On the bifurcation theory of the Ginzburg-Landau equations

TL;DR

This paper develops a bifurcation theory approach to construct new solutions to the Ginzburg-Landau equations on closed manifolds, revealing novel solutions on nontrivial line bundles.

Contribution

It introduces a bifurcation method to find nonminimal, irreducible solutions on manifolds with trivial first cohomology, a first in this context.

Findings

Constructed new solutions on closed manifolds of arbitrary dimension

Characterized bifurcation points via eigenvalues of a Laplace-type operator

First examples of such solutions on nontrivial line bundles

Abstract

We construct nonminimal and irreducible solutions to the Ginzburg-Landau equations on closed manifolds of arbitrary dimension with trivial first real cohomology. Our method uses bifurcation theory where the "bifurcation points" are characterized by the eigenvalues of a Laplace-type operator. To our knowledge these are the first such examples on nontrivial line bundles.