and scaling behavior.\
If both $ a_1 $ and $ a_2 $ are nonzero, we can set $ |a_1|=|a_2|=1 $ without loss of generality by the following scale transformation: $$\begin{gathered}
        \bar{x}=\frac{x}{\xi},\ \bar{t}=\frac{t}{\xi^2},\  \bar{\phi}(\bar{x},\bar{t})=\frac{1}{\eta}\phi(x,t), \\
        \xi=\frac{\sqrt{|a_2|}}{|a_1|},\ \eta=\sqrt{\frac{|a_1|}{|a_2|}}.
    \end{gathered}$$ So we performed numerical calculations by setting $ (a_1,a_2)=(-1,1)$. Note that $ \rho_\infty $ cannot be normalized to be unity if we choose $ \xi $ and $ \eta $ as the above. Another choice of $ \eta $ is possible to normalize $ \rho_\infty=1 $, but in this case either $ a_1 $ or $ a_2 $ cannot be normalized.\
Using the dark soliton solution (\[eq:dscqnls\]), we numerically solved the stationary Bogoliubov equation (\[eq:bogocqnls\]), and constructed the solution with the asymptotic form (\[eq:uvasymptotic\]). Figure \[fig:RC\] shows the reflection coefficient with $ \rho_\infty=0.45 $ for various soliton velocities. We can observe that the zero-energy phonon transmits perfectly, unless the soliton velocity is equal to the critical one. When the soliton velocity comes close to the critical value, the slope of the reflection coefficient becomes very steep, and at the critical state, the perfect transmission eventually vanishes. The approximate expression (\[eq:mainrR\]) is good for sufficiently low excitation energy. Figure \[fig:scaledR\] shows the scaling behavior 