Instability of point defects in a two-dimensional nematic liquid crystal model

We study a class of symmetric critical points in a variational $2D$ Landau - de Gennes model where the state of nematic liquid crystals is described by symmetric traceless $3\times 3$ matrices. These critical points play the role of topological point defects carrying a degree $\frac k 2$ for a nonzero integer $k$. We prove existence and study the qualitative behavior of these symmetric solutions. Our main result is the instability of critical points when $k\neq \pm 1, 0$.