determinants $$f=|D|, \eqno(3.1a)$$ $$g=\begin{vmatrix} D & {\bf z}^T\\ {1\over \rho^2}{\bf z}_t^* & 1\end{vmatrix}=|D|+{1\over \rho^2}\begin{vmatrix} D & {\bf z}^T\\ {\bf z}_t^* & 0\end{vmatrix}. \eqno(3.1b)$$ Here, $D$ is an $N\times N$ matrix and ${\bf z}$ and ${\bf z}_t$ are $N$-component row vectors defined below and the symbol $T$ denotes the transpose: $$D=(d_{jk})_{1\leq j,k\leq N}, \quad d_{jk}=\delta_{jk}+{\kappa-{\rm i}p_j\over p_j+p_k^*}\,z_jz_k^*, 
\quad z_j={\rm exp}\left(p_jx+{\kappa\rho^2\over p_j}t+{1\over p_j+{\rm i}\kappa}\,\tau+\zeta_{j0}\right), \eqno(3.2a)$$ $${\bf z}=(z_1, z_2, ..., z_N), \quad {\bf z}_t=\left({\kappa\rho^2z_1\over p_1}, {\kappa\rho^2z_2\over p_2}, ..., {\kappa\rho^2z_N\over p_N}\right),  \eqno(3.2b)$$ where $p_j$ are complex parameters satisfying the constraints $$(p_j+{\rm i}\kappa)(p_j^*-{\rm i}\kappa)={1+\kappa\rho^2\over \kappa\rho^2}p_jp_j^*,\quad j=1, 2, ..., N, \eqno(3.2c)$$ $\zeta_{j0}\ (j=1, 2, ..., N)$ are arbitrary complex parameters, $\delta_{jk}$ is kronecker’s delta and $\tau$ is an auxiliary variable.*]{}

The dark $N$-soliton solution is parameterized by $2N$ complex parameters $p_j$ and $\zeta_{j0}\ (j=1, 2, ..., N)$. The parameters $p_j$ determine the amplitude and velocity of the solitons whereas the parameters $\zeta_{j0}$ determine the phase of the solitons. As opposed to the bright soliton case explored in I, however, the real and imaginary parts of $p_j$ are not independent because of the constraints (3.2c). Actually, it may be parameterized either by the velocity of the $j$th soliton or by a single angular variable, as will see 