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Dynamical instability of symmetric vortices. (English) Zbl 0994.35029

From the introduction: We study the dynamic instability of the radial solutions of the Ginzburg-Landau equations in \(\mathbb R^2\). \[ \begin{cases} \text{curl}^2A+ {i\over 2}(\overline\varphi D\varphi-\varphi\overline {D\varphi})=0,\\ -D^2\varphi +{\lambda\over 2}\bigl(|\varphi|^2-1\bigr) \varphi= 0,\end{cases} \] where \(\varphi:\mathbb R^2\to \mathbb C\) is the Higgs field, or condensed wave function \((|\varphi |^2\) is proportional to the local density of Cooper pairs), and \(A\) is the gauge potential 1-form (it can also be seen as the vector potential of the magnetic field). The covariant derivative is \(D\varphi= \nabla\varphi- iA\varphi\), with \(i=\sqrt{-1}\). The electric field is absent in the stationary model, and \(H=\text{curl} A\) is the magnetic field. The dimensionless coupling constant \(\lambda\) is positive, \(\lambda<1\) corresponding to superconductors of type I and \(\lambda>1\) to those of type II.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
82D55 Statistical mechanics of superconductors
35A15 Variational methods applied to PDEs
35B35 Stability in context of PDEs
35R25 Ill-posed problems for PDEs
58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
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