We address a general system of nonlinear Dirac equations in (1+1) dimensions and prove nonexisten... more We address a general system of nonlinear Dirac equations in (1+1) dimensions and prove nonexistence of classical self-similar blowup solutions in the space of bounded functions. While this argument does not exclude the possibility of finite-time blowup, it still suggests that smooth solutions to the nonlinear Dirac equations in (1+1) dimensions do not develop self-similar singularities in a finite time. In the particular case of the cubic Dirac equations, we characterize (unbounded) self-similar solutions in the closed analytical form.
We propose the Chern-Simons gauged sigma model from R 1+2 into the hyperbolic plane H 2. We seek ... more We propose the Chern-Simons gauged sigma model from R 1+2 into the hyperbolic plane H 2. We seek a static configuration of this model and derive self-dual equations. We also establish some existence results for solutions of the self-dual equations under appropriate boundary conditions near ∞.
The semi-nonrelativistic limit of Chern–Simons–Higgs system is considered. We obtain several unif... more The semi-nonrelativistic limit of Chern–Simons–Higgs system is considered. We obtain several uniform estimates with respect to the light speed c and show the convergence of the solution of Chern–Simons–Higgs system to the solution of Chern–Simons–Schrödinger equations.
The semi-nonrelativistic limit of Chern–Simons–Higgs system is considered. We obtain several unif... more The semi-nonrelativistic limit of Chern–Simons–Higgs system is considered. We obtain several uniform estimates with respect to the light speed c and show the convergence of the solution of Chern–Simons–Higgs system to the solution of Chern–Simons–Schrödinger equations.
Journal of Mathematical Analysis and Applications, 2010
The initial value problem of the Chern-Simons-Dirac equations in one space dimension is studied. ... more The initial value problem of the Chern-Simons-Dirac equations in one space dimension is studied. We prove the existence of global solution and investigate asymptotic behaviors.
We derive the self-dual equations of the generalized Jackiw-Pi model with the self-dual potential... more We derive the self-dual equations of the generalized Jackiw-Pi model with the self-dual potential (1.7). We also show that the finite energy solution of the generalized Jackiw-Pi model with special self-dual potential (1.9) satisfies the self-dual system which can be transformed into the Liouville type equation.
We propose a gauge-invariant system of the Chern-Simons-Schrödinger type on a one-dimensional lat... more We propose a gauge-invariant system of the Chern-Simons-Schrödinger type on a one-dimensional lattice. By using the spatial gauge condition, we prove local and global well-posedness of the initial-value problem in the space of square summable sequences for the scalar field. We also study the existence region of the stationary bound states, which depends on the lattice spacing and the nonlinearity power. A major difficulty in the existence problem is related to the lack of variational formulation of the stationary equations. Our approach is based on the implicit function theorem in the anti-continuum limit and the solvability constraint in the continuum limit.
Journal of Mathematical Analysis and Applications, 2020
Abstract By making use of local energy conservation, we estimate the L ∞ norm of the first deriva... more Abstract By making use of local energy conservation, we estimate the L ∞ norm of the first derivative of the solution to the Maxwell-gauged O ( 3 ) sigma model in R 1 + 1 . As an application of the bound, we obtain the growth of the H 2 norm of the solution. Quadratic polynomial growth is obtained by improving exponential growth. A similar idea can be applied to the Maxwell–Klein–Gordon equations in R 1 + 1 .
We prove global existence of solutions to Chern-Simons-Higgs equations under the gauge conditionA... more We prove global existence of solutions to Chern-Simons-Higgs equations under the gauge conditionA1=0. We also find stationary solutions.
We study the Schrodinger-Lohe model. Making use of the principal fundamental matrix \begin{docume... more We study the Schrodinger-Lohe model. Making use of the principal fundamental matrix \begin{document}$ Y $\end{document} of linear ODEs with variable coefficients, the coupled nonlinear Schrodinger-Lohe system is transformed into the decoupled linear Schrodinger equations. The boundedness of \begin{document}$ Y $\end{document} is shown for the case of complete synchronization. We also study the cases where the principal fundamental matrices can be derived explicitly.
