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Freezing Traveling and Rotating Waves in Second Order Evolution Equations

  • Wolf-Jürgen BeynEmail author
  • Denny Otten
  • Jens Rottmann-Matthes
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 205)

Abstract

In this paper we investigate the implementation of the so-called freezing method for second order wave equations in one and several space dimensions. The method converts the given PDE into a partial differential algebraic equation which is then solved numerically. The reformulation aims at separating the motion of a solution into a co-moving frame and a profile which varies as little as possible. Numerical examples demonstrate the feasability of this approach for semilinear wave equations with sufficient damping. We treat the case of a traveling wave in one space dimension and of a rotating wave in two space dimensions. In addition, we investigate in arbitrary space dimensions the point spectrum and the essential spectrum of operators obtained by linearizing about the profile, and we indicate the consequences for the nonlinear stability of the wave.

Keywords

Systems of damped wave equations Traveling waves Rotating waves Freezing method Second order evolution equations Point spectra Essential spectra 

Mathematics Subject Classification:

35K57 35Pxx 65Mxx (35Q56 47N40 65P40) 

Notes

Acknowledgements

We gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 701 and CRC 1173.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Wolf-Jürgen Beyn
    • 1
    Email author
  • Denny Otten
    • 1
  • Jens Rottmann-Matthes
    • 2
  1. 1.Department of MathematicsBielefeld UniversityBielefeldGermany
  2. 2.Institute for AnalysisKarlsruhe Institute of TechnologyKarlsruheGermany

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