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 $w(x)=(w^+, w^-): { \mathbb R}^2 \rightarrow\mathbb{ C}^2$ of degree pair $(1, 1)$ 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.
Categories: math.AP