1. Introduction



2. A brief description of the modified $\left(\tfrac{G^{\prime} }{G}\right)$
-expansion scheme
• | Postulation 1: calculate m using the rule of the homogeneous analysis in ( |
• | Postulation 2: the modified $\left(\tfrac{G^{\prime} }{G}\right)$ $\begin{eqnarray}h(\xi )=\displaystyle \sum _{i=-m}^{m}{A}_{i}{{\rm{\Theta }}}^{i},\end{eqnarray}$ where ${\rm{\Theta }}=\left(\tfrac{{G}^{{\prime} }}{G}+\tfrac{\lambda }{2}\right)$ ![]() $\begin{eqnarray}{G}^{{\prime\prime} }+\lambda {G}^{{\prime} }+\mu G=0,\end{eqnarray}$ where Ai, $\delta$ and $\mu$ are free parameters. From ( $\begin{eqnarray}{\rm{\Theta }}^{\prime} =b-{{\rm{\Theta }}}^{2},\end{eqnarray}$ where $b=\tfrac{{\lambda }^{2}-4\mu }{4}$ ![]() If b > 0, then $\begin{eqnarray}{\rm{\Theta }}=\sqrt{b}\tanh (\sqrt{b}\xi ),\end{eqnarray}$ $\begin{eqnarray}{\rm{\Theta }}=\sqrt{b}\coth (\sqrt{b}\xi ),\end{eqnarray}$ If b = 0, then $\begin{eqnarray}{\rm{\Theta }}=\displaystyle \frac{1}{\xi },\end{eqnarray}$ If b < 0, then $\begin{eqnarray}{\rm{\Theta }}=-\sqrt{-b}\tan (\sqrt{-b}\xi ),\end{eqnarray}$ $\begin{eqnarray}{\rm{\Theta }}=\sqrt{-b}\cot (\sqrt{-b}\xi ).\end{eqnarray}$ |
• | Postulation 3: inserting ( |
3. Soliton solutions for the model system related to ISWs and LWs


• | Result 1: $q=\pm \tfrac{\sqrt{4\mu {r}^{2}-2{p}^{2}-{\lambda }^{2}{r}^{2}}}{2}$ $\begin{eqnarray*}\begin{array}{rcl}{E}_{11}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\\ & & \times \ \left.\coth \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right],\\ {n}_{11}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\\ & & \times \ {\left.\coth \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right]}^{2}.\\ {E}_{12}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\\ & & \times \ \left.\tanh \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right],\\ {n}_{12}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\end{array}\end{eqnarray*}$ $\begin{eqnarray*}\begin{array}{rcl} & & \times \ {\left.\tanh \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right]}^{2}.\\ {E}_{13}(x,t) & = & -{{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\\ & & \times \ \left.\cot \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right],\\ {n}_{13}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[-\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\\ & & \times \ {\left.\cot \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right]}^{2}.\\ {E}_{14}(x,t) & = & -{{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\end{array}\end{eqnarray*}$ $\begin{eqnarray*}\begin{array}{rcl} & & \times \ \left.\tan \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right],\\ {n}_{14}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\\ & & \times \ {\left.\tan \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right]}^{2}.\\ {E}_{15}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r({\lambda }^{2}-4\mu )}{4}\right.\\ & & \times \ \left.\displaystyle \frac{1}{({rx}+{st})}\right],\\ {n}_{15}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r({\lambda }^{2}-4\mu )}{4}\right.\\ & & \times \ {\left.\displaystyle \frac{1}{({rx}+{st})}\right]}^{2}.\end{array}\end{eqnarray*}$ |
• | Result 2: $q=\pm \tfrac{\sqrt{4\mu {r}^{2}-2{p}^{2}-{\lambda }^{2}{r}^{2}}}{2}$ $\begin{eqnarray*}\begin{array}{rcl}{E}_{21}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\\ & & \times \ \left.\tanh \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right],\\ {n}_{21}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\\ & & \times \ {\left.\tanh \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right]}^{2}.\\ {E}_{22}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\\ & & \times \ \left.\coth \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right],\\ {n}_{22}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\\ & & \times \ {\left.\coth \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right]}^{2}.\\ {E}_{23}(x,t) & = & -{{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\\ & & \times \ \left.\tan \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right],\end{array}\end{eqnarray*}$ $\begin{eqnarray*}\begin{array}{rcl}{n}_{23}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[-\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\\ & & \times \ {\left.\tan \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right]}^{2}.\end{array}\end{eqnarray*}$ $\begin{eqnarray*}\begin{array}{rcl}{E}_{24}(x,t) & = & -{{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\\ & & \times \ \left.\cot \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right],\\ {n}_{24}(x,t) & = & -\displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\\ & & \times \ {\left.\cot \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right]}^{2}.\\ {E}_{25}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r({\lambda }^{2}-4\mu )}{4}\right.\\ & & \times \ \left.\displaystyle \frac{1}{({rx}+{st})}\right],\\ {n}_{25}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r({\lambda }^{2}-4\mu )}{4}\right.\\ & & \times \ {\left.\displaystyle \frac{1}{({rx}+{st})}\right]}^{2}.\end{array}\end{eqnarray*}$ |
