side becomes zero by (2.2)-(2.4). This yields (2.10). $\Box$

In view of proposition 2.2, the proof of the dark $N$-soliton solution is completed if one can prove any three equations among the three bilinear equations (2.2)-(2.4) and a trilinear (2.10). We will see later in section 3 that the proof of (2.4) is not easy to perform and hence we prove (2.10) instead.

In this section, we show that the tau functions $f$ and $g$ representing the dark $N$-soliton solution admit the compact determinantal expressions. This statement is proved by an elementary calculation using the basic formulas for determinants. We first prove that the proposed dark $N$-soliton solution solves the bilinear equations (2.2) and (2.3) and then the trilinear equation (2.10) in place of (2.4). The implication of the equation (2.10) will be discussed in conjunction with the dark $N$-soliton solution of the derivative NLS equation. Last, we perform the linear stability analysis of the plane wave solution (1.2) and provide a criterion for the stability.

[*3.1. Dark $N$-soliton solution*]{}

The main result in this paper is given by the following theorem:

[**Theorem 3.1.**]{} [*The dark $N$-soliton solution of the system of bilinear equations (2.2)-(2.4) is expressed by the following 