Existence, asymptotic behaviors, and high-dimensional uniqueness of topological solutions to the skew-symmetric Chern-Simons system on lattice graphs

In this paper, we consider the topological solutions to the skew-symmetric Chern-Simons system on lattice graphs: $$\{\begin{array}{rlr}{\displaystyle \mathrm{\Delta }u}& {\displaystyle =\lambda {\mathrm{e}}^{\upsilon }({\mathrm{e}}^u-1)+4\pi \sum _{j=1}^{k_1}{m}_j{\delta }_{p_j},\mathrm{\Delta }\upsilon }& {\displaystyle =\lambda {\mathrm{e}}^u({\mathrm{e}}^{\upsilon }-1)+4\pi \sum _{j=1}^{k_2}{n}_j{\delta }_{q_j},}\end{array}$$ here, $\lambda \in {\mathbb{R}}_{+}$, $k_1$ 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 ${\delta }_p$ denotes the Dirac mass at vertex $p$. Write $$g=4\pi \sum _{j=1}^{k_1}{m}_j{\delta }_{p_j},h=4\pi \sum _{j=1}^{k_2}{n}_j{\delta }_{q_j},B=4\pi \sum _{j=1}^{k_1}{m}_j+4\pi \sum _{j=1}^{k_2}{n}_j\mathrm{.}$$ 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 $\lambda \to 0_{+}$ and $\lambda \to +\mathrm{\infty }$, and finally prove the uniqueness of the topological solutions when the dimension of lattice graph ${\mathbb{Z}}^n$ is large enough or $\lambda$ is large enough.