Hostname: page-component-7dd5485656-wjfd9 Total loading time: 0 Render date: 2025-10-31T12:20:26.311Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

Information

Type
Papers
Information
European Journal of Applied Mathematics , Volume 26 , Issue 1 , February 2015 , pp. 33 - 60
Copyright
Copyright © Cambridge University Press 2014 

References

[1]Ginzburg, V. L. & Landau, L. D. (1965) On the theory of superconductivity. J. ETP 20, 1064–1082.Google Scholar
[2]Chapman, S. J., Howison, S. D. & Ockendon, J. R. (1992) Macroscopic models for superconductivity. SIAM Rev. 34 (4), 529–560.Google Scholar
[3]Marcus, P. M. (1964) Exact solution of the Ginzburg-Landau equations for slabs in tangential magnetic fields. Rev. Mod. Phys. 36, 294–299.CrossRefGoogle Scholar
[4]Odeh, F. (1967) Existence and bifurcation theorems for the ginzburglandau equations. J. Math. Phys. 8 (12), 2351–2356.CrossRefGoogle Scholar
[5]Seydel, R. (1983) Branch switching in bifurcation problems for ordinary differential equations. Numer. Math. 41 (1), 93–116.Google Scholar
[6]Yang, Y. S. (1990) Boundary value problems of the Ginzburg-Landau equations. Proc. R. Soc. Edinburgh A 114 (3–4), 355–365.Google Scholar
[7]Wang, S. & Yang, Y. S. (1992) Symmetric superconducting states in thin films. SIAM J. Appl. Math. 52 (3), 614–629.CrossRefGoogle Scholar
[8]Hastings, S. P., Kwong, M. K. & Troy, W. C. (1996) The existence of multiple solutions for a Ginzburg-Landau type model of superconductivity. Eur. J. Appl. Math. 7 (6), 559–574.CrossRefGoogle Scholar
[9]Aftalion, A. (1997) On the minimizers of the Ginzburg-Landau energy for high kappa: The one-dimensional case. Eur. J. Appl. Math. 8 (4), 331–345.CrossRefGoogle Scholar
[10]Aftalion, A. & Chapman, S. J. (1999) Asymptotic analysis of the bifurcation diagram for symmetric one-dimensional solutions of the Ginzburg-Landau equations. Eur. J. Appl. Math. 10 (5), 477–495.Google Scholar
[11]Aftalion, A. & Chapman, S. J. (2000) Asymptotic analysis of a secondary bifurcation of the one-dimensional Ginzburg-Landau equations of superconductivity. SIAM J. Appl. Math. 60 (4), 1157–1176.Google Scholar
[12]Dancer, E. N. & Hastings, S. P. (2000) On the global bifurcation diagram for the one-dimensional Ginzburg-Landau model of superconductivity. Eur. J. Appl. Math. 11 (3), 271–291.Google Scholar
[13]Aftalion, A. & Troy, W. C. (1999) On the solutions of the one-dimensional Ginzburg-Landau equations for superconductivity. Physica D 132 (1–2), 214–232.CrossRefGoogle Scholar
[14]Hastings, S. P. & Troy, W. C. (1999) There are asymmetric minimizers for the one-dimensional Ginzburg-Landau model of superconductivity. SIAM J. Math. Anal. 30 (1), 1–18.CrossRefGoogle Scholar
[15]Aftalion, A. & Troy, W. C. (2000) On the uniqueness of solutions of the Ginzburg-Landau system for thin films. Eur. J. Appl. Math. 11 (4), 365–380.Google Scholar
[16]Aftalion, A. & Du, Q. (2002) The bifurcation diagrams for the Ginzburg-Landau system of superconductivity. Physica D 163 (1-2), 94–105.Google Scholar
[17]Du, Y. & Pan, X. B. (2005) Multiple states and hysteresis for type I superconductors. J. Math. Phys. 46 (7), 073301, 34.Google Scholar
[18]Du, Q. (2005) Numerical approximations of the Ginzburg-Landau models for superconductivity. J. Math. Phys. 46 (9), 095109, 22.Google Scholar
[19]Du, Q., Gunzburger, M. D. & Peterson, J. S. (1992) Analysis and approximation of the Ginzburg-Landau model of superconductivity. SIAM Rev. 34 (1), 54–81.CrossRefGoogle Scholar
[20]Day, S., Lessard, J.-P. & Mischaikow, K. (2007) Validated continuation for equilibria of PDEs. SIAM J. Numer. Anal. 45 (4), 1398–1424.Google Scholar
[21]van den Berg, J. B., Mireles-James, J. D., Lessard, J.-P. & Mischaikow, K. (2011) Rigorous numerics for symmetric connecting orbits: even homoclinics of the Gray-Scott equation. SIAM J. Math. Anal. 43 (4), 1557–1594.Google Scholar
[22]van den Berg, J. B. & Lessard, J.-P. (2008) Chaotic braided solutions via rigorous numerics: chaos in the Swift-Hohenberg equation. SIAM J. Appl. Dyn. Syst. 7 (3), 988–1031.Google Scholar
[23]Gameiro, M. & Lessard, J.-P. (2010) Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs. J. Differ. Equ. 249 (9), 2237–2268.CrossRefGoogle Scholar
[24]van den Berg, J. B., Lessard, J.-P. & Mischaikow, K. (2010) Global smooth solution curves using rigorous branch following. Math. Comput. 79 (271), 1565–1584.Google Scholar
[25]Lessard, J.-P. & Reinhardt, C. (2014) Rigorous numerics for nonlinear differential equations using chebyshev series. SIAM J. Numer. Anal. 52 (1), 1–22.Google Scholar
[26]Moore, R. E. (1966) Interval Analysis, Prentice-Hall Inc., Englewood Cliffs, N.J.Google Scholar
[27]Boyd, J. P. (2001) Chebyshev and Fourier Spectral Methods, 2nd edition, Dover Publications Inc., Mineola, NY.Google Scholar
[28]Gameiro, M. & Lessard, J.-P. (2013) Efficient rigorous numerics for higher-dimensional PDEs via one-dimensional estimates. SIAM J. Numer. Anal. 51 (4), 2063–2087.Google Scholar
[29]Breden, M., Lessard, J.-P. & Vanicat, M. (2013) Global bifurcation diagrams of steady states of systems of PDEs via rigorous numerics: A 3-component reaction-diffusion system. Acta Appl. Math. 128, 113–152.CrossRefGoogle Scholar
[30]Gameiro, M., Lessard, J.-P. && Mischaikow, K. (2008) Validated continuation over large parameter ranges for equilibria of PDEs. Math. Comput. Simul. 79 (4), 1368–1382.Google Scholar
[31]Rump, S. M. (1999) INTLAB - INTerval LABoratory. In: Csendes, T. (ed.), Developments in Reliable Computing, Kluwer Academic Publishers, Dordrecht, pages 77–104. http://www.ti3.tu-harburg.de/rump/.Google Scholar
[32]Arioli, G. & Koch, H. (2010) Computer-assisted methods for the study of stationary solutions in dissipative systems, applied to the Kuramoto-Sivashinski equation. Arch. Ration. Mech. Anal. 197 (3), 1033–1051.Google Scholar
[33]Correc, A. & Lessard, J.-P. MATLAB codes to perform the proofs. http://archimede.mat.ulaval.ca/jplessard/GinzburgLandauGoogle Scholar