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

Abstract

We study the linear stability of entire radial solutions u(re(i theta)) = f(r)e(i theta), with positive increasing profile f(r), to the anisotropic Ginzburg-Landau equation -Delta u - delta(partial derivative(x)+i partial derivative(y))(2)(u) over bar = (1 -vertical bar u vertical bar(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 is an element of (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 not a direct consequence of stability with respect to lower Fourier modes. In particular, in the case where delta approximate to -1, lower modes are stable and yet higher modes are unstable.