Self-similar solutions to nonlinear Dirac systems (1) and (2) are constructed. As an application,... more Self-similar solutions to nonlinear Dirac systems (1) and (2) are constructed. As an application, we obtain nonuniqueness of strong solution in super-critical space C([0, T ]; H s (R)) (s < 0) to the system (1) which is L 2 (R) scaling critical equations. Therefore the well-posedness theory breaks down in Sobolev spaces of negative order.
We address a general system of nonlinear Dirac equations in (1+1) dimensions and prove nonexisten... more We address a general system of nonlinear Dirac equations in (1+1) dimensions and prove nonexistence of classical self-similar blowup solutions in the space of bounded functions. While this argument does not exclude the possibility of finite-time blowup, it still suggests that smooth solutions to the nonlinear Dirac equations in (1+1) dimensions do not develop self-similar singularities in a finite time. In the particular case of the cubic Dirac equations, we characterize (unbounded) self-similar solutions in the closed analytical form.
We study the initial value problem of the exponential wave equation in R n+1 for small initial da... more We study the initial value problem of the exponential wave equation in R n+1 for small initial data. We shows, in the case of n = 1, the global existence of solution by applying the formulation of first order quasilinear hyperbolic system which is weakly linearly degenerate. When n ≥ 2, a vector field method is applied to show the stability of a trivial solution φ = 0.
Journal of Hyperbolic Differential Equations, 2017
The conformal identity for the solution of the Chern–Simons gauged wave equations is derived. As ... more The conformal identity for the solution of the Chern–Simons gauged wave equations is derived. As applications of it, we show some decay estimates and the nonexistence of nontrivial solutions.
Journal of Mathematical Analysis and Applications, 2015
We prove global existence of solutions to space-time monopole equations in one space dimension. O... more We prove global existence of solutions to space-time monopole equations in one space dimension. Observing the null structure of the system, we first prove local existence in L p by applying a bilinear estimate. From a priori L p estimates, we can extend the local solution to the global one. For the critical case L 1 , the concentration of the solution is denied to show the time global solvability.
We study wave function synchronization of the Schrödinger–Lohe model, which describes the dynamic... more We study wave function synchronization of the Schrödinger–Lohe model, which describes the dynamics of the ensemble of coupled quantum Lohe oscillators with infinite states. To do this, we first derive a coupled system of ordinary differential equations for the L x 2 L^2_x inner products between distinct wave functions. For the same one-body potentials, we show that the inner products of two wave functions converge to unity for some restricted class of initial data, so complete wave function synchronization emerges asymptotically when the dynamical system approach is used. Moreover, for the family of one-body potentials consisting of real-value translations of the same base potential, we show that the inner products for a two-oscillator system follow the motion of harmonic oscillators in a small coupling regime, and then as the coupling strength increases, the inner products converge to constant values; this behavior yields convergence toward constant values for the L x 2 L^2_x diffe...
Communications of the Korean Mathematical Society, 2016
We obtain solution formulas for some nonlocal heat equations. As a simple application, we prove f... more We obtain solution formulas for some nonlocal heat equations. As a simple application, we prove finite time blow-up of the solution.
Communications in Partial Differential Equations, 2016
In this paper, we address the problem of local well-posedness of the Chern-Simons-Dirac (CSD) and... more In this paper, we address the problem of local well-posedness of the Chern-Simons-Dirac (CSD) and the Chern-Simons-Higgs (CSH) equations in the Lorenz gauge for low regularity initial data. One of our main contributions is the uncovering of a null structure of (CSD). Combined with the standard machinery of X s,b spaces, we obtain local well-posedness of (CSD) for initial data aµ, ψ ∈ H 1/4+ǫ x. Moreover, it is observed that the same techniques applied to (CSH) lead to a quick proof of local-wellposedness for initial data aµ ∈ H 1/4+ǫ x , (φ, ∂tφ) ∈ H 3/4+ǫ x ×H −1/4+ǫ x , which improves the previous result of [18].
ABSTRACT We study finite time blow-up solutions of the ChernSimonsSchrödinger system. In partic... more ABSTRACT We study finite time blow-up solutions of the ChernSimonsSchrödinger system. In particular, explicit blow-up solutions are constructed by observing the pseudo-conformal invariance and finding solutions of the self-dual equations.
