Dynamics of Ginzburg-Landau vortices
type: Item
instance of: scholarly article
title: Dynamics of Ginzburg-Landau vortices
zbMATH Open document ID: 0923.35167
author: Robert L. Jerrard
author: H. Mete Soner
published in: Archive for Rational Mechanics and Analysis
publication date: 1999-05-19T00:00:00Z
review text: The authors study the asymptotics of the sequence of complex valued solutions \(u^\varepsilon(x,t)\) in the limit \(\varepsilon\to 0\) of the system \[ \partial u^\varepsilon/\partial t-\Delta u^\varepsilon= \varepsilon^{-2} u^\varepsilon(1-| u^\varepsilon|^2)\quad\text{in }\Omega\times (0,\infty), \] \[ u^\varepsilon(x,t)= g(x)\quad\text{on }\partial\Omega\times (0,\infty). \] Here \(\Omega\subset \mathbb{R}^2\) is an open bounded subset and \(g\) is a given function with \(| g|= 1\). The system provides a gradient flow of the functional \[ I^\varepsilon(w)= \int_\Omega (\textstyle{{1\over 2}}|\nabla w|^2+ \varepsilon^{-2} \textstyle{{1\over 4}}(1-| w|^2))dx \] which serves for the main technical tool. The most interesting geometrical results concern the behaviour of vortices (the zeroes of solutions). Assuming that initially there are \(N\) isolated vortices with degree \(\pm 1\), then, in the limit, these vortices persist and satisfy a system of ordinary differential equations of the kind \[ {d\over dt} y^i(t)= -2d_i\Biggl((\nabla\varphi(y^i(t), \vec y(t)))^\perp+ \sum_{m\neq i} d_m {y^m(t)- y^i(t)\over| y^m(t)- y^i(t)|^2}\Biggr), \] where \(d_i\in\{\pm 1\}\), \(\varphi\) is a solution of a Dirichlet problem (which cannot be stated here), \(\perp\) denotes an orthogonal vector. All the proofs are given with details and are completed by numerous comments.
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35K57
zbMATH DE Number: 1176732
zbMATH Keywords: reaction diffusion system
zbMATH Keywords: nodal set
zbMATH Keywords: vortex
zbMATH Keywords: Ginzburg-Landau model
reviewed by: Jan Chrastina
Wikidata QID: Q57635978
MaRDI profile type: Publication
full work available at URL: s002050050085
OpenAlex ID: W1967688238
DOI: 10.1007/S002050050085
description: scientific article; zbMATH DE number 1176732