Vortex Condensation in General U(1)×U(1) Abelian Chern-Simons Model on a flat torus
Authors: Huang Hsin-Yuan 
Published: September 2021

<p style='text-indent:20px;'>In this paper, we study an elliptic system arising from the U(1)<inline-formula><tex-math id="M2">\begin{document}$ \times $\end{document}</tex-math></inline-formula>U(1) Abelian Chern-Simons Model[<xref ref-type="bibr" rid="b25">25</xref>,<xref ref-type="bibr" rid="b37">37</xref>] of the form</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE123"> \begin{document}$ \begin{equation} \left\{\begin{split} \Delta u = &amp;\lambda \left(a(b-a)e^{u}-b(b-a)e^{v}+a^2e^{2u} -abe^{2v}+b(b-a)e^{u+v}\right)\\ &amp; +4\pi \sum\limits_{j = 1}^{k_1}m_j\delta_{p_j}, \\ \Delta v = &amp;\lambda \left(-b(b-a)e^{u}+a(b-a)e^{v}-abe^{2u} +a^2e^{2v}+b(b-a)e^{u+v}\right)\\ &amp; +4\pi \sum\limits_{j = 1}^{k_2}n_j\delta_{q_j}, \end{split}\right. \quad\quad\quad\quad (1)\end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>which are defined on a parallelogram <inline-formula><tex-math id="M3">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^2 $\end{document}</tex-math></inline-formula> with doubly periodic boundary conditions. Here, <inline-formula><tex-math id="M5">\begin{document}$ a $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ b $\end{document}</tex-math></inline-formula> are interaction constants, <inline-formula><tex-math id="M7">\begin{document}$ \lambda&gt;0 $\end{document}</tex-math></inline-formula> is related to coupling constant, <inline-formula><tex-math id="M8">\begin{document}$ m_j&gt;0(j = 1,\cdots,k_1) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ n_j&gt;0(j = 1,\cdots,k_2) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ \delta_{p} $\end{document}</tex-math></inline-formula> is the Dirac measure, <inline-formula><tex-math id="M11">\begin{document}$ p $\end{document}</tex-math></inline-formula> is called vortex point. Concerning the existence results of this system over <inline-formula><tex-math id="M12">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>, only the cases <inline-formula><tex-math id="M13">\begin{document}$ (a,b) = (0,1) $\end{document}</tex-math></inline-formula>[<xref ref-type="bibr" rid="b28">28</xref>] and <inline-formula><tex-math id="M14">\begin{document}$ a&gt;b&gt;0 $\end{document}</tex-math></inline-formula>[<xref ref-type="bibr" rid="b14">14</xref>] were studied in the literature. The solvability of this system (1) is still an open problem as regards other parameters <inline-formula><tex-math id="M15">\begin{document}$ (a,b) $\end{document}</tex-math></inline-formula>. We show that the system (1) admits topological solutions provided <inline-formula><tex-math id="M16">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> is large and <inline-formula><tex-math id="M17">\begin{document}$ b&gt;a&gt;0 $\end{document}</tex-math></inline-formula> Our arguments are based on a iteration scheme and variational formulation.</p>