  Two effectively one-dimensional parallel coupled Bose-Einstein condensates in
the presence of external potentials are studied. The system is modelled by
linearly coupled Gross-Pitaevskii equations. In particular, grey-soliton-like
solutions representing analogues of superconducting Josephson fluxons as well
as coupled dark solitons are discussed. Theoretical approximations based on
variational formulations are derived. It is found that the presence of a
magnetic trap can destabilize the fluxon analogues. However, stabilization is
possible by controlling the effective linear coupling between the condensates.
