Formal analytical solutions for the Gross–Pitaevskii equation

C. Trallero-Giner, Julio C. Drake-Perez, V. López-Richard, Joseph L. Birman
2008 Physica D : Non-linear phenomena  
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 range 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
more » ... accumulated error function we have performed a quantitative analysis with 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.
doi:10.1016/j.physd.2008.02.017 fatcat:g6r5hfk5yfdcvkpokscstcg3gq