We find that depending on the sign of $\kappa$ and that of the real part of the complex amplitude parameter, the solution can be classified into two types, i.e., the dark and bright solitons. The latter soliton may be termed “anti-dark soliton” since the background field is nonzero. However, we use a term “bright soliton” throughout the paper. We demonstrate that regardless the sign of $\kappa$, the bright soliton has a limiting profile of algebraic type (or an algebraic bright soliton) whereas an algebraic dark soliton appears only if $\kappa<0$. We then analyze the asymptotic behavior of the two-soliton solution and derive the explicit formulas for the phase shift in terms of the amplitude parameters of solitons. In particular, we address the interaction between a dark soliton and a bright soliton as well as that of two dark solitons. Last, the similar asymptotic analysis to that of the two-soliton solution is performed for the general dark $N$-soliton solution. Section 5 is devoted to concluding remarks.

In this section, we develop a direct method of solution for constructing dark soliton solutions of the FL equation (1.1) under the boundary condition (1.2). In particular, we show that it can be transformed to 