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Coexistence of nontrivial solutions of the one-dimensional Ginzburg-Landau equation: A computer-assisted proof

Published online by Cambridge University Press:  08 October 2014

ANAÏS CORREC  and
JEAN-PHILIPPE LESSARD
ANAÏS CORREC
Affiliation:
Université Laval, Département de Mathématiques et de Statistique, 1045 avenue de la Médecine, Québec, QC, G1V 0A6, Canada emails: jean-philippe.lessard@mat.ulaval.ca, anais.correc.1@ulaval.ca
JEAN-PHILIPPE LESSARD
Affiliation:
Université Laval, Département de Mathématiques et de Statistique, 1045 avenue de la Médecine, Québec, QC, G1V 0A6, Canada emails: jean-philippe.lessard@mat.ulaval.ca, anais.correc.1@ulaval.ca
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Abstract

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In this paper, Chebyshev series and rigorous numerics are combined to compute solutions of the Euler-Lagrange equations for the one-dimensional Ginzburg-Landau model of superconductivity. The idea is to recast solutions as fixed points of a Newton-like operator defined on a Banach space of rapidly decaying Chebyshev coefficients. Analytic estimates, the radii polynomials and the contraction mapping theorem are combined to show existence of solutions near numerical approximations. Coexistence of as many as seven nontrivial solutions is proved.

Keywords

Ginzburg-Landau equationboundary value problemscoexistence of nontrivial solutionsrigorous numericsChebyshev seriescontraction mapping theorem
Type
Papers
Information
European Journal of Applied Mathematics , Volume 26 , Issue 1 , February 2015 , pp. 33 - 60
Copyright
Copyright © Cambridge University Press 2014 

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