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Amplitude equation for the generalized Swift–Hohenberg equation with noise

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Abstract

We derive an amplitude equation for a stochastic partial differential equation of Swift–Hohenberg type under the simplifying assumption that the noise acts uniformly on the whole system. Due to the natural separation of timescales, solutions are well approximated by a stochastic differential equation, the so called amplitude equation, describing the evolution of the dominant pattern. Although the slow dominant modes are not forced directly, via the nonlinearity the noise gets transmitted through the system to those modes, too, and multiplicative noise appears in the amplitude equation. Moreover, additional linear and cubic terms appear due to averaging. This leads to either noise-induced stabilization or destabilization effects in the dominating pattern.

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References

  1. Blömker D.: Amplitude Equations for Stochastic Partial Differential Equations. Interdisciplinary Mathematical Sciences, Vol. 3. World Scientific, Singapore (2007)

    Google Scholar 

  2. Blömker D., Hairer M.: Multiscale expansion of invariant measures for SPDEs. Commun. Math. Phys. 251(3), 515–555 (2004)

    Article  MATH  Google Scholar 

  3. Blömker D., Hairer M., Pavliotis G.A.: Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities. Nonlinearity 20, 1–5 (2007)

    Article  Google Scholar 

  4. Blömker D., Mohammed W.W.: Amplitude equation for SPDEs with quadratic nonlinearities. Electron. J. Probab. 14, 2527–2550 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. Blömker D., Mohammed W.W.: Amplitude equations for SPDEs with cubic nonlinearities, stochastics. Int. J. Probab. Stochast. Process. 85(2), 181–215 (2013)

    Article  MATH  Google Scholar 

  6. Blömker D., Mohammed W.W., Nolde C., Wöhrl F.: Numerical study of amplitude equations for SPDEs with degenerate forcing. Int. J. Comput. Math. 89(18), 2499–2516 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Burke J., Dawes J.H.P.: Localized states in an extended Swift–Hohenberg equation. SIAM J. Appl. Dyn. Syst. 11(1), 261–284 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  8. Burke J., Knobloch E.: Localized states in the generalized Swift–Hohenberg equation. Phys. Rev. E 73, 056211 (2006)

    Article  MathSciNet  Google Scholar 

  9. Collet P., Eckmann J.-P.: The time dependent amplitude equation for the Swift–Hohenberg problem. Commun. Math. Phys. 132, 139–153 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cross M.C., Hohenberg P.C.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 581–1112 (1993)

    Article  Google Scholar 

  11. Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and Its Applications, vol. 44. Cambridge University Press, Cambridge (1992)

  12. Hutt A., Longtin A., Schimansky-Geier L.: Additive global noise delays turing bifurcations. Phys. Rev. Lett. 98, 230601 (2007)

    Article  Google Scholar 

  13. Hutt A., Longtin A., Schimansky-Geier L.: Additive noise-induced turing transitions in spatial systems with application to neural fields and the Swift–Hohenberg equation. Phys. D Nonlinear Phenom. 237(6), 755–773 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hilali M.F., Métens S., Borckmans P., Dewel G.: Pattern selection in the generalized Swift–Hohenberg model. Phys. Rev. E 51, 2046–2052 (1995)

    Article  Google Scholar 

  15. Hutt A.: Additive noise may change the stability of nonlinear systems. EPL (Eur. Lett.) 84(3), 34003 (2008)

    Article  MathSciNet  Google Scholar 

  16. Kalliadasis, S., Pavliotis, G., Pradas, M., Schmuck, M.: A new mode reduction strategy for the generalized Kuramoto–Sivashinsky equation Preprint (2011)

  17. Kalliadasis S., Pavliotis G., Pradas M., Schmuck M.: A new stochastic mode reduction strategy for dissipative systems. Phys. Rev. Lett. 110, 244101 (2013)

    Article  Google Scholar 

  18. Kirrmann P., Schneider G., Mielke A.: The validity of modulation equations for extended systems with cubic nonlinearities. Proc. Royal Soc. Edinb. Sect. A. 122, 459–490 (1992)

    MathSciNet  Google Scholar 

  19. Mielke A., Schneider G., Ziegra A.: Comparison of inertial manifolds and application to modulated systems. Math. Nachr. 214, 53–69 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  20. Mohammed W.W., Blömker D., Klepel K.: Modulation equation for stochastic Swift–Hohenberg equation. SIAM J. Math. Anal. 45(1), 14–30 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  21. Pazy A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44. Springer, New York (1983)

    Book  Google Scholar 

  22. Roberts A.J.: A step towards holistic discretisation of stochastic partial differential equations. ANZIAM J. 45(E), C1–C15 (2003)

    Google Scholar 

  23. Schneider G.: The validity of generalized Ginzburg–Landau equations. Math. Methods Appl. Sci. 19(9), 717–736 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  24. Swift J., Hohenberg P.C.: Hydrodynamic fluctuations at the convective instability. Phys. Rev. A 15, 319–328 (1977)

    Article  Google Scholar 

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Authors and Affiliations

  1. Institut für Mathematik, Universität Augsburg, 86135, Augsburg, Germany

    Konrad Klepel & Dirk Blömker

  2. Department of Mathematics, Faculty of Science, Mansoura university, Mansoura, Egypt

    Wael W. Mohammed

Authors
  1. Konrad Klepel
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  2. Dirk Blömker
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  3. Wael W. Mohammed
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Correspondence to Dirk Blömker.

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Klepel, K., Blömker, D. & Mohammed, W.W. Amplitude equation for the generalized Swift–Hohenberg equation with noise. Z. Angew. Math. Phys. 65, 1107–1126 (2014). https://doi.org/10.1007/s00033-013-0371-8

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  • DOI: https://doi.org/10.1007/s00033-013-0371-8

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