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

Grudzien, CJ, Bridges, TJ and Jones, CKRT (2016) Geometric phase in the Hopf bundle and the stability of non-linear waves Physica D: Nonlinear Phenomena.

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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 S2n−1⊂CnS2n−1⊂Cn. 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 Way (2009). We provide a detailed proof of the relationship between the phase and eigenvalues for dynamical systems defined on C2C2 and sketch the proof of the method of geometric phase for CnCn and its generalization to boundary-value problems. Implementing the numerical method, modified from Way (2009), we conclude with open questions inspired from the results.

Item Type: Article
Subjects : subj_Mathematics
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
AuthorsEmailORCID
Grudzien, CJUNSPECIFIEDUNSPECIFIED
Bridges, TJUNSPECIFIEDUNSPECIFIED
Jones, CKRTUNSPECIFIEDUNSPECIFIED
Date : 13 April 2016
Identification Number : 10.1016/j.physd.2016.04.005
Copyright Disclaimer : © 2016. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
Related URLs :
Depositing User : Symplectic Elements
Date Deposited : 05 May 2016 11:23
Last Modified : 05 May 2016 11:23
URI: http://epubs.surrey.ac.uk/id/eprint/810634

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