  We theoretically investigate the droplets formation in a tightly trapped
one-dimensional dipolar gas of bosonic atoms. When the strength of the dipolar
interaction becomes sufficiently attractive compared to the contact one, we
show how a solitonic-like density profile evolves into a liquid-like droplet on
increasing the number of particles in the trap. The incipient gas-liquid
transition is also signalled by a steep increase of the breathing mode and a
change in sign of the chemical potential. Upon a sudden release of the trap,
varying the number of trapped atoms and the scattering length, the numerical
solution of a time-dependent generalized Gross-Pitaevskii equation shows either
an evaporation of the cloud, the formation of a single self-bound droplet or a
fragmentation in multiple droplets.These results can be probed with lanthanides
atoms and help in characterizing the effect of the dipolar interaction in a
quasi-one-dimensional geometry.
