critical velocity of the dark soliton, i.e., $ P'(p_c)=0 $. The expansion of $ P $ near $ p=p_c $ gives $$\begin{aligned}
        P(p) &\simeq P(p_c)+\frac{1}{2}P''(p_c)(p-p_c)^2+\dotsb, \\
        \rightarrow |P_p| &= |P'(p)| \simeq |2P''(p_c)(P(p)-P(p_c))|^{1/2}.
    \end{aligned}$$ Therefore we obtain $$\begin{aligned}
        \frac{\epsilon}{|P_p|} \simeq \frac{\epsilon}{|2P''(p_c)(P(p)-P(p_c))|^{1/2}}. \label{eq:energyscale}
    \end{aligned}$$ This is an expected scaling behavior from the normal form of saddle-node bifurcation[@GuckenheimerHolmes; @PhamBrachet], if we regard the renormalized momentum $ P $ as a parameter of normal form.

Comparison with Numerical Results in CQNLS System {#subsec:cqnls}
-------------------------------------------------

In this subsection, we numerically verify the analytical results of the preceding subsection in the CQNLS system. We first derive the expressions for the dark soliton solution and the renormalized momentum in Subsec. \[subsubsec:darksolitoncqnls\], and solve the scattering problems of linearized excitations for (i) the purely cubic case in Subsec. \[sec:purecubic\], (ii) the purely quintic case in Subsec. \[sec:purequintic\], and (iii) the case where a non-trivial critical velocity exists in Subsec. \[sec:criticalcase\].

### Dark soliton solution and renormalized momentum {#subsubsec:darksolitoncqnls}

In the CQNLS system, the nonlinear term is defined by $$\begin{aligned}
        U(\rho) &= a_1\rho^2+a_2\rho^3, \\
        F(\rho)&=U'(\rho) = 2a_1\rho+3a_2\rho^2,
    \end{aligned}$$ and the NLS equation (\[eq:nls\]) has the cubic-quintic nonlinearity: $$\begin{aligned}
        \mathrm{i}\partial_t\phi=-\partial_x^2\phi+2a_1|\phi|^2\phi+3a_2|\phi|^4\phi.
    \end{aligned}$$ The stationary linearized equation, i.e., the stationary Bogoliubov equation 