  Motivated by recent experimental and theoretical progresses of the magnetic
properties of iron-based superconductors, we provide a comprehensive analysis
of the spin-1 bilinear-biquadratic (BBQ) model on the square lattice. Using
variational approach in the mean-field level, we identify the existence of
various magnetic phases, including conventional spin dipolar orderings (ferro-
or antiferromagnet), novel quadrupolar (spin nematic) orderings and mixed
dipolar-quadrupolar orderings. In contrast to the usual Heisenberg model, the
elementary excitations of the spin-1 BBQ model are described by the flavor-wave
theory within the SU(3) representation. By fitting the experimental spin-wave
dispersion, we determine the refined exchange couplings corresponding to the
collinear antiferromagnetic iron pnictides. We also present the dynamic
structure factors of both spin dipolar and quadrupolar components with
connections to the future experiments.
