A kinetic approach to Bose-Einstein condensates: self-phase modulation   and Bogoliubov oscillations
  A kinetic approach to the Bose-Einstein condensates (BECs) is proposed, based on the Wigner-Moyal equation (WME). In the quasi-classical limit, the WME reduces to the particle number conservation equation. Two examples of application are: i) a self-phase modulation of a BE condensate beam where we show that part of the beam is decelerated and eventually stops as a result of the gradient of the effective self-potential; ii) the derivation of a kinetic dispersion relation for sound waves in the BECs, including a collisionless Landau damping. 
J. T. Mendonca, R. Bingham and P. K. Shukla