Jongmin Han

Self-dual Chern–Simons Vortices on Bounded Domains

In this Letter we consider the Abelian Chern–Simons vortices on a bounded simply connected domain. We establish the existence of solutions for the self-duality equations. We prove the uniqueness of solutions when all the vortex points are equal and the domain is star-shaped. We also show the radial symmetry of solutions on balls centered at the vortex point.

Existence and properties of radial solutions in the self-dual Chern-Simons O(3) sigma model

In this paper, we study the self-dual equations arising from the Chern-Simons gauged O(3) sigma model with symmetric potential. We prove the existence of radially symmetric solutions of the reduced elliptic equation having topological and nontopological boundary conditions.

Nontopological bare solutions in the relativistic self-dual Maxwell–Chern–Simons–Higgs model

In this paper we prove the existence of the radially symmetric nontopological bare solutions in the relativistic self-dual Maxwell–Chern–Simons–Higgs model. We also verify the Chern–Simons limit for those solutions.

DYNAMIC BIFURCATION OF THE PERIODIC SWIFT-HOHENBERG EQUATION

In this paper we study the dynamic bifurcation of the Swift- Hohenberg equation on a periodic cell = ( L,L). It is shown that the equations bifurcates from the trivial solution to an attractor Awhen the control parametercrosses the critical value. In the odd periodic case, Ais homeomorphic to S1 and consists of eight singular points and their connecting orbits. In the periodic case, Ais homeomorphic to S1, and contains a torus and two circles which consist of singular points. Fluid motion driven by the thermal gradients is common in nature, especially in geophysical flows such as the atmosphere, the oceans, the mantle of the earth, and the interior of stars. A typical model for fluid convection is the Rayleigh- Benard convection describing a fluid placed between flat horizontal plates such that the lower plate is maintained at a temperature above the upper plate temperature. Due to the thermal expansion, the fluid near the lower plate is less dense and become unstable in the gravitational field. Eventually, we encounter an instability at a finite wave length giving a spatio-temporal pattern formation. The mathematical model for the Rayleigh-Benard convection comes from the equation of fluid dynamics in the Boussinesq approximation which involves the Navier-Stokes equations coupled with the temperature equation. In 1977, Swift and Hohenberg derived in (14) that when the Rayleigh number is near the onset of the convection, the Rayleigh-Benard convection model may be approximated by the following Swift-Hohenberg equation (SHE)

Topological Solutions in the Self-dual Chern–Simons–Higgs Theory in a Background Metric

In this Letter we show the existence of topological multi-vortex solutions in the self-dual Chern–Simons–Higgs theory in a background metric which interpolates flat spacetime and cylinder smoothly.

ASYMPTOTIC LIMITS FOR THE SELF-DUAL CHERN-SIMONS CP(1) MODEL

In this paper we study the asymptotics for the energy density in the self-dual Chern-Simons CP(1) model. When the sequence of corresponding multivortex solutions converges to the topological limit, we show that the field configurations saturating the energy bound converges to the limit function. Also, we show that the energy density tends to be concentrated at the vortices and antivortices as the Chern-Simons coupling constant goes to zero.

RADIAL SYMMETRY OF TOPOLOGICAL ONE- VORTEX SOLUTIONS IN THE MAXWELL-CHERN-SIMONS-HIGGS MODEL

In this paper we show the radial symmetry of topological one-vortex solutions in the Maxwell-Chern-Simons-Higgs Model.

Multiplicity for Self-Dual Condensate Solutions in the Maxwell-Chern-Simons O(3) Sigma Model

In this paper, we study multiple existence of solutions of an elliptic system on a flat torus arising from the self-dual Maxwell-Chern-Simons O(3) sigma model. Using the Leray-Schauder degree, we find an explicit range of κ and q for which we have at least two vortex-antivortex solutions. For vortex solutions, we provide a complete picture of the range of κ and q for multiple existence of solutions.

Maxwell-Chern-Simons vortices on compact surfaces: Nonequivalence of the first and the second order equations

In this paper we study the Maxwell-Chern-Simons-Higgs and the Chern-Simons-Higgs vortices on a compact Riemann surface. We establish the existence of a solution of the static Maxwell-Chern-Simons-Higgs vortex equations, which is a minimizer of the static energy functional. This shows the nonequivalence of the first and the second order Maxwell-Chern-Simons-Higgs vortex equations. The nonequivalence is also proved for the Chern-Simons-Higgs vortices by verifying the Chern-Simons limit.

Dynamical Bifurcation of the Generalized Swift–Hohenberg Equation

In this paper, we prove that the generalized Swift–Hohenberg equation bifurcates from the trivial states to an attractor as the control parameter α passes through critical points. The bifurcation is divided into two groups according to the dimension of the center manifolds. We show that the bifurcated attractor is homeomorphic to S1 or S3 and it contains invariant circles of static solutions. We provide a criterion on the quadratic instability parameter μ which determines the bifurcation to be supercritical or subcritical.