We address a general system of nonlinear Dirac equations in (1+1) dimensions and prove nonexisten... more We address a general system of nonlinear Dirac equations in (1+1) dimensions and prove nonexistence of classical self-similar blowup solutions in the space of bounded functions. While this argument does not exclude the possibility of finite-time blowup, it still suggests that smooth solutions to the nonlinear Dirac equations in (1+1) dimensions do not develop self-similar singularities in a finite time. In the particular case of the cubic Dirac equations, we characterize (unbounded) self-similar solutions in the closed analytical form.
We propose the Chern-Simons gauged sigma model from R 1+2 into the hyperbolic plane H 2. We seek ... more We propose the Chern-Simons gauged sigma model from R 1+2 into the hyperbolic plane H 2. We seek a static configuration of this model and derive self-dual equations. We also establish some existence results for solutions of the self-dual equations under appropriate boundary conditions near ∞.
The semi-nonrelativistic limit of Chern–Simons–Higgs system is considered. We obtain several unif... more The semi-nonrelativistic limit of Chern–Simons–Higgs system is considered. We obtain several uniform estimates with respect to the light speed c and show the convergence of the solution of Chern–Simons–Higgs system to the solution of Chern–Simons–Schrödinger equations.
The semi-nonrelativistic limit of Chern–Simons–Higgs system is considered. We obtain several unif... more The semi-nonrelativistic limit of Chern–Simons–Higgs system is considered. We obtain several uniform estimates with respect to the light speed c and show the convergence of the solution of Chern–Simons–Higgs system to the solution of Chern–Simons–Schrödinger equations.
Journal of Mathematical Analysis and Applications, 2010
The initial value problem of the Chern-Simons-Dirac equations in one space dimension is studied. ... more The initial value problem of the Chern-Simons-Dirac equations in one space dimension is studied. We prove the existence of global solution and investigate asymptotic behaviors.
We derive the self-dual equations of the generalized Jackiw-Pi model with the self-dual potential... more We derive the self-dual equations of the generalized Jackiw-Pi model with the self-dual potential (1.7). We also show that the finite energy solution of the generalized Jackiw-Pi model with special self-dual potential (1.9) satisfies the self-dual system which can be transformed into the Liouville type equation.
We propose a gauge-invariant system of the Chern-Simons-Schrödinger type on a one-dimensional lat... more We propose a gauge-invariant system of the Chern-Simons-Schrödinger type on a one-dimensional lattice. By using the spatial gauge condition, we prove local and global well-posedness of the initial-value problem in the space of square summable sequences for the scalar field. We also study the existence region of the stationary bound states, which depends on the lattice spacing and the nonlinearity power. A major difficulty in the existence problem is related to the lack of variational formulation of the stationary equations. Our approach is based on the implicit function theorem in the anti-continuum limit and the solvability constraint in the continuum limit.
Journal of Mathematical Analysis and Applications, 2020
Abstract By making use of local energy conservation, we estimate the L ∞ norm of the first deriva... more Abstract By making use of local energy conservation, we estimate the L ∞ norm of the first derivative of the solution to the Maxwell-gauged O ( 3 ) sigma model in R 1 + 1 . As an application of the bound, we obtain the growth of the H 2 norm of the solution. Quadratic polynomial growth is obtained by improving exponential growth. A similar idea can be applied to the Maxwell–Klein–Gordon equations in R 1 + 1 .
We prove global existence of solutions to Chern-Simons-Higgs equations under the gauge conditionA... more We prove global existence of solutions to Chern-Simons-Higgs equations under the gauge conditionA1=0. We also find stationary solutions.
We study the Schrodinger-Lohe model. Making use of the principal fundamental matrix \begin{docume... more We study the Schrodinger-Lohe model. Making use of the principal fundamental matrix \begin{document}$ Y $\end{document} of linear ODEs with variable coefficients, the coupled nonlinear Schrodinger-Lohe system is transformed into the decoupled linear Schrodinger equations. The boundedness of \begin{document}$ Y $\end{document} is shown for the case of complete synchronization. We also study the cases where the principal fundamental matrices can be derived explicitly.
