Dimensional sine-Gordon Research Articles

Canonical reduction of self-dual Yang-Mills theory to sine-Gordon and Liouville theories

We present the (constrained) canonical reduction of four dimensional self-dual Yang-Mills theory to two dimensional sine-Gordon and Liouville field theories: The Dirac brackets in the latter two theories are derived from the reduction of the Poisson brackets in the former, at the classical level.

New class of running-wave solutions of the (2+1)-dimensional sine-Gordon equation

A new class of running-wave solutions of the (2+1)-dimensional sine-Gordon equation is investigated. The obtained waves require two spatial dimensions for their propagation, i.e. they generalize solutions of the (2+0)-dimensional sine-Gordon equation. The parameters of the waves strongly depend on the wave amplitude and there exist forbidden areas for the wavenumber and frequency. The obtained solutions describe a new class of Josephson waves whose velocity is smaller than the Swihart velocity. If omega =0 the running waves are reduced to the self-consistent phase, current and magnetic field distributions in a large two-dimensional Josephson junction. The self-restriction coefficient for the Josephson current corresponding to one of the structures is calculated.

Charge-unbinding transition in the two-dimensional classical Coulomb-plasma model by use of a variational Gaussian wave functional.

The charge-unbinding transitions in the two-dimensional classical Coulomb-plasma model are analyzed by use of the equivalence between the two-dimensional Coulomb plasma and (1+1)-dimensional quantum sine-Gordon models. The grand canonical potential and the screening length are obtained from the ground states of the (1+1)-dimensional quantum sine-Gordon model as derived by use of a variational Gaussian wave functional. We have shown that the charge-unbinding transition at small fugacity is a Kosterlitz-Thouless transition, while at large fugacity it is a first-order transition. The dependence of the potential on the mass is similar to that of the phenomenological Landau theory.

Variational Gaussian wave functional and the (1+1)-dimensional sine-Gordon model.

The (1+1)-dimensional sine-Gordon model is studied in the framework of the variational Gaussian wave functionals. The renormalized mass depends on two dimensionless parameters. In the phase diagram a first-order phase transition will replace the second-order phase transition for large bare mass. The spectrum of the two-particle states consists of a continuum of scattering states and the isolated values connected to the excitonlike bound states. The energy of the bound states and the phase shifts of the scattering states are explicitly calculated.

A method for generating analytical solutions of the nonlinear Klein-Gordon equation

A simple method for generating solutions of the nonlinear Klein-Gordon equation is presented. The obtained solutions depend on arbitrary functions. The method can be applied also to a more general class of partial differential equations. A physical application of a solution of the (2+1)-dimensional sine-Gordon equation obtained by this method is proposed.

First-order transition in the dense two-dimensional classical Coulomb gas.

The phase diagram of the two-dimensional classical neutral Coulomb gas (CG) at finite temperature has been investigated via the mapping to the ground-state properties of the (1+1)-dimensional quantum sine-Gordon model. On the basis of self-consistent-field theory recently proposed by the authors for the quantum sine-Gordon model, we show that the CG undergoes two distinct classes of Kosterlitz-Thouless phase transitions between the dielectric gas and conductive plasma: one is the usual continuous transition of infinite order at very small fugacity, the other is a first-order discontinuous transition starting at the fugacity ${\mathit{z}}^{\mathrm{*}}$=0.08 and the temperature ${\mathit{T}}^{\mathrm{*}}$=0.25. Our results further corroborate the existence of a first-order phase transition in the dense CG system predicted by Minnhagen's self-consistent linear screening renormalization theory and Monte Carlo simulations.

Stimulated waves of electric and magnetic fields in a two-dimensional semi-infinite Josephson junction

A two-dimensional semi-infinite Josephson junction without damping is considered. Its interaction with an external oscillating electromagnetic field in the form of a running wave with a phase velocity equal to the Swihart velocity is investigated. The results obtained are based on an exact solution of the (2+1)-dimensional sine-Gordon equation, depending on an arbitrary function. The boundary conditions on the interface are provided by an external time-varying electric field consistent with the exact solution. Under these conditions, an electromagnetic structure arises inside the junction. It is shown how the existence of this formation may be proved experimentally. A method for measuring the Swihart velocity is proposed.

A (2+1)-dimensional extension for the sine-Gordon equation

Starting from the breaking soliton equation, the author obtains a new integrable equation in 2+1 dimensions. Though the equation has no exchange symmetry of the space variables x and y, the model reduces back to the known (1+1)-dimensional sine-Gordon (or Liouville) equation.

On (2+1)-dimensional Ermakov systems

Ermakov-type systems in 2+1 dimensions are introduced. Multiwave solutions of a (2+1)-dimensional Pinney equation and a modulated (2+1)-dimensional sine-Gordon equation are thereby constructed.

Light-front quantization of the sine-Gordon model.

It is shown how to modify the canonical light-front quantization of the (1+1)-dimensional sine-Gordon model such that the zero-mode problem of light-front quantization is avoided. The canonical sine-Gordon Lagrangian is replaced by an effective Lagrangian which does not lead to divergences as [ital k][sup +]=([ital k][sup 0]+[ital k][sup 1])/ [radical]2 [r arrow]0. After canonically quantizing the effective Lagrangian, one obtains the effective light-front Hamiltonian which agrees with the naive light-front (LF) Hamiltonian, up to one additional renormalization. The spectrum of the effective LF Hamiltonian is determined using discrete light-cone quantization and agrees with results from equal-time quantization.

