One-dimensional sine-Gordon Research Articles

On some solutions of the two-dimensional sine-Gordon equation

There exists an approach for finding of exact solutions of the two-dimensional sine-Gordon equation. Using this approach three classes of solutions have been found. One of the classes consists of running wave solutions which are a generalization of the solutions of the one-dimensional sine-Gordon equation. This class of solutions is studied here.

Instantons in spherical model thermodynamics

The instanton thermodynamics of a spherical model analogous to the soliton thermodynamics of one-dimensional sine-Gordon andφ 4-models is constructed. Decomposition of the system phase volume integral into a sum of contributions corresponding to the thermal fluctuations above the basic and instanton vacua is obtained and all the components of this sum are found. It appears that fluctuations above instanton vacua are Gaussian at all temperature. It is shown that the phase transition temperature in the spherical model can be found from the Kosterlitz-Thouless criterion: in the high-temperature phase the instanton configurations become thermodynamically favorable. The obtained results are exact and are naturally formulated in terms of singularity theory.

Tunnel creation of kink-antikink pairs from an inhomogeneous vacuum by an external field.

The problem of under-barrier creation of kink-antikink pairs by a constant background field is considered within the framework of the one-dimensional sine-Gordon and ${\mathrm{\ensuremath{\varphi}}}^{4}$ models with an inserted periodic lattice of local defects, which renders the true vacuum of the models spatially inhomogeneous. Two types of defects are considered, viz., the ones generating the inhomogeneous vacuum containing dipoles or quadrupoles pinned at sites of the lattice. A specific feature of the pair production in such a system is that one or both members of the pair appear on the mass shell in a state pinned at a site. The energy of a pinned kink is lower than that of a free kink, which results in a substantially shorter under-barrier trajectory and, hence, gives rise to a strong growth of the pair-production rate. In the absence of the background field the dipole inhomogeneous vacuum is double degenerate in its polarization, which gives rise to kinks of the second kind\char22{}the ones reversing the polarization. Mass, charge, and an exponent determining the kink-antikink tunnel-creation rate are found for these kinks.

Parametric instabilities in the discrete sine-Gordon equation

The one-dimensional sine-Gordon equation is an example of an exactly solvable nonlinear partial differential equation. When discretized on a periodic lattice in space and in time, it corresponds to a lattice of pendula coupled by linear springs. We show that the discretized system is unstable to a parametric mode when the frequency of time discretization is sufficiently small, and we obtain the instability condition and growth rate. By considering the effects of two finite amplitude modes, we also obtain the nonlinear saturation of the instability. We also examine how solutions of the exact sine-Gordon equation behave under this map, both in and away from the parametrically unstable regime.

Parametrically Driven One-Dimensional Sine-Gordon Equation and Chaos

Abstract The chaotic behaviour of the parametrically driven one-dimensional sine-Gordon equation with periodic boundary conditions is studied. The initial condition is u(x, 0) = ƒ(x), ut (x, 0) = 0 where ƒ is the breather solution of the one-dimensional sine-Gordon equation at t = 0. We vary the amplitude of the driving force, the frequency of the driving force and the damping constant. For appropriate values of the driving force, frequency and damping constant chaotic behaviour with respect to the time-evolution of w(x = fixed, t) can be found. The space structure u(t = fixed, x) changes with increasing driving force from a zero mode structure to a breather-like structure consisting of a few modes.

Phase-pulling and breather instability in an AC driven-damped one-dimensional sine-Gordon system

Abstract An AC-driven damped sine-Gordon system, in the nonlinear Schrodinger regime, may converge into various nontrivial space-time attractors. One of them is breather riding on top of flat profile, both locked to the driver frequency. Using collective-coordinate ansatz for this attractor, and a multiple-scale perturbation technique, we study the location and nature of its instability. Among the various characteristics, this study includes understanding the role of the breather-background coupling, the structure of the associated unstable modes, the scaling structure of the instability boundary, etc. Most strikingly we find the instability mechanism and the above features to be strongly sensitive to the asymptotic phase-difference between the spatially localized breather and the spatially extended uniform background. We call this phenomenon “phase-pulling”.

Phase-pulling and breather instability in an AC-driven damped one-dimensional sine-Gordon system

An AC-driven damped sine-Gordon system, in the nonlinear Schrödinger regime, may converge into various nontrivial space-time attractors. One of them is breather riding on top of flat profile, both locked to the driver frequency. Using collective-coordinate ansatz for this attractor, and a multiple-scale perturbation technique, we study the location and nature of its instability. Among the various characteristics, this study includes understanding the role of the breather-background coupling, the structure of the associated unstable modes, the scaling structure of the instability boundary, etc. Most strikingly we find the instability mechanism and the above features to be strongly sensitive to the asymptotic phase-difference between the spatially localized breather and the spatially extended uniform background. We call this phenomenon “phase-pulling”.

Pattern competition as origin of intermittency-type chaos

Abstract The one-dimensional sine-Gordon equation, in the presence of driving and damping, is examined numerically. For increasing driver strength the system exhibits an infinite sequence of period doublings leading to chaos. The first bifurcation, however, occurs after a narrow regime characterized by intermittency-type chaos and quasi-periodic oscillations. The origin of the intermittency-type chaos in this system is a competition between two spatial patterns described by the presence of one or two breather-like modes, respectively.

Sine-Gordon kinks on a discrete lattice. II. Static properties.

The differences in the static properties between the discrete and continuum kink solutions of the one-dimensional sine-Gordon equation are found with use of a discretized Hamiltonian formalism which we have developed previously. The difference in the kink profile is treated as a static dressing of the continuum soliton. We point out some of the salient features of the static dressing and their dependence on the kink-size parameter ${l}_{0}$. We find that the dressing has important effects on dynamical quantities such as the kink pinning frequency and the depth of the Peierls-Nabarro potential. When the static dressing is included, the actual depth and curvature of the potential are found to be approximately twice the value obtained using