  We consider one-dimensional bosonic chains with a repulsive boson-boson
interaction that decays exponentially on large length-scales. This model
describes transport of Cooper-pairs in a Josepshon junction array, or transport
of magnetic flux quanta in quantum-phase-slip ladders, i.e. arrays of
superconducting wires in a ladder-configuration that allow for the coherent
tunnelling of flux quanta. In the low-frequency, long wave-length regime these
chains can be mapped to an effective model of a one-dimensional elastic field
in a disordered potential. The onset of transport in these systems, when biased
by external voltage, is described by the standard depinning theory of elastic
media in disordered pinning potentials. We numerically study the regimes that
are of relevance for quantum-phase-slip ladders. These are (i) very short
chains and (ii) the regime of weak disorder. For chains shorter than the
typical pinning length, i.e., the Larkin length, the chains reach a saturation
regime where the depinning voltage does not depend on the decay length of the
repulsive interaction. In the regime of weak disorder we find an emergent
correlation length-scale that depends on the disorder strength. For arrays
shorter than this length the onset of transport is similar to the clean arrays,
i.e., is due to the penetration of solitons into the array. We discuss the
depinning scenarios for longer arrays in this regime.