Virtual-pion fluctuations around the bare skyrmion

Pion fluctuations around the skyrmion are quantized by Dirac's method. In the large- N c limit, where N c is the number of colors, the O( N c 0) quantum correction to the skyrmion mass that comes from the virtual-pion fluctuations is shown to be negative. A perturbative expansion using the usual Feynman-Dyson technique but with noncovariant pion propagators is developed to evaluate the quantum correction. As a result of delicate cancellations, it is a good approximation to keep just the leading-order terms in the perturbation series. This approximation gives rise to the pion decay constant F π = 172 MeV, to be compared with the experimental value of 186 MeV. The axial coupling constant g A is also improved considerably. An application of the method to the (1 + 1)-dimensional kink model and sine-Gordon model supports our results.

On localized solitonic solutions of a (2+1)-dimensional sine-Gordon system

Exponentially decaying solutions of an integrable sine-Gordon system in 2+1 dimensions are presented. A simple superposition formula is given.

A (2+1)-dimensional sine-Gordon system: its auto-Bäcklund transformation

An integrable (2+1)-dimensional sine-Gordon equation wherein the spatial variables occur in a symmetric manner was recently introduced by Konopelchenko and Rogers. It represents a (2+1)-dimensional extension of the classical sine-Gordon equation analogous to the Nizhnik-Novikov-Veselov generalization of the Korteweg-de Vries equation and the Davey-Stewartson generalization of the nonlinear Schrödinger equation. Here, an auto-Bäcklund transformation is constructed via a generalized Darboux-Levi approach both for the (2+1)-dimensional sine-Gordon equation and an integrable coupled system in which it is embedded. An iterated version of the auto-Bäcklund transformation is presented along with a class of exact solutions.

A method for generating exact solutions of the nonlinear Klein–Gordon equation

A simple method for generating the exact solutions of the nonlinear Klein–Gordon equation is proposed. The solutions obtained depend on two arbitrary functions and are in the form of running waves. An application of one of the solutions for the (2 + 1) – dimensional sine-Gordon equation is proposed. It concerns the selective properties of a two-dimensional semi-infinite Josephson junction with regard to an external electromagnetic field in the form of running waves with a phase velocity equal to the Swihart velocity. A method for measuring the Swihart velocity is presented.

The Finite Temperature Effects on the Coleman Phase Transition in Double Sine-Gordon Model11The project supported by National Natural Science Foundation of China.

By means of the finite temperature Gaussian effective potential (FTGEP) method, the - and -dimensional quantum double sine-Gordon (DsG) field theories are studied nonperturbatively for finite temperature. The explicitly FTGEP is obtained for both - and -dimensional DsG models. The temperature-dependent Coleman phase transition conditions obtained by us would reduce to the known results when . The alternative explanation of the Coleman phase transition at finite temperature is that for fixed coupling constant there is a critical temperature (whose value depends on ) such that the theory becomes unstable when the temperature of the system exceeds the critical value .

RENORMALIZATION OF THE SINE-GORDON MODEL AND NONCONSERVATION OF THE KINK CURRENT

We renormalize the (1+1)-dimensional sine-Gordon model by placing it on a Euclidean lattice, and study the renormalization group flow. We start with a compactified theory with controllable vortex activity. In the continuum limit the theory has a phase in which the kink current is anomalous, with divergence given by the vortex density. The phase structure is quite complicated. Roughly speaking, the system is normal for small coupling T. At the Kosterlitz-Thouless point T=π/2, the current can become anomalous. At the Coleman point T=8π, either the current becomes anomalous or the theory becomes trivial.

Exact effective potential for a scalar source coupled to the sine-Gordon model: Test of effective potentials for composite nucleons.

Bethe ansatz methods are used to obtain an exact expression for the energy density for a system consisting of the (1+1)-dimensional sine-Gordon model coupled to an external scalar source. This method exploits the equivalence between the sine-Gordon model and the massive Thirring model. These calculations allow one to test various loop-expansion approximations to the effective potential for the scalar. Such calculations are of relevance to fundamental issues regarding quantum hadrodynamics.

Deforming some special solutions of the sine-Gordon equation to that of the double sine-Gordon equation

Some special types of solutions in the high dimensional sine-Gordon equation can be deformed to that of the double sine-Gordon equation by solving an ordinary differential equation or using a pure algebraic deformation relation.

Transverse wall instabilities on a driven, damped two-dimensional lattice system.

Analytic and numerical studies are presented for the onset and saturation of transverse patterns occurring on walls propagating in a two-dimensional, longitudinally discrete, underdamped, dc-driven Frenkel-Kontorova model. For sufficiently weak damping, transverse instabilities occur at a series of critical field thresholds, where the wall velocity is of intermediate magnitude. Typically, the transverse patterns saturate as counterpropagating kink-antikink pairs, whose nucleation supports the longitudinal wall propagation. The wall dynamics in a D-dimensional lattice is governed approximately by a damped drive (D-1)-dimensional sine-Gordon equation.

Quantum Brownian Motion of Solitons in the Sine-Gordon Model

The effect of quantum random force on the damped motion of a soliton in the one dimensional sine-Gordon model is investigated by the numerical simulation using quantum Langevin equation. It is found that the mean square displacement of the center of the soliton depends on time t as A t α (0<α<1) where A and α depend on the strength of dissipation. The qualitative feature of the result is also applicable to other more realistic soliton bearing systems such as spin chains or polymers.