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

For the coupled Ginzburg-Landau system in \begin{document}$ {\mathbb R}^2 $\end{document} style='text-indent:20px;'> \begin{document}$ \begin{align*} \begin{cases} -\Delta w^+ +\Big[A_+\big(|w^+|^2-{t^+}^2\big)+B\big(|w^-|^2-{t^-}^2\big)\Big]w^+ = 0, \\ w^- +\Big[A_-\big(|w^-|^2-{t^-}^2\big)+B\big(|w^+|^2-{t^+}^2\big)\Big]w^- \end{cases} \end{align*} $\end{document} style='text-indent:20px;'>with following constraints for constant coefficients id="FE2"> A_+, A_->0,\ B^2<A_+A_-,\ t^+, t^->0, style='text-indent:20px;'>the radially symmetric solution id="M2">\begin{document}$ w(x) (w^+, w^-): \rightarrow\mathbb{C}^2 of degree pair id="M3">\begin{document}$ (1, 1) was given by A. Alama and Q. Gao J. Differential Equations 255 (2013), 3564-3591. We will concern its linearized operator id="M4">\begin{document}$ {\mathcal L} around id="M5">\begin{document}$ w prove non-degeneracy result under one more assumption id="M6">\begin{document}$ B<0 $\end{document}: kernel id="M7">\begin{document}$ is spanned functions id="M8">\begin{document}$ \frac{\partial w}{\partial{x_1}} id="M9">\begin{document}$ w}{\partial{x_2}} a natural Hilbert space. As an application result, solvability theory id="M10">\begin{document}$ be given.