  The induction of spin density wave (SDW) and charge density wave (CDW)
orderings in the mixed state of high $T_c$ superconductors (HTS) is
investigated by using the self-consistent Bogoliubov-de Gennes equations based
upon an effective model Hamiltonian with competing SDW and d-wave
superconductivity interactions. For optimized doping sample, the modulation of
the induced SDW and its associated CDW is determined by the vortex lattice and
their patterns obey the four-fold symmetry. By deceasing doping level, both SDW
and CDW show quasi-one dimensional like behavior, and the CDW has a period just
half that of the SDW along one direction. From the calculation of the local
density of states (LDOS), we found that the majority of the quasi-particles
inside the vortex core are localized. All these results are consistent with
several recent experiments on HTS.
