Landau damping of transverse quadrupole oscillations of an elongated   Bose-Einstein condensate
  We study the interaction between low-lying transverse collective oscillations and thermal excitations of an elongated Bose-Einstein condensate by means of perturbation theory. We consider a cylindrically trapped condensate and calculate the transverse elementary excitations at zero temperature by solving the linearized Gross-Pitaevskii equations in two dimensions. We use them to calculate the matrix elements between thermal excited states coupled with the quasi-2D collective modes. The Landau damping of transverse collective modes is investigated as a function of temperature. At low temperatures, the damping rate due to the Landau decay mechanism is in agreement with the experimental data for the decay of the transverse quadrupole mode, but it is too small to explain the slow experimental decay of the transverse breathing mode. The reason for this discrepancy is discussed. 
M. Guilleumas and L.P. Pitaevskii