  We study spontaneous symmetry breaking in a system of two parallel
quasi-one-dimensional traps, equipped with optical lattices (OLs) and filled
with a Bose-Einstein condensate (BEC). The cores are linearly coupled by
tunneling. Analysis of the corresponding system of linearly coupled
Gross-Pitaevskii equations (GPEs) reveals that spectral bandgaps of the single
GPE split into subgaps. Symmetry breaking in two-component BEC solitons is
studied in cases of the attractive (AA) and repulsive (RR) nonlinearity in both
traps; the mixed situation, with repulsion in one trap and attraction in the
other (RA), is considered too. In all the cases, stable asymmetric solitons are
found, bifurcating from symmetric or antisymmetric ones (and destabilizing
them), in the AA and RR systems, respectively. In either case, bi-stability is
predicted, with a nonbifurcating stable branch, either antisymmetric or
symmetric, coexisting with asymmetric ones. Solitons destabilized by the
bifurcation tend to rearrange themselves into their stable asymmetric
counterparts. The impact of a phase mismatch, between the OLs in the two cores
is also studied. Also considered is a related model, for a binary BEC in a
single-core trap with the OL, assuming that the two species (representing
different spin states of the same atom) are coupled by linear interconversion.
In that case, the symmetry-breaking bifurcations in the AA and RR models switch
their character, if the inter-species nonlinear interaction becomes stronger
than the intra-species nonlinearity.
