$ and a phase difference $ \delta $ are calculated as $$\begin{aligned}
        N &= -\frac{2}{\sqrt{a_2}}\tanh^{-1}\left[ \frac{\sqrt{a_2}(\rho_\infty-\rho_0)}{\kappa} \right],\label{eq:cqnlsN} \\
        \delta &= 2 \tan^{-1}\left[ \frac{p(\rho_\infty-\rho_0)}{\rho_0\kappa} \right]. \label{eq:cqnlsdelta}
    \end{aligned}$$ From them we can calculate the renormalized momentum $ -P=pN+\rho_\infty \delta $. An example is shown in Fig. \[fig:renP\]. The case where the unstable region exists is analyzed in Subsec. \[sec:criticalcase\] in detail.

![\[fig:renP\](Color online) Plot of renormalized momentum $ -P/\rho_\infty=\delta+pN/\rho_\infty $ in CQNLS. Here we set $ (a_1,a_2)=(-1,1) $. The values of $ \rho_\infty $ of each curve are set, from top to bottom, $ \rho_\infty=0.55,\, 0.52,\, 0.502,\, 0.5,\, 0.498,\, 0.48,\, \text{ and } 0.45 $, respectively. The dark solitons are stable in the regions of the solid lines, while unstable in the regions of the dashed lines. The critical points are marked by black dots. The unstable soliton appears when $ \rho_\infty<-a_1/(2a_2)=0.5 $. ](renP102.eps)

### Purely cubic case {#sec:purecubic}

As a first example, let us consider the case $ a_2=0 $, i.e., the nonlinearity is purely cubic. As mentioned in the Introduction, the NLS equation is integrable in this case and the Bogoliubov phonons are reflectionless for any energy. Let us see that our analytical result Eq. (\[eq:mainrR\]) is consistent with these 