On the global bifurcation diagram for the one-dimensional Ginzburg Landau model of superconductivity

TL;DR

This paper presents new global bifurcation results for the one-dimensional Ginzburg-Landau model, establishing the existence of asymmetric solutions connecting to the normal state and providing simplified proofs for local bifurcations.

Contribution

It introduces a proof of a branch of asymmetric solutions in a parameter range, connecting symmetric solutions to the normal state, and simplifies existing bifurcation proofs without detailed asymptotics.

Findings

Existence of asymmetric solution branches connecting to the normal state.

Simplified proofs for local bifurcation results.

Discussion of an error in Odeh's previous work.

Abstract

Some new global results are given about solutions to the boundary value problem for the Euler-Lagrange equations for the Ginzburg-Landau model of a one-dimensional superconductor. The main advance is a proof that in some parameter range there is a branch of asymmetric solutions connecting the branch of symmetric solutions to the normal state. Also, simplified proofs are presented for some local bifurcation results of Bolley and Helffer. These proofs require no detailed asymptotics for the solutions of the linear equations. Finally, an error in Odeh's work on this problem is discussed.