Skip to main content
SpringerLink
Log in

Propagating fronts and the center manifold theorem

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We prove the existence of propagating front solutions for the Swift-Hohenberg equation (SH). Using the center manifold theorem we reduce the problem to a two dimensional system of ordinary differential equations. They describe stationary solutions and front solutions of the partial differential equation (SH). We identify the well-known “amplitude equation” as the lowest order approximation to the equation of motion on the center manifold.

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

  • [A] Arnold, V.: Ordinary differential equations. Cambridge, MA: MIT Press 1973

    Google Scholar 

  • [AW] Aronson, D., Weinberger, H.: Multidimensional nonlinear diffusion arising in Population Genetics. Adv. Math.30, 33 (1978)

    Article  Google Scholar 

  • [CE1] Collet, P., Eckmann, J.-P.: The existence of dendritic fronts. Commun. Math. Phys.107, 39–92 (1986)

    Article  Google Scholar 

  • [CE2] Collet, P., Eckmann, J.-P.: Instabilities and fronts in extended systems, Princeton, NJ: Princeton University Press 1990

    Google Scholar 

  • [CM1] Carr, C., Muncaster, R. G.: The application of centre manifolds to amplitude expansions. I. Ordinary differential equations. J. Differ. Eqs.50, 260–279 (1983)

    Article  Google Scholar 

  • [CM2] Carr, C., Muncaster, R. G.: The application of centre manifolds to amplitude expansions. II. Infinite dimensional Problems. J. Differ. Eqs.50, 280–288 (1983)

    Article  Google Scholar 

  • [HPS] Hirsch, M., Pugh, C., Shub, M.: Invariant Manifolds: Lecture Notes in Mathematics vol. 583. Berlin Heidelberg, New York: Springer 1977

    Google Scholar 

  • [K] Kirchgässner, K.: Nonlinearly resonant surface waves and Homoclinic bifurcation. Adv. Appl. Mech.26, 135–181 (1988)

    Google Scholar 

  • [LW] de la Llave, R., Wayne, C. E.: Whiskered and Low Dimensional Tori in Nearly Integrable Hamiltonian Systems. Preprint (1989)

  • [M] Mielke, A.: Reduction of Quasilinear Elliptic Equations in Cyclindrical Domains with Applications. Math. Meth. Appl. Sci.10, 51–66 (1988)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de Physique Théorique, Université de Genève, CH-1211, Geneva 4, Switzerland

    J. -P. Eckmann

  2. Department of Mathematics, Pennsylvania State University, 16802, University Park, PA, USA

    C. E. Wayne

Authors
  1. J. -P. Eckmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. E. Wayne
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by A. Jaffe

Rights and permissions

Reprints and permissions

About this article

Cite this article

Eckmann, J.P., Wayne, C.E. Propagating fronts and the center manifold theorem. Commun.Math. Phys. 136, 285–307 (1991). https://doi.org/10.1007/BF02100026

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02100026

Keywords

Search

Navigation