  We analyze in detail the expansion of a 1D Bose gas after removing the axial
confinement. We show that during its one-dimensional expansion the density of
the Bose gas does not follow a self-similar solution, but on the contrary, it
asymptotically approaches a Tonks-Girardeau profile. Our analysis is based on a
nonlinear Schr\"odinger equation with variable nonlinearity whose validity is
discussed for the expansion problem, by comparing with an exact Bose-Fermi
mapping for the case of an initial Tonks-Girardeau gas. For this case, the gas
is shown to expand self-similarly, with a different similarity law compared to
the one-dimensional Thomas-Fermi condensate.
