  We investigate the Lieb-Liniger gas initially prepared in an
out-of-equilibrium state that is Gaussian in terms of the phonons. Because the
phonons are not exact eigenstates of the Hamiltonian, the gas relaxes to a
stationary state at very long times. Thanks to integrability, that stationary
state needs not be a thermal state. We characterize the stationary state of the
gas after relaxation and compute its phonon population distribution.
Technically, this follows from the mapping between the exact eigenstates of the
Lieb-Liniger Hamiltonian and those of a non-interacting Fermi gas -- a mapping
provided by the Bethe equations -- , as well as on bosonization formulas valid
in the low-energy sector of the Hilbert space. We apply our results to the case
where the initial state is an excited coherent state for a single phonon mode,
and we compare them to exact results obtained in the hard-core limit.
