  In neutron star matter, there exist $^{1}S_{0}$ superfluids in lower density
in the crust while $^{3}P_{2}$ superfluids are believed to exist at higher
density deep inside the core. In the latter, depending on the temperature and
magnetic field, either the uniaxial nematic phase, the D$_{2}$-biaxial nematic
phase, or the D$_{4}$-biaxial nematic phase appears. In this paper, we discuss
a mixture of the $^{1}S_{0}$ and $^{3}P_{2}$ superfluids and find their
coexistence. Adopting the loop expansion and the weak-coupling approximation
for the interaction between two neutrons, we obtain the Ginzburg-Landau (GL)
free energy in which both of the $^{1}S_{0}$ and $^{3}P_{2}$ condensates are
taken into account by including the coupling terms between them. We analyze the
GL free energy and obtain the phase diagram for the temperature and magnetic
field. We find that the $^{1}S_{0}$ superfluid excludes the $^{3}P_{2}$
superfluid completely in the absence of magnetic field, they can coexist for
weak magnetic fields, and the $^{1}S_{0}$ superfluid is expelled by the
$^{3}P_{2}$ superfluid at strong magnetic fields, thereby proving the
robustness of $^{3}P_{2}$ superfluid against the magnetic field. We further
show that the D$_{4}$-BN phase covers the whole region of the $^{3}P_{2}$
superfluidity as a result of the coupling term, in contrast to the case of a
pure $^{3}P_{2}$ superfluid studied before in which the D$_{4}$-BN phase is
realized only under strong magnetic fields. Thus, the D$_{4}$-BN phase is
topologically the most interesting phase, e.g., admitting half-quantized
non-Abelian vortices relevant not only in magnetars but also in ordinary
neutron stars.
