Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Stability of $n$-vortices in the Ginzburg-Landau equation

Author: James Coleman
Journal: Proc. Amer. Math. Soc. 128 (2000), 1567-1569
MSC (2000): Primary 35Q55
Published electronically: February 7, 2000
MathSciNet review: 1751311
Full-text PDF

Abstract | References | Similar Articles | Additional Information


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 [Enhancements On Off] (What's this?)

  • 1. F. Bethuel, H. Brézis and F. Hélein, Ginzburg-Landau Vortices, Birkhauser, 1994. MR 95c:58044
  • 2. X. Chen, C.M. Elliott and T. 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 95j:35208
  • 3. P. Hagan, Spiral waves in reaction-diffusion equations, SIAM J. of Appl. Math. 42 (1982), no. 4, 762-786. MR 84c:92069
  • 4. R.M. Hervé and M. Hervé, Etude qualitative des solutions réelles d'une équation différentielle liée à l'équation de Ginzburg-Landau, Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994), no. 4, 427-440. MR 95g:35194
  • 5. Y.M. Ovchinnikov and I.M. Sigal, Ginzburg-Landau equation I. Static vortices, Partial Differential Equations and their Applications, 199-220, CRM Publications, 1997. MR 98k:35179
  • 6. M.H. Protter and H.F. Weinberger, Maximum Principles in Differential Equations, Springer, 1984. MR 86f:35034
  • 7. I. Shafrir, Remarks on solutions of $-\Delta u = (1-\vert u \vert^2)u$ in $\mathbf{R}^2$, C.R. Acad. Sci. Paris Sér I Math. 318 (1994), no. 4, 327-331. MR 95c:35091
  • 8. M. Struwe, Variational Methods, Springer, 1990. MR 92b:49002

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35Q55

Retrieve articles in all journals with MSC (2000): 35Q55

Additional Information

James Coleman
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3

Received by editor(s): April 15, 1999
Published electronically: February 7, 2000
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2000 American Mathematical Society