There are asymmetric minimizers for the one-dimensional Ginzburg-Landau model of superconductivity
We study a boundary value problem associated with a system of two second-order differential equations with cubic nonlinearity which model a film of superconductor material subjected to a tangential magnetic field. We show that for an appropriate range of parameters there are asymmetric solutions and only trivial symmetric solutions. We then correct an error of the authors in [Nonlinear Problems in Applied Mathematics, SIAM, Philadelphia, PA, 1996, pp. 150--158] and show that the associated energy function is negative for the asymmetric solutions. Since the energy is zero for the trivial symmetric solution, it follows that a global minimizer of the energy is asymmetric. This property resolves a conjecture of Marcus [Rev. Mod. Phys., 36 (1964), pp. 294--299].