  We study the behaviour of a Modica-Mortola phase transition type problem with
a non-homogeneous Neumann boundary condition. According to the parameters of
the problem, this leads to the existence of either one component occupying most
of the condensate with an outer boundary layer containing the other component,
or to many interfaces, on a periodic pattern. This is related to the striped
behaviour of a two component Bose-Einstein condensate with spin orbit coupling
in one dimension. We prove that minimizers of the full Gross-Pitaevskii energy
in 1D converge, in the Thomas-Fermi limit of strong intra-component
interaction, to those of the simplified Modica-Mortola problem we have studied
in the first part.
