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The validity of modulation equations for extended systems with cubic nonlinearities

Published online by Cambridge University Press:  14 November 2011

Pius Kirrmann ,
Guido Schneider  and
Alexander Mielke
Pius Kirrmann
Affiliation:
Mathematisches Institut A, Universität Stuttgart, Pfaffenwaldring 57, W-7000 Stuttgart, Germany
Guido Schneider
Affiliation:
Mathematisches Institut A, Universität Stuttgart, Pfaffenwaldring 57, W-7000 Stuttgart, Germany
Alexander Mielke
Affiliation:
Mathematisches Institut A, Universität Stuttgart, Pfaffenwaldring 57, W-7000 Stuttgart, Germany
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Synopsis

Modulation equations play an essential role in the understanding of complicated systems near the threshold of instability. Here we show that the modulation equation dominates the dynamics of the full problem locally, at least over a long time-scale. For systems with no quadratic interaction term, we develop a method which is much simpler than previous ones. It involves a careful bookkeeping of errors and an estimate of Gronwall type.

As an example for the dissipative case, we find that the Ginzburg–Landau equation is the modulation equation for the Swift–Hohenberg problem. Moreover, the method also enables us to handle hyperbolic problems: the nonlinear Schrodinger equation is shown to describe the modulation of wave packets in the Sine–Gordon equation.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 122 , Issue 1-2 , 1992 , pp. 85 - 91
Copyright
Copyright © Royal Society of Edinburgh 1992

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References

1Calogero, F. and Eckhaus, W.. Nonlinear evolution equations, rescalings, model PDEs and their integrability: I + II. Inverse Problems 3 (1987), 229–262; 4 (1988) 11–33.CrossRefGoogle Scholar
2Collet, P. and Eckmann, J.-P.. The time dependent amplitude equation for the Swift-Hohenberg problem. Comm. Math. Physics 132 (1990), 139–153.CrossRefGoogle Scholar
3DiPrima, R. C., Eckhaus, W. and Segel, L. A.. Non-linear wave-number interaction in near-critical two-dimensional flows. Fluid Mech. 49 (1971), 705–744.CrossRefGoogle Scholar
4Iooss, G., Mielke, A. and Demay, Y.. Theory of steady Ginzburg-Landau equation in hydrodynamic stability problems. European J. Mech. B/Fluids 3 (1989), 229–268.Google Scholar
5Kalyakin, L. A.. Long wave asymptotics, integrable equations as asymptotic limits of non-linear systems. Russ. Math. Surveys 44 (1) (1989), 3–42.CrossRefGoogle Scholar
6Kato, T.. Perturbation Theory for Linear Operators (Berlin: Springer, 1966).Google Scholar
7Krol, M. S.. On a Galerkin-averaging method for weakly nonlinear wave equations. Math. Methods Applied Sci. 11 (1989), 649–664.CrossRefGoogle Scholar
8Mielke, A.. Reduction of PDEs on domains with several unbounded directions. J. Appl. Math. Phys. (ZAMP) 43 (1992).CrossRefGoogle Scholar
9Newell, A. and Whitehead, J.. Finite bandwidth, finite amplitude convection. J. Fluid Mech. 38 (1969), 279–303.CrossRefGoogle Scholar
10Stroucken, A. and Verhulst, F.. The Galerkin-averaging method for nonlinear, undamped continuous systems. Math. Methods Applied Sci. 9 (1986), 520–549.CrossRefGoogle Scholar
11van Harten, A.. On the validity of Ginzburg-Landau's equation. J. Nonlinear Science 1 (1991), 397–422.CrossRefGoogle Scholar