  We report Dirac monopoles with polar-core vortex induced by spin-orbit
coupling in ferromagnetic Bose-Einstein condensates, which are attached to two
nodal vortex lines along the vertical axis. These monopoles are more stable in
the time scale of experiment and can be detected through directly imaging
vortex lines. When the strength of spin-orbit coupling increases, Dirac
monopoles with vortex can be transformed into those with square lattice. In the
presence of spin-orbit coupling, increasing the strength of interaction can
induce a cyclic phase transition from Dirac monopoles with polar-core vortex to
those with Mermin-Ho vortex. The spin-orbit coupled Bose-Einstein condensates
not only provide a new unique platform for investigating exotic monopoles and
relevant phase transitions, but also can preserve stable monopoles after a
quadrupole field is turned off.
