The dispersion relation of a dark soliton

Abstract

The energy-velocity relation of a dark soliton is usually derived by its exact solution, which has been used to explain the kinetic motion of the dark soliton widely in many-body physical systems. We perform a variational method to re-derive the dispersion relation, with the consideration that the number of particles of the dark soliton could be conserved. The re-derived dispersion relation is completely different from that given by the exact dark soliton solution. The validity of these two dispersion relations is tested by observing the motion of the dark soliton when we drive impurity atoms that coupled with the soliton. The results suggest that the dispersion relation given by the exact solution usually works better than the one with particle number conservation. This motivates us to reveal that density waves (carrying particle transport) are generated during the acceleration process of a dark soliton, in addition to the previously known sound waves (only carrying energy transport). We further show that the density wave emissions of dark solitons can be inhibited by increasing the impurity atom number, which is trapped by the dark soliton through nonlinear coupling. The discussion is meaningful for investigating and understanding the kinetic motion of dark solitons in many different circumstances.