  We analyze free expansion of a trapped one-dimensional Bose gas after a
sudden release from the confining trap potential. By using the stationary phase
and local density approximations, we show that the long-time asymptotic density
profile and the momentum distribution of the gas are determined by the initial
distribution of Bethe rapidities (quasimomenta) and hence can be obtained from
the solutions to the Lieb-Liniger equations in the thermodynamic limit. For
expansion from a harmonic trap, and in the limits of very weak and very strong
interactions, we recover the self-similar scaling solutions known from the
hydrodynamic approach. For all other power-law traps and arbitrary interaction
strengths, the expansion is not self-similar and shows strong dependence of the
density profile evolution on the trap anharmonicity. We also characterize
dynamical fermionization of the expanding cloud in terms of correlation
functions describing phase and density fluctuations.
