Correspondence between dark solitons and the type II excitations of   Lieb-Liniger model
  A one-dimensional model of bosons with repulsive short-range interactions, solved analytically by Lieb and Liniger many years ago, predicts existence of two branches of elementary excitations. One of them represents Bogoliubov phonons, the other, as suggested by some authors, might be related to dark solitons. On the other hand, it has been already demonstrated within a framework of the classical field approximation that quasi-one-dimensional interacting Bose gas at equilibrium exhibits excitations which are phonons and dark solitons. By showing that statistical distributions of dark solitons obtained within the classical field approximation match the distributions of quasiparticles of the second kind derived from fully quantum description we demonstrate that type II excitations in the Lieb-Liniger model are, indeed, quantum solitons. 
T. Karpiuk, T. Sowinski, M. Gajda, K. Rzazewski, M. Brewczyk
cond-mat.quant-gas