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Phase dynamics in the real Ginzburg‐Landau equation
Author(s) -
Melbourne Ian,
Schneider Guido
Publication year - 2004
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200310129
Subject(s) - mathematics , diffusion equation , dynamics (music) , diffusion , phase (matter) , mathematical physics , mathematical analysis , physics , quantum mechanics , economy , acoustics , economics , service (business)
Spatially periodic equilibria A ( X, T ) = √1 − q 2 e iqX + iϕ 0are the locally preferred planform for the Ginzburg‐Landau equation ∂ T A = ∂ 2 X A + A − A | A | 2 . To describe the global spatial behavior, an evolution equation for the local wave number q can be derived formally. The local wave number q satisfies approximately a so called phase diffusion equation ∂ τ q = ∂ 2 ξ h ( q ). It is the purpose of this paper to explain the extent to which the phase diffusion equation is valid by proving estimates for this formal approximation. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)