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