Stability of point defects of degree $\pm \frac 1 2$ in a two-dimensional nematic liquid crystal model

TL;DR

This paper analyzes the stability of degree ±1/2 topological defects in 2D nematic liquid crystals, showing stability for charge ±1 and establishing uniqueness of radial profiles using monotonicity and cooperative system properties.

Contribution

It demonstrates the stability of degree ±1/2 defects in a 2D nematic liquid crystal model and proves uniqueness of their radial profiles, extending previous results to these specific charges.

Findings

Degree ±1/2 defects are stable in the model.

Radial profiles satisfying sign invariance are unique.

Monotonicity and cooperative system properties are key to the proofs.

Abstract

We study $k$-radially symmetric solutions corresponding to topological defects of charge $\frac{k}{2}$ for integer $k \neq 0$ in the Landau-de Gennes model describing liquid crystals in two-dimensional domains. We show that the solutions whose radial profiles satisfy a natural sign invariance are stable when $|k| = 1$ (unlike the case $|k|>1$ which we treated before). The proof crucially uses the monotonicity of the suitable components, obtained by making use of the cooperative character of the system. A uniqueness result for the radial profiles is also established.