### Abstract

The linear stability of the travelling wave solutions of a general reaction-diffusion system is in-vestigated. The spectrum of the corresponding second order differential operator L is studied. The problem is reduced to an asymptotically autonomous first order linear system. The relation between the spectrum of L and the corresponding first order system is dealt with in detail. The first order system is investigated using exponential dichotomies. A self-contained short presentation is shown for the study of the spectrum, with elementary proofs. An algorithm is given for the determination of the exact position of the essential spectrum. The Evans function method for determining the isolated eigenvalues of L is also presented. The theory is illustrated by three examples: a single travelling wave equation, a three variable combustion model and the generalized KdV equation.

Original language | English |
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Pages (from-to) | 1-19 |

Number of pages | 19 |

Journal | Electronic Journal of Qualitative Theory of Differential Equations |

Publication status | Published - Oct 19 2003 |

### Fingerprint

### Keywords

- Evans function
- Exponential dichotomies
- Linear stability of travelling waves

### ASJC Scopus subject areas

- Analysis

### Cite this

**On the structure of spectra of travelling waves.** / Simon, Peter L.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - On the structure of spectra of travelling waves

AU - Simon, Peter L.

PY - 2003/10/19

Y1 - 2003/10/19

N2 - The linear stability of the travelling wave solutions of a general reaction-diffusion system is in-vestigated. The spectrum of the corresponding second order differential operator L is studied. The problem is reduced to an asymptotically autonomous first order linear system. The relation between the spectrum of L and the corresponding first order system is dealt with in detail. The first order system is investigated using exponential dichotomies. A self-contained short presentation is shown for the study of the spectrum, with elementary proofs. An algorithm is given for the determination of the exact position of the essential spectrum. The Evans function method for determining the isolated eigenvalues of L is also presented. The theory is illustrated by three examples: a single travelling wave equation, a three variable combustion model and the generalized KdV equation.

AB - The linear stability of the travelling wave solutions of a general reaction-diffusion system is in-vestigated. The spectrum of the corresponding second order differential operator L is studied. The problem is reduced to an asymptotically autonomous first order linear system. The relation between the spectrum of L and the corresponding first order system is dealt with in detail. The first order system is investigated using exponential dichotomies. A self-contained short presentation is shown for the study of the spectrum, with elementary proofs. An algorithm is given for the determination of the exact position of the essential spectrum. The Evans function method for determining the isolated eigenvalues of L is also presented. The theory is illustrated by three examples: a single travelling wave equation, a three variable combustion model and the generalized KdV equation.

KW - Evans function

KW - Exponential dichotomies

KW - Linear stability of travelling waves

UR - http://www.scopus.com/inward/record.url?scp=3042576519&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=3042576519&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:3042576519

SP - 1

EP - 19

JO - Electronic Journal of Qualitative Theory of Differential Equations

JF - Electronic Journal of Qualitative Theory of Differential Equations

SN - 1417-3875

ER -