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Amplitude Equations for Patterns

  • L.M. Pismen
Part of the Springer Series in Synergetics book series (SSSYN)

Abstract

A typical dispersion relation λ(κ) in a spatially extended system just beyond a bifurcation point may have one of the forms shown in Fig. 4.1. In both cases, a narrow band of wavenumbers adjacent to the maximum of the dispersion relation (which may be reached either at κ = 0 or at a finite κ = κ0) becomes unstable. Since the dispersion relation should be, generically, parabolic near the maximum, and the leading eigenvalue can be assumed to depend linearly on a chosen bifurcation parameter, say, μ, the width of the excited band scales as the square root of the deviation from the bifurcation point μμ0. A spectral band of a finite width can be modeled by allowing the amplitude to change on an extended spatial scale, large compared to either κ–1 0 or any “natural” length scale characteristic to the underlying system.

Keywords

Wave Vector Domain Wall Phase Equation Amplitude Equation Phase Gradient 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer 2006

Authors and Affiliations

  • L.M. Pismen
    • 1
  1. 1.Department of Chemical EngineeringTechnion - Israel Institute of Technology Technion CityHaifaIsrael

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