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Nonlocalized Modulation of Periodic Reaction Diffusion Waves: Nonlinear Stability

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Abstract

Extending results of Johnson and Zumbrun showing stability under localized (L 1) perturbations, we show that spectral stability implies nonlinear modulational stability of periodic traveling-wave solutions of reaction diffusion systems under small perturbations consisting of a nonlocalized modulation plus a localized perturbation. The main new ingredient is a detailed analysis of linear behavior under modulational data \({\bar{u}^{\prime}(x)h_{0}(x)}\), where \({\bar{u}}\) is the background profile and h 0 is the initial modulation.

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Author information

Authors and Affiliations

  1. University of Kansas, Lawrence, KS, 66045, USA

    Mathew A. Johnson

  2. Université Lyon I, Villeurbanne, France

    Pascal Noble & L. Miguel Rodrigues

  3. Indiana University, Bloomington, IN, 47405, USA

    Kevin Zumbrun

Authors
  1. Mathew A. Johnson
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  2. Pascal Noble
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  3. L. Miguel Rodrigues
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  4. Kevin Zumbrun
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Corresponding author

Correspondence to Kevin Zumbrun.

Additional information

Communicated by A. Bressan

Research was partially supported by an NSF Postdoctoral Fellowship under NSF grant DMS-0902192.

Research was partially supported by the French ANR Project no. ANR-09-JCJC-0103-01.

Stay in Bloomington was supported by French ANR project no. ANR-09-JCJC-0103-01.

Research was partially supported under NSF grant no. DMS-0300487.

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Johnson, M.A., Noble, P., Rodrigues, L.M. et al. Nonlocalized Modulation of Periodic Reaction Diffusion Waves: Nonlinear Stability. Arch Rational Mech Anal 207, 693–715 (2013). https://doi.org/10.1007/s00205-012-0573-9

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  • DOI: https://doi.org/10.1007/s00205-012-0573-9

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