ISSN:
1432-1467
Keywords:
modulation equations
;
Ginzburg-Landau approximation
;
nonlinear partial differential equations on unbounded domains
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
,
Physics
Notes:
Summary Modulation equations play an essential role in the understanding of complicated systems near the threshold of instability. For scalar parabolic equations for which instability occurs at nonzero wavelength, we show that the associated Ginzburg-Landau equation dominates the dynamics of the nonlinear problem locally, at least over a long timescale. We develop a method which is simpler than previous ones and allows initial conditions of lower regularity. It involves a careful handling of the critical modes in the Fourier-transformed problem and an estimate of Gronwall's type. As an example, we treat the Kuramoto-Shivashinsky equation. Moreover, the method enables us to handle vector-valued problems [see G. Schneider (1992)].
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02430625
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