  The emergence of a Bose-glass region in a quasi one-dimensional
Bose-Einstein-condensed gas in a harmonic trapping potential with an additional
delta-correlated disorder potential at zero temperature is studied using three
approaches. At first, the corresponding time-independent Gross-Pitaevskii
equation is numerically solved for the condensate wave function, and disorder
ensemble averages are evaluated. In particular, we analyse quantitatively the
emergence of mini-condensates in the local minima of the random potential,
which occurs for weak disorder preferentially at the border of the condensate,
while for intermediate disorder strength this happens in the trap centre.
Second, in view of a more detailed physical understanding of this phenomenon,
we extend a quite recent non-perturbative approach towards the weakly
interacting dirty boson problem, which relies on the Hartree-Fock theory and is
worked out on the basis of the replica method, from the homogeneous case to a
harmonic confinement. Finally, in the weak disorder regime we also apply the
Thomas-Fermi approximation, while in the intermediate disorder regime we
additionally use a variational ansatz in order to describe analytically the
numerically observed redistribution of the fragmented mini-condensates with
increasing disorder strength.
