A_{b \beta},~ \alpha, \beta=1, 2, \ldots, M,\\
c_{\alpha a}\, A_{a \beta}&=&\bar{c}_{\beta b} \bar{A}_{b \alpha},~ \alpha, \beta=1,2, \ldots, M,\end{aligned}$$ so that $\rho=\bar{\rho}$.

Nonlocal Reductions
===================

Let $v_{\alpha}^{\varepsilon}=v_{\alpha}(\varepsilon_{1} t, \varepsilon_{2} x)$ where $\varepsilon_{1}^2=\varepsilon_{2}^2=1$. Then there are also two different types of nonlocal reductions.

[**(a)**]{} Let $v_{a}=A_{a \alpha}\,v_{\alpha}^{\varepsilon}$, then (\[coupledmKdVN\]) consistently reduce to

$$\label{eqn3}
\mu v_{\alpha,t}+3 \rho \,v_{\alpha,x}+v_{\alpha,xxx}=0,~~\alpha=1, 2, \ldots, M,$$

where $$\rho=c_{\alpha \beta} v_{\alpha} v_{\beta}+ 2c_{\alpha a} A_{a \beta} v_{\alpha} v_{\beta}^{\varepsilon}+c_{ab}\,A_{a \beta} A_{b \alpha} v_{\alpha}^{\varepsilon}\,v_{\beta}^{\varepsilon}$$ with the constraints $$\begin{aligned}
c_{\alpha \beta}&=&c_{ab} A_{b \alpha}\,A_{a \beta},~  \alpha, \beta=1, 2, \ldots, M,\\
c_{\alpha a}A_{a \beta}&=&c_{\beta b}\,A_{b \alpha}, ~ \alpha, \beta=1, 2, \ldots, M,\\
\varepsilon_{1} \varepsilon_{2}&=&1,\end{aligned}$$ so that $\rho=\rho^{\varepsilon}$.

[**(b)**]{} Let $v_{a}=A_{a \alpha}\,\bar{v}_{\alpha}^{\varepsilon}$, then (\[coupledmKdVN\]) consistently reduce to $$\label{eqn4}
\mu v_{\alpha,t}+3 \rho \,v_{\alpha,x}+v_{\alpha,xxx}=0,~~\alpha=1, 2, \ldots, M,$$ where $$\rho=c_{\alpha \beta} v_{\alpha} v_{\beta}+ 2c_{\alpha a} A_{a \beta} v_{\alpha} \bar{v}_{\beta}^{\varepsilon}+c_{ab}\,A_{a \alpha} a_{b \beta} \bar{v}_{\alpha}^{\varepsilon}\,\bar{v}_{\beta}^{\varepsilon}$$ with the constraints $$\begin{aligned}
 \bar{c}_{\alpha \beta}&=&c_{ab} A_{b \alpha}\,A_{a \beta}, ~ \alpha, \beta=1, 2, \ldots, M,\\
c_{\alpha a}A_{a \beta}&=&\bar{c}_{\beta b}\bar{A}_{b \alpha}, ~\alpha, \beta=1, 2 ,\ldots, M,\\
\bar{\mu} \varepsilon_{1} \varepsilon_{2}&=&\mu,\end{aligned}$$ so that $\rho=\bar{\rho}^{\varepsilon}$.

Hirota Bilinear Method for Coupled HI-mKdV\
Systems
===========================================

Let $$\displaystyle v_i=\frac{g_i}{f}, \quad i=1, 2, \ldots, N$$ in (\[coupledmKdVN\]). Then Hirota bilinear form of (\[coupledmKdVN\]) can be found as $$\begin{aligned}
&(\mu D_t+D_x^2-3\lambda_i )\{g_i\cdot f\}=0,\\
&(D_x^2-\lambda_i)\{f\cdot 