Uniqueness of symmetric vortex solutions in the Ginzburg-Landau model superconductivity
The authors consider the Ginzburg-Landau system of the form \[ \begin{cases} -f''-{1\over r} f'+{(d- S)^2\over r^2} f= x^2(1- f^2)\cdot f,\\ - S''+{1\over r} S'= (d- S) f^2,\end{cases}\tag{1} \] where \(f\geq 0\), \(f(r)\), \(S(r)\to 0\), as \(r\to 0\), and \(f(r)\to 1\), \(S(r)\to d\) as \(r\to\infty\). A solution \((f,S)\) of (1) is said to be ``admissible'' if \(f\geq 0\) for all \(r\geq 0\), \(S(0)= 0\), and if the Ginzbug-Landau free energy is finite. The main result is the following theorem: Let \(d\in\mathbb{Z}\setminus \{0\}\) and \(x\in\mathbb{R}\) with \(x^2\geq 2d^2\). Then there exists a unique admissible solution to the Ginzburg-Landau equations (1). The authors show that any such solution is a nondegenerate relative minimizer of the free energy functional constrained to a convex set, then uses a version of the Mountain Pass Theorem to derive a contraction, should there be more than one solution.
scientific article; zbMATH DE number 1367852