Entire vortex solutions of negative degree for the anisotropic Ginzburg-Landau system

TL;DR

This paper establishes the existence of vortex solutions with negative degree in an anisotropic Ginzburg-Landau model for liquid crystals, using energy minimization in symmetric classes for small anisotropy.

Contribution

It proves the existence of entire vortex solutions with negative degree in an anisotropic Ginzburg-Landau system, extending beyond the isotropic case and employing symmetry-based energy minimization.

Findings

Existence of solutions with prescribed negative degree for small anisotropy.

Solutions cannot be reduced to radial equations due to anisotropy.

Method involves energy minimization within symmetric classes.

Abstract

The anisotropic Ginzburg-Landau system \[ \Delta u+\delta\, \nabla (\mathrm{div}\: u) +\delta\, \mathrm{curl}^*(\mathrm{curl}\: u)=(|u|^2-1) u, \] for $u\colon\mathbb R^2\to\mathbb R^2$ and $\delta\in (-1,1)$, models the formation of vortices in liquid crystals. We prove the existence of entire solutions such that $|u(x)|\to 1$ and $u$ has a prescribed topological degree $d\leq -1$ as $|x|\to\infty$, for small values of the anisotropy parameter $|\delta| < \delta_0(d)$. Unlike the isotropic case $\delta=0$, this cannot be reduced to a one-dimensional radial equation. We obtain these solutions by minimizing the anisotropic Ginzburg-Landau energy in an appropriate class of equivariant maps, with respect to a finite symmetry subgroup.