  We study the quantum phase transition of the 1D weakly interacting Bose gas
in the presence of disorder. We characterize the phase transition as a function
of disorder and interaction strengths, by inspecting the long-range behavior of
the one-body density matrix as well as the drop in the superfluid fraction. We
focus on the properties of the low-energy Bogoliubov excitations that drive the
phase transition, and find that the transition to the insulator state is marked
by a diverging density of states and a localization length that diverges as a
power-law with power 1. We draw the phase diagram and we observe that the
boundary between the superfluid and the Bose glass phase is characterized by
two different algebraic relations. These can be explained analytically by
considering the limiting cases of zero and infinite disorder correlation
length.
