The method of the geometric phase in the Hopf bundle as a reformulation of the Evans function for reaction diffusion equations Public Deposited

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  • March 21, 2019
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  • Grudzien, Colin
    • Affiliation: College of Arts and Sciences, Department of Mathematics
Abstract
  • This thesis develops a stability index for the travelling waves of non-linear reaction diffusion equations using the geometric phase induced on the Hopf bundle, an odd dimensional sphere realized in an arbitrary complex vector space. 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 a wave or, time invariant, coherent state. The stability of such a state can be determined by the existence of eigenvalues of positive real part for the linear operator associated to it. The method of geometric phase for locating and counting eigenvalues as demonstrated in this thesis is inspired by the numerical results in Way's Dynamics in the ``Hopf bundle, the geometric phase and implications for dynamical systems,'' but it diverges on several important points. This thesis develops a detailed proof of the relationship between the phase and eigenvalues for dynamical systems defined in a simple case and sketches the proof of the generalized method of geometric phase for arbitrary systems on unbounded domains and its generalization to boundary-value problems. In addition it establishes novel links between the geometric phase generated in the Hopf bundle, and an equivalent phase generated by a path in the Stiefel bundle. A demonstration of the numerical method is included for a simple bistable equation, and the Hocking-Stewartson Pulse of the Complex Ginzburg-Landau equation. These examples highlight the novel features of this formulation of the winding of the Evans function, namely the use of either the stable or unstable manifold, and the dependence on the wave parameter for the eigenvalue calculation. The continuous accumulation of the eigenvalue count is exhibited with a characteristic phase change, depending on the wave parameter. This thesis concludes with a discussion of open questions arising from the numerical implementation, regarding the phase transition, its link to the underlying wave structure and the possible formulation of the method of geometric phase with respect to a phase generated on the Stiefel bundle.
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  • In Copyright
Advisor
  • Van Vleck, Erik
  • Eberlein, Patrick
  • Jones, Christopher K. R. T.
  • Rimanyi, Richard
  • Marzuola, Jeremy
Degree
  • Doctor of Philosophy
Degree granting institution
  • University of North Carolina at Chapel Hill Graduate School
Graduation year
  • 2016
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