A unified model for an external trap in a cigar-shaped Bose-Einstein condensate (Q2927699)

The set of exact analytic solutions within an amplitude-phase ansatz is considered for the normalized one-dimensional Gross-Pitaevskii equation (GPE), which models coherent many-body states resulting in the emergence of a cigar-shaped Bose-Einstein condensate in an external trapping potential. The basic procedure connects a shape of the function representing the non-linear multiplier at the \(|\psi|^2\) term with the corresponding potential. Among the exactly solvable examples are harmonic, double-well, Pösch-Teller, Morse, Toda lattice and power-law potentials. The last part of the work discusses and illustrates a matter wave dynamics for some of them.