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On Non-topological Solutions of the $\textbf {A}_2$ and $\textbf {B}_2$ Chern-Simons System

About this Title

Weiwei Ao, Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong, Chang-Shou Lin, Taida Institute of Mathematics, Center for Advanced study in Theoretical Science, National Taiwan University, Taipei, Taiwan and Juncheng Wei, Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2 and Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong

Publication: Memoirs of the American Mathematical Society
Publication Year: 2016; Volume 239, Number 1132
ISBNs: 978-1-4704-1543-3 (print); 978-1-4704-2747-4 (online)
DOI: https://doi.org/10.1090/memo/1132
Published electronically: June 30, 2015
MSC: Primary 35J60; Secondary 35B10, 58J37

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Table of Contents

Chapters

  • 1. Introduction
  • 2. Proof of Theorem in the $\textbf {A}_2$ case
  • 3. Proof of Theorem in the $\textbf {B}_2$ case
  • 4. Appendix

Abstract

For any rank 2 of simple Lie algebra, the relativistic Chern-Simons system has the following form: \begin{equation*}\left \lbrace \begin {array}{@{}l@{\quad }l@{}}\Delta u_1+(\sum _{i=1}^2K_{1xi}e^{u_i} -\sum _{i=1}^2\sum _{j=1}^2e^{u_i}K_{1i}e^{u_j}K_{ij})=4\pi \displaystyle \sum _{j=1}^{N_1}\delta _{p_j}\\ \Delta u_2+(\sum _{i=1}^2K_{2i}e^{u_i}-\sum _{i=1}^2\sum _{j=1}^2e^{u_i}K_{2i}e^{u_j}K_{ij})=4\pi \displaystyle \sum _{j=1}^{N_2}\delta _{q_j} \end{array} \right . \quad \mathrm {in}\; \mathbb {R}^2, \end{equation*} where $K$ is the Cartan matrix of rank $2$. There are three Cartan matrix of rank 2: $\textbf {A}_2$, $\textbf {B}_2$ and $\textbf {G}_2$. A long-standing open problem for this equation is the question of the existence of non-topological solutions. In this paper, we consider the $\textbf {A}_2$ and $\textbf {B}_2$ case. We prove the existence of non-topological solutions under the condition that either $N_2\sum _{j=1}^{N_1} p_j=N_1\sum _{j=1}^{N_2} q_j$ or $N_2\sum _{j=1}^{N_1} p_j \not = N_1\sum _{j=1}^{N_2} q_j$ and $N_1, N_2 >1, |N_1-N_2| \not = 1$. We solve this problem by a perturbation from the corresponding $\textbf {A}_2$ and $\textbf {B}_2$ Toda system with one singular source.

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