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Geometric phase in the Hopf bundle and the stability of non-linear waves. (English) Zbl 1415.35149

Summary: We develop a stability index for the traveling waves of non-linear reaction-diffusion equations using the geometric phase induced on the Hopf bundle \(S^{2 n - 1} \subset \mathbb{C}^n\). 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 R. Way [Dynamics in the Hopf bundle, the geometric phase and implications for dynamical systems. Guildford, UK: University of Surrey. (PhD Thesis) (2009)]. 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 [loc. cit.], we conclude with open questions inspired from the results.

MSC:

35K45 Initial value problems for second-order parabolic systems
35K57 Reaction-diffusion equations
57R22 Topology of vector bundles and fiber bundles
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