Stability analysis of orbital modes for a generalized Lane-Emden equation

TL;DR

This paper analyzes the stability of orbital modes in a generalized Lane-Emden equation, revealing conditions for stable atomic structures in ultracold atomic clouds and providing numerical solutions consistent with the theoretical findings.

Contribution

It introduces a stability analysis for a generalized Lane-Emden equation, identifying specific conditions for stability related to the adiabatic index and providing numerical solutions.

Findings

Stable structures occur only for even n in the adiabatic index.

Instability and divergence occur for odd n.

Numerical solutions confirm the stability analysis.

Abstract

We present a stability analysis of the standard nonautonomous systems type for a recently introduced generalized Lane-Emden equation which is shown to explain the presence of some of the structures observed in the atomic spatial distributions of magnetically-trapped ultracold atomic clouds. A Lyapunov function is defined which helps us to prove that stable spatial structures in the atomic clouds exist only for the adiabatic index $\gamma=1+1/n$ with even $n$. In the case when $n$ is odd we provide an instability result indicating the divergence of the density function for the atoms. Several numerical solutions, which according to our stability analysis are stable, are also presented.