• 대한수학회지
• /
• 제61권1호
• /
• pp.133-147
• /
• 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, δ_ps is the Dirac mass at the vertex p_s, and p₁, p₂, . . . , p_N 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.

• 대한수학회지
• /
• 제37권6호
• /
• pp.977-989
• /
• 2000

In this paper we construct a standing solitary wave solution with prescribed total electric charge to the planer Chern-Simons gauged nonlinear Schrodinger equation with an external electromagnetic field by using a variations method.

• 대한수학회논문집
• /
• 제12권2호
• /
• pp.305-309
• /
• 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.

• 대한수학회지
• /
• 제58권6호
• /
• pp.1327-1345
• /
• 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 → ℝ^k 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 ${\mid}\vec{\;l\;}{\mid}{\in}\{0,1\}$.

• 대한수학회논문집
• /
• 제33권3호
• /
• pp.871-888
• /
• 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.

• 대한수학회보
• /
• 제54권3호
• /
• pp.737-746
• /
• 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.