Formal analytical solutions for the Gross-Pitaevskii equation

TL;DR

This paper derives formal analytical solutions for the Gross-Pitaevskii equation, exploring different interaction regimes, and compares these solutions with established methods to identify their applicable ranges.

Contribution

It provides a unified analytical framework for the GPE, including solutions for various interaction strengths and a quantitative comparison with existing approximation methods.

Findings

Analytical solutions are obtained as functions of the non-linear parameter Λ.

Bright soliton solutions accurately reproduce the GPE wave function for Λ < -9.

The study establishes the universal range of Λ where each solution is valid.

Abstract

Considering the Gross-Pitaevskii integral equation we are able to formally obtain an analytical solution for the order parameter $\Phi (x)$ and for the chemical potential $\mu $ as a function of a unique dimensionless non-linear parameter $\Lambda $. 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 $\Lambda $. 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 $\Lambda $-values where each solution can be easily implemented. In particular we showed that for $\Lambda <-9$, the bright soliton function reproduces the exact solution of GPE wave function.