We see from (3.36) that if $a_1>0$, then $|q_1|$ takes the form of a dark soliton whereas if $a_1<0$, it becomes a bright soliton on a constant background $u=\rho$.

Let $A_d$ and $A_b$ be the amplitudes of the dark and bright solitons, respectively with respective to the background. The amplitude-velocity relations follow from (3.36) and (3.38). They read $$A_d=\rho-\left|\sqrt{c_1-\kappa-2\rho^2}-\sqrt{\kappa+\rho^2}\right|, \eqno(3.39a)$$ $$A_b=\sqrt{c_1-\kappa-2\rho^2}+\sqrt{\kappa+\rho^2}-\rho. \eqno(3.39b)$$ The detailed analysis for the case $\kappa>0$ has been undertaken in \[21\]. To sum up, the solution has been shown to exhibit the spiky modulation of the amplitude and phase. It also has been demonstrated that the bright soliton reduces to an algebraic soliton for both limits $c_1\rightarrow c_{\rm max}$ and $c_1\rightarrow c_{\rm min}$ whereas the algebraic dark soliton never exists. In the case $\kappa<0$ which has not been treated in \[21\], however, a careful inspection of (3.36) and (3.38) reveals that the algebraic bright and dark solitons are produced in the limit $c_1\rightarrow c_{\rm max}$ and $c_1\rightarrow c_{\rm min}$, respectively. The latter new feature is pointed out here for the first time.

[**Remark 3.3.**]{} Using the result obtained in proposition 3.1, we can construct the dark $N$-soliton solution of the modified NLS equation $${\rm i}q_t+q_{xx}+\mu |q|^2q+{\rm i}\gamma(|q|^2q)_x=0,\quad 