solution of the system (\[localiN=4\]) becomes $$\begin{aligned}
\displaystyle
&v_1=\frac{4e^{\frac{1}{2}x+\frac{85}{24}t}(36e^{\frac{9}{4}t}+e^{3x})}{144e^{\frac{35}{6}t}+16e^{3x+\frac{43}{12}t}+144e^{2x+\frac{14}{3}t}
+144e^{x+\frac{23}{4}t}+e^{4x+\frac{7}{2}t}}\\
&v_2=\frac{36e^{\frac{3}{2}x+\frac{37}{8}t}(4e^{\frac{1}{12}t}+e^{x})}{144e^{\frac{35}{6}t}+16e^{3x+\frac{43}{12}t}+144e^{2x+\frac{14}{3}t}
+144e^{x+\frac{23}{4}t}+e^{4x+\frac{7}{2}t}}.\end{aligned}$$ These solutions are finite and bounded for $t\geq 0$. The graphs of the functions $v_1$ and $v_2$ are given in Figure 1(a) and Figure 1(b) respectively.

**ii.** $v_4=a_1\bar{v}_2$, $v_3=b_1\bar{v}_1$, $a_1$ and $b_1$ are constants. To have a consistent reduction of the system (\[N=4a\])-(\[N=4d\]) we must have $$\mu=\bar{\mu},\quad  \rho=\bar{\rho}.$$ The relation $\rho=\bar{\rho}$ is satisfied when $$\label{N=4localiicond}
1)\, c_{12}=\bar{c}_{34}\bar{a}_1\bar{b}_1,\quad 2)\, c_{14}a_1=\bar{c}_{23}\bar{b}_1,\quad 3)\, c_{24}a_1=\bar{c}_{24}\bar{a}_1, \quad 4)\, c_{13}b_1=\bar{c}_{13}\bar{b}_1.$$ The reduced system is $$\begin{aligned}
&\mu v_{1,t}+6[c_{12}v_1v_2+c_{13}b_1|v_1|^2+c_{14}a_1v_1\bar{v}_2+\bar{c}_{14}\bar{a}_1\bar{v}_1v_2+c_{24}a_1|v_2|^2+
\bar{c}_{12}\bar{v}_1\bar{v}_2]v_{1,x}+v_{1,xxx}=0,\nonumber\\
&\mu v_{2,t}+6[c_{12}v_1v_2+c_{13}b_1|v_1|^2+c_{14}a_1v_1\bar{v}_2+\bar{c}_{14}\bar{a}_1\bar{v}_1v_2+c_{24}a_1|v_2|^2+
\bar{c}_{12}\bar{v}_1\bar{v}_2]v_{2,x}+v_{2,xxx}=0,\label{localiiN=4}\end{aligned}$$ where the conditions (\[N=4localiicond\]) are satisfied and $a\in \mathbb{R}$.

