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arXiv · 0708.3840

Formal analytical solutions for the Gross-Pitaevskii equation

C. Trallero-Giner·Julio C. Drake-Perez·V. Lopez-Richard·Joseph L. Birman
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Abstract

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 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.

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C. Trallero-Giner, Julio C. Drake-Perez, V. Lopez-Richard, Joseph L. Birman. 2007-08-28. Formal analytical solutions for the Gross-Pitaevskii equation. https://doi.org/10.1016/j.physd.2008.02.017

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