Formal analytical solutions for the Gross–Pitaevskii equation

Considering the Gross–Pitaevskii integral equation we are able to formally obtain an analytical solution for the order parameter Φ(x)Φ(x) and for the chemical potential μμ as a function of a unique dimensionless non-linear parameter ΛΛ. We report solutions for different ranges of values for the repulsive and the attractive non-linear interactions in the condensate. Also, we study a bright soliton-like variational solution for the order parameter for positive and negative values of ΛΛ. Introducing an accumulated error function we have performed a quantitative analysis with respect to other well-established methods as: the perturbation theory, the Thomas–Fermi approximation, and the numerical solution. This study gives a very useful result establishing the universal range of the ΛΛ-values where each solution can be easily implemented. In particular, we showed that for Λ<−9Λ<−9, the bright soliton function reproduces the exact solution of GPE wave function.