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 ) and for the chemical potential μ as a function of a unique dimensionless non-linear parameter Λ . 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 Λ . 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 Λ -values where each solution can be easily implemented. In particular, we showed that for Λ < − 9 , the bright soliton function reproduces the exact solution of GPE wave function.