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

Abstract

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 $\Omega$ is a flat 2-dimensional torus ${\mathbb{T}}^2$ or ${\mathbb{R}}^2$, $\epsilon >0$ is a coupling parameter, and ${\delta }_p$ denotes the Dirac measure concentrated at $p$. In this paper, we prove that, when the coupling parameter $\epsilon$ 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.

19pages

References in corpus (3)

• Non-Abelian Chern-Simons Vortices
• Non-Abelian Chern-Simons vortices with generic gauge groups
• Fractional and semi-local non-Abelian Chern-Simons vortices