reflection coefficient $ R $ of linearized excitations for various soliton velocities in the purely quintic system. Here we set $ (a_1,a_2)=(0,1) $ and $ \rho_\infty=1 $. The Landau velocity is given by $ p_{\text{L}}=\sqrt{3}\rho_\infty=\sqrt{3} $. Parabolic curves represent the theoretical approximate expression (\[eq:mainrR\]) with (\[eq:quinticcoeff\]).](quinticRC002.eps)

The energy-dependence of the reflection coefficient $ R $ of linearized excitations is obtained by solving the Bogoliubov equation (\[eq:bogocqnls\]) numerically, and the results are shown in Fig. \[fig:quinticRC\]. We can verify that the expression (\[eq:mainrR\]) with (\[eq:quinticcoeff\]) is valid for low-energy region. From this figure we can also see that the soliton with zero velocity is the strongest scatterer. It is intuitively clear since the shape of the soliton becomes shallower and wider if the velocity of the soliton increases. However, this intuitive understanding is not always correct, as the integrable cubic case in Subsec. \[sec:purecubic\] and the instability-induced anomaly in Subsec. \[sec:criticalcase\] illustrate.

### The case with $ a_1<0 $ and $ a_2>0 $ {#sec:criticalcase}

Finally, we consider the case with $ a_1<0 $ and $ a_2>0 $, which is most interesting from the viewpoint of critical phenomena, since the soliton can become unstable and the reflection coefficient can show the singular 