论文标题
Chern-Simons系统解决方案的存在和渐近行为和有限图上方程
Existence and asymptotic behaviors of solutions to Chern-Simons systems and equations on finite graphs
论文作者
论文摘要
在本文中，我们考虑了$ \ text {u}（1）（1）\ times \ text {u}（1）$ Abelian Chern-Simons模型\ begin {eqnarray*} \ left \ left { ＆=λ\ left（a（b-a）\ mathrm {e}^u-b（b-a）\ mathrm {e}^{\ upsilon}+a^2 \ math rm {e}^{2u} -ab \ m \ 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} \ right）+4π\ sum \ limits_ {j = 1}^{k_2}n_jΔ__{q_j}，\ end {aligned} \ right。 \ end {eqnarray*}在有限图上。 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 $p$. 我们建立迭代方案并证明解决方案的存在。我们还开发了一种新方法，以获取解决方案的渐近行为，因为$λ$变为无限。此方法也适用于Chern-Simons System $$ \ left \ {\ begin {Aligned}ΔU＆=λ\ mathrm {e}^{\ upsilon}（\ mathrm {e} +4π\ sum \ limits_ {j = 1}^{k_1}m_jΔ__{p_j}，\\ δ\ upsilon＆=λ\ mathrm {e}^{u}（\ mathrm {e}^{\ upSilon} -1）+4π\ sum \ sum \ limits_ { $$和经典的Chern-Simons方程$$ΔU=λ\ Mathrm {e}^u（\ Mathrm {e}^u-1）+4π\ sum \ sum \ limits_ {
In this paper, we consider a system of equations arising from the $\text{U}(1)\times \text{U}(1)$ Abelian Chern-Simons model \begin{eqnarray*}\left\{\begin{aligned} Δu &=λ\left(a(b-a)\mathrm{e}^u-b(b-a)\mathrm{e}^{\upsilon}+a^2\mathrm{e}^{2u}-ab\mathrm{e}^{2\upsilon}+b(b-a)\mathrm{e}^{u+\upsilon} \right)+4π\sum\limits_{j=1}^{k_1}m_jδ_{p_j},\\ Δ\upsilon&=λ\left(-b(b-a)\mathrm{e}^u+a(b-a)\mathrm{e}^{\upsilon}-ab\mathrm{e}^{2u}+a^2\mathrm{e}^{2\upsilon}+b(b-a)\mathrm{e}^{u+\upsilon} \right)+4π\sum\limits_{j=1}^{k_2}n_jδ_{q_j}, \end{aligned} \right. \end{eqnarray*} on finite graphs. Here $λ>0$, $b>a>0$, $m_j>0\, (j=1,2,\cdot\cdot\cdot,k_1)$, $n_j>0\,(j=1,2,\cdot\cdot\cdot,k_2)$, $δ_{p}$ is the Dirac delta mass at vertex $p$. We establish the iteration scheme and prove existence of solutions. We also develop a new method to get the asymptotic behaviors of solutions as $λ$ goes to infinity. This method is also applicable to the Chern-Simons system $$\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}.$$
