  We consider a model of two tunnel-coupled one-dimensional Bose gases with
hard-wall boundary conditions. Bosonizing the model and retaining only the most
relevant interactions leads to a decoupled theory consisting of a quantum
sine-Gordon model and a free boson, describing respectively the antisymmetric
and symmetric combinations of the phase fields. We go beyond this description
by retaining the perturbation with the next smallest scaling dimension. This
perturbation carries conformal spin and couples the two sectors. We carry out a
detailed investigation of the effects of this coupling on the non-equilibrium
dynamics of the model. We focus in particular on the role played by spatial
inhomogeneities in the initial state in a quantum quench setup.
