Stability of 𝑛-vortices in the Ginzburg-Landau equation

Coleman, James

doi:10.1090/S0002-9939-00-05695-1

Stability of $n$-vortices in the Ginzburg-Landau equation
HTML articles powered by AMS MathViewer

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