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The Validity of Generalized Ginzburg‐Landau Equations
Author(s) -
Schneider Guido
Publication year - 1996
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/(sici)1099-1476(199606)19:9<717::aid-mma792>3.0.co;2-z
Subject(s) - mathematics , mathematical analysis , fourier transform , bifurcation , amplitude , stable manifold , exponential growth , manifold (fluid mechanics) , center manifold , fisher's equation , hopf bifurcation , partial differential equation , physics , integro differential equation , quantum mechanics , nonlinear system , first order partial differential equation , mechanical engineering , engineering
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 infinite‐dimensional 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.