On the mathematical description of the effective behaviour of one-dimensional Bose-Einstein condensates with defects

Bose-Einstein condensation and the related topic of Gross-Pitaevskii equation have become an important source of models and problems in mathematical physics and analysis. In particular, in the last decade, the interest in low-dimensional systems that evolve through the nonlinear Schroedinger equation has undergone an impressive growth. The reason is twofold: on the one hand, effectively one-dimensional Bose-Einstein condensates are currently realized, and the investigation on their dynamics isnowadays a well-developed field for experimentalists. On the other hand, in contrast to its higher-dimensional analogous,the one-dimensional nonlinear Schroedinger equation allows explicit solutions, that simplify remarkably the analysis. The recentliterature reveals an increasing interest for the dynamics ofnonlinear systems in the presence of so-called defects, namelymicroscopic scatterers, which model the presence of impurities.We review here some recent achievements on such systems, withparticular attention to the cases of the ``Dirac's delta'' and ``delta prime'' defects. We give rigorous definitions, recall and comment on known results for the delta case, and introduce new results for the delta prime case. The latter system turns out to be richer and interesting since it produces a bifurcation with symmetry breaking in the ground state.Our purpose lies mainly on collecting and conveying results, so proofs are not included.