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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.14538 |
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| _version_ | 1866908545380777984 |
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| author | Liu, Honggang |
| author_facet | Liu, Honggang |
| contents | In this paper, we consider the topological solutions to the skew-symmetric Chern-Simons system on lattice graphs: $$\left\{\begin{aligned} Δu &=λ\mathrm{e}^{\upsilon}(\mathrm{e}^{u}-1)+4π\sum\limits_{j=1}^{k_1}m_jδ_{p_j}, Δ\upsilon&=λ\mathrm{e}^{u}(\mathrm{e}^{\upsilon}-1)+4π\sum\limits_{j=1}^{k_2}n_jδ_{q_j}, \end{aligned} \right. $$ here, $λ\in\mathbb{R}_+$, $k_1$ and $k_2$ are two positive integers, $m_j\in\mathbb{N}\, (j=1,2,\cdot\cdot\cdot,k_1)$, $n_j\in\mathbb{N}\,(j=1,2,\cdot\cdot\cdot,k_2)$, and $δ_{p}$ denotes the Dirac mass at vertex $p$. Write $$g=4π\sum_{j=1}^{k_1}m_jδ_{p_j},\ h=4π\sum_{j=1}^{k_2}n_jδ_{q_j},\ B = 4π\sum_{j=1}^{k_1}m_j + 4π\sum_{j=1}^{k_2}n_j.$$ For any fixed $g,h$, we prove the existence of the topological solutions to the systems, then obtain the asymptotic behaviors of topological solutions as $λ\rightarrow 0_+$ and $λ\rightarrow +\infty$, and finally prove the uniqueness of the topological solutions when the dimension of lattice graph $\mathbb{Z}^n$ is large enough or $λ$ is large enough. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_14538 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence, asymptotic behaviors, and high-dimensional uniqueness of topological solutions to the skew-symmetric Chern-Simons system on lattice graphs Liu, Honggang Analysis of PDEs In this paper, we consider the topological solutions to the skew-symmetric Chern-Simons system on lattice graphs: $$\left\{\begin{aligned} Δu &=λ\mathrm{e}^{\upsilon}(\mathrm{e}^{u}-1)+4π\sum\limits_{j=1}^{k_1}m_jδ_{p_j}, Δ\upsilon&=λ\mathrm{e}^{u}(\mathrm{e}^{\upsilon}-1)+4π\sum\limits_{j=1}^{k_2}n_jδ_{q_j}, \end{aligned} \right. $$ here, $λ\in\mathbb{R}_+$, $k_1$ and $k_2$ are two positive integers, $m_j\in\mathbb{N}\, (j=1,2,\cdot\cdot\cdot,k_1)$, $n_j\in\mathbb{N}\,(j=1,2,\cdot\cdot\cdot,k_2)$, and $δ_{p}$ denotes the Dirac mass at vertex $p$. Write $$g=4π\sum_{j=1}^{k_1}m_jδ_{p_j},\ h=4π\sum_{j=1}^{k_2}n_jδ_{q_j},\ B = 4π\sum_{j=1}^{k_1}m_j + 4π\sum_{j=1}^{k_2}n_j.$$ For any fixed $g,h$, we prove the existence of the topological solutions to the systems, then obtain the asymptotic behaviors of topological solutions as $λ\rightarrow 0_+$ and $λ\rightarrow +\infty$, and finally prove the uniqueness of the topological solutions when the dimension of lattice graph $\mathbb{Z}^n$ is large enough or $λ$ is large enough. |
| title | Existence, asymptotic behaviors, and high-dimensional uniqueness of topological solutions to the skew-symmetric Chern-Simons system on lattice graphs |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2509.14538 |