Skip to main content
SpringerLink
Log in

Computer-Assisted Methods for the Study of Stationary Solutions in Dissipative Systems, Applied to the Kuramoto–Sivashinski Equation

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We develop some computer-assisted techniques for the analysis of stationary solutions of dissipative partial differential equations, of their stability, and of their bifurcation diagrams. As a case study, these methods are applied to the Kuramoto–Sivashinski equation. This equation has been investigated extensively, and its bifurcation diagram is well known from a numerical point of view. Here, we rigorously describe the full graph of solutions branching off the trivial branch, complete with all secondary bifurcations, for parameter values between 0 and 80. We also determine the dimension of the unstable manifold for the flow at some stationary solution in each branch.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Kuramoto Y., Tsuzuki T.: Persistent propagation of concentration waves in dissipative media far from thermal equilibrium. Progr. Theor. Phys. 55, 365–369 (1976)

    ADS  Google Scholar 

  2. Sivashinsky G.I.: Nonlinear analysis of hydrodynamic instability in laminal flames— I. Derivation of basic equations. Acta Astr. 4, 1177–1206 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  3. Kevrekidis J.G., Nicolaenko B., Scovel J.C.: Back in the saddle again: a computer assisted study of the Kuramoto–Sivashinsky equation. SIAM J. Appl. Math. 50, 760–790 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  4. Jolly M.S., Kevrekidis J.G., Titi E.S.: Approximate inertial manifolds for the Kuramoto–Sivashinsky equation: analysis and computations. Physica D 44, 38–60 (1990)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. Ilyashenko Y.S.: Global analysis of the phase portrait for the Kuramoto–Sivashinski equation. J. Dyn. Differ. Equ. 4, 585–615 (1992)

    Article  MathSciNet  Google Scholar 

  6. Collet P., Eckmann J.-P., Epstein H., Stubbe J.: A global attracting set for the Kuramoto-Sivashinsky equation. Commun. Math. Phys. 152, 203–214 (1993)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  7. Grujić Z.: Spatial analyticity on the global attractor for the Kuramoto–Sivashinsky equation. J. Dyn. Differ. Equ. 12, 217–228 (2000)

    Article  MATH  Google Scholar 

  8. Zgliczyński P., Mischaikow K.: Rigorous numerics for partial differential equations: the Kuramoto–Sivashinsky equation. Found. Comp. Math. 1, 255–288 (2001)

    MATH  Google Scholar 

  9. Zgliczyński P., Mischaikow K.: Towards a rigorous steady states bifurcation diagram for the Kuramoto–Sivashinsky equation—a computer assisted rigorous approach. Preprint, available at http://www.ii.uj.edu.pl/~zgliczyn/papers/ks/bifks.pd, 2003

  10. Arioli G., Koch H., Terracini S.: Two novel methods and multi-mode periodic solutions for the Fermi-Pasta-Ulam model. Comm. Math. Phys. 255, 1–19 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. Pazy A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983)

    MATH  Google Scholar 

  12. Bates, P.W., Lu, K., Zeng, C.: Existence and persistence of invariant manifolds for semiflows in Banach space. Mem. Amer. Math. Soc. 135(645) (1998)

  13. de la Llave R.: A smooth center manifold theorem which applies to some ill-posed partial differential equations with unbounded nonlinearities. J. Dyn. Diff. Eq. 21, 371–415 (2009)

    Article  MATH  Google Scholar 

  14. Dunford N., Schwartz J.T.: Linear Operators. Part I: General Theory. Wiley- Interscience, New Edition (1988)

    MATH  Google Scholar 

  15. The GNU NYU Ada 9X Translator. Available at ftp://cs.nyu.edu/pub/gnat and many other places

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica and MOX, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milan, Italy

    Gianni Arioli

  2. Department of Mathematics, University of Texas at Austin, Austin, TX, 78712, USA

    Hans Koch

Authors
  1. Gianni Arioli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hans Koch
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gianni Arioli.

Additional information

Communicated by G. Friesecke

This work was supported in part by MIUR project “Metodi variazionali ed Equazioni Differenziali Non Lineari”.

This work was supported in part by the National Science Foundation under Grants No. DMS-0088935 and DMS-0322962.

Electronic Supplementary Material

The Below is the Electronic Supplementary Material.

ESM 1 (TAR 420 kb)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Arioli, G., Koch, H. Computer-Assisted Methods for the Study of Stationary Solutions in Dissipative Systems, Applied to the Kuramoto–Sivashinski Equation. Arch Rational Mech Anal 197, 1033–1051 (2010). https://doi.org/10.1007/s00205-010-0309-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-010-0309-7

Keywords

Search

Navigation