Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Uniqueness of topological solutions of self-dual Chern-Simons equation
  with collapsing vortices   We consider the following Chern-Simons equation, \begin{equation} \label{0.1}
\Delta u+\frac 1{\varepsilon^2} e^u(1-e^u)=4\pi\sum_{i=1}^N
\delta_{p_i^\varepsilon},\quad \text{in}\quad \Omega, \end{equation} where
$\Omega$ is a 2-dimensional flat torus, $\varepsilon>0$ is a coupling parameter
and $\delta_p$ stands for the Dirac measure concentrated at $p$. In this paper,
we proved that the topological solutions of \eqref{0.1} are uniquely determined
by the location of their vortices provided the coupling parameter $\varepsilon$
is small and the collapsing velocity of vortices $p_i^\varepsilon$ is slow
enough or fast enough comparing with $\varepsilon$. This extends the uniqueness
results of Choe \cite{Choe2005} and Tarantello \cite{Tarantello2007}.
Meanwhile, for any topological solution $\psi$ defined in $\mathbb R^2$ whose
linearized operator is non-degenerate, we construct a sequence topological
solutions $u_\varepsilon$ of \eqref{0.1} whose asymptotic limit is exactly
$\psi$ after rescaling around $0$. A consequence is that non-uniqueness of
topological solutions in $\mathbb R^2$ implies non-uniqueness of topological
solutions on torus with collapsing vortices.

26
The categories of this paper are: bayes-an
12
The categories of this paper are: nucl-th
5
The categories of this paper are: gr-qc
34
The categories of this paper are: chem-ph
13
The categories of this paper are: quant-ph
7
The categories of this paper are: hep-th
1
The categories of this paper are: cs
14
The categories of this paper are: math-ph
19
The categories of this paper are: eess
16
The categories of this paper are: stat
17
The categories of this paper are: q-fin
9
The categories of this paper are: nlin
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The categories of this paper are: atom-ph
15
The categories of this paper are: nucl-ex
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The categories of this paper are: comp-gas
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The categories of this paper are: q-alg