Self-Dual Vortices in Chern–Simons Hydrodynamics

The classical theory of a nonrelativistic charged particle interacting with a U(1) gauge field is reformulated as the Schrödinger wave equation modified by the de Broglie–Bohm nonlinear quantum potential. The model is gauge equivalent to the standard Schrödinger equation with the Planck constant for the deformed strength of the quantum potential and to the pair of diffusion–antidiffusion equations for the strength . Specifying the gauge field as the Abelian Chern–Simons (CS) one in 2+1 dimensions interacting with the nonlinear Schrödinger (NLS) field (the Jackiw–Pi model), we represent the theory as a planar Madelung fluid, where the CS Gauss law has the simple physical meaning of creation of the local vorticity for the fluid flow. For the static flow when the velocity of the center-of-mass motion (the classical velocity) is equal to the quantum velocity (generated by the quantum potential velocity of the internal motion), the fluid admits an N-vortex solution. Applying a gauge transformation of the Auberson–Sabatier type to the phase of the vortex wave function, we show that deformation parameter , the CS coupling constant, and the quantum potential strength are quantized. We discuss reductions of the model to 1+1 dimensions leading to modified NLS and DNLS equations with resonance soliton interactions.     

Minimizers near the first critical field for the nonself-dual Chern–Simons–Higgs energy

The Chern–Simons–Higgs energy serves as a model for high temperature superconductivity. We show the existence of weak solutions to the CSH equations that are minimizers of the CSH energy. The solutions are vortexless for an applied magnetic field h _ex below the critical field strength, whereas vortices appear when h _ex exceeds the critical field strength. D. Spirn was supported in part by NSF grants DMS-0510121 and DMS-0707714. X. Yan was supported in part by NSF grants DMS-0700966 and DMS-0401048.     

Topological multivortex solutions for the Chern–Simons system with two Higgs particles

In this paper, we prove the uniqueness of topological multivortex solutions to the self-dual abelian Chern–Simons model if either the Chern–Simons coupling parameter is sufficiently small or sufficiently large. In addition, we also establish the sharp region of the flux for nontopological solutions with a single vortex point.     

Low-energy vortex dynamics in the self-dual Chern–Simons–Higgs model

The relativistic Chern–Simons–Higgs theory finds application in anyonic superconductivity and contains topological vortices whose dynamics are poorly understood. The gauge fields are defined by a set of nonlinear constraint equations that can be accurately solved with effective Green’s functions, spectral methods, and a discretization scheme using lattice gauge techniques. Simulations show that low-energy two-vortex interactions are elastic with final scattering angles sensitive to vortex velocity; furthermore, vortex pairs form rotating breather states for certain impact parameters. In this study, a function that reproduces scattering angles in the adiabatic limit for nontangential collisions is presented. Simulation results are discussed in the context of analytical methods that extract vortex dynamics from low-energy effective Lagrangians, and a numerical method to calculate the effective Lagrangian is suggested. The numerical techniques used can be applied to the study of other Chern–Simon theories.     

On the Chern–Simons–Higgs vortex equation without gauge fields

This paper is concerned with the nonself-dual Chern–Simons–Higgs model on R²${\mathbb{R}}^2$ with vanishing gauge fields. We prove the existence of radial solutions with the topological boundary condition, and the nonexistence of radial solutions with the nontopological boundary condition. We also establish the asymptotic properties of solutions and derive the quantization of the potential energy.     

Asymptotic Behavior of the Chern–Simons Higgs 6-th Theory

We analyze the asymptotic behavior of solutions of the Chern–Simons Higgs 6-th model introduced by Hong–Kim–Pac and Jackiw–Weinberg.     

Casimir effect for Chern–Simons layers in the vacuum

We solve the diffraction problem for electromagnetic waves on a planar (2+1)-dimensional layer with a given Chern–Simons action. The Casimir energy of a system of two parallel planar Chern–Simons layers is expressed in terms of the coefficients of reflection from separate layers.     