Self-similar solutions to nonlinear Dirac systems (1) and (2) are constructed. As an application,... more Self-similar solutions to nonlinear Dirac systems (1) and (2) are constructed. As an application, we obtain nonuniqueness of strong solution in super-critical space C([0, T ]; H s (R)) (s < 0) to the system (1) which is L 2 (R) scaling critical equations. Therefore the well-posedness theory breaks down in Sobolev spaces of negative order.
We address a general system of nonlinear Dirac equations in (1+1) dimensions and prove nonexisten... more We address a general system of nonlinear Dirac equations in (1+1) dimensions and prove nonexistence of classical self-similar blowup solutions in the space of bounded functions. While this argument does not exclude the possibility of finite-time blowup, it still suggests that smooth solutions to the nonlinear Dirac equations in (1+1) dimensions do not develop self-similar singularities in a finite time. In the particular case of the cubic Dirac equations, we characterize (unbounded) self-similar solutions in the closed analytical form.
We study the initial value problem of the exponential wave equation in R n+1 for small initial da... more We study the initial value problem of the exponential wave equation in R n+1 for small initial data. We shows, in the case of n = 1, the global existence of solution by applying the formulation of first order quasilinear hyperbolic system which is weakly linearly degenerate. When n ≥ 2, a vector field method is applied to show the stability of a trivial solution φ = 0.
Journal of Hyperbolic Differential Equations, 2017
The conformal identity for the solution of the Chern–Simons gauged wave equations is derived. As ... more The conformal identity for the solution of the Chern–Simons gauged wave equations is derived. As applications of it, we show some decay estimates and the nonexistence of nontrivial solutions.
Journal of Mathematical Analysis and Applications, 2015
We prove global existence of solutions to space-time monopole equations in one space dimension. O... more We prove global existence of solutions to space-time monopole equations in one space dimension. Observing the null structure of the system, we first prove local existence in L p by applying a bilinear estimate. From a priori L p estimates, we can extend the local solution to the global one. For the critical case L 1 , the concentration of the solution is denied to show the time global solvability.
We study wave function synchronization of the Schrödinger–Lohe model, which describes the dynamic... more We study wave function synchronization of the Schrödinger–Lohe model, which describes the dynamics of the ensemble of coupled quantum Lohe oscillators with infinite states. To do this, we first derive a coupled system of ordinary differential equations for the L x 2 L^2_x inner products between distinct wave functions. For the same one-body potentials, we show that the inner products of two wave functions converge to unity for some restricted class of initial data, so complete wave function synchronization emerges asymptotically when the dynamical system approach is used. Moreover, for the family of one-body potentials consisting of real-value translations of the same base potential, we show that the inner products for a two-oscillator system follow the motion of harmonic oscillators in a small coupling regime, and then as the coupling strength increases, the inner products converge to constant values; this behavior yields convergence toward constant values for the L x 2 L^2_x diffe...
Communications of the Korean Mathematical Society, 2016
We obtain solution formulas for some nonlocal heat equations. As a simple application, we prove f... more We obtain solution formulas for some nonlocal heat equations. As a simple application, we prove finite time blow-up of the solution.
Communications in Partial Differential Equations, 2016
In this paper, we address the problem of local well-posedness of the Chern-Simons-Dirac (CSD) and... more In this paper, we address the problem of local well-posedness of the Chern-Simons-Dirac (CSD) and the Chern-Simons-Higgs (CSH) equations in the Lorenz gauge for low regularity initial data. One of our main contributions is the uncovering of a null structure of (CSD). Combined with the standard machinery of X s,b spaces, we obtain local well-posedness of (CSD) for initial data aµ, ψ ∈ H 1/4+ǫ x. Moreover, it is observed that the same techniques applied to (CSH) lead to a quick proof of local-wellposedness for initial data aµ ∈ H 1/4+ǫ x , (φ, ∂tφ) ∈ H 3/4+ǫ x ×H −1/4+ǫ x , which improves the previous result of [18].
ABSTRACT We study finite time blow-up solutions of the ChernSimonsSchrödinger system. In partic... more ABSTRACT We study finite time blow-up solutions of the ChernSimonsSchrödinger system. In particular, explicit blow-up solutions are constructed by observing the pseudo-conformal invariance and finding solutions of the self-dual equations.
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