Uniqueness of topological solutions of self-dual Chern-Simons equation with collapsing vortices

Original · EN

We consider the following Chern-Simons equation, equation 0.1 Δu+ 1ε² eᵘ(1-eᵘ)=4π∑ᵢ₌₁ⁿ δₚᵢε, in Ω, equation 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 Choe2005 and Tarantello Tarantello2007. Meanwhile, for any topological solution ψ defined in R² whose linearized operator is non-degenerate, we construct a sequence 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.