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

Abstract

We study the behavior 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 domain with an outer boundary layer containing the other component, or to many interfaces, on a periodic pattern. This is related to the striped behavior 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 behave, in the Thomas-Fermi limit of strong intra-component interaction, like those of the simplified Modica-Mortola problem we have studied in the first part.