A Model for Vortex Nucleation in the Ginzburg–Landau Equations

Abstract

This paper studies questions related to the dynamic transition between local and global minimizers in the Ginzburg–Landau theory of superconductivity. We derive a heuristic equation governing the dynamics of vortices that are close to the boundary, and of dipoles with small inter-vortex separation. We consider a small random perturbation of this equation and study the asymptotic regime under which vortices nucleate.

Appendix: Annihilation Times of the Heuristic ODE

We devote this appendix to proving Proposition 2.2, estimating the annihilation times of the heuristic equation (2.1). A direct calculation shows that when $\lambda > 0$ the solution to (2.1) is given by

for some constant C. Here $W_0$ is the principal branch of the Lambert W function. We recall (see Corless et al. (1996), or Section 4.13 in Olver et al. (2010)) that $W_0$ is defined by the functional relation

$$\begin{aligned} W_0(z e^{z}) = z \end{aligned}$$

when $z \geqslant -1$.

Using the initial data $a_\varepsilon (0) = \varepsilon ^\alpha$, we find

Annihilation occurs when $W_0 = -1$ which is precisely when

Substituting C above gives

and hence

Since by assumption, dividing both sides by , Eq. (2.5) follows.

It remains to prove (2.5) when $\lambda = 0$. In this case, the exact solution to (2.1) is given by

and hence

(5.8)

for any $\varepsilon > 0$. This concludes the proof of Proposition 2.2.