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# memo_has_moved_text();On Non-topological Solutions of the ${\bf A}_2$ and ${\bf B}_2$ Chern-Simons System

Weiwei Ao, Chang-Shou Lin and Juncheng Wei

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: http://dx.doi.org/10.1090/memo/1132
Published electronically: June 30, 2015

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Chapters

• Chapter 1. Introduction
• Chapter 2. Proof of Theorem 1.1 in the ${\bf A}_2$ case
• Chapter 3. Proof of Theorem 1.1 in the ${\bf B}_2$ case
• Chapter 4. Appendix

### Abstract

For any rank 2 of simple Lie algebra, the relativistic Chern-Simons system has the following form:

* where $K$ is the Cartan matrix of rank $2$. There are three Cartan matrix of rank 2: ${\bf A}_2$, ${\bf B}_2$ and ${\bf G}_2$. A long-standing open problem for this equation is the question of the existence of non-topological solutions. In this paper, we consider the ${\bf A}_2$ and ${\bf B}_2$ 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 ${\bf A}_2$ and ${\bf B}_2$ Toda system with one singular source.