Domain walls in the coupled Gross-Pitaevskii equations with the harmonic potential

We study the existence and variational characterization of steady states in a coupled system of Gross--Pitaevskii equations modeling two-component Bose-Einstein condensates with the magnetic field trapping. The limit with no trapping has been the subject of recent works where domain walls have been constructed and several properties, including their orbital stability have been derived. Here we focus on the full model with the harmonic trapping potential and characterize minimizers according to the value of the coupling parameter $\gamma$. We first establish a rigorous connection between the two problems in the Thomas-Fermi limit via $\Gamma$-convergence. Then, we identify the ranges of $\gamma$ for which either the symmetric states $(\gamma < 1)$ or the uncoupled states $(\gamma > 1)$ are minimizers. Domain walls arise as minimizers in a subspace of the energy space with a certain symmetry for some $\gamma > 1$. We study bifurcation of the domain walls and furthermore give numerical illustrations of our results.