  Magnetism in the FeAs stoichiometric compounds and its interplay with
superconductivity in vortex states are studied by self-consistently solving the
BdG equations based on a two-orbital model with including the on-site
interactions between electrons in the two orbitals. It is revealed that for the
parent compound, magnetism is caused by the strong Hund's coupling, and the
Fermi surface topology aids to select the spin-density-wave (SDW) pattern. The
superconducting (SC) order parameter with
$s_{\pm}=\Delta_{0}\cos(k_{x})\cos(k_{y})$ symmetry is found to be the most
favorable pairing for both the electron- and hole-doped cases, while the local
density-of-states (LDOS) exhibits the characteristic of nodal gap for the
former and full gap for the latter. In the vortex state, the emergence of the
field-induced SDW depends on the strength of the Hund's coupling and the
Coulomb repulsions. The field-induced SDW gaps the finite energy contours on
the electron and hole pocket sides, leading to the dual structures with one
reflecting the SC pairing and the other being related to the SDW order. These
features can be discernable in STM measurements for identifying the interplay
between the field-induced SDW order and the SC order around the core region.

Superconductivity