similar problem to that posed in this paper has been studied recently and an explicit formula for the dark $N$-soliton solution have been presented by an ingenious application of the Bäcklund transformation between solutions of the FL equation and the Ablowitz-Ladik hierarchy \[7\]. Nevertheless, the detailed analysis of the soliton solutions has not been undertaken as yet.

An exact method of solution employed here which is sometimes called the direct method \[8\] or the bilinear transformation method \[9\] is a powerful tool for analyzing soliton equations and differs from the method used in \[7\]. Once the equation under consideration is transformed to a system of bilinear equations, the standard technique in the bilinear formalism is applied to obtain soliton solutions. A novel feature of the bilinearization of the FL equation is that one of the bilinear equations can be replaced by a [*trilinear*]{} equation, as already demonstrated in I. The same situation happens in the current dark soliton problem. However, the resulting trilinear equation will be used essentially in the process of performing the proof of the dark $N$-soliton solution.

This paper is organized as follows. In section 2, we bilinearize the FL equation under the boundary condition (1.2). We 