  In a previous paper [Phys. Rev. B 85,195144 (2012)], variational Monte Carlo
method (based on Gutzwiller projected states) was generalized to $S=1$ systems.
This method provided very good trial ground states for the gapped phases of
$S=1$ bilinear-biquadratic (BLBQ) Heisenberg chain. In the present paper, we
extend the approach to study the low-lying elementary excitations in $S=1$
chains. We calculate the one-magnon and two-magnon excitation spectra of the
BLBQ Heisenberg chain and the results agree very well with recent data in
literature. In our approach, the difference of the excitation spectrum between
the Haldane phase and the dimer phase (such as the even/odd size effect) can be
understood from their different topology of corresponding mean field theory. We
especially study the Takhtajan-Babujian critical point. Despite the fact that
the `elementary excitations' are spin-1 magnons which are different from the
spin-1/2 spinons in Bethe solution, we show that the excitation spectrum,
critical exponent ($\eta=0.74$) and central charge ($c=1.45$) calculated from
our theory agree well with Bethe ansatz solution and conformal field theory
predictions.
