Over the last several years, the boson-Hubbard Hamiltonian has been widely used as an effective model of the physics of ultra-cold optically trapped atoms. In this paper we will review its equilibrium properties, in one dimension, both in the absence and in the presence of an external potential. Then we will describe some new results for the visibility V of the interference pattern obtained after the free expansion of a gas. We will show that the evolution of V exhibits kinks with increasing interaction strength U/t, replicating phenomena seen experimentally. The behavior of the density profiles n i correlates well with the features in V. These profiles reveal an unexpected 'freezing' - a range of interaction strengths in which they remain unchanged. Finally, we show that V exhibits an increase with U/t signaling when a central n i = 2 Mott region melts and coherence develops between two superfluid regions with local densities 1 < n i < 2.