  Ultracold gases are a versatile platform to simulate condensed matter
physics, as virtually any parameter is experimentally tunable. In particular,
highly anisotropic traps allow the realization of low-dimensional systems,
where the role of quantum fluctuations is enhanced. I investigate the
paradigmatic model of a one-dimensional Bose gas with contact interactions,
a.k.a the Lieb-Liniger model, using powerful analytical tools such as Bethe
Ansatz, Conformal Field Theory and the Tomonaga-Luttinger liquid formalism, as
well as the Bose-Fermi mapping. The effect of an additional harmonic trap in
the longitudinal direction is studied within the Local Density Approximation.
These methods allow to investigate with high accuracy its ground-state
thermodynamics of the gas, its excitation spectra and correlation functions,
from the k-body local correlations to the momentum distribution and the
dynamical structure factor. These quantities characterize the coherence and
superfluidity of the gas. In particular, I study the drag force exerted by a
weak gaussian barrier stirred into the gas within linear response theory. Then,
I investigate the appearance of a multimode structure obtained by releasing a
transverse trapping, and the dimensional crossover in the limit of an infinite
number of transverse modes.
