Differential and Integral Equations

Classification of symmetric vortices for the Ginzburg-Landau equation

Myrto Sauvageot

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This work proposes a description of the set of symmetric vortices, defined as specific solutions of the Ginzburg-Landau equations for a superconducting cylinder with applied magnetic field. It is conducted through a two parameters shooting procedure which relates the behaviour of a symmetric vortex at the center to its behaviour at the boundary. The main result is that, for a given degree $d$, the set of parameters for which such a "shooting" leads to a "response" -- i.e., admissible values for the radius $\bar r$ of the cylinder and the intensity $h$ of the magnetic field -- is a bounded subset in $\mathbb R^2$. This shows in particular that, for large intensities of the applied magnetic field, normal states do not appear as a limit of superconducting vortices of given degree, and that symmetric vortices are not equilibrium states of the system for too large or too low intensities of the applied magnetic field. Moreover, a simpler proof for the existence of bifurcations (a model for phase transitions) from the normal state to superconducting states, as studied in [11], is provided.

Article information

Differential Integral Equations, Volume 19, Number 7 (2006), 721-760.

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]
Secondary: 34B15: Nonlinear boundary value problems 35A15: Variational methods 35J55 47J15: Abstract bifurcation theory [See also 34C23, 37Gxx, 58E07, 58E09] 47J30: Variational methods [See also 58Exx] 58E50: Applications 82D55: Superconductors


Sauvageot, Myrto. Classification of symmetric vortices for the Ginzburg-Landau equation. Differential Integral Equations 19 (2006), no. 7, 721--760. https://projecteuclid.org/euclid.die/1356050347

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