Uniqueness of topological solutions of self-dual Chern-Simons equation with collapsing vortices
Journal
Journal of Differential Equations
Journal Volume
259
Journal Issue
5
Pages
1819-1840
Date Issued
2015
Author(s)
Huang, G.
Abstract
We consider the following Chern-Simons equation,(0.1)δu+1ε2eu(1-eu)=4π∑i=1Nδpiε,inΩ, where Ω is a 2-dimensional flat torus, ε>0 is a coupling parameter and δp stands for the Dirac measure concentrated at p. In this paper, we proved that the topological solutions of (0.1) are uniquely determined by the location of their vortices provided the coupling parameter ε is small and the collapsing velocity of vortices piε is slow enough or fast enough comparing with ε. This extends the uniqueness results of Choe [5] and Tarantello [22]. Meanwhile, for any topological solution ψ defined in R2 whose linearized operator is non-degenerate, we construct a sequence of topological solutions uε of (0.1) whose asymptotic limit is exactly ψ after rescaling around 0. A consequence is that non-uniqueness of topological solutions in R2 implies non-uniqueness of topological solutions on torus with collapsing vortices.
Type
journal article