q=q(x,t), \eqno(3.40)$$ under the boundary condition $$q\rightarrow \rho\,{\rm exp}\left\{{\rm i}\left(\kappa x-\omega^{\prime\prime} t+\psi^{(\pm)}\right)\right\}, \quad x\rightarrow \pm\infty, \eqno(3.41)$$ where $\omega^{\prime\prime}=\kappa^2-\mu\rho^2+\gamma\kappa\rho^2$ and $\mu$ and $\gamma$ are real constants. To show this, we apply the gauge transformation $$q={\rm exp}\left[{\mu\over\gamma}\tilde x+\left({\mu\over\gamma}\right)^2\tilde t\right]\tilde q, \quad x=\tilde x+{2\mu\over\gamma}\,\tilde t, \quad t=\tilde t, \eqno(3.42)$$ to equation (3.40) and see that it can be recast to the derivative NLS equation ${\rm i}\tilde q_{\tilde t}+\tilde q_{\tilde x\tilde x}+\gamma (|\tilde q|^2\tilde q)_{\tilde x}=0$, which coincides with equation (3.27) with the identification $\tilde q=q,\ \tilde x=x,\ \tilde t=t$ and $\gamma=2$. The dark $N$-soliton solution of the equation (3.40) then takes the form $$q= \rho\,{\rm e}^{{\rm i}(\kappa x-\omega^{\prime\prime} t)}\,{{f^\prime}^*h^\prime\over {f^\prime}^2}, \eqno(3.43a)$$ where the tau functions $f^\prime$ and $h^\prime$ are given respectively by $$f^\prime=\left|\left(\delta_{jk}+{\kappa-{\mu\over\gamma}-{\rm i}p_j\over p_j+p_k^*}\,z_jz_k^*\right)_{1\leq j,k\leq N}\right|,\eqno(3.43b)$$ $$h^\prime=\left|\left(\delta_{jk}-{\kappa-{\mu\over\gamma}-{\rm i}p_j\over p_j+p_k^*}{p_j\over p_k^*}\,z_jz_k^*\right)_{1\leq j,k\leq N}\right|,\eqno(3.43c)$$ with $$z_j={\rm exp}\left[p_jx\!+\!\{{\rm i}p_j^2-(2\kappa+\gamma\rho^2)p_j\}t+\zeta_{j0}\right]. \eqno(3.43d)$$ The constraints for $p_j$ become $$p_jp_j^*={\gamma\rho^2\over 2}\left\{\kappa-{\mu\over\gamma}-{\rm i}(p_j-p_j^*)\right\},\quad j=1, 2, ..., N. \eqno(3.44)$$ The complex parameters $p_j$ exist only if the condition $\gamma\left(\kappa-{\mu\over\gamma}+{\gamma\rho^2\over 2}\right)>0$ is satisfied.

The following two special cases are worth remarking. The case $\mu=0$ and $\gamma=2$ reduces to the result given by proposition 3.1. On the other hand, in the limit $\gamma\rightarrow 0$ while $\mu$ being fixed, we first replace $z_j$ by 