If we use the Type 1 approach by checking the equalities $$\begin{aligned}
\displaystyle
&v_4=a_1\bar{v}_2(t,x) \Rightarrow \frac{g_4}{f}=a_1\frac{\bar{g}_2}{\bar{f}} \Rightarrow g_4=a_1\bar{g}_2, f=\bar{f},\\
&v_3=b_1\bar{v}_1(t,x) \Rightarrow \frac{g_3}{f}=b_1\frac{\bar{g}_1}{\bar{f}} \Rightarrow g_3=b_1\bar{g}_1, f=\bar{f},\end{aligned}$$ we obtain the conditions on the parameters of one-soliton solution of the system (\[localiiN=4\]) as $$1)\, k_4=\bar{k}_2,\quad 2)\, k_3=\bar{k}_1,\quad 3)\, e^{\delta_4}=a_1e^{\bar{\delta}_2},\quad 4)\, e^{\delta_3}=b_1e^{\bar{\delta}_1},$$ besides the conditions (\[N=4localiicond\]). With all of these constraints one-soliton solution of the system (\[localiiN=4\]) becomes $$\displaystyle
v_1=\frac{g_1}{f},\quad v_2=\frac{g_2}{f}$$ where $$\begin{aligned}
&g_1=e^{k_1x-\frac{k_1^3}{\mu}t
+\delta_1}+\frac{(k_1-k_2)(k_1-\bar{k}_1)(k_2-\bar{k}_1)}{(k_1+k_2)
(k_1+\bar{k}_1)(k_2+\bar{k}_1)}\frac{\bar{c}_{14}\bar{a}_1}
{(k_2^2-\bar{k}_1^2)}e^{(k_1+k_2+\bar{k}_1)x-\frac{k_1^3+k_2^3+\bar{k}_1^3}{\mu}t+\delta_1+\delta_2+\bar{\delta}_1}\nonumber\\
&+\frac{(k_1-k_2)(k_1-\bar{k}_2)(k_2-\bar{k}_2)}{(k_1+k_2)
(k_1+\bar{k}_2)(k_2+\bar{k}_2)}\frac{c_{24}a_1}
{(k_2^2-\bar{k}_2^2)}e^{(k_1+k_2+\bar{k}_2)x-\frac{k_1^3+k_2^3+\bar{k}_2^3}{\mu}t+\delta_1+\delta_2+\bar{\delta}_2}\nonumber\\
&+\frac{(k_1-\bar{k}_1)(k_1-\bar{k}_2)(\bar{k}_1-\bar{k}_2)}{(k_1+\bar{k}_1)
(k_1+\bar{k}_2)(\bar{k}_1+\bar{k}_2)}\frac{\bar{c}_{12}}
{(\bar{k}_1^2-\bar{k}_2^2)}e^{(k_1+\bar{k}_1+\bar{k}_2)x-\frac{k_1^3+\bar{k}_1^3+\bar{k}_2^3}{\mu}t+\delta_1+\bar{\delta}_1+\bar{\delta}_2},\end{aligned}$$ $$\begin{aligned}
&g_2=e^{k_2x-\frac{k_2^3}{\mu}t
+\delta_2}-\frac{(k_1-k_2)(k_1-\bar{k}_1)(k_2-\bar{k}_1)}{(k_1+k_2)
(k_1+\bar{k}_1)(k_2+\bar{k}_1)}\frac{c_{13}b_1}
{(k_1^2-\bar{k}_1^2)}e^{(k_1+k_2+\bar{k}_1)x-\frac{k_1^3+k_2^3+\bar{k}_1^3}{\mu}t+\delta_1+\delta_2+\bar{\delta}_1}\nonumber\\
&-\frac{(k_1-k_2)(k_1-\bar{k}_2)(k_2-\bar{k}_2)}{(k_1+k_2)
(k_1+\bar{k}_2)(k_2+\bar{k}_2)}\frac{c_{14}a_1}
{(k_1^2-\bar{k}_2^2)}e^{(k_1+k_2+\bar{k}_2)x-\frac{k_1^3+k_2^3+\bar{k}_2^3}{\mu}t+\delta_1+\delta_2+\bar{\delta}_2}\nonumber\\
&+\frac{(k_2-\bar{k}_1)(k_2-\bar{k}_2)(\bar{k}_1-\bar{k}_2)}{(k_2+\bar{k}_1)
(k_2+\bar{k}_2)(\bar{k}_1+\bar{k}_2)}\frac{\bar{c}_{12}}
{(\bar{k}_1^2-\bar{k}_2^2)}e^{(k_2+\bar{k}_1+\bar{k}_2)x-\frac{k_2^3+\bar{k}_1^3+\bar{k}_2^3}{\mu}t+\delta_2+\bar{\delta}_1+\bar{\delta}_2},\end{aligned}$$ and $$\begin{aligned}
&f=1+\frac{c_{12}}{(k_1+k_2)^2}e^{(k_1+k_2)x-\frac{(k_1^3+k_2^3)}{\mu}t+\delta_1+\delta_2}
+\frac{\bar{a}_1\bar{c}_{14}}{(k_2+\bar{k}_1)^2}e^{(k_2+\bar{k}_1)x-\frac{(k_2^3+\bar{k}_1^3)}{\mu}t+\delta_2+\bar{\delta}_1}\nonumber\\
&+\frac{b_1c_{13}}{(k_1+\bar{k}_1)^2}e^{(k_1+\bar{k}_1)x-\frac{(k_1^3+\bar{k}_1^3)}{\mu}t+\delta_1+\bar{\delta}_1}
+\frac{a_1c_{24}}{(k_2+\bar{k}_2)^2}e^{(k_2+\bar{k}_2)x-\frac{(k_2^3+\bar{k}_2^3)}{\mu}t+\delta_2+\bar{\delta}_2}\nonumber\\
&+\frac{a_1c_{14}}{(k_1+\bar{k}_2)^2}e^{(k_1+\bar{k}_2)x-\frac{(k_1^3+\bar{k}_2^3)}{\mu}t+\delta_1+\bar{\delta}_2}
+\frac{\bar{c}_{12}}{(\bar{k}_1+\bar{k}_2)^2}e^{(\bar{k}_1+\bar{k}_2)x-\frac{(\bar{k}_1^3+\bar{k}_2^3)}{\mu}t+\bar{\delta}_1+\bar{\delta}_2}\nonumber\\
&+a_1b_1\frac{(k_1-k_2)(k_1-\bar{k}_1)(k_1-\bar{k}_2)(k_2-\bar{k}_1)(k_2-\bar{k}_2)(\bar{k}_1-\bar{k}_2)}
{(k_1+k_2)(k_1+\bar{k}_1)(k_1+\bar{k}_2)(k_2+\bar{k}_1)(k_2+\bar{k}_2)(\bar{k}_1+\bar{k}_2)}[\frac{c_{12}\bar{c}_{12}}
{a_1b_1(k_1^2-k_2^2)(\bar{k}_1^2-\bar{k}_2^2)}\nonumber\\
&-\frac{c_{13}c_{24}}{(k_1^2-\bar{k}_1^2)(k_2^2-\bar{k}_2^2)}+\frac{c_{14}c_{23}}{(k_1^2-\bar{k}_2^2)(k_2^2-\bar{k}_1^2)}]
e^{(k_1+k_2+\bar{k}_1+\bar{k}_2)x-\frac{k_1^3+k_2^3+\bar{k}_1^3+\bar{k}_2^3}{\mu}t+\delta_1+\delta_2+\bar{\delta}_1+\bar{\delta}_2}.\end{aligned}$$

When 