  In atomic Bose-Einstein condensates in optical lattices, mean-field energy
can support the existence of period-doubled density waves, which are similar to
Bloch waves but have the double periodicity of the underlying lattice
potentials. However, they are dynamically unstable. Here, we propose to use the
spin-orbit coupling to stabilize the period-doubled density waves. The
stabilization mechanism is revealed to relate to interaction-induced
spontaneous symmetry breaking of the spin-flip parity symmetry.
