### 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 S^{2n−1}⊂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.

Language | English (US) |
---|---|

Pages | 4-18 |

Number of pages | 15 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 334 |

DOIs | |

State | Published - Nov 1 2016 |

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### Keywords

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

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Condensed Matter Physics

### Cite this

*Physica D: Nonlinear Phenomena*,

*334*, 4-18. DOI: 10.1016/j.physd.2016.04.005

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

Research output: Contribution to journal › Article

*Physica D: Nonlinear Phenomena*, vol 334, pp. 4-18. DOI: 10.1016/j.physd.2016.04.005

}

TY - JOUR

T1 - Geometric phase in the Hopf bundle and the stability of non-linear waves

AU - Grudzien,Colin J.

AU - Bridges,Thomas J.

AU - Jones,Christopher K.R.T.

PY - 2016/11/1

Y1 - 2016/11/1

N2 - 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.

AB - 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.

KW - Evans function

KW - Geometric dynamics

KW - Stability analysis

KW - Steady states

KW - Traveling waves

UR - http://www.scopus.com/inward/record.url?scp=84969916758&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84969916758&partnerID=8YFLogxK

U2 - 10.1016/j.physd.2016.04.005

DO - 10.1016/j.physd.2016.04.005

M3 - Article

VL - 334

SP - 4

EP - 18

JO - Physica D: Nonlinear Phenomena

T2 - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

ER -