f\}=\sum_{j, k=1, \ldots, N } c_{jk}g_jg_k.\end{aligned}$$ For simplicity let us take $\lambda_i=0$, $i=1, 2, \ldots, N$. Hence we have $$\begin{aligned}
&(\mu D_t+D_x^2)\{g_i\cdot f\}=0, \, i=1, 2,\ldots, N, \\
&D_x^2\{f\cdot f\}=\sum_{j, k=1, \ldots, N } c_{jk}g_jg_k\end{aligned}$$ as Hirota bilinear form of (\[coupledmKdVN\]). In [@IH], multi-soliton solution of (\[coupledmKdVN\]) which has $M_i$ solitons for $v_i$, $i=1,2,\ldots, N$, respectively is expressed by pfaffians as $$\begin{aligned}
\label{g_if}
&g_i=\mathrm{pf}(d_0,a_1,\ldots,a_L,b_1,\ldots,b_L,\beta_i)\\
&f=\mathrm{pf}(a_1,\ldots,a_L,b_1,\ldots,b_L)\end{aligned}$$ for $i=1,2,\ldots, N$ and $L=M_1+M_2+\ldots+M_N$. Here the elements of pfaffians are defined as $$\begin{aligned}
&\mathrm{pf}(d_n,a_i)=\frac{\partial^n}{\partial x^n}e^{\theta_i}=k_i^ne^{\theta_i},\, \theta_i=k_ix-k_i^3t+\alpha_i^0\nonumber\\
&\mathrm{pf}(a_i,a_m)=\frac{k_i-k_m}{k_i+k_m}e^{\theta_i+\theta_m}\nonumber\\
&\mathrm{pf}(b_i,b_m)=-\frac{c_{jk}}{k_i^2-k_m^2},\, b_i\in B_j,\, b_m\in B_k \label{pf1}\end{aligned}$$ $$\begin{aligned}
\label{pf2}
\mathrm{pf}(a_i,b_m)=\left\{ \begin{array}{cc}
                1, & \hspace{5mm} \mathrm{if}\quad  i=m \\
                0, & \hspace{5mm} \mathrm{if}\quad i\neq m
                                \end{array} \right.\end{aligned}$$ $$\begin{aligned}
\label{pf3}
\mathrm{pf}(b_m,\beta_i)=\left\{ \begin{array}{cc}
                1, & \hspace{5mm} \mathrm{if}\quad b_m\in B_i \\
                0, & \hspace{5mm} \mathrm{if}\quad b_m \notin B_i
\end{array} \right.\end{aligned}$$ and $\mathrm{pf}(\mathrm{otherwise})=0$. Here the class of the sets $B_i$ of letters chosen out of $\{b_1,\ldots, b_L\}$, $i=1, 2, \ldots, N$, satisfies the following condition $$\begin{aligned}
&M_i=\mathrm{number}\, \mathrm{of}\, \mathrm{elements}\, \mathrm{in}\, \mathrm{the}\, \mathrm{set} B_i\nonumber\\
& B_i \bigcap B_j=\emptyset,\,\mathrm{if}\, i\neq j\nonumber\\
&\displaystyle \bigcup\limits_{i=1}^N B_i=\{b_1,\ldots,b_L \}.\label{pf4}\end{aligned}$$

N=2 Coupled HI-mKdV System
==========================

The system (\[coupledmKdVN\]) for $N=2$ is $$\begin{aligned}
& \mu v_{1,t}+3\rho v_{1,x}+v_{1,xxx}=0,\label{N=2a}\\
& \mu v_{2,t}+3\rho v_{2,x}+v_{2,xxx}=0,\label{N=2b}\end{aligned}$$ where $$\rho= 2c_{12}v_1v_2.$$ The corresponding Hirota bilinear form is $$\begin{aligned}
&(\mu 