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Archive for Rational Mechanics and Analysis

, Volume 207, Issue 2, pp 693–715 | Cite as

Nonlocalized Modulation of Periodic Reaction Diffusion Waves: Nonlinear Stability

  • Mathew A. Johnson
  • Pascal Noble
  • L. Miguel Rodrigues
  • Kevin ZumbrunEmail author
Article

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.

Keywords

Periodic Wave Reaction Diffusion Nonlinear Stability Localize Perturbation Spectral Stability 
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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References

  1. 1.
    Barker, B., Johnson, M., Noble, P., Rodrigues, M., Zumbrun, K.: Witham averaged equations and modulational stability of periodic solutions of hyperbolic-parabolic balance laws. Proceedings and seminars, Centre de Mathématiques de l’École polytechnique; Conference proceedings. Journées equations aux derivées partielles, Port d’Albret, France, 2010, (to appear)Google Scholar
  2. 2.
    Barker B., Johnson M., Noble P., Rodrigues M., Zumbrun K.: Stability of periodic Kuramoto–Sivashinsky waves. Appl. Math. Lett. 25(5), 824–829 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Barker, B., Johnson, M., Noble, P., Rodrigues, M., Zumbrun, K.: Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto–Sivashinsky equation. Preprint, 2012Google Scholar
  4. 4.
    Doelman, A., Sandstede, B., Scheel, A., Schneider, G.: The dynamics of modulated wave trains. Mem. Am. Math. Soc. 199(934), viii+105 pp. ISBN: 978-0-8218-4293-5 (2009)Google Scholar
  5. 5.
    Henry D.: Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics. Springer, Berlin (1981)Google Scholar
  6. 6.
    Hoff D., Zumbrun K.: Asymptotic behavior of multi-dimensional scalar viscous shock fronts. Indiana Univ. Math. J. 49, 427–474 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Johnson, M., Noble, P., Rodrigues, L.M., Zumbrun, K.: Nonlocalized modulation of periodic reaction diffusion waves: the Whitham equation. Arch. Rational Mech. Anal. (to appear)Google Scholar
  8. 8.
    Johnson, M., Noble, P., Rodrigues, L.M., Zumbrun, K.: Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations. In preparationGoogle Scholar
  9. 9.
    Johnson M., Zumbrun K.: Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations. Annales de l’Institut Henri Poincaré- Analyse non linéaire 28(4), 471–483 (2011)MathSciNetADSzbMATHCrossRefGoogle Scholar
  10. 10.
    Johnson M., Zumbrun K.: Nonlinear stability of periodic traveling waves of viscous conservation laws in the generic case. J. Differ. Equ. 249(5), 1213–1240 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Johnson M., Zumbrun K., Noble P.: Nonlinear stability of viscous roll waves. SIAM J. Math. Anal. 43(2), 557–611 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kato T.: Perturbation Theory for Linear Operators. Springer, Berlin (1985)Google Scholar
  13. 13.
    Noble, P., Rodrigues, M.: Whitham’s equations for modulated roll-waves in shallow flows. Unpublished manuscript, arXiv:1011.2296 (2010)Google Scholar
  14. 14.
    Noble, P., Rodrigues, M.: Whitham’s modulation equations and stability of periodic wave solutions of the generalized Kuramoto-Sivashinsky equations. Indiana Univ.Math. J. (to appear)Google Scholar
  15. 15.
    Oh M., Zumbrun K.: Stability of periodic solutions of viscous conservation laws with viscosity-1. Analysis of the Evans function. Arch. Rational Mech. Anal. 166(2), 99–166 (2003)MathSciNetADSzbMATHCrossRefGoogle Scholar
  16. 16.
    Oh, M., Zumbrun, K.: Stability and asymptotic behavior of traveling-wave solutions of viscous conservation laws in several dimensions. Arch. Rational Mech. Anal. 196(1), 1–20 (2010); Erratum: Arch. Rational Mech. Anal. 196(1), 21–23 (2010)Google Scholar
  17. 17.
    Oh M., Zumbrun K.: Low-frequency stability analysis of periodic traveling-wave solutions of viscous conservation laws in several dimensions. Z. Anal. Anwend. 25(1), 1–21 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Prüss J.: On the spectrum of C 0-semigroups. Trans. Am. Math. Soc. 284(2), 847–857 (1984)zbMATHCrossRefGoogle Scholar
  19. 19.
    Sandstede B., Scheel A., Schneider G., Uecker H.: Diffusive mixing of periodic wave trains in reaction-diffusion systems. J. Differ. Equ. 252(5), 3541–3574 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Schneider, G.: Nonlinear diffusive stability of spatially periodic solutions—abstract theorem and higher space dimensions. Proceedings of the International Conference on Asymptotics in Nonlinear Diffusive Systems (Sendai, 1997), pp. 159–167. Tohoku Mathematical Publication, Vol. 8. Tohoku University, Sendai, 1998Google Scholar
  21. 21.
    Schneider G.: Diffusive stability of spatial periodic solutions of the Swift-Hohenberg equation (English. English summary). Commun. Math. Phys. 178(3), 679–702 (1996)ADSzbMATHCrossRefGoogle Scholar
  22. 22.
    Serre D.: Spectral stability of periodic solutions of viscous conservation laws: large wavelength analysis. Commun. Partial Differ. Equ. 30(1–3), 259–282 (2005)zbMATHCrossRefGoogle Scholar
  23. 23.
    Texier B., Zumbrun K.: Relative Poincaré-Hopf bifurcation and galloping instability of traveling waves. Methods Appl. Anal. 12(4), 349–380 (2005)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Texier B., Zumbrun K.: Galloping instability of viscous shock waves. Physica D 237, 1553–1601 (2008)MathSciNetADSzbMATHCrossRefGoogle Scholar
  25. 25.
    Texier B., Zumbrun K.: Nash-Moser iteration and singular perturbations. Ann. Inst. H. Poincaré Anal. Non Linaire 28(4), 499–527 (2011)MathSciNetADSzbMATHCrossRefGoogle Scholar
  26. 26.
    Uecker H.: Diffusive stability of rolls in the two-dimensional real and complex Swift-Hohenberg equation. Commun. Partial Differ. Equ. 24(11–12), 2109–2146 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Zumbrun K.: Refined wave-tracking and nonlinear stability of viscous Lax shocks. Methods Appl. Anal. 7, 747–768 (2000)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Zumbrun K.: Instantaneous shock location and one-dimensional nonlinear stability of viscous shock waves. Quart. Appl. Math. 69(1), 177–202 (2011)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Zumbrun K.: Conditional stability of unstable viscous shocks. J. Differ. Equ. 247(2), 648–671 (2009)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Mathew A. Johnson
    • 1
  • Pascal Noble
    • 2
  • L. Miguel Rodrigues
    • 2
  • Kevin Zumbrun
    • 3
    Email author
  1. 1.University of KansasLawrenceUSA
  2. 2.Université Lyon IVilleurbanneFrance
  3. 3.Indiana UniversityBloomingtonUSA

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