• | Result 3: $q=\pm \sqrt{\tfrac{{\lambda }^{2}{r}^{2}-4\mu {r}^{2}-{p}^{2}}{2}}$ $\begin{eqnarray*}\begin{array}{rcl}{E}_{31}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\\ & & \times \ \tanh \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\\ & & -\ \left.\displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\right.\\ & & \times \ \left.\coth \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right],\\ {n}_{31}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\\ & & \times \ \tanh \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\\ & & -\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\end{array}\end{eqnarray*}$ $\begin{eqnarray*}\begin{array}{rcl} & & \times \ {\left.\coth \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right]}^{2}.\\ {E}_{32}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\\ & & \times \ \coth \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\\ & & -\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\\ & & \times \ \left.\tanh \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right],\end{array}\end{eqnarray*}$ $\begin{eqnarray*}\begin{array}{rcl}{n}_{32}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\\ & & \times \ \coth \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\\ & & -\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\\ & & \times \ {\left.\tanh \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right]}^{2}.\\ {E}_{33}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\\ & & \times \ \tan \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\\ & & -\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\\ & & \times \ \left.\cot \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right],\end{array}\end{eqnarray*}$ $\begin{eqnarray*}\begin{array}{rcl}{n}_{33}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\\ & & \times \ \tan \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\\ & & -\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\\ & & \times \ {\left.\cot \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right]}^{2}.\\ {E}_{34}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\end{array}\end{eqnarray*}$ $\begin{eqnarray*}\begin{array}{rcl} & & \times \ \cot \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\\ & & -\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\\ & & \times \ \left.\tan \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right],\\ {n}_{34}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\\ & & \times \ \cot \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\\ & & -\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\\ & & \times \ {\left.\tan \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right]}^{2}.\end{array}\end{eqnarray*}$ $\begin{eqnarray*}\begin{array}{rcl}{E}_{35}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r({\lambda }^{2}-4\mu )}{4}\right.\\ & & \times \ \displaystyle \frac{1}{({rx}+{st})}\\ & & -\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{8{f}_{1}}\\ & & \times \ \left.\displaystyle \frac{1}{({rx}+{st})}\right],\\ {n}_{35}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r({\lambda }^{2}-4\mu )}{4}\right.\\ & & \times \ \displaystyle \frac{1}{({rx}+{st})}\\ & & -\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{8{f}_{1}}\\ & & \times \ {\left.\displaystyle \frac{1}{({rx}+{st})}\right]}^{2}.\end{array}\end{eqnarray*}$ |
• | Result 4: $q=\pm \sqrt{\tfrac{8\mu {r}^{2}-2{\lambda }^{2}{r}^{2}-{p}^{2}}{2}}$ $\begin{eqnarray*}\begin{array}{rcl}{E}_{41}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\\ & & \times \ \tanh \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\\ & & +\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\\ & & \times \ \left.\coth \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right],\end{array}\end{eqnarray*}$ $\begin{eqnarray*}\begin{array}{rcl}{n}_{41}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\\ & & \times \ \tanh \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\\ & & +\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\\ & & \times \ {\left.\coth \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right]}^{2}.