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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2211.04237 |
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| _version_ | 1866910312951709696 |
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| author | Hou, Songbo Kong, Xiaoqing |
| author_facet | Hou, Songbo Kong, Xiaoqing |
| contents | In this paper, we consider a system of equations arising from the $\text{U}(1)\times \text{U}(1)$ Abelian Chern-Simons model \begin{eqnarray*}\left\{\begin{aligned} Δu &=λ\left(a(b-a)\mathrm{e}^u-b(b-a)\mathrm{e}^{\upsilon}+a^2\mathrm{e}^{2u}-ab\mathrm{e}^{2\upsilon}+b(b-a)\mathrm{e}^{u+\upsilon} \right)+4π\sum\limits_{j=1}^{k_1}m_jδ_{p_j},\\ Δ\upsilon&=λ\left(-b(b-a)\mathrm{e}^u+a(b-a)\mathrm{e}^{\upsilon}-ab\mathrm{e}^{2u}+a^2\mathrm{e}^{2\upsilon}+b(b-a)\mathrm{e}^{u+\upsilon} \right)+4π\sum\limits_{j=1}^{k_2}n_jδ_{q_j}, \end{aligned} \right. \end{eqnarray*} on finite graphs. Here $λ>0$, $b>a>0$, $m_j>0\, (j=1,2,\cdot\cdot\cdot,k_1)$, $n_j>0\,(j=1,2,\cdot\cdot\cdot,k_2)$, $δ_{p}$ is the Dirac delta mass at vertex $p$.
We establish the iteration scheme and prove existence of solutions. We also develop a new method to get the asymptotic behaviors of solutions as $λ$ goes to infinity. This method is also applicable to the Chern-Simons system $$\left\{\begin{aligned} Δu &=λ\mathrm{e}^{\upsilon}(\mathrm{e}^{u}-1) +4π\sum\limits_{j=1}^{k_1}m_jδ_{p_j},\\ Δ\upsilon&=λ\mathrm{e}^{u}(\mathrm{e}^{\upsilon}-1)+4π\sum\limits_{j=1}^{k_2}n_jδ_{q_j}, \end{aligned} \right. $$ and the classical Chern-Simons equation $$ Δu=λ\mathrm{e}^u(\mathrm{e}^u-1)+4π\sum\limits_{j=1}^{N}δ_{p_j}.$$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_04237 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Existence and asymptotic behaviors of solutions to Chern-Simons systems and equations on finite graphs Hou, Songbo Kong, Xiaoqing Analysis of PDEs 35J47, 05C22 In this paper, we consider a system of equations arising from the $\text{U}(1)\times \text{U}(1)$ Abelian Chern-Simons model \begin{eqnarray*}\left\{\begin{aligned} Δu &=λ\left(a(b-a)\mathrm{e}^u-b(b-a)\mathrm{e}^{\upsilon}+a^2\mathrm{e}^{2u}-ab\mathrm{e}^{2\upsilon}+b(b-a)\mathrm{e}^{u+\upsilon} \right)+4π\sum\limits_{j=1}^{k_1}m_jδ_{p_j},\\ Δ\upsilon&=λ\left(-b(b-a)\mathrm{e}^u+a(b-a)\mathrm{e}^{\upsilon}-ab\mathrm{e}^{2u}+a^2\mathrm{e}^{2\upsilon}+b(b-a)\mathrm{e}^{u+\upsilon} \right)+4π\sum\limits_{j=1}^{k_2}n_jδ_{q_j}, \end{aligned} \right. \end{eqnarray*} on finite graphs. Here $λ>0$, $b>a>0$, $m_j>0\, (j=1,2,\cdot\cdot\cdot,k_1)$, $n_j>0\,(j=1,2,\cdot\cdot\cdot,k_2)$, $δ_{p}$ is the Dirac delta mass at vertex $p$. We establish the iteration scheme and prove existence of solutions. We also develop a new method to get the asymptotic behaviors of solutions as $λ$ goes to infinity. This method is also applicable to the Chern-Simons system $$\left\{\begin{aligned} Δu &=λ\mathrm{e}^{\upsilon}(\mathrm{e}^{u}-1) +4π\sum\limits_{j=1}^{k_1}m_jδ_{p_j},\\ Δ\upsilon&=λ\mathrm{e}^{u}(\mathrm{e}^{\upsilon}-1)+4π\sum\limits_{j=1}^{k_2}n_jδ_{q_j}, \end{aligned} \right. $$ and the classical Chern-Simons equation $$ Δu=λ\mathrm{e}^u(\mathrm{e}^u-1)+4π\sum\limits_{j=1}^{N}δ_{p_j}.$$ |
| title | Existence and asymptotic behaviors of solutions to Chern-Simons systems and equations on finite graphs |
| topic | Analysis of PDEs 35J47, 05C22 |
| url | https://arxiv.org/abs/2211.04237 |