Integrability of an inhomogeneous nonlinear Schroedinger equation in Bose\u2013Einstein condensates and fiber optics

In this paper, we investigate the integrability of an inhomogeneous nonlinear Schr\uf6dinger equation, which has several applications in many branches of physics, as in Bose-Einstein condensates and fiber optics. The main issue deals with Painlev\ue9 property (PP) and Liouville integrability for a nonlinear Schr\uf6dinger-type equation. Solutions of the integrable equation are obtained by means of the Darboux transformation. Finally, some applications on fiber optics and Bose-Einstein condensates are proposed (including Bose-Einstein condensates in three-dimensional in cylindrical symmetry)