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A scattering problem (or more precisely, a transmission-reflection problem)of linearized excitations in the presence of a dark soliton is considered in aone-dimensional nonlinear Schr\"odinger system with a general nonlinearity: $\mathrm{i}\partial_t \phi = -\partial_x^2 \phi + F(|\phi|^2)\phi $. If thesystem is interpreted as a Bose-Einstein condensate, the linearized excitationis a Bogoliubov phonon, and the linearized equation is the Bogoliubov equation.We exactly prove that the perfect transmission of the zero-energy phonon issuppressed at a critical state determined by Barashenkov's stability criterion[Phys. Rev. Lett. 77, (1996) 1193.], and near the critical state, theenergy-dependence of the reflection coefficient shows a saddle-node typescaling law. The analytical results are well supported by numerical calculationfor cubic-quintic nonlinearity. Our result gives an exact example of scalinglaws of saddle-node bifurcation in time-reversible Hamiltonian systems. As aby-product of the proof, we also give all exact zero-energy solutions of theBogoliubov equation and their finite energy extension.

Soliton-phonon scattering problem in 1D nonlinear Schr\"odinger systems with general nonlinearity