  We study the formation of a quasi-condensate in a nearly one dimensional,
weakly interacting trapped atomic Bose gas. We show that a Hartree Fock
(mean-field) approach fails to explain the presence of the quasi-condensate in
the center of the cloud: the quasi-condensate appears through an
interaction-driven cross-over and not a saturation of the excited states.
Numerical calculations based on Bogoliubov theory give an estimate of the
cross-over density in agreement with experimental results.
