t,\varepsilon_2 x),\, \varepsilon_i^2=1, i=1, 2,$$ as $$1)\, k_2=\varepsilon_2 \bar{k}_1,\quad 2)\, e^{\delta_2}=a_1e^{\delta_1},$$ besides the conditions (\[condnonlocaliiN=2\]).

Note that since we have given one-soliton solutions of the local and nonlocal reduced equations for $N=2$ in [@GurPek2], we are not presenting them here also.

N=4 Coupled HI-mKdV System
==========================

The system (\[coupledmKdVN\]) for $N=4$ is $$\begin{aligned}
& \mu v_{1,t}+3\rho v_{1,x}+v_{1,xxx}=0,\label{N=4a}\\
& \mu v_{2,t}+3\rho v_{2,x}+v_{2,xxx}=0,\label{N=4b}\\
& \mu v_{3,t}+3\rho v_{3,x}+v_{3,xxx}=0,\label{N=4c}\\
& \mu v_{4,t}+3\rho v_{4,x}+v_{4,xxx}=0,\label{N=4d}\end{aligned}$$ where $$\rho= 2(c_{12}v_1v_2+c_{13}v_1v_3+c_{14}v_1v_4+c_{23}v_2v_3+c_{24}v_2v_4+c_{34}v_3v_4),$$ and $\mu$ is a constant. The corresponding Hirota bilinear form is $$\begin{aligned}
&(\mu D_t+D_x^3)\{g_i\cdot f\}=0,\quad i=1, 2, 3, 4,\\
&D_x^2\{f\cdot f\}= 2(c_{12}g_1g_2+c_{13}g_1g_3+c_{14}g_1g_4+c_{23}g_2g_3+c_{24}g_2g_4+c_{34}g_3g_4).\end{aligned}$$

One-Soliton Solution of N=4 Coupled HI-mKdV System
--------------------------------------------------

If we consider the solution given by (\[g\_if\]) with the pfaffian elements (\[pf1\])-(\[pf3\]) under the condition (\[pf4\]), the solution which has one-soliton for every $\displaystyle v_i=\frac{g_i}{f}$ i.e., $M_i=1$, $i=1, 2, 3, 4$ so $L=M_1+M_2+M_3+M_4=4$ with $B_i=\{b_i\}$, $i=1, 2, 3, 4$ is expressed by $$\begin{aligned}
&g_i=\mathrm{pf}(d_0,a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4,\beta_i),\, i=1, 2, 3, 4,\\
&f=\mathrm{pf}(a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4),\end{aligned}$$ which are explicitly written as $$\begin{aligned}
& g_1=e^{\theta_1}+\alpha_{12}\alpha_{13}\alpha_{23}\beta_{23}e^{\theta_1+\theta_2+\theta_3}
+\alpha_{12}\alpha_{14}\alpha_{24}\beta_{24}e^{\theta_1+\theta_2+\theta_4}+\alpha_{13}\alpha_{14}\alpha_{34}\beta_{34}e^{\theta_1+\theta_3+\theta_4},\\
& g_2=e^{\theta_2}-\alpha_{12}\alpha_{13}\alpha_{23}\beta_{13}e^{\theta_1+\theta_2+\theta_3}-
\alpha_{12}\alpha_{14}\alpha_{24}\beta_{14}e^{\theta_1+\theta_2+\theta_4}+\alpha_{23}\alpha_{24}\alpha_{34}\beta_{34}e^{\theta_2+\theta_3+\theta_4},\\
& g_3=e^{\theta_3}+\alpha_{12}\alpha_{13}\alpha_{23}\beta_{12}e^{\theta_1+\theta_2+\theta_3}
-\alpha_{13}\alpha_{14}\alpha_{34}\beta_{14}e^{\theta_1+\theta_3+\theta_4}-\alpha_{23}\alpha_{24}\alpha_{34}\beta_{24}e^{\theta_2+\theta_3+\theta_4},\\
&f=1+\alpha_{12}\beta_{12}e^{\theta_1+\theta_2}+\alpha_{13}\beta_{13}e^{\theta_1+\theta_3}+\alpha_{14}\beta_{14}e^{\theta_1+\theta_4}
+\alpha_{23}\beta_{23}e^{\theta_2+\theta_3}+\alpha_{24}\beta_{24}e^{\theta_2+\theta_4}\nonumber\\
&\hspace{1cm}+\alpha_{34}\beta_{34}e^{\theta_3+\theta_4}+\alpha_{12}\alpha_{13}\alpha_{14}\alpha_{23}\alpha_{24}\alpha_{34}[\beta_{12}\beta_{34}-\beta_{13}\beta_{24}
+\beta_{14}\beta_{23}]e^{\theta_1+\theta_2+\theta_3+\theta_4},\end{aligned}$$ where $\theta_i=k_ix-\frac{k_i^3}{\mu}t+\delta_i$, $\displaystyle \alpha_{ij}=\frac{k_i-k_j}{k_i+k_j}$, and $\displaystyle \beta_{ij}=\frac{c_{ij}}{k_i^2-k_j^2}$ for $i,j=1,2,3,4$. Here $k_i, \delta_i$, $i=1, 2, 3, 4$ are arbitrary constants.

Local and Nonlocal Reductions for N=4
=====================================

The coupled HI-mKdV 