  While the properties and the shape of the ground state of a gas of ultracold
bosons are well understood in harmonic potentials, they remain for a large part
unknown in the case of random potentials. Here, we use the
localization-landscape (LL) theory to study the properties of the solutions to
the Gross-Pitaevskii equation (GPE) in one-dimensional (1D) speckle potentials.
In the cases of attractive interactions, we find that the LL allows one to
predict the position of the localization center of the ground state (GS) of the
GPE. For weakly repulsive interactions, we point out that the GS of the
quasi-1D GPE can be understood as a superposition of a finite number of
single-particle states, which can be computed by exploiting the LL. For
intermediate repulsive interactions, we introduce a Thomas-Fermi-like approach
for the GS which holds in the smoothing regime, well beyond the usual
approximation involving the original potential. Moreover, we show that, in the
Lifshitz glass regime, the particle density and the chemical potential can be
well estimated by the LL. Our approach can be applied to any positive-valued
random potential endowed with finite-range correlations and can be generalized
to higher-dimensional systems.
