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

TL;DR

This paper analyzes how linear excitations, specifically phonons, scatter off dark solitons in a 1D nonlinear Schrödinger system with general nonlinearity, revealing critical behavior and scaling laws at stability thresholds.

Contribution

It provides an exact proof of suppressed zero-energy phonon transmission at a critical state and characterizes the associated saddle-node scaling law in a general nonlinear Schrödinger framework.

Findings

Perfect transmission of zero-energy phonons is suppressed at the critical state.

Near the critical state, the reflection coefficient exhibits saddle-node scaling.

Numerical calculations support the analytical results for cubic-quintic nonlinearity.

Abstract

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 Schr\"odinger system with a general nonlinearity: $ \mathrm{i}\partial_t \phi = -\partial_x^2 \phi + F(|\phi|^2)\phi $. 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 [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.