  We show that quantum solitons in the Lieb-Liniger Hamiltonian are precisely
the yrast states. We identify such solutions clearly with Lieb's type II
excitations from weak to strong interactions, clarifying a long-standing
question of the physical meaning of this excitation branch. We demonstrate that
the metastable quantum phase transition previously found in mean field analysis
of the weakly-interacting Lieb-Liniger Hamiltonian [Phys. Rev. A {\bf 79},
063616 (2009)] extends into the medium- to strongly-interacting regime of a
periodic one-dimensional Bose gas. Our methods are exact diagonalization,
finite-size Bethe ansatz, and the boson-fermion mapping in the Tonks-Girardeau
limit.
