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

TL;DR

This paper proves that for a skew-symmetric Chern-Simons system on a 2D domain, the topological vortex solutions are uniquely determined by vortex locations when the coupling parameter is small, using bubbling analysis and linearized equations.

Contribution

It establishes the uniqueness of topological vortex solutions based solely on vortex point locations for small coupling parameters in a skew-symmetric Chern-Simons system.

Findings

Topological solutions are uniquely determined by vortex points for small b5.

Bubbling analysis confirms solution uniqueness.

Non-degeneracy of linearized equations is key to the proof.

Abstract

Consider the following skew-symmetric Chern-Simons system \begin{equation*}\left \{ \begin{split} &\Delta u_{1}+\frac{1}{\varepsilon^2} e^{u_{2}}(1-e^{u_{1}})=4\pi \sum^{N_1}_{j=1}\delta_{p_{j,1}}\\ &\Delta u_{2}+\frac{1}{\varepsilon^2} e^{u_{1}}(1-e^{u_{2}})=4\pi \sum^{N_2}_{j=1}\delta_{p_{j,2}} \end{split}\right.\quad\text{ in }\quad\Omega, \end{equation*} where $\Omega$ is a flat 2-dimensional torus $\mathbb{T}^2$ or $\mathbb{R}^2$, $\varepsilon> 0$ is a coupling parameter, and $\delta_p$ denotes the Dirac measure concentrated at $p$. In this paper, we prove that, when the coupling parameter $\varepsilon$ 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.