{#sec:mainresult}
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In this section we present the main results of this paper and verify them by numerical study of CQNLS equation. The proof will be given in Secs. \[sec:proof1\] and \[sec:proof2\].

Main result {#subsec:mainresult}
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In the scattering problem of linearized excitations defined in Subsec. \[sec:defofsp\], the amplitude of the reflected component $ r $ in Eq. (\[eq:uvasymptotic\]) is given by the following Padé approximant-like form: $$\begin{aligned}
        r = \frac{- \mathrm{i}\epsilon(d+d_1P_p)+O(\epsilon^2)}{a P_p - \mathrm{i}\epsilon(b+b_1P_p)+O(\epsilon^2)}. \label{eq:mainr}
    \end{aligned}$$ Here $ X_p:= \partial X/\partial p $ and $$\begin{aligned}
        a &= 4p_{\text{L}}\rho_\infty, \\
        b &= (N+pN_p)^2+(p_{\text{L}}N_p)^2, \\
        d &= (N+pN_p)^2-(p_{\text{L}}N_p)^2, \\
        b_1 &= N-\frac{p_{\text{L}}^2+p^2}{p_{\text{L}}^2-p^2}\widetilde{N}, \\
        d_1 &= N+\widetilde{N}
    \end{aligned}$$ with $$\begin{aligned}
        \widetilde{N}:=p \frac{\partial N}{\partial p}-\rho_\infty\frac{\partial N}{\partial \rho_\infty}.
    \end{aligned}$$ From (\[eq:mainr\]), the energy dependence of the reflection coefficient $ R $ (\[eq:defofR\]) becomes $$\begin{aligned}
        R = \begin{cases} \displaystyle \left( \frac{d+d_1P_p}{aP_p} \right)^2\epsilon^2+O(\epsilon^4) & (P_p\ne0) \\ \displaystyle \left( \frac{d}{b} \right)^2+O(\epsilon^2) & (P_p=0). \end{cases} \label{eq:mainrR}
    \end{aligned}$$ When $ P_p \ne0$, the zero-energy phonon transmits perfectly: $ \lim_{\epsilon\rightarrow0}R=0 $. On the other hand, when the soliton velocity reaches a critical value, i.e. $ P_p=0 $, this perfect transmission disappears. We note that the rational form expression (\[eq:mainr\]) makes it possible to unify the description of low-energy behaviors in 