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

Abstract

The anisotropic Ginzburg-Landau system \[ Δu+δ\, \nabla (\mathrm{div}\: u) +δ\, \mathrm{curl}^*(\mathrm{curl}\: u)=(|u|^2-1) u, \] for $u:{\mathbb{R}}^2\to {\mathbb{R}}^2$ 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 $d\le -1$ 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.

References in corpus (1)

• Bifurcation analysis in a frustrated nematic cell