On the structure of spectra of travelling waves

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2 Citations (Scopus)

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 languageEnglish
Pages (from-to)1-19
Number of pages19
JournalElectronic Journal of Qualitative Theory of Differential Equations
Publication statusPublished - Oct 19 2003

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First-order System
Wave equations
Traveling Wave
Linear systems
Generalized KdV Equation
Evans Function
Exponential Dichotomy
Essential Spectrum
Linear Stability
Traveling Wave Solutions
Reaction-diffusion System
Combustion
Differential operator
Wave equation
Linear Systems
Eigenvalue
Model

Keywords

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

ASJC Scopus subject areas

  • Analysis

Cite this

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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.",
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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.

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