  We study the domain walls which form when Bose condensates acquire a
double-well dispersion. Experiments have observed such domain walls in
condensates driven across a $\mathbb{Z}_2$ symmetry-breaking phase transition
in a shaken optical lattice. We derive a generic model to describe the
dispersion and to compute the wavefunctions and energies of the domain walls.
We find two distinct regimes which demand different physical pictures. In the
weak coupling regime, where interactions are weak compared to the kinetic
energy barrier, "density wave domain walls" form that support an extended
density wave and a series of phase steps. These features can be understood as
the quantum interference between domains with distinct momenta. In the strong
coupling regime where interaction dominates, the system forms "phase domain
walls" which have the minimum width allowed by the uncertainty principle and
suppressed density modulation. Analytic results for the domain wall
wavefunctions are obtained in the two regimes. The energy of domain walls
behaves similarly to that of topological defects in paradigmatic field
theories.
