holds at $t=0$. On the other hand, the expression in the parentheses on the right-hand side is just $P_3^\prime$ defined by (3.25c) and hence it is equal to (3.26b). This fact and (3.14) lead, after applying the formula (3.6), to $$\begin{aligned}
g_xf-gf_x+{\rm i}\kappa fg &={\rm i}\kappa\Bigl\{|D|-|D({\bf z}^*;{\bf z})|\Bigr\}\Bigl\{|D|+{1\over\rho^2}|D({\bf z}_t^*;{\bf z})|-{{\rm i}\over \kappa\rho^2}|D({\bf z}_t^*;{\bf z}_x)|\Bigr\} \notag \\
                           &={\rm i}\kappa f^*\begin{vmatrix} D & \left(z_j-{{\rm i}p_j\over\kappa}z_j\right)_{1\leq j\leq N}^T \\ \left({\kappa\over p_k^*}z_k^*\right)_{1\leq k\leq N} & 1\end{vmatrix}.     \notag\end{aligned}$$ Multiplying the $(N+1)$th column of the determinant by $\kappa z_k^*/p_k^*$ and subtracting it from the $k$th column for $k=1, 2, ..., N$, one finds that the above expression becomes ${\rm i}\kappa f^*h$. Consequently, $$q={\rm i}\kappa \rho\,{\rm e}^{{\rm i}\kappa x}\,{f^*h\over f^2}\Bigg|_{t=0}.$$ If one replaces $q$ by ${\rm i}q$ and $\rho$ by $\rho/\kappa$, respectively and introduces the time dependence appropriately, one arrives at (3.29) with (3.30). The constraints (3.30c) follow from (3.2c) by the above replacement of $\rho$. The complex parameters $p_j$ subjected to the constraints (3.30c) exist only if the condition $\kappa+\rho^2>0$ is satisfied. $\Box$

It is instructive to perform the bilinearization of the derivative NLS equation under the boundary condition (3.28). This provides an alternative way to construct the dark $N$-soliton solution given by proposition 3.1, as 