  We study a system of penetrable bosons on a line, focusing on the
high-density/weak-interaction regime, where the ground state is, to a good
approximation, a condensate. Under compression, the system clusterizes at zero
temperature, i.e., particles gather together in separate, equally populated
bunches. We compare predictions from the Gross-Pitaevskii (GP) equation with
those of two distinct variational approximations of the single-particle state,
written as either a sum of Gaussians or the square root of it. Not only the
wave functions in the three theories are similar, but also the phase-transition
density is the same for all. In particular, clusterization occurs together with
the softening of roton excitations in GP theory. Compared to the latter theory,
Gaussian variational theory has the advantage that the mean-field energy
functional is written in (almost) closed form, which enables us to extract the
phase-transition and high-density behaviors in fully analytic terms. We also
compute the superfluid fraction of the clustered system, uncovering its exact
behavior close, as well as very far away from, the transition.
