# Stability Analysis of a Bose-Einstein Condensate Trapped in a Generic Potential

## TL;DR

This paper analyzes the stability of a Bose-Einstein condensate in a spherical power law potential using dynamical systems, revealing that stability depends critically on the potential's exponent.

## Contribution

It provides a dynamical systems framework for understanding the stability of Bose-Einstein condensates in generic power law traps, highlighting the impact of the potential's exponent.

## Key findings

- Stability for exponents ≤ 2
- Instability increases for exponents > 2
- Large exponents lead to particle loss from the condensate

## Abstract

We investigate the dynamical behavior of the Gross-Pitaevskii equation for a Bose-Einstein condensate trapped in a spherical power law potential restricted to the repulsive case, from the dynamical system formalism point of view. A five-dimensional dynamical system is found (due the symmetry of the Gross-Pitaevskii equation interacting with a potential), where the Thomas-Fermi approximation constrains the parameter space of solutions. We show that for values of the power law exponent equal or smaller than 2 the system seems to be stable. However, when the corresponding exponent is bigger than 2, the instability of the system grows when the power law exponent grows, indicating that large values of the aforementioned parameter can be related to a loss in the number of particles from the condensed state. This fact can be used also to show that the stability conditions of the condensate are highly sensitive to the exponent associated with the external potential.