  We study a two-dimensional array of coupled one-dimensional (1D) tubes of
interacting bosons. Such systems can be produced by loading ultra-cold atoms in
anisotropic optical lattices. We investigate the effects of coupling the tubes
via hopping of the bosons (i.e. Josephson coupling). In the absence of a
periodic potential along the tubes, or when such potential is incommensurate
with the boson density, the system undergoes a transition from an array of
incoherent Tomonaga-Luttinger liquids at high temperature to an anisotropic
Bose-Einstein condensate (BEC), at low temperature. We determine the transition
temperature and long wave-length excitations of the BEC. In addition to the
usual gapless (Goldstone) mode found in standard superfluids, we also find a
gapped mode associated with fluctuations of the amplitude of the order
parameter. When a commensurate periodic potential is applied along the tubes,
they can become 1D Mott insulators. Intertube hopping leads to a deconfinement
quantum phase transition between the 1D Mott insulators and the anisotropic
BEC. We also take into account the finite size of the gas tubes as realized in
actual experiments. We map out the phase diagram of the quasi-1D lattice and
compare our results with the existing experiments on such systems.
