  We explore if the phase separation or coexistence of the spin density wave
(SDW) and superconductivity (SC) states has any relation to the
incommensurability of the SDW in the Fe pnictide superconductors. A systematic
method of determining the phase separation or coexistence was employed by
computing the anisotropy coefficient $\beta$ from the the 4th order terms of
the Ginzburg--Landau (GL) expansion of the free energy close to the
tricritical/tetracritical point. It was complemented by the self-consistent
numerical iterations of the gap equations to map out the boundaries between the
phase separation and coexistence of the SDW and SC phases, and between
commensurate (C) and incommensurate (IC) SDW in the temperature--doping plane.
Our principal results for the sign reversed $s$-wave pairing SC, in terms of
the multicritical temperature, $T_c$, the phase separation/coexistence boundary
between the SDW and SC, $T^*$, and the boundary between C/IC SDW, $T_M^*$, are:
(a) IC-SDW and SC coexist for $T_c < T^*$ and phase separate otherwise, (b) SDW
takes the C form for $T_c>T_M^*$ and IC form for $T_c<T_M^*$, and (c) the
thermodynamic first order phase transition intervenes in between the C-SDW and
IC-SDW boundary for large $T_M^0$, where $T_M^0$ is the SDW transition
temperature at zero doping, $T^*=0.35 ~T_M^0$ and $T_M^*=0.56\ T_M^0$. The
intervention makes the phase diagram more complicated than previously reported.
By contrast no coexistence was found for the equal sign pairing SC. These
results will be compared with the experimental reports in the Fe pnictide
superconductors.
