by expanding Eqs (\[eq:cqnlsN\]) and (\[eq:cqnlsdelta\]) with respect to $ a_2 $, but the expression is not so simple. In this case, one can derive an approximate formula valid not only for small energy but for arbitrary energy by the method given in Refs. [@Muryshev; @SinhaChernyKovrizhinBrand].

### Purely quintic case {#sec:purequintic}

Next, we treat the purely (self-defocusing) quintic case. As already mentioned, the quintic NLS equation is known to describe the dynamics of the Tonks-Girardeau gas[@Kolomeisky].\
Without loss of generality, we can set $ a_1=0 $ and $ a_2=1 $. Though we can also set $ \rho_\infty=1 $, we keep it for a moment because we need the differentiation of $ \rho_\infty $ to calculate the reflection coefficient (\[eq:mainrR\]). Eqs. (\[eq:cqnlsN\]) and (\[eq:cqnlsdelta\]) are reduced to $$\begin{aligned}
        N &= -\tanh^{-1}\left[ \frac{\sqrt{3(1-y^2)}}{2} \right], \\
        \delta &= 2 \tan^{-1}\left[ \frac{1-3y^2+\sqrt{1+3y^2}}{3y\sqrt{1-y^2}} \right] \\
        \text{with}\quad y &:= \frac{p}{p_{\text{L}}} = \frac{1}{\sqrt{3}}\frac{p}{\rho_\infty}.
    \end{aligned}$$ From them one can plot the renormalized momentum $ -P/\rho_\infty=\sqrt{3}yN+\delta $ and can show that $ -P_p<0 $ always holds, i.e., the soliton is always stable. One can also obtain the coefficient of $ \epsilon^2 $ in Eq. (\[eq:mainrR\]) as follows: $$\begin{aligned}
        -\frac{d+d_1P_p}{aP_p} &= \frac{1}{4\sqrt{3}\rho_\infty^2}\frac{\gamma\tanh^{-1}\gamma}{\gamma+2(1-\gamma^2)\tanh^{-1}\gamma}, \label{eq:quinticcoeff} \\
         \gamma&:=\frac{\sqrt{3(1-y^2)}}{2}.    \end{aligned}$$

![\[fig:quinticRC\](Color online) Energy-dependence of 