×

zbMATH — the first resource for mathematics

Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity. (English) Zbl 0814.34039
Summary: We study the dynamics of vortices in time-dependent Ginzburg-Landau theories in the asymptotic limit when the vortex core size is much smaller than the inter-vortex distance. We derive reduced systems of ODEs governing the evolution of these vortices. We then extend these to study the dynamics of vortices in extremely type-II superconductors. Dynamics of vortex lines is also considered. For the simple Ginzburg-Landau equation without the magnetic field, we find that the vortices are stationary in the usual diffusive scaling, and obey remarkably simple dynamic laws when time is speeded up by a logarithmic factor. For columnar vortices in superconductors, we find a similar dynamic law with a potential which is screened by the current. For curved vortex lines in superconductors, we find that the vortex lines move in the direction of the normal with a speed proportional to the curvature. Comparisons are made with the previous results of John Neu.

MSC:
34E05 Asymptotic expansions of solutions to ordinary differential equations
82D55 Statistical mechanical studies of superconductors
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Berger, M. S.; Chen, Y. Y., Symmetric vortices for the Ginzburg-Landau equations of superconductivity and the nonlinear desingularization phenomenon, J. Funct. Anal., 82, 259-295, (1989) · Zbl 0685.46051
[2] Bulaevskii, L. N.; Ledvij, M.; Kogan, V. G., Vortices in layered superconductors with Josephson coupling, Phys. Rev. B, 46, 366, (1992)
[3] F. Bethuel, H. Brezis and F. Helein, Ginzburg-Landau Vortices (Birkhauser), to appear.
[4] Bethuel, F.; Riviere, T., Vortices for a variational problem related to super-conductivity, (1993), preprint
[5] Callegari, A. J.; Ting, L., Motion of a curved vortex filament with decaying vortical cone and axial velocity, SIAM J. Appl. Mathl., 35, 148-175, (1978) · Zbl 0395.76024
[6] Creswick, J.; Morrison, N., On the dynamics of quantum vortices, Phys. Lett. A, 76, 267, (1980)
[7] De Gennes, P. G., Superconductivity of metals and alloys, (1966), Benjamin New York · Zbl 0138.22801
[8] Dorsey, A., Vortex motion and the Hall effect in type-II superconductors: a time-dependent Ginzburg-Landau theory approach, Phys. Rev. B, 46, 8376-8392, (1992)
[9] W. E, Dynamics of vortices in superconductors, (Proc. First World Congress of Nonlinear Analysts, Tampa, (1992))
[10] Enomoto, Y.; Katsumi, K., Numerical study of magnetic flux lines in random media, (1992), Nagoya University, Preprint
[11] P. Fife and L.A. Peletier, On the location of defects in stationary solutions of the Ginzburg-Landau equation in \(R\)^2, Preprint, Quart. J. Appl. Math., to appear. · Zbl 0848.35042
[12] Fukuyama, H.; Ebisawa, H.; Tsuzuki, T., Prog. Theor. Phys., 46, 1028, (1971)
[13] Klein, R.; Majda, A., Self-stretching of a perturbed vortex filament, I, Physica D, 49, 323-352, (1991) · Zbl 0738.35063
[14] Nelson, D., Vortex line fluctuation in superconductors from elementary quantum mechanics, (1993), Harvard University, Preprint
[15] Neu, J., Vortex dynamics of complex scalar fields, Physica D, 43, 385-406, (1990) · Zbl 0711.35024
[16] J. Neu, Vortex Dynamics in Superconductors, unpublished.
[17] Peres, L.; Rubenstein, J., Vortex dynamics in U(1) Ginzburg-Landau models, Physica D, 64, 299-309, (1993) · Zbl 0772.35069
[18] Pismen, L. M.; Rubenstein, J., Motion of vortex lines in the Ginzburg-Landau model, Physica D, 47, 353-360, (1991) · Zbl 0728.35090
[19] Rubenstein, J., Self-induced motion of line defects, Quart. J. Appl. Math., Vol. XLIX, 1-9, (1991)
[20] Tinkham, M., Introduction to superconductivity, (1975), McGraw-Hill
[21] Wang, R., Existence, uniqueness and stability of symmetric vortices for Ginzburg-Landau equations, (1993), Harvard University, Preprint
[22] Weinstein, M., On the vortex solutions of some nonlinear scalar field equations, Rocky Mountain J. Math., 21, 821-827, (1991) · Zbl 0732.35093
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.