the soliton is determined by $ \partial P/\partial v=0 $, where $ v $ is a velocity of the soliton and $ P $ is a renormalized momentum. The existence of the critical velocity lower than the sound velocity and the separation of stable and unstable regions are similar to the phenomena of superflows against a potential barrier. Therefore, we can expect some scaling behavior near the critical state. Furthermore, in the present case, the preserved translational symmetry of the fundamental equation makes it possible to access the problem analytically.\
In this paper, we solve the scattering problem of linearized phonon excitations, and exactly show the following: (i) At the critical state determined by Barashenkov’s criterion[@Barashenkov], the transparency of the zero-energy phonon is suppressed, and only partial transmission occurs. (ii) Near the critical state, the energy-dependence of the reflection coefficient of low-energy phonons shows saddle-node scaling behavior, regarding the renormalized momentum as a parameter of a normal form of saddle-node bifurcation. The obtained analytical results are well confirmed by comparison with the numerical results of CQNLS equation. Our result gives an exact example of scaling laws of saddle-node bifurcation in time-reversible Hamiltonian systems. The proof is based on the exact 