NLS equation, the spatial part of the Lax pair associated with the former equation coincides with that of the latter equation with an identification $q=u_x\ [2, 5]$. This observation enables us to obtain the dark $N$-soliton solution of the derivative NLS equation $${\rm i}q_t+q_{xx}+2{\rm i}(|q|^2q)_x=0,\quad q=q(x,t), \eqno(3.27)$$ under the boundary condition $$q\rightarrow \rho\,{\rm exp}\left\{{\rm i}\left(\kappa x-\omega^\prime t+\psi^{(\pm)}\right)\right\}, \quad x\rightarrow \pm\infty,  \eqno(3.28)$$ where $\omega^\prime=\kappa^2+2\kappa\rho^2$ and $\psi^{(\pm)}$ are real phase constants. In particular, we establish the following proposition:

[**Proposition 3.1.**]{} [*The dark $N$-soliton solution of the derivative NLS equation (3.27) subjected to the boundary condition (3.28) is given in terms of the tau functions $f$ and $h$ by $$q=\rho\,{\rm e}^{{\rm i}(\kappa x-\omega^\prime t)}\,{hf^*\over f^2}, \eqno(3.29a)$$ with $$f=|D|, \quad h=|H|. \eqno(3.29b)$$ Here, $D$ and $H$ are $N\times N$ matrices defined respectively by $$D=(d_{jk})_{1\leq j,k\leq N}, \quad d_{jk}=\delta_{jk}+{\kappa-{\rm i}p_j\over p_j+p_k^*}\,z_jz_k^*, 
\quad z_j={\rm exp}\left[p_jx\!+\!\{{\rm i}p_j^2-2(\kappa+\rho^2)p_j\}t+\zeta_{j0}\right], \eqno(3.30a)$$ $$H=(h_{jk})_{1\leq j,k\leq N}, \quad h_{jk}=\delta_{jk}-{\kappa-{\rm i}p_j\over p_j+p_k^*}\,{p_j\over p_k^*}\,z_jz_k^*, \eqno(3.30b)$$ where $p_j$ are complex parameters satisfying the constraints $$p_jp_j^*=\rho^2\{\kappa-{\rm i}(p_j-p_j^*)\},\quad j=1, 2, ..., N, \eqno(3.30c)$$ and $\zeta_{j0}\ (j=1, 2, ..., N)$ are arbitrary complex parameters.*]{}

[**Proof.**]{} The correspondence between $q$ and $u_x$ mentioned above implies that the relation $$q=u_x={\partial\over\partial x}\left(\rho\,{\rm e}^{{\rm i}\kappa x}\,{g\over f}\right)=\rho\,{\rm e}^{{\rm i}\kappa x}{1\over f^2}(g_xf-gf_x+{\rm i}\kappa fg),$$ 