Geometric phase in the Hopf bundle and the stability of non-linear waves

Highlights: • A geometric re-formulation of the Evans function for reaction–diffusion equation is presented. • Holonomy and a geometric phase are introduced into the formulation of the linearization. • The winding of a geometric phase is connected to the number of eigenvalues of the linearization. • It is proved that the geometric phase is asymptotically equal to the Chern number. • A numerical demonstration of the method is provided. We develop a stability index for the traveling waves of non-linear reaction–diffusion equations using the geometric phase induced on the Hopf bundle $S^{2n-1}\subset C^n$. This can be viewed as an alternative formulation of the winding number calculation of the Evans function, whose zeros correspond to the eigenvalues of the linearization of reaction–diffusion operators about the wave. The stability of a traveling 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 Way (2009). We provide a detailed proof of the relationship between the phase and eigenvalues for dynamical systems defined on $C^2$ and sketch the proof of the method of geometric phase for $C^n$ and its generalization to boundary-value problems. Implementing the numerical method, modified from Way (2009), we conclude with open questions inspired from the results.