On the stability of radial solutions to an anisotropic Ginzburg-Landau equation

We study the linear stability of entire radial solutions $u(r{e}^{i\theta })=f(r)e^{i\theta }$, with positive increasing profile $f(r)$, to the anisotropic Ginzburg-Landau equation $$-\mathrm{\Delta }u-\delta ({\mathrm{\partial }}_x+i{\mathrm{\partial }}_y{)}^2\bar{u}=(1-\mathrm{\mid }u{\mathrm{\mid }}^2)u,-1<\delta <1,$$ which arises in various liquid crystal models. In the isotropic case $\delta =0$, Mironescu showed that such solution is nondegenerately stable. We prove stability of this radial solution in the range $\delta \in ({\delta }_1,0]$ for some $-1<{\delta }_1<0$, and instability outside this range. In strong contrast with the isotropic case, stability with respect to higher Fourier modes is \emph{not} a direct consequence of stability with respect to lower Fourier modes. In particular, in the case where $\delta \approx -1$, lower modes are stable and yet higher modes are unstable.