Perturbative stability of catenoidal soap films

TL;DR

This paper analyzes the stability of catenoidal soap films between rings, demonstrating azimuthally asymmetric stability and symmetric instability through analytical, semi-analytical, and numerical methods, including the Asymptotic Iteration Method.

Contribution

It introduces a semi-analytical approach using AIM to determine eigenvalues and eigenfunctions, confirming stability results and providing visualizations of perturbation evolution.

Findings

Azimuthally asymmetric perturbations are stable.

Symmetric perturbations lead to instability.

Semi-analytical AIM results agree with analytical and experimental findings.

Abstract

The perturbative stability of catenoidal soap films formed between parallel, equal radii, coaxial rings is studied using analytical and semi-analytical methods. Using a theorem on the nature of eigenvalues for a class of Sturm--Liouville operators, we show that for the given boundary conditions, azimuthally asymmetric perturbations are stable, while symmetric perturbations lead to an instability--a result demonstrated in Ben Amar et. al [7] using numerics and experiment. Further, we show how to obtain the lowest real eigenvalue of perturbations, using the semi-analytical Asymptotic Iteration Method (AIM). Conclusions using AIM support the analytically obtained result as well as the results in [7]. Finally, we compute the eigenfunctions and show, pictorially, how the perturbed soap film evolves in time.