On the other hand, for the dark $N$-soliton solution given by (2.1), this limiting procedure is not relevant because of the singular nature of the dispersion relation.

Last, we shall briefly describe the properties of the one-soliton solution for the purpose of comparison with those of the one-soliton solution of the FL equation. Introducing the new real parameters $a_1$ and $b_1$ by $p_1=a_1+{\rm i}b_1$, the square of the modulus of the one-soliton solution from (3.29) and (3.30) with $N=1$ can be written in the form $$|q_1|^2=\rho^2-{2a_1^2\,{\rm sgn}\, a_1\over \sqrt{a_1^2+(\kappa+b_1)^2}}\,{1\over \cosh\,2(\theta_1+\delta_1)+{(\kappa+b_1){\rm sgn}\, a_1\over \sqrt{a_1^2+(\kappa+b_1)^2}}}. \eqno(3.36a)$$ with $$\theta_1=a_1(x+c_1t)+\theta_{10},\qquad c_1=2(b_1+\kappa+\rho^2),\qquad {\rm e}^{4\delta_1}={a_1^2+(\kappa+b_1)^2\over 4a_1^2}, \eqno(3.36b)$$ where ${\rm sgn}\, a_1$ denotes the sign of $a_1$, i.e., $a_1=1$ for $a_1>0$ and $a_1=-1$ for $a_1<0$, and $\theta_{10}$ is a real constant. The constraint (3.30c) then becomes $$a_1^2+b_1^2=\rho^2(2b_1+\kappa). \eqno(3.37)$$ Using (3.36b) and (3.37), the parameters $a_1$ and $b_1$ are expressed in terms of the velocity $c_1$ of the soliton as $$a_1^2={1\over 4}\left(c_{\rm max}-c_1\right)\left(c_1-c_{\rm min}\right),\qquad b_1={c_1\over 2}-\kappa-\rho^2, \qquad c_{\rm min}<c_1<c_{\rm max},   \eqno(3.38a)$$ where $$c_{\rm max}=2(\kappa+2\rho^2)+2\rho\sqrt{\kappa+\rho^2},\qquad  c_{\rm min}=2(\kappa+2\rho^2)-2\rho\sqrt{\kappa+\rho^2}. \eqno(3.38b)$$ One must impose the condition $\kappa+\rho^2> 0$ to assure the existence of soliton solutions. Recall that this condition coincides with a criterion for the stability of the plane wave (3.28) \[20\]. 