$N$-soliton solution, we first define the matrices associated with the dark $N$-soliton solution and then provide some basic formulas for determinants. Although these formulas have been used extensively in I, we reproduce them for convenience.

The following bordered matrices appear frequently in our analysis: $$D({\bf a}; {\bf b})=\begin{pmatrix} D & {\bf b}^T\\ {\bf a} & 0\end{pmatrix},\quad
D({\bf a},{\bf b};{\bf c},{\bf d})=\begin{pmatrix} D &{\bf c}^T & {\bf d}^T \\ {\bf a} & 0 &0\\
                                                                            {\bf b} & 0& 0 \end{pmatrix}, \eqno(3.4)$$ where ${\bf a}, {\bf b}, {\bf c}$ and [**d**]{} are $N$ component row vectors. Let $D_{jk}$ be the cofactor of the element $d_{jk}$. The following formulas are well known in the theory of determinants \[10\]: $${\partial\over\partial x}|D|=\sum_{j,k=1}^N{\partial d_{jk}\over\partial x}D_{jk}, \eqno(3.5)$$ $$\begin{vmatrix} D & {\bf a}^T\\ {\bf b} & z\end{vmatrix}=|D|z-\sum_{j,k=1}^ND_{jk}a_jb_k,  \eqno(3.6)$$ $$|D({\bf a}, {\bf b}; {\bf c}, {\bf d})||D|= |D({\bf a}; {\bf c})||D({\bf b}; {\bf d})|-|D({\bf a}; {\bf d})||D({\bf b}; {\bf c})|. \eqno(3.7)$$ The formula (3.5) is the differentiation rule of the determinant and (3.6) is the expansion formula for a bordered determinant with respect to the last row and last column. The formula (3.7) is Jacobi’s identity. The proof of lemmas described below is based on the above three formulas 