Some remarks concerning symmetry-breaking for the Ginzburg-Landau equation

TL;DR

This paper investigates the correlation term's role in symmetry-breaking for the Ginzburg-Landau equation, disproving a conjecture that it is a multiple of π/4 by showing it always vanishes, impacting vortex configuration analysis.

Contribution

It proves that the correlation coefficient always vanishes, challenging previous conjectures and providing new insights into vortex symmetry-breaking in the Ginzburg-Landau equation.

Findings

Correlation coefficient always vanishes.

Disproves the conjecture that it is a multiple of π/4.

Implications for vortex symmetry analysis.

Abstract

The correlation term, introduced in [13] to describe the interaction between very far apart vortices, governs symmetry-breaking for the Ginzburg-Landau equation in R^2 or bounded domains. It is a homogeneous function of degree (-2), and then for 2\pi/N-symmetric vortex configurations can be expressed in terms of the so-called correlation coefficient. Ovchinnikov and Sigal [13] have computed it in few cases and conjectured its value to be an integer multiple of \pi/4. We will disprove this conjecture by showing that the correlation coefficient always vanishes, and will discuss some of its consequences.