Appendix
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Here we present local and nonlocal reductions, and one-soliton solution of the system (\[coupledmKdVN\]) for $N=3$. Note that even this system has reductions to systems of two equations, if we require Type 1 one-soliton solution [@GurPek2], all the systems of two equations reduce to single well-known equations.

N=3 Coupled HI-mKdV System
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The system (\[coupledmKdVN\]) for $N=3$ is $$\begin{aligned}
& \mu v_{1,t}+3\rho v_{1,x}+v_{1,xxx}=0,\label{N=3a}\\
& \mu v_{2,t}+3\rho v_{2,x}+v_{2,xxx}=0,\label{N=3b}\\
& \mu v_{3,t}+3\rho v_{3,x}+v_{3,xxx}=0,\label{N=3c}\end{aligned}$$ where $$\rho= 2(c_{12}v_1v_2+c_{13}v_1v_3+c_{23}v_2v_3).$$ The corresponding Hirota bilinear form is $$\begin{aligned}
&(\mu D_t+D_x^3)\{g_i\cdot f\}=0,\quad i=1, 2, 3, \\
&D_x^2\{f\cdot f\}= 2(c_{12}g_1g_2+c_{13}g_1g_3+c_{23}g_2g_3).\end{aligned}$$

### One-Soliton Solution of N=3 Coupled HI-mKdV System

Similar to $N=2$ case we take the solution given by (\[g\_if\]) with the pfaffian elements (\[pf1\])-(\[pf3\]) under the condition (\[pf4\]). The solution which has one-soliton for every $\displaystyle v_i=\frac{g_i}{f}$, $i=1, 2, 3$, i.e., $M_j=1$, $j=1, 2, 3$ so $L=M_1+M_2+M_3=3$ with $B_j=\{b_j\}$, $j=1, 2, 3$ is expressed by $$\begin{aligned}
&g_i=\mathrm{pf}(d_0,a_1,a_2,a_3,b_1,b_2,b_3,\beta_i),\, i=1,2,3,\\
&f=\mathrm{pf}(a_1,a_2,a_3,b_1,b_2,b_3),\end{aligned}$$ which are explicitly given as $$\begin{aligned}
& g_1=-e^{\theta_1}-\alpha_{13}\alpha_{12}\alpha_{23}\beta_{23}e^{\theta_1+\theta_2+\theta_3},\\
& g_2=-e^{\theta_2}+\alpha_{13}\alpha_{12}\alpha_{23}\beta_{13}e^{\theta_1+\theta_2+\theta_3},\\
& g_3=-e^{\theta_3}-\alpha_{13}\alpha_{12}\alpha_{23}\beta_{12}e^{\theta_1+\theta_2+\theta_3},\\
&f=-1-\alpha_{12}\beta_{12}e^{\theta_1+\theta_2}-\alpha_{13}\beta_{13}e^{\theta_1+\theta_3}-\alpha_{23}\beta_{23}e^{\theta_2+\theta_3},\end{aligned}$$ 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 $i,j=1,2,3$. Here $k_i, \delta_i$, $i=1, 2, 3$ are arbitrary constants.

Local and Nonlocal Reductions for N=3
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The coupled HI-mKdV system for $N=3$ given by (\[N=3a\])-(\[N=3c\]) has 