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

Colin J. Grudzien, Thomas J. Bridges, Christopher K.R.T. Jones

Research output: Contribution to journalArticle

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Abstract

We develop a stability index for the traveling waves of non-linear reaction–diffusion equations using the geometric phase induced on the Hopf bundle S2n−1⊂Cn. 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 C2 and sketch the proof of the method of geometric phase for Cn and its generalization to boundary-value problems. Implementing the numerical method, modified from Way (2009), we conclude with open questions inspired from the results.

LanguageEnglish (US)
Pages4-18
Number of pages15
JournalPhysica D: Nonlinear Phenomena
Volume334
DOIs
StatePublished - Nov 1 2016

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bundles
eigenvalues
traveling waves
dynamical systems
linear operators
reaction-diffusion equations
linearization
boundary value problems
counting
formulations
operators

Keywords

  • Evans function
  • Geometric dynamics
  • Stability analysis
  • Steady states
  • Traveling waves

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Condensed Matter Physics

Cite this

Geometric phase in the Hopf bundle and the stability of non-linear waves. / Grudzien, Colin J.; Bridges, Thomas J.; Jones, Christopher K.R.T.

In: Physica D: Nonlinear Phenomena, Vol. 334, 01.11.2016, p. 4-18.

Research output: Contribution to journalArticle

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