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 Schrodinger 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 is nowadays a well-developed field for experimentalists. On the other hand, in contrast to its higher-dimensional analogous, the one-dimensional nonlinear Schrodinger equation allows explicit solutions, that simplify remarkably the analysis. The recent literature reveals an increasing interest for the dynamics of nonlinear systems in the presence of so-called defects, namely microscopic scatterers, which model the presence of impurities. We review here some recent achievements on such systems, with particular 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.