On the non-degeneracy of radial vortex solutions for a coupled Ginzburg-Landau system

For the coupled Ginzburg-Landau system in $ {\mathbb R}^2 $

$ \begin{align*} \begin{cases} -\Delta w^+ +\Big[A_+\big(|w^+|^2-{t^+}^2\big)+B\big(|w^-|^2-{t^-}^2\big)\Big]w^+ = 0, \\ -\Delta w^- +\Big[A_-\big(|w^-|^2-{t^-}^2\big)+B\big(|w^+|^2-{t^+}^2\big)\Big]w^- = 0, \end{cases} \end{align*} $

with following constraints for the constant coefficients

$ A_+, A_->0,\ B^2<A_+A_-,\ t^+, t^->0, $

the radially symmetric solution $ w(x) = (w^+, w^-): {\mathbb R}^2 \rightarrow\mathbb{C}^2 $ of degree pair $ (1, 1) $ was given by A. Alama and Q. Gao in J. Differential Equations 255 (2013), 3564-3591. We will concern its linearized operator $ {\mathcal L} $ around $ w $ and prove the non-degeneracy result under one more assumption $ B<0 $: the kernel of $ {\mathcal L} $ is spanned by the functions $ \frac{\partial w}{\partial{x_1}} $ and $ \frac{\partial w}{\partial{x_2}} $ in a natural Hilbert space. As an application of the non-degeneracy result, a solvability theory for the linearized operator $ {\mathcal L} $ will be given.