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Bounded solutions for an ordinary differential system from the Ginzburg–Landau theory

Part of: Elliptic equations and systems Boundary value problems Spectral theory and eigenvalue problems

Published online by Cambridge University Press:  14 August 2020

Anne Beaulieu
Anne Beaulieu*
Affiliation:
LAMA, Univ. Paris Est Creteil, Univ. Gustave Eiffel, UPEM, CNRS, F-94010, Créteil, France ([email protected])
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Abstract

In this paper, we look at a linear system of ordinary differential equations as derived from the two-dimensional Ginzburg–Landau equation. In two cases, it is known that this system admits bounded solutions coming from the invariance of the Ginzburg–Landau equation by translations and rotations. The specific contribution of our work is to prove that in the other cases, the system does not admit any bounded solutions. We show that this bounded solution problem is related to an eigenvalue problem.

Keywords

Ginzburg–Landau equationslinearized operator

MSC classification

Primary: 34B40: Boundary value problems on infinite intervals
Secondary: 35J60: Nonlinear elliptic equations 35P15: Estimation of eigenvalues, upper and lower bounds
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 3378 - 3408
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Beaulieu, Anne. Some remarks on the linearized operator about the radial solution for the Ginzburg-Landau equation. Nonlinear Anal. 54 (2003), 1079–1119.CrossRefGoogle Scholar
Beaulieu, Anne, The kernel of the linearized Ginzburg-Landau operator, preprint 2018, HAL Id: hal-01724551.Google Scholar
Bethuel, Fabrice, Brezis, Haïm and Hélein, Frédéric. Ginzburg-Landau vortices (Boston, Basel, Berlin: Birkhaüser, 1994).CrossRefGoogle Scholar
Hervé, Rose-Marie and Hervé, Michel. Étude qualitative des solutions réelles d'une équation différentielle liée à l’équation de Ginzburg-Landau. (French) Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994), 427–440.CrossRefGoogle Scholar
Lieb, E. H. and Loss, M.. Symmetry of the Ginzburg-Landau minimizer in a disc. Math. Res. Lett. 1 (1994), 701–715.CrossRefGoogle Scholar
Lin, Tai-Chia. The stability of the radial solution to the Ginzburg-Landau equation. Commun. Partial Differ. Equ. 22 (1997), 619–632.Google Scholar
Lin, Tai-Chia. Spectrum of the linearized operator for the Ginzburg-Landau equation. Electron. J. Differ. Equ. (2000), 25.Google Scholar
Mironescu, Petru. On the stability of radial solutions of the Ginzburg-Landau equation. J. Funct. Anal. 130 (1995), 334–344.CrossRefGoogle Scholar
Pacard, Frank and Rivière, Tristan. Linear and nonlinear aspects of vortices. The Ginzburg-Landau model. Progress in Nonlinear Differential Equations and their Applications, vol. 39 (Boston, MA: Birkhaüser Boston, Inc., 2000).Google Scholar