The exactly solvable Lieb-Liniger model of interacting bosons in
one-dimension has attracted renewed interest as current experiments with
ultra-cold atoms begin to probe this regime. Here we numerically solve the
equations arising from the Bethe ansatz solution for the exact many-body wave
function in a finite-size system of up to twenty particles for attractive
interactions. We discuss the novel features of the solutions, and how they
deviate from the well-known string solutions [H. B. Thacker, Rev. Mod. Phys.\
\textbf{53}, 253 (1981)] at finite densities. We present excited state string
solutions in the limit of strong interactions and discuss their physical
interpretation, as well as the characteristics of the quantum phase transition
that occurs as a function of interaction strength in the mean-field limit.
Finally we compare our results to those of exact diagonalization of the
many-body Hamiltonian in a truncated basis. We also present excited state
solutions and the excitation spectrum for the repulsive 1D Bose gas on a ring.Comment: 13 pages, 12 figure
