One dimensional phase transition problem modelling striped spin orbit coupled Bose-Einstein condensates

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.