Finding Eigenvalues the Rupert Way

TL;DR

This paper introduces a geometric phase-based method to determine the stability of travelling waves in reaction diffusion equations by locating eigenvalues through a novel winding number approach, offering an alternative to traditional Evans function calculations.

Contribution

It develops a new geometric phase method for eigenvalue analysis of reaction diffusion operators, generalizing previous work and providing detailed proofs and numerical implementations.

Findings

The geometric phase method accurately locates eigenvalues in $ ext{C}^2$ systems.

The approach generalizes to higher dimensions and boundary-value problems.

Numerical results demonstrate the effectiveness of the method.

Abstract

We develop a stability index for the travelling waves of non-linear reaction diffusion equations using the geometric phase induced on the Hopf bundle $S^{2n-1} \subset \mathbb{C}^n$. This can be viewed as an alternative formulation of the winding number calculation of the Evans function, whose zeroes correspond to the eigenvalues of the linearization of reaction diffusion operators about the wave. The stability of a travelling wave can be determined by the existence of eigenvalues of positive real part for the linear operator. Our method of geometric phase for locating and counting eigenvalues is inspired by the numerical results in Way's Dynamics in the Hopf bundle, the geometric phase and implications for dynamical systems. We provide a detailed proof of the relationship between the phase and eigenvalues for dynamical systems defined on $\mathbb{C}^2$ and sketch the proof of the method of geometric phase for $\mathbb{C}^n$ and its generalization to boundary-value problems. Implementing the numerical method, modified from Way's work, we conclude with open questions inspired from the results.