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

For the following 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, \[3mm] -\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 constraints $A_{+},A_{-}>0$, $B<0$, $B^2<A_{+}{A}_{-}$ and $t^{+},t^{-}>0$, we will concern its linearized operator $\mathcal{L}$ around the radially symmetric solution $w(x)=(w^+, w^-): {\mathbb R}^2 \rightarrow$\mathbb{C}$^2$ofdegreepair$(1, 1)$ and prove the non-degeneracy result: the kernel of $\mathcal{L}$ is spanned by $\big{\frac{\partial w}{\partial{x_1}}, \frac{\partial w}{\partial{x_2}}\big}$ 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.