Finding Eigenvalues the Rupert Way

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.