Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-27T12:02:51.019Z Has data issue: false hasContentIssue false
  • English
  • Français

Justification of mean-field coupled modulation equations

Published online by Cambridge University Press:  14 November 2011

Guido Schneider
Guido Schneider
Affiliation:
IfAM, Universität Hannover, D-30167 Hannover, Germany
Get access Rights & Permissions [Opens in a new window]

Abstract

We are interested in reflection symmetric (x↦–x) evolution problems on the infinite line. In the systems which we have in mind, a trivial ground state loses stability and bifurcates into a temporally oscillating, spatial periodic pattern. A famous example of such a system is the Taylor-Couette problem in the case of strongly counter-rotating cylinders. In this paper, we consider a system of coupled Kuramoto–Shivashinsky equations as a model problem for such a system. We are interested in solutions which are slow modulations in time and in space of the bifurcating pattern. Multiple scaling analysis is used in the existing literature to derive mean-field coupled Ginzburg–Landau equations as approximation equations for the problem. The aim of this paper is to give exact estimates between the solutions of the coupled Kuramoto–Shivashinsky equations and the associated approximations.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 3 , 1997 , pp. 639 - 650
Copyright
Copyright © Royal Society of Edinburgh 1997

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Bollerman, P.. On the validity of amplitude equations. (Thesis, University of Utrecht, 1996).Google Scholar
2Bollerman, P., A. van Harten and G. Schneider. On the justification of the Ginzburg—Landau approximation. In Nonlinear Dynamics and Pattern Formation in the Natural Environment, eds Doelman, A. and van Harten, A., 20–36 (Harlow: Longman, 1995).Google Scholar
3Chossat, P. and Iooss, G.. The Taylor—Couette Problem, Applied Mathematical Sciences 102 (New York: Springer, 1994).CrossRefGoogle Scholar
4Collet, P. and Eckmann, J.-P.. The time dependent amplitude equation for the Swift–Hohenberg problem. Comm. Math. Phys. 132 (1990), 139–53.CrossRefGoogle Scholar
5Coullet, P., Fauve, S. and Tirapegui, E.. Large-scale instability of nonlinear standing waves. J. Phys. Lett. 46(1985), L787–L791.CrossRefGoogle Scholar
6Dangelmayr, G.. Ginzburg–Landau description of waves in extended systems (Preprint, Universitat Tübingen, 1993); in Dynamics of Nonlinear Waves in Dissipative Systems: Reduction, Bifurcation, and Stability, Pitman Research Notes in Mathematics Series (Harlow: Longman, 1996).Google Scholar
7Eckhaus, W.. The Ginzburg–Landau equation is an attractor. J. Nonlinear Sci. 3 (1993), 329–48.CrossRefGoogle Scholar
8Eckhaus, W. and Shepeleva, A.. Attraction properties of the Ginzburg–Landau manifold (Preprint 878, University of Utrecht, 1994).Google Scholar
9Henry, D.. Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics 840 (Berlin: Springer, 1981).CrossRefGoogle Scholar
10Kirrmann, P., Schneider, G. and Mielke, A.. The validity of modulation equations for extended systems with cubic nonlinearities. Proc. Roy. Soc. Edinburgh Sect. A 122 (1992), 85–91.CrossRefGoogle Scholar
11Knobloch, E.. Nonlocal amplitude equations. In Pattern Formation in Complex Dissipative Systems, eds Kai, S., 263–74 (Singapore: World Scientific, 1992).Google Scholar
12Knobloch, E. and DeLuca, J.. Amplitude equations for travelling wave correction. Nonlinearity 3 (1990), 975–80.CrossRefGoogle Scholar
13Knobloch, E. and Gibbon, J. D.. Coupled NLS equations for counter propagating waves in systems with reflectional symmetry. Phys. Lett. A 154 (1991), 353–6.CrossRefGoogle Scholar
14Matkowsky, B. J. and Volpert, V.. Coupled Ginzburg–Landau equations in gasless combustion. Phys. D 54(1992), 203–19.Google Scholar
15Mielke, A. and Schneider, G.. Attractors for modulation equations on unbounded domains—existence and comparison. Nonlinearity 8 (1995), 743–68.CrossRefGoogle Scholar
16Newell, A. and Whitehead, J.. Finite bandwidth, finite amplitude convection. J. Fluid Mech. 38 (1969), 279–303.CrossRefGoogle Scholar
17Pereira, C. A. and Vega, J. M.. On the pulsating instability of two-dimensional flames. European J. Appl. Math. 3 (1992), 55–73.CrossRefGoogle Scholar
18Pierce, R. D. and Knobloch, E.. On the modulational stability of traveling and standing waves. Phys. Fluids 6 (1994), 1177–90.CrossRefGoogle Scholar
19Pierce, R. D. and Wayne, C. E.. On the validity of mean–field amplitude equations for counterpropagating wavetrains. Nonlinearity 8 (1995), 769–79.CrossRefGoogle Scholar
20Schneider, G.. A new estimate for the Ginzburg—Landau approximation on the real axis. J. Nonlinear Sci. 4(1994), 23–34.CrossRefGoogle Scholar
21Schneider, G.. Error estimates for the Ginzburg–Landau approximation. J. Appl. Math. Phys. (ZAMP) 45(1994), 433–57.CrossRefGoogle Scholar
22Schneider, G.. Global existence via Ginzburg–Landau formalism and pseudo-orbits of Ginzburg–Landau approximations. Comm. Math. Phys. 164 (1994), 157–79.CrossRefGoogle Scholar
23Schneider, G.. Analyticity of Ginzburg–Landau modes. J. Differential Equations 121 (1995), 233–57.CrossRefGoogle Scholar
24Schneider, G.. The long wave limit for a Boussinesq equation (IfAM Preprint, Universitat Hannover, 1995).Google Scholar
25van Harten, A.. On the validity of Ginzburg–Landau's equation. J. Nonlinear Sci. 1 (1991), 397–422.CrossRefGoogle Scholar
26Vega, J. M.. On the amplitude equations arising at the onset of the oscillatory instability. SIAM J. Math. Anal. 24 (1993), 603–17.CrossRefGoogle Scholar