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

Abstract

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 \[ -Δu -δ(\partial_x+i\partial_y)^2\bar u =(1-|u|^2)u,\quad -1<δ<1, \] 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.