as in Section 3, \(\alpha=0.2\), \(\beta=0.16\), \(\gamma=0.05\). First, assume that \(\tau=0\). Choose \(k=3,\ l=2\), admissible by (44) the left part. Then, we have: \(A_{1}\approx 4.4883,\ A_{2}\approx 0.88891,\ \omega\approx 3.9211\). For these coefficients the wave is  
\[+\frac{\alpha\gamma}{\beta}\bigg{(}\frac{1}{8}\int(\eta_{y}^{2}+ \eta\eta_{yy})\,dx+\frac{1}{8}\eta\int\eta_{yy}\,dx+\eta_{y}\int\eta_{y}\,dx -\frac{1}{2}\eta_{x}\int(\int\eta_{yy}\,dx)\,dx\bigg{)}=0.\]7  
In the second example with \(\tau=0\), we choose \(k=7,\ l=7\), admissible by (44) the right part. Then, we have \(A_{1}\approx 22.6018,\ A_{2}\approx 0.150415,\ \omega\approx 17.155\). For these coefficients, the wave is  
\[+\frac{\alpha\gamma}{\beta}\bigg{(}\frac{1}{8}\int(\eta_{y}^{2}+ \eta\eta_{yy})\,dx+\frac{1}{8}\eta\int\eta_{yy}\,dx+\eta_{y}\int\eta_{y}\,dx -\frac{1}{2}\eta_{x}\int(\int\eta_{yy}\,dx)\,dx\bigg{)}=0.\]8  
Does the (2+1)-dimensional Gardner equation in two dimensions (21) admit table-top soliton solutions for shallow water waves? The answer is yes. Remember that these solutions appear when \(A_{2}\to 0\). In the considered example, when \(\alpha=0.2\), \(\beta=0.16\), \(\gamma=0.05\), we can find such \(k,l\), which provide small \(A_{2}\to 0\). For instance, with \(k\ =\ 7,l\ =\ 6.69886\), we have \(A_{1}\approx 22.8619,\ A_{2}\approx 0.000608695,\ \omega\approx 17.0767\). The wave is  
\[+\frac{\alpha\gamma}{\beta}\bigg{(}\frac{1}{8}\int(\eta_{y}^{2}+ \eta\eta_{yy})\,dx+\frac{1}{8}\eta\int\eta_{yy}\,dx+\eta_{y}\int\eta_{y}\,dx -\frac{1}{2}\eta_{x}\int(\int\eta_{yy}\,dx)\,dx\bigg{)}=0.\]9  
The graphical presentation of these waves is shown in Figure 1:0 The 3D graphics present three snapshots (overlapped) of the waves' (47)-(49) profiles. The 2D graphic shows the profiles of these three waves plotted in rotated coordinates. In all three cases, \(t=0,2,4\), respectively. The waves exhibit very different amplitudes, widths, and velocities. Therefore, there is a wide family of possible Gardner solitons for the case of shallow water waves, which are related to the case when the surface tension effects can be neglected (\(\tau\approx 0\)). There are also interesting, physically relevant cases when \(\tau\) is close to \(\frac{1}{3}\), corresponding to waves on thin fluid layer surfaces. Similar to before, we limited our examples to positive \(k,l\). Then, when \(\tau<\frac{1}{3}\) the condition \(A_{2}^{2}>0\) requires  
Figure 1:1  
Figure 1:2Consider a case when \(\tau=0.2\) and, as in other examples, \(\alpha=0.2\), \(\beta=0.16\), \(\gamma=0.05\). Take \(k=10\) and \(l=27\), admissible by (51). Then, \(A_{1}=9.97322\), \(A_{2}=0.875716\) and \(\omega=25.57\).