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The Validity of Generalized Ginzburg-Landau Equations

✍ Scribed by Guido Schneider

Publisher
John Wiley and Sons
Year
1996
Tongue
English
Weight
1014 KB
Volume
19
Category
Article
ISSN
0170-4214

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✦ Synopsis

Communicated by W. Eckhaus

We consider parabolic systems defined on cylindrical domains close to the threshold of instability, in which the Fourier modes with positive growth rates are concentrated at a non-zero critical wave number. In particular, we consider systems for which a so-called Ginzburg-Landau equation can be derived. Due to the presence of continuous spectrum, classical bifurcation theory is not available to describe bifurcating solutions. Thus, we consider a modified system with artificial spectral gap, which possesses an infinitedimensional centre manifold. The amplitude equation on this manifold is called a generalized Ginzburg-Landau equation. From previous work [18] it is known that the Fourier modes are exponentially concentrated at integer multiples of the critical wave number. Hence, the error made by this modification is exponentially small in powers of the bifurcation parameter. The approximations obtained via the generalized Ginzburg-Landau equation are valid on a much longer time scale than those obtained by using the classical Ginzburg-Landau equation as an amplitude equation.

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