Soliton–phonon scattering problem in 1D nonlinear Schrödinger systems with general nonlinearity

A scattering problem (or more precisely, a transmission–reflection problem) of linearized excitations in the presence of a dark soliton is considered in a one-dimensional nonlinear Schrödinger system with a general nonlinearity: i ∂ t ϕ = − ∂ x 2 ϕ + F ( | ϕ | 2 ) ϕ . If the system is interpreted as a Bose–Einstein condensate, the linearized excitation is a Bogoliubov phonon, and the linearized equation is the Bogoliubov equation. We exactly prove that the perfect transmission of the zero-energy phonon is suppressed at a critical state determined by Barashenkov’s stability criterion [I.V. Barashenkov, Stability criterion for dark solitons, Phys. Rev. Lett. 77, (1996) 1193.], and near the critical state, the energy-dependence of the reflection coefficient shows a saddle–node type scaling law. The analytical results are well supported by numerical calculation for cubic-quintic nonlinearity. Our result gives an exact example of scaling laws of saddle–node bifurcation in time-reversible Hamiltonian systems. As a by-product of the proof, we also give all exact zero-energy solutions of the Bogoliubov equation and their finite energy extension.