\\ {E}_{42}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\\ & & \times \ \coth \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\\ & & +\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\\ & & \times \ \left.\tanh \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right],\end{array}\end{eqnarray*}$ $\begin{eqnarray*}\begin{array}{rcl}{n}_{42}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r\sqrt{{\lambda }^{2}-4\mu }}{2}\right.\\ & & \times \ \coth \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\\ & & +\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\\ & & \times \ {\left.\tanh \left\{\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}({rx}+{st})\right\}\right]}^{2}.\\ {E}_{43}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\\ & & \times \ \tan \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\\ & & +\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\\ & & \times \ \left.\cot \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right],\end{array}\end{eqnarray*}$ $\begin{eqnarray*}\begin{array}{rcl}{n}_{43}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\\ & & \times \ \tan \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\\ & & +\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\end{array}\end{eqnarray*}$ $\begin{eqnarray*}\begin{array}{rcl} & & \times \ {\left.\cot \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right]}^{2}.\\ {E}_{44}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\\ & & \times \ \cot \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\\ & & +\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\\ & & \times \ \left.\tan \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right],\end{array}\end{eqnarray*}$ $\begin{eqnarray*}\begin{array}{rcl}{n}_{44}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r\sqrt{4\mu -{\lambda }^{2}}}{2}\right.\\ & & \times \ \cot \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\\ & & +\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{4{f}_{1}\sqrt{4\mu -{\lambda }^{2}}}\\ & & \times \ {\left.\tan \left\{\displaystyle \frac{\sqrt{4\mu -{\lambda }^{2}}}{2}({rx}+{st})\right\}\right]}^{2}.\\ {E}_{45}(x,t) & = & {{\rm{e}}}^{{\rm{i}}({px}+{qt})}\left[\displaystyle \frac{{f}_{1}r({\lambda }^{2}-4\mu )}{4}\right.\\ & & \times \ \displaystyle \frac{1}{({rx}+{st})}\\ & & +\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{8{f}_{1}}\\ & & \times \ \left.\displaystyle \frac{1}{({rx}+{st})}\right],\\ {n}_{45}(x,t) & = & \displaystyle \frac{2}{{p}^{2}-1}\left[\displaystyle \frac{{f}_{1}r({\lambda }^{2}-4\mu )}{4}\right.\\ & & \times \ \displaystyle \frac{1}{({rx}+{st})}\\ & & +\ \displaystyle \frac{r({p}^{2}{\lambda }^{2}-4{p}^{2}\mu -{\lambda }^{2}+4\mu )}{8{f}_{1}}\\ & & \times \ {\left.\displaystyle \frac{1}{({rx}+{st})}\right]}^{2}.\end{array}\end{eqnarray*}$ |
3.1. Physical and graphical explanation of constructed solutions
Figure 1. Graphical descriptions of the solution E11(x, t) under the values p = 2, q = 0.5, r = 2, $\mu$ = 1, $\delta$ = 3 and t = 0.01 for 2D graphics. |
Figure 2. Graphical descriptions of the solution n11(x, t) under the values p = 2, q = 0.5, r = 2, $\mu$ = 1, $\delta$ = 3 and t = 0.01 for 2D graphics. |
Figure 3. Graphical descriptions of the solution E13(x, t) when p = 2, q = 0.5, r = 2, $\mu$ = 2, $\delta$ = 2 and for 2D graphics t = 0.01. |
Figure 4. Graphical descriptions of the solution n13(x, t) when p = 2, q = 0.5, r = 2, $\mu$ = 2, $\delta$ = 2 and for 2D graphics t = 0.01. |
Figure 5. Graphical descriptions of the solution E14(x, t) when p = 2, q = 0.5, r = 2, $\mu$ = 2, $\delta$ = 2 and for 2D graphics t = 0.01. |
Figure 6. Graphical descriptions of the solution n14(x, t) when p = 2, q = 0.5, r = 2, $\mu$ = 2, $\delta$ = 2 and for 2D graphics t = 0.01. |
Figure 7. Graphical descriptions of the solution E23(x, t) when p = 2, q = 0.5, r = 2, $\mu$ = 2, $\delta$ = 2 and for 2D graphics t = 0.01. |
Figure 8. Graphical descriptions of the solution n23(x, t) when p = 2, q = 0.5, r = 2, $\mu$ = 2, $\delta$ = 2 and for 2D graphics t = 0.01. |
Figure 9. Graphical descriptions of the solution E24(x, t) when p = 2, q = 0.5, r = 2, $\mu$ = 2, $\delta$ = 2 and for 2D graphics t = 0.01. |
Figure 10. Graphical descriptions of the solution n24(x, t) when p = 2, q = 0.5, r = 2, $\mu$ = 2, $\delta$ = 2 and for 2D graphics t = 0.01. |
Figure 11. Graphical descriptions of the solution E45(x, t) when p = 2, q = 0.5, r = 2, $\mu$ = 2, $\delta$ = 2 and for 2D graphics t = 0.01. |
Figure 12. Graphical descriptions of the solution n45(x, t) when p = 2, q = 0.5, r = 2, $\mu$ = 2, $\delta$ = 2 and for 2D graphics t = 0.01. |
4. Conclusion

