be proved in the same way if one uses (3.5), (3.6) and the relation ${\bf z}_{xt}=\kappa\rho^2{\bf z}$ as well as some basic properties of determinants. $\Box$

The complex conjugate expressions of the tau functions $f$ and $g$ and their derivatives are expressed as follows:

[**Lemma 3.2.**]{}

$$f^*=|D|-{\rm i}|D({\bf z}^*;{\bf z})|,  \eqno(3.14)$$ $$f_t^*=-{\rm i}|D({\bf z}^*;{\bf z}_t)|- {1\over \rho^2}|D({\bf z}_t^*;{\bf z}_t)|+{\rm i\over \rho^2}|D({\bf z}_t^*,{\bf z}^*;{\bf z}_t,{\bf z})|, \eqno(3.15)$$ $$g^*=|D|-{\rm i}|D({\bf z}^*;{\bf z})|+{1\over \rho^2}|D({\bf z}^*;{\bf z}_t)|. \eqno(3.16)$$ [**Proof.**]{} It follows from (3.2a) that $d_{jk}^*=d_{kj}+{\rm i}z_j^*z_k$ or in the matrix form, $D^*= D^T+{\rm i}(z_jz_k^*)_{1\leq j,k\leq N}^T$. Since $|D^T|=|D|$, one has $$f^*=|D+{\rm i}(z_jz_k^*)_{1\leq j,k\leq N}|=\begin{vmatrix} D & {\bf z}^T\\ -{\rm i}{\bf z}^* & 1\end{vmatrix}.$$ Applying formula (3.6) to the right-hand side, formula (3.14) follows immediately. Formulas (3.15) and (3.16) can be derived in the same way. $\Box$

[*3.4. Proof of the dark $N$-soliton solution*]{}

[*3.4.1. Proof of (2.2)*]{}

Let $$P_1=D_tf\cdot f^*-{\rm i}\rho^2(gg^*-ff^*). \eqno(3.17)$$ Substituting (3.8), (3.9) and (3.14)-(3.16) into (3.17), most terms are canceled, leaving the following three terms $$P_1={{\rm i}\over \rho^2}\Bigl\{-|D||D({\bf z}_t^*,{\bf z}^*;{\bf z}_t,{\bf z})|+|D({\bf z}^*;{\bf z})||D({\bf z}_t^*;{\bf z}_t)|
-|D({\bf z}_t^*;{\bf z})||D({\bf z}^*;{\bf z}_t)|\Bigr\}.$$ This expression becomes zero by Jacobi’s identity. $\Box$

[*3.4.2. Proof of (2.3)*]{}

Instead of proving (2.3) directly, we differentiate (2.2) 