Existence of charged vortices in a Maxwell–Chern–Simons model

In this paper, we prove the existence of charged vortex solitons in a Maxwell–Chern–Simons model. We establish the main existence theorem by a constrained minimization method applied on an indefinite action functional which is induced from the original field-theoretical Lagrangian. We also show that the solutions obtained are smooth.     

Existence of doubly periodic vortices in a generalized Chern–Simons model

We establish an existence theorem for the doubly periodic vortices in a generalized self-dual Chern–Simons model. We show that there exists a critical value of the coupling parameter such that there exist self-dual doubly periodic vortex solutions for the generalized self-dual Chern–Simons equation if and only if the coupling parameter is less than or equal to the value. The energy, magnetic flux, and electric charge associated to the field configurations are all specifically quantized. By the solutions obtained for this generalized self-dual Chern–Simons equation we can also construct doubly periodic vortex solutions to a related generalized self-dual Abelian Higgs equation.     

One Loop Approximation of the Chern–Simons Integral

It is proven that the one loop approximation of the Wilson line integral in a perturbative SU(2) Chern–Simons theory is localized around the critical point in the large level.     

Chern–Simons Invariants of Torus Links

We compute the vacuum expectation values of torus knot operators in Chern–Simons theory, and we obtain explicit formulae for all classical gauge groups and for arbitrary representations. We reproduce a known formula for the HOMFLY invariants of torus knots and links, and we obtain an analogous formula for Kauffman invariants. We also derive a formula for cable knots. We use our results to test a recently proposed conjecture that relates HOMFLY and Kauffman invariants.     

Chern–Simons solitons in quantum potential

The self-dual Chern–Simons solitons under the influence of the quantum potential are considered. The single-valuedness condition for an arbitrary integer number N⩾0 of solitons leads to quantization of Chern–Simons coupling constant κ=m(e²/g), and the integer strength of quantum potential s=1−m². As we show, the Jackiw–Pi model corresponds to the first member (m=1) of our hierarchy of the Chern–Simons gauged nonlinear Schrödinger models, admitting self-dual solitons. New types of exponentially localized Chern–Simons solitons for the Bloch electrons near the hyperbolic energy band boundary are found.     

Global wellposedness for a one-dimensional Chern–Simons–Dirac system in Lp

The global wellposedness in L^p(?) for the Chern–Simons–Dirac equation in the 1+1 space and time dimension is discussed. We consider two types of quadratic nonlinearity: the null case and the non-null case. We show the time global wellposedness for the Chern–Simon–Dirac equation in the framework of L^p(?), where 1≤p≤∞ for the null case. For the scaling critical case, p = 1, mass concentration phenomena of the solutions may occur in considering the time global solvability. We invoke the Delgado–Candy estimate which plays a crucial role in preventing concentration phenomena of the global solution. Our method is related to the original work of Candy (2011), who showed the time global wellposedness for the single Dirac equation with cubic nonlinearity in the critical space L²(?).     

Analytical Issues in the Construction of Self-dual Chern–Simons Vortices

In this note we discuss the solvability of Liouville-type systems in presence of singular sources, which arise from the study of non-abelian Chern Simons vortices in Gauge Field Theory and their asymptotic behaviour (for limiting values of the physical parameters). This investigation has contributed towards the understanding of singular PDE ’s in Mean Field form, in connection to surfaces with conical singularities, sharp Moser–Trudinger and log(HLS)-inequalities, bubbling phenomena and point-wise profile estimates in terms of Harnack type inequalities. We shall emphasise mostly the physical impact of the rigorous mathematical results established and mention several of the remaining open problems.     

Standing waves for the Chern–Simons–Schrödinger systems without (AR) condition

In this paper we study the nonlinear Chern–Simons–Schrödinger systems with an external potential. We show the existence, non-existence, and multiplicity of standing waves to this problem with asymptotically linear nonlinearities, which do not hold the Ambrosetti–Rabinowitz condition.