known facts.\
Without loss of generality, we can set $ a_1=1 $. The dark soliton solution (\[eq:dscqnls\]) is then reduced to $$\begin{aligned}
        \phi = \mathrm{e}^{\mathrm{i}px}\left( p+\mathrm{i}\kappa\tanh \kappa x \right), \quad \kappa = \sqrt{\rho_\infty-p^2}.
    \end{aligned}$$ The exact solution of the linearized equation, i.e., the Bogoliubov equation is given by the squared Jost solution[@ChenChenHuang; @Kovrizhin]: $$\begin{aligned}
        u &= \mathrm{e}^{\mathrm{i}(k_j+p)x}\left( \mathrm{i}\kappa\tanh \kappa x+\frac{k_j}{2}+\frac{\epsilon}{2k_j} \right)^2, \\
        v &= \mathrm{e}^{\mathrm{i}(k_j-p)x}\left( \mathrm{i}\kappa\tanh \kappa x+\frac{k_j}{2}-\frac{\epsilon}{2k_j} \right)^2,
    \end{aligned}$$ where $ k_j $s are given by the roots of the dispersion relation Eq. (\[eq:uniformdisp\]) with $ F'(\rho_\infty)=2 $.\
From the above explicit expression, it is obvious that the phonons are reflectionless. Therefore, the coefficient of $ \epsilon^2 $ in Eq. (\[eq:mainrR\]) must vanish. Let us check it. For the cubic case, it follows that $$\begin{aligned}
        N=-2\kappa,\quad \delta = 2 \tan^{-1}\frac{\kappa}{p},
    \end{aligned}$$ by taking the limit $ a_2\rightarrow0 $ of Eqs. (\[eq:cqnlsN\]) and (\[eq:cqnlsdelta\]). With the use of them, we can show $$\begin{gathered}
        P_p = -N-pN_p-\rho_\infty \delta_{p}=4\kappa, \\
        d=4(\kappa^2-3p^2),\quad d_1=\frac{-\kappa^2+3p^2}{\kappa}.
    \end{gathered}$$ Thus we obtain $ d+d_1P_p=0 $, as expected. We also note that the soliton is always stable since $ -P_p<0 $ for all velocities.\
It is also possible to discuss the reflection properties when the quintic term is small 