On the non-degeneracy of radial vortex solutions for a coupled Ginzburg-Landau system
| Autor: | Jun Yang, Lipeng Duan |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
Physics
Degree (graph theory) Computer Science::Information Retrieval Applied Mathematics Operator (physics) Hilbert space 01 natural sciences Degeneracy (graph theory) 010101 applied mathematics Combinatorics Kernel (algebra) symbols.namesake Mathematics - Analysis of PDEs symbols Symmetric solution FOS: Mathematics Discrete Mathematics and Combinatorics 0101 mathematics Fredholm alternative Ginzburg landau Analysis Analysis of PDEs (math.AP) |
| DOI: | 10.48550/arxiv.1905.00347 |
| Popis: | For the coupled Ginzburg-Landau system in \begin{document}$ {\mathbb R}^2 $\end{document} \begin{document}$ \begin{align*} \begin{cases} -\Delta w^+ +\Big[A_+\big(|w^+|^2-{t^+}^2\big)+B\big(|w^-|^2-{t^-}^2\big)\Big]w^+ = 0, \\ -\Delta w^- +\Big[A_-\big(|w^-|^2-{t^-}^2\big)+B\big(|w^+|^2-{t^+}^2\big)\Big]w^- = 0, \end{cases} \end{align*} $\end{document} with following constraints for the constant coefficients \begin{document}$ A_+, A_->0,\ B^2 0, $\end{document} the radially symmetric solution \begin{document}$ w(x) = (w^+, w^-): {\mathbb R}^2 \rightarrow\mathbb{C}^2 $\end{document} of degree pair \begin{document}$ (1, 1) $\end{document} was given by A. Alama and Q. Gao in J. Differential Equations 255 (2013), 3564-3591. We will concern its linearized operator \begin{document}$ {\mathcal L} $\end{document} around \begin{document}$ w $\end{document} and prove the non-degeneracy result under one more assumption \begin{document}$ B : the kernel of \begin{document}$ {\mathcal L} $\end{document} is spanned by the functions \begin{document}$ \frac{\partial w}{\partial{x_1}} $\end{document} and \begin{document}$ \frac{\partial w}{\partial{x_2}} $\end{document} in a natural Hilbert space. As an application of the non-degeneracy result, a solvability theory for the linearized operator \begin{document}$ {\mathcal L} $\end{document} will be given. |
| Databáze: | OpenAIRE |
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