A unified model for an external trap in a cigar-shaped Bose-Einstein condensate

In recent years, the study of cigar shaped Bose-Einstein condensate (BEC) under a variety of external confinements has attracted a great deal of attention, from both theoretical and experimental researchers. We report a unified model for obtaining explicit solutions under various kinds of physically relevant space-and time-modulated external traps and nonlinearities for the cigar shaped BEC. Our novel mechanism paves the way to investigate the system for a family of potential functions unified as a physical parameter of the system. We apply and illustrate our results for a number of exactly solvable quantum mechanical potentials; harmonic, double-well, Posch-Teller, Morse, Toda lattice and power-law as some of the applications of our model. Expressions of the condensate density are provided for these potentials. In addition, as an application of our model, we have illustrated the condensate dynamics for harmonic, double-well and Posch-Teller potentials. In the presence of loss/gain for a Posch-Teller potential, the condensate density shows an interesting collapse and revival.