## 3 Soliton Solutions to (2+1)-Dimensional Fifth-Order KdV Equation  
In [11], we found that the form of soliton and periodic solutions to the (2+1)-dimensional KdV and extended KdV equations remain the same as those of (1+1)-dimensional equations. This inspired us to search for solutions to the (2+1)-dimensional fifth-order KdV equation in the same form as solutions for the (1+1)-dimensional fifth-order KdV equation. Assuming travelling wave solutions in the form (1), that is, \(\eta(kx+ly-\omega t)=\eta(x^{\prime}-\omega t)\), one can write Equation (17) as  
\[\eta_{t} +\bigg{(}k+\frac{\gamma}{2\beta}\frac{l^{2}}{k}-\frac{\gamma^{2} l^{4}}{8\beta^{2}k^{3}}\bigg{)}\eta_{x^{\prime}}+\frac{3\alpha}{2}k\,\eta\eta_{x^{ \prime}}+\bigg{(}\frac{\beta}{6}k^{3}+\frac{\gamma}{4}kl^{2}\bigg{)}(1-3\tau) \eta_{3x^{\prime}}\] \[+\beta^{2}\bigg{(}\frac{19-30\tau-45\tau^{2}}{360}\bigg{)}k^{5}\, \eta_{5x^{\prime}}=0. \tag{22}\]  
Equation (22) is in the form of a one-dimensional fifth-order KdV equation in a fixed reference frame, see Equation (31) of [14]. Therefore, we will look for solutions to Equation (17) in the form analogous to solutions of the one-dimensional fifth-order KdV equation. So, we postulate travelling wave solutions to (17) as  
\[\eta(x,y,t)=A_{0}+A_{4}\,\text{sech}(kx+ly-\omega t)^{4}=A_{0}+A_{4}\,\text{ sech}(\xi)^{4}, \tag{23}\]  
where \(\xi=kx+ly-\omega t\). For \(\eta\), in the form (23), Equation (17) becomes \[\frac{A_{4}}{90k^{3}\beta^{2}}\Big{[}-45\Big{(}4\beta^{2}k^{4}(3\alpha A _{0}+2)-8\beta^{2}k^{3}\omega+4\beta\gamma k^{2}l^{2}-\gamma^{2}l^{4}\Big{)}\] \[-4\beta^{2}k^{4}\text{sech}^{4}(\xi)\Big{(}135\alpha A_{2}-94\beta^ {2}k^{4}\Big{(}45\tau^{2}+30\tau-19\Big{)}\Big{)} \tag{24}\] \[+256\beta^{4}k^{8}\Big{(}45\tau^{2}+30\tau-19\Big{)}\tanh^{4}(\xi) +480\beta^{2}k^{4}(3\tau-1)\tanh^{2}(\xi)\Big{(}2\beta k^{2}+3\gamma l^{2}\Big{)}\] \[-4\text{sech}^{2}(\xi)\Big{(}262\beta^{4}k^{8}(45\tau^{2}+30\tau-1 9)\tanh^{2}(\xi)+105\beta^{2}k^{4}(3\tau-1)(2\beta k^{2}+3\gamma l^{2})\Big{)} \Big{]}=0.\]  
This equation, when reduced to a single elliptic function, becomes  
\[\frac{A_{4}}{90k^{3}\beta^{2}}\Big{(}C_{0}+C_{2}\,\text{sech}^{2}(\xi)+C_{4}\, \text{sech}^{4}(\xi)\Big{)}=0. \tag{25}\]  
By equating \(C_{i}\) to zero, we obtain three equations for the parameters of the solution (23). \[C_{0}= -180\beta^{2}k^{4}\Big{(}3\alpha A_{0}+8\gamma l^{2}(1-3\tau)+2 \Big{)}+256\beta^{4}k^{8}\Big{(}45\tau^{2}+30\tau-19\Big{)}\] \[+960\beta^{3}k^{6}(3\tau-1)+360\beta^{2}k^{3}\omega-180\beta\gamma k ^{2}l^{2}+45\gamma^{2}l^{4}=0, \tag{26}\] \[+\beta^{2}\bigg{(}\frac{19-30\tau-45\tau^{2}}{360}\bigg{)}k^{5}\, \eta_{5x^{\prime}}=0. \tag{22}\]0 (27) \[+\beta^{2}\bigg{(}\frac{19-30\tau-45\tau^{2}}{360}\bigg{)}k^{5}\, \eta_{5x^{\prime}}=0. \tag{22}\]1  
Then, from (28)  
\[+\beta^{2}\bigg{(}\frac{19-30\tau-45\tau^{2}}{360}\bigg{)}k^{5}\, \eta_{5x^{\prime}}=0. \tag{22}\]2  
and from (27)  
\[+\beta^{2}\bigg{(}\frac{19-30\tau-45\tau^{2}}{360}\bigg{)}k^{5}\, \eta_{5x^{\prime}}=0. \tag{22}\]3  
Equation (26), with \(l^{2}\) given by (30), provides  
\[+\beta^{2}\bigg{(}\frac{19-30\tau-45\tau^{2}}{360}\bigg{)}k^{5}\, \eta_{5x^{\prime}}=0. \tag{22}\]4 \[+\beta^{2}\bigg{(}\frac{19-30\tau-45\tau^{2}}{360}\bigg{)}k^{5}\, \eta_{5x^{\prime}}=0. \tag{22}\]5  
We see, that the (2+1)-dimensional fifth-order KdV Equation (17) admits the family of soliton solutions in the form (23). From a mathematical point of view, \(A_{0}\) can be arbitrary. The physical soliton solution, however, requires \(A_{0}=0\) for the volume of the liquid to be the same as that of the undisturbed liquid. Equation (30) restricts the admissible parameters \(k,l\) as \(l\) must be real. By definition, see [11], \(\beta>0,\gamma>0\), and the bond number \(\tau\geq 0\).