Derivation of the time-dependent Ginzburg-Landau equation on the line
type: Item
instance of: scholarly article
title: Derivation of the time-dependent Ginzburg-Landau equation on the line
zbMATH Open document ID: 0889.35044
published in: Journal of Nonlinear Science
publication date: 1998-05-13T00:00:00Z
review text: The author proposes a rigorous derivation of the one-dimensional time-dependent Ginzburg-Landau (GL) equation; previously, rigorous derivations were elaborated only for its stationary (time-independent) version. The starting points are nonlinear second-order evolutional PDEs and PDE systems, such the Swift-Hohenberg equation (including not only cubic, but also quadratic nonlinear terms), and reaction-diffusion equations (the so-called Brusselator model). By means of the Lyapunov-Schmidt reduction technique and circumventing a known difficulty related to the contribution from the coninuum spectrum, the author derives time-dependent GL equations that, actually, are not PDEs, but rather partial pseudodifferential equations, i.e., those that contain \(x\)-derivatives of all orders. A formal truncation transforms the pseudodifferential equation into the traditional GL one, but the truncation procedure is not derived rigorously in this work. Some results obtained in the paper admit generalizations for multidimensional GL equations.
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35R50
zbMATH DE Number: 1129276
zbMATH Keywords: pseudodifferential equation
zbMATH Keywords: Lyapunov-Schmidt reduction
zbMATH Keywords: Swift-Hohenberg equation
zbMATH Keywords: Brusselator model
zbMATH Keywords: formal truncation
author: Ian Melbourne
reviewed by: B. A. Malomed
MaRDI profile type: Publication
full work available at URL: s003329900041
OpenAlex ID: W2019806348
DOI: 10.1007/S003329900041
description: scientific article; zbMATH DE number 1129276