• Title/Summary/Keyword: Chern-Simons equation

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EXISTENCE OF SOLUTIONS TO A GENERALIZED SELF-DUAL CHERN-SIMONS EQUATION ON FINITE GRAPHS

  • Yuanyang Hu
    • Journal of the Korean Mathematical Society
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    • v.61 no.1
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    • pp.133-147
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    • 2024
  • Let G = (V, E) be a connected finite graph. We study the existence of solutions for the following generalized Chern-Simons equation on G $${\Delta}u={\lambda}e^u(e^u-1)^5+4{\pi}\sum_{s=1}^{N}\delta_{ps}$$, where λ > 0, δps is the Dirac mass at the vertex ps, and p1, p2, . . . , pN are arbitrarily chosen distinct vertices on the graph. We show that there exists a critical value $\hat{\lambda}$ such that when λ > $\hat{\lambda}$, the generalized Chern-Simons equation has at least two solutions, when λ = $\hat{\lambda}$, the generalized Chern-Simons equation has a solution, and when λ < $\hat{\lambda}$, the generalized Chern-Simons equation has no solution.

EXISTENCE OF A MULTIVORTEX SOLUTION FOR ${SU(N)_g}{\times}U(1)_l$ CHERN-SIMONS MODEL IN ${R^2}/{Z^2}$

  • Yoon, Jai-Han
    • Communications of the Korean Mathematical Society
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    • v.12 no.2
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    • pp.305-309
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    • 1997
  • In this paper we prove the existence of a special type of multivortex solutions of $SU (N)_g \times U(1)_l$ Chern-Simons model. More specifically we prove existence of solutions of the self-duality equations for $(\Phi(x), j =1, \cdots, N$ has the same zeroes. In this case we find that the equation can be reduced to the single semilinear elliptic partial differential equations studied by Caffarelli and Yang.

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A LIOUVILLE THEOREM OF AN INTEGRAL EQUATION OF THE CHERN-SIMONS-HIGGS TYPE

  • Chen, Qinghua;Li, Yayun;Ma, Mengfan
    • Journal of the Korean Mathematical Society
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    • v.58 no.6
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    • pp.1327-1345
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    • 2021
  • In this paper, we are concerned with a Liouville-type result of the nonlinear integral equation of Chern-Simons-Higgs type $$u(x)=\vec{\;l\;}+C_{\ast}{{\displaystyle\smashmargin{2}{\int\nolimits_{\mathbb{R}^n}}}\;{\frac{(1-{\mid}u(y){\mid}^2){\mid}u(y){\mid}^2u(y)-\frac{1}{2}(1-{\mid}u(y){\mid}^2)^2u(y)}{{\mid}x-y{\mid}^{n-{\alpha}}}}dy.$$ Here u : ℝn → ℝk is a bounded, uniformly continuous function with k ⩾ 1 and 0 < α < n, $\vec{\;l\;}{\in}\mathbb{R}^k$ is a constant vector, and C* is a real constant. We prove that ${\mid}\vec{\;l\;}{\mid}{\in}\{0,\frac{\sqrt{3}}{3},1\}$ if u is the finite energy solution. Further, if u is also a differentiable solution, then we give a Liouville type theorem, that is either $u{\rightarrow}\vec{\;l\;}$ with ${\mid}\vec{\;l\;}{\mid}=\frac{\sqrt{3}}{3}$, when |x| → ∞, or $u{\equiv}\vec{\;l\;}$, where ${\mid}\vec{\;l\;}{\mid}{\in}\{0,1\}$.

NONRELATIVISTIC LIMIT OF CHERN-SIMONS GAUGED FIELD EQUATIONS

  • Chae, Myeongju;Yim, Jihyun
    • Communications of the Korean Mathematical Society
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    • v.33 no.3
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    • pp.871-888
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    • 2018
  • We study the nonrelativistic limit of the Chern-Simons-Dirac system on ${\mathbb{R}}^{1+2}$. As the light speed c goes to infinity, we first prove that there exists an uniform existence interval [0, T] for the family of solutions ${\psi}^c$ corresponding to the initial data for the Dirac spinor ${\psi}_0^c$ which is bounded in $H^s$ for ${\frac{1}{2}}$ < s < 1. Next we show that if the initial data ${\psi}_0^c$ converges to a spinor with one of upper or lower component zero in $H^s$, then the Dirac spinor field converges, modulo a phase correction, to a solution of a linear $Schr{\ddot{o}}dinger$ equation in C([0, T]; $H^{s^{\prime}}$) for s' < s.

EXISTENCE THEOREM FOR NON-ABELIAN VORTICES IN THE AHARONY-BERGMAN-JAFFERIS-MALDACENA THEORY

  • Zhang, Ruifeng;Zhu, Meili
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.3
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    • pp.737-746
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    • 2017
  • In this paper, we discuss the existence theorem for multiple vortex solutions in the non-Abelian Chern-Simons-Higgs field theory developed by Aharony, Bergman, Jafferis, and Maldacena, on a doubly periodic domain. The governing equations are of the BPS type and derived by Auzzi and Kumar in the mass-deformed framework labeled by a continuous parameter. Our method is based on fixed point method.