  We study the integrable model of one-dimensional bosons with contact
repulsion. In the limit of weak interaction, we use the microscopic
hydrodynamic theory to obtain the excitation spectrum. The statistics of
quasiparticles changes with the increase of momentum. At lowest momenta good
quasiparticles are fermions, while at higher momenta they are Bogoliubov
bosons, in accordance with recent studies. In the limit of strong interaction,
we analyze the exact solution and find exact results for the spectrum in terms
of the asymptotic series. Those results undoubtedly suggest that fermionic
quasiparticle excitations actually exist at all momenta for moderate and strong
interaction, and also at lowest momenta for arbitrary interaction. Moreover, at
strong interaction we find highly accurate analytical results for several
relevant quantities of the Lieb-Liniger model.
