We investigate the Landau damping of the collective mode in a quasi-one-dimensional repulsive Bose-Einstein condensate by using the self-consistent time-dependent Hartree-Fock-Bogoliubov approximation. We put forward a new method to calculate the Landau damping rate of the collective mode in the condensate and discuss the dependence of the Landau damping on temperature, on transverse trapping frequency, on atom number in the condensate, and on length of the system. Different from the usual calculation method for the three dimension system, our new calculation method is an interactive one by considering the practical relaxation of the elementary excitation. With little approximation, our theoretical calculation results agree with the experimental ones. Comparing with the usual calculation method, our theory is helpful to deduce the inter-particle interactions in damping phenomenon.

The Landau damping and frequency-shift of monopole mode in an elongated-rubidium Bose-Einstein condensate are investigated by using the time-dependent Hartree-Fock-Bogoliubov approximation. Improving the previous approach, We have taken into account the practical relaxations of elementary excitations and the orthogonal relation among them. With such an approach, we provide a new calculation formula for Landau damping rate and frequency-shift. In addition, our previous method of eliminating the divergence in three-mode coupling matrix elements is also improved by zeroing the kinetic energy at the condensate boundary instead of minimizing the ground-state energy. Based on these improvements, both the Landau damping rate and the frequency-shift of the monopole mode are analytically calculated and their temperature dependences are also discussed. And all the theoretical results are in agree meat with experimental data.