Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-18T07:07:25.557Z Has data issue: false hasContentIssue false
  • English
  • Français

Shooting method for vortex solutions of a complex-valued Ginzburg–Landau equation

Published online by Cambridge University Press:  14 November 2011

Xinfu Chen ,
Charles M. Elliott  and
Tang Qi
Xinfu Chen
Affiliation:
Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, U.S.A.
Charles M. Elliott
Affiliation:
School of Mathematical and Physical Sciences, University of Sussex, Brighton, BN1 9QH, U.K.
Tang Qi
Affiliation:
School of Mathematical and Physical Sciences, University of Sussex, Brighton, BN1 9QH, U.K.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we study all the stationary solutions of the form u(r)einθ to the complex-valued Ginzburg–Landau equation on the complex plane: here (r, θ) are the polar coordinates, and n is any real number. In particular, we show that there exists a unique solution which approaches to a nonzero constant as r → ∞.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 6 , 1994 , pp. 1075 - 1088
Copyright
Copyright © Royal Society of Edinburgh 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Berger, M. S. and Chen, Y. Y.. Symmetric vortices for the Ginzburg-Landau equations of superconductivity and the nonlinear desingularization phenomenon. J. Fund. Anal. 82 (1989), 259–295.CrossRefGoogle Scholar
2Bethuel, F., Brezis, H. and Helein, F.. Limite singuliere pour la minimisation de fonctionnelles du type Ginzburg-Landau. C. R. Acad. Sci. Paris Ser. 1 Math. 314 (1992), 891–895.Google Scholar
3Elliott, C. M., Matano, H. and Tang, Qi. Zeros of a complex Ginzburg-Landau order parameter with applications to superconductivity. European J. Appl. Math, (to appear).Google Scholar
4Greenberg, J. M.. Spiral waves for A-co systems. SIAM J. Appl. Math. 39 (1980), 301–309.CrossRefGoogle Scholar
5McLeod, J. B., Stuart, C. A. and Troy, W. C.. An exact reduction of Maxwell's equation. In Progress in Nonlinear Differential Equations: Nonlinear Diffusion Equations and Their Equilibrium States, 3, eds Lloyd, N. G., Ni, W. M., Peletier, L. A. and Serrin, J., 391–405 (Boston: Birkhauser, 1992).CrossRefGoogle Scholar
6McLeod, K. and Serrin, J.. Uniqueness of positive radial solutions of Δu + f(U) = 0 in Rn. Arch. Rational Mech. Anal. 99 (1987), 115–145.CrossRefGoogle Scholar
7McLeod, K., Troy, W. C. and Weissler, F. B.. Radial solutions of Δu + f(u) = 0 with prescribed number of zeros. J. Differential Equations (to appear).Google Scholar
8Neu, J. C.. Vortices in complex scalar fields. Phys. D 43 (1990), 385–406.Google Scholar
9Peres, L. and Rubinstein, J.. Vortex dynamics in U(l) Ginzburg-Landau models, Phys. D 64 (1993), 299–309.CrossRefGoogle Scholar
10Pisman, L. M. and Rubinstein, J.. Motion of vortex lines in the Ginzburg-Landau model (preprint).Google Scholar