both critical and non-critical cases. If we use a simple Taylor series, the singular behavior at the critical velocity state cannot be expressed.\
The above (\[eq:mainrR\]) is one good result valid for any soliton velocity, even if it is far from the critical state. However, when the velocity comes close to the critical value, i.e., $ P_p $ comes close to zero, we can derive a more powerful scaling law as below. Let us assume that coefficients of $ \epsilon^n \ (n\ge2) $ in (\[eq:mainr\]) are all finite in the limit $ P_p\rightarrow0 $. and take the limit $ \epsilon\rightarrow 0 $ and $ P_p\rightarrow0 $ with a constraint $ \epsilon/P_p=\text{fix} $. We then obtain $$\begin{aligned}
        r \rightarrow \frac{-\mathrm{i}d(\epsilon/P_p)}{a-\mathrm{i}b(\epsilon/P_p)},
    \end{aligned}$$ and in the same limit, the *universal* form of reflection coefficient $$\begin{aligned}
        \lim_{\substack{\epsilon\rightarrow0,\, P_p\rightarrow0,\\ \epsilon/P_p:\text{fix}}} R = \frac{d^2(\epsilon/P_p)^2}{a^2+b^2 (\epsilon/P_p)^2} \label{eq:scaledR}
    \end{aligned}$$ follows. Here, the values of $ a, b, $ and $ d $ in the critical state must be substituted when we use Eq. (\[eq:scaledR\]). We remark that Eq. (\[eq:scaledR\]) contains only $ p $-derivatives, whereas the expression before taking the scaling limit contains $ \rho_\infty $-derivatives in addition to $ p $-derivatives.\
Let $ p_c $ be a 