of reflection coefficient $ R $. Here, based on Eq. (\[eq:energyscale\]), the horizontal axis is chosen to be the scaled energy $ \tilde{\epsilon}=\epsilon/\sqrt{2P''(p_c)(P(p)-P(p_c))} $. We can see that if the soliton velocity is close to the critical one, numerically calculated points are well fitted to the universal curve (\[eq:scaledR\]). Thus, the theoretical results are well confirmed in this example.

![\[fig:RC\](Color online) Energy-dependence of reflection coefficient $ R $ of linearized excitations for various soliton velocities. Here we set $ (a_1,a_2)=(-1,1) $ and $ \rho_\infty=0.45 $. The critical velocity of the dark soliton is given by $ p_c = (0.206597\dots)\times p_{\text{L}}$. (See the lowest curve of Fig. \[fig:renP\].) A reflection coefficient of zero-energy phonon at the critical state is given by $ (d/b)^2\simeq 0.5718 $. Parabolic curves represent the theoretical approximate expression (\[eq:mainrR\]).](RC103.eps)

![\[fig:scaledR\](Color online) Scaling behavior of reflection coefficient $ R $. Here we set $ (a_1,a_2)=(-1,1) $ and $ \rho_\infty=0.45 $. $ \tilde{\epsilon}:=\epsilon/\sqrt{2P''(p_c)(P(p)-P(p_c))} $ is a scaled excitation energy. “Theory” represents the universal form of reflection coefficient (\[eq:scaledR\]).](scaledR002.eps)

Proof – Step 1: Exact Zero-Energy Solutions {#sec:proof1}
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In this and the next section, we prove the main result. This section is particularly devoted to the construction of exact zero-energy 