Stationary states of the Gross-Pitaevskii equation with linear counterpart

TL;DR

This paper investigates stationary solutions of the Gross-Pitaevskii equation, connecting them to linear Schrödinger eigenfunctions, providing analytical and numerical insights into their existence, form, and boundedness across different non-linearity regimes and dimensions.

Contribution

It introduces an existence condition for stationary solutions based on the ratio of chemical potential to linear eigenvalue, and provides approximate solutions valid in strong non-linearity limits.

Findings

Solutions resemble soliton chains in 1D depending on non-linearity sign.

Norm of solutions is bounded for negative non-linearity in dimensions > 1.

Analytical and numerical support for the existence condition.

Abstract

We study the stationary solutions of the Gross-Pitaevskii equation that reduce, in the limit of vanishing non-linearity, to the eigenfunctions of the associated Schr\"odinger equation. By providing analytical and numerical support, we conjecture an existence condition for these solutions in terms of the ratio between their proper frequency (chemical potential) and the corresponding linear eigenvalue. We also give approximate expressions for the stationary solutions which become exact in the opposite limit of strong non-linearity. For one-dimensional systems these solutions have the form of a chain of dark or bright solitons depending on the sign of the non-linearity. We demonstrate that in the case of negative non-linearity (attractive interaction) the norm of the solutions is always bounded for dimensions greater than one.