  We consider a one-dimensional model of a two-component Bose-Einstein
condensate in the presence of periodic external potentials of opposite signs,
acting on the two species. The interaction between the species is attractive,
while intra-species interactions may be attractive too [the system of the
right-bright (BB) type], or of opposite signs in the two components [the
gap-bright (GB) model]. We identify the existence and stability domains for
soliton complexes of the BB and GB types. The evolution of unstable solitons
leads to the establishment of oscillatory states. The increase of the strength
of the nonlinear attraction between the species results in symbiotic
stabilization of the complexes, despite the fact that one component is centered
around a local maximum of the respective periodic potential.
