Dynamical transition from a quasi-one dimensional Bose-Einstein condensate to a Tonks-Girardeau gas

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.