Abstract
We study the dynamic transition of the Swift-Hohenberg equation (SHE) when linear multiplicative noise acting on a finite set of modes of the dominant linear flow is introduced. Existence of a stochastic flow and a local stochastic invariant manifold for this stochastic form of SHE are both addressed in this work. We show that the approximate reduced system corresponding to the invariant manifold undergoes a stochastic pitchfork bifurcation, and obtain numerical evidence suggesting that this picture is a good approximation for the full system as well.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Birnir, B.: The Kolmogorov–Obukhov Theory of Turbulence: A Mathematical Theory of Turbulence. Springer Briefs in Mathematics. Springer, New York (2013)
Budd, C., Kuske, R.: Localized periodic patterns for the non-symmetric generalized swift-hohenberg equation. Phys. D 208, 73–95 (2005)
Caraballo, T., Crauel, H., Langa, J., Robinson, J.: The effect of noise on the chafee-infante equation: a nonlinear case study. Proc. Am. Math. Soc. 135, 373–382 (2007)
Caraballo, T., Duan, J., Lu, K., Schmalfuss, B.: Invariant manifolds for random and stochastic partial differential equations. Adv. Nonlinear Stud. 10(1), 23–52 (2010)
Caraballo, T., Langa, J., Robinson, J.: A stochastic pitchfork bifurcation in a reaction-diffusion equation. Proc. R. Soc. Lond. A 457, 2041–2061 (2001)
Chekroun, M.D., Liu, H., Wang, S.: Stochastic parameterizing manifolds: applicaion to stochastic transitions in spdes, in preparation
Chekroun, M.D., Liu, H., Wang, S.: Approximation of Stochastic Invariant Manifolds: Stochastic Manifolds for Nonlinear SPDEs I. Springer Briefs in Mathematics. Springer, New York (2015)
Chekroun, M.D., Liu, H., Wang, S.: Stochastic Parameterizing Manifolds and Non-Markovian Reduced Equations: Stochastic Manifolds for Nonlinear SPDEs II. Springer Briefs in Mathematics. Springer, New York (2015)
Chekroun, M.D., Simonnet, E., Ghil, M.: Stochastic climate dynamics: random attractors and time-dependent invariant measures. Phys. D 21, 1685–1700 (2011)
Collet, P., Eckmann, J.: Instabilities and Fronts in Extended Systems. Princeton Series in Physics. Princeton University Press, Princeton (1990)
Crauel, H., Debussche, A., Flandoli, F.: Random attractors. J. Dyn. Differ. Equ. 9(2), 307–341 (1997)
Cross, M., Hohenberg, P.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112 (1993)
Duan, J., Wang, W.: Effective Dynamics of Stochastic Partial Differential Equations. Elsevier, Amsterdam (2014)
Engel, K.J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, vol. 194. Springer, New York (2000)
Hadamard, J.: Sur l’iteration et les solutions asymptotiques des equations differentielles. Bull. Soc. Math. France 29, 224–228 (1901)
Han, J., Hsia, C.: Dynamical bifurcation of the two dimensional swift-hohenberg equation with odd periodic condition. Discrete Contin. Dyn. Syst. Ser. B 17(7), 2431–2449 (2012)
Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840. Springer, New York (1981)
Hilali, M.F., Mtens, S., Borckmans, P., Dewel, G.: Pattern selection in the generalized swift-hohenberg model. Phys. Rev. E 51, 2046–2052 (1995)
Klebaner, F.: Introduction to Stochastic Calculus with Applications, 2nd edn. Imperial College Press, London (2005)
Liapunov, A.: Problème géneral de la stabilité du mouvement. Princeton University Press, Princeton (1947)
Ma, T., Wang, S.: Phase Transition Dynamics. Springer, New York (2014)
Mao, X.: Stochastic Differential Equations and Applications, 2nd edn. Horwood Publishing Limited, Chichester (2008)
Mohammad, S., Zhang, T., Zhao, H.: The stable manifold theorem for semilinear stochastick evolution equations and stochastic partial differential equations. Mem. Am. Math. Soc. 196(917), 1–105 (2008)
Oksendal, B.: Stochastic Differential Equations: An Introduction with Apllications, 6th edn. Springer, Berlin (2013)
Perron, O.: Über stabilität und asymptotisches verhalten der integrale von differentialgleichungssystemen. Math. Z. 29, 129–160 (1928)
Swift, J., Hohenberg, P.C.: Hydrodynamic fluctuations at the convective instability. Phys. Rev. A 15, 319–328 (1977)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
All authors declare that they have no conflict of interest.
Additional information
Communicated by S. Friedlander
This research is supported in part by the National Science Foundation (NSF) Grant DMS-1515024, and by the Office of Naval Research (ONR) Grant N00014-15-1-2662. The authors would also like to thank Professor Shouhong Wang for his suggestions and advice.
Rights and permissions
About this article
Cite this article
Hernández, M., Ong, K.W. Stochastic Swift-Hohenberg Equation with Degenerate Linear Multiplicative Noise. J. Math. Fluid Mech. 20, 1353–1372 (2018). https://doi.org/10.1007/s00021-018-0368-3
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00021-018-0368-3