  We study the density-wave states of quasi-one-dimensional atomic gas mixture
of one- and two-component boson and fermion using the mean-field approximation.
Owing to the Peierls instability in the quasi-one-dimensional fermion system,
the ground state of the system shows the fermion density wave and the periodic
Bose-Einstein condensation induced by the boson-fermion interatomic
interaction. For the two-component fermions, two density waves appear in these
components, and the phase difference between them distinguishes two types of
ground states, the in-phase and the out-phase density-waves. In this paper, a
self-consistent method in the mean-field approximation is presented to treat
the density-wave states in boson-fermion mixture with two-component fermions.
  From the analysis of the effective potential and the interaction energies
calculated by this method, the density-waves are shown to appear in the ground
state, which are in-phase or out-phase depending on the strength of the
inter-fermion interaction. It is also shown that the periodic Bose-Einstein
condensate coexists with the in-phase density-wave of fermions, but, in the
case of the out-phase one, only the uniform condensate appears. The phase
diagram of the system is given for the effective coupling constants.
