There are Asymmetric Minimizers for the One-Dimensional Ginzburg-Landau Model of Superconductivity

TL;DR

This paper demonstrates the existence of asymmetric minimizers in a boundary value problem modeling superconducting films under magnetic fields, revealing that asymmetric solutions can have lower energy than symmetric ones.

Contribution

It introduces the existence of asymmetric solutions in the Ginzburg-Landau model, showing they are global energy minimizers, which was not previously established.

Findings

Asymmetric solutions exist for certain parameters.

Asymmetric solutions have negative energy.

Symmetric solutions have zero energy.

Abstract

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 {\it asymmetric} solutions, and only trivial {\it symmetric} solutions. We then show that the associated energy function is negative for the asymmetric solutions, and zero for the trivial symmetric solution. It follows that a global minimizer of the energy is asymmetric.