Excited states in the Thomas-Fermi limit: a variational approach

TL;DR

This paper develops a variational approach to analyze excited states, specifically dark solitons, in Bose-Einstein condensates within the Thomas-Fermi limit, providing analytical and numerical insights into their equilibrium and oscillation behaviors.

Contribution

It introduces a variational method to characterize dark soliton equilibria and oscillations in the semi-classical limit of the Gross-Pitaevskii equation, extending to multiple solitons.

Findings

Analytical expressions for dark soliton equilibrium positions.

Predictions of oscillation frequencies around equilibria.

Numerical validation for 2- and 3-soliton configurations.

Abstract

Excited states of Bose--Einstein condensates are considered in the semi-classical (Thomas-Fermi) limit of the Gross--Pitaevskii equation with repulsive inter-atomic interactions and a harmonic potential. The relative dynamics of dark solitons (density dips on the localized condensate) with respect to the harmonic potential and to each other is approximated using the averaged Lagrangian method. This permits a complete characterization of the equilibrium positions of the dark solitons as a function of the chemical potential parameter. It also yields an analytical handle on the oscillation frequencies of dark solitons around such equilibria. The asymptotic predictions are generalized for an arbitrary number of dark solitons and are corroborated by numerical computations for 2- and 3-soliton configurations.