then show that one of the resulting bilinear equations can be replaced by a trilinear equation. In section 3, we present the dark $N$-soliton solution of the bilinear equations. It has a simple structure expressed in terms of certain determinants. Subsequently, we perform the proof of the dark $N$-soliton solution using an elementary theory of determinants in which Jacobi’s identity will play a central role. As already noted, the proof of the trilinear equation turns out to be a core in the analysis. In accordance with the relation between the FL equation and the derivative NLS equation at the level of the Lax representation, we also demonstrate that the dark $N$-soliton solution obtained here yields the dark $N$-soliton solution of the derivative NLS equation by replacing simply the time dependence of the solution. As in the case of the defocusing NLS equation subjected to nonvanishing boundary conditions, it is necessary for the existence of dark solitons that the asymptotic state given by (1.2) must be stable. Hence, we perform the linear stability analysis of the plane wave solution (1.2) and provide a criterion for the stability. In section 4, we first investigate the properties of the one-soliton solution in detail. 