Stability of $n$-vortices in the Ginzburg-Landau equation
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- by James Coleman PDF
- Proc. Amer. Math. Soc. 128 (2000), 1567-1569 Request permission
Abstract:
We consider the class of $n$-vortex solutions to the time-independent Ginzburg-Landau equation on $\mathbf {R}^2$. We prove an inequality governing the solutions of a particular boundary value problem. This inequality is crucial for an elementary proof by Ovchinnikov and Sigal that such $n$-vortices are unstable in the case $\vert n \vert \ge 2$.References
- Fabrice Bethuel, Haïm Brezis, and Frédéric Hélein, Ginzburg-Landau vortices, Progress in Nonlinear Differential Equations and their Applications, vol. 13, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1269538, DOI 10.1007/978-1-4612-0287-5
- Xinfu Chen, Charles M. Elliott, and Tang Qi, Shooting method for vortex solutions of a complex-valued Ginzburg-Landau equation, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), no. 6, 1075–1088. MR 1313190, DOI 10.1017/S0308210500030122
- Patrick S. Hagan, Spiral waves in reaction-diffusion equations, SIAM J. Appl. Math. 42 (1982), no. 4, 762–786. MR 665385, DOI 10.1137/0142054
- Rose-Marie Hervé and Michel Hervé, Étude qualitative des solutions réelles d’une équation différentielle liée à l’équation de Ginzburg-Landau, Ann. Inst. H. Poincaré C Anal. Non Linéaire 11 (1994), no. 4, 427–440 (French, with English and French summaries). MR 1287240, DOI 10.1016/S0294-1449(16)30182-2
- Yuri N. Ovchinnikov and Israel M. Sigal, Ginzburg-Landau equation. I. Static vortices, Partial differential equations and their applications (Toronto, ON, 1995) CRM Proc. Lecture Notes, vol. 12, Amer. Math. Soc., Providence, RI, 1997, pp. 199–220. MR 1479248, DOI 10.1090/crmp/012/16
- Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Springer-Verlag, New York, 1984. Corrected reprint of the 1967 original. MR 762825, DOI 10.1007/978-1-4612-5282-5
- Itai Shafrir, Remarks on solutions of $-\Delta u=(1-|u|^2)u$ in $\textbf {R}^2$, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), no. 4, 327–331 (English, with English and French summaries). MR 1267609
- Michael Struwe, Variational methods, Springer-Verlag, Berlin, 1990. Applications to nonlinear partial differential equations and Hamiltonian systems. MR 1078018, DOI 10.1007/978-3-662-02624-3
Additional Information
- James Coleman
- Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3
- Email: coleman@math.utoronto.ca
- Received by editor(s): April 15, 1999
- Published electronically: February 7, 2000
- Communicated by: David S. Tartakoff
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 1567-1569
- MSC (2000): Primary 35Q55
- DOI: https://doi.org/10.1090/S0002-9939-00-05695-1
- MathSciNet review: 1751311