Uniqueness of topological multi-vortex solutions for a skew-symmetric Chern-Simons system

Consider the following skew-symmetric Chern-Simons system \begin{equation*}\left \{ \begin{split} &Δu_{1}+\frac{1}{\varepsilon^2} e^{u_{2}}(1-e^{u_{1}})=4π\sum^{N_1}_{j=1}δ_{p_{j,1}}\\ &Δu_{2}+\frac{1}{\varepsilon^2} e^{u_{1}}(1-e^{u_{2}})=4π\sum^{N_2}_{j=1}δ_{p_{j,2}} \end{split}\right.\quad\text{ in }\quadΩ, \end{equation*} where $Ω$ is a flat 2-dimensional torus $\mathbb{T}^2$ or $\mathbb{R}^2$, $\varepsilon> 0$ is a coupling parameter, and $δ_p$ denotes the Dirac measure concentrated at $p$. In this paper, we prove that, when the coupling parameter $\varepsilon$ is small, the topological type solutions to the above system are uniquely determined by the location of their vortex points. This result follows by the bubbling analysis and the non-degency of linearized equations.

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