  We study the properties of a Bose-Einstein condensate (BEC)-impurity mixture
at finite temperature employing the time dependent Hartree-Fock Bogoliubov
(TDHFB) theory which is a set of coupled nonlinear equations of motion for the
condensate and its normal and anomalous fluctuations on the one hand, and for
impurity on the other. The numerical solutions of these equations in the static
quasi-1D regime show that the thermal cloud and the anomalous density are
deformed as happens to the condensate and the impurity becomes less localized
at nonzero temperatures. Effects of the BEC fluctuations on the self-trapping
state are studied in homogeneous weakly interacting BEC-impurity at low
temperature. The self-trapping threshold is also determined in such a system.
The formation of solitons in the BEC-impurity mixture at finite temperature is
investigated. Our formalism shows several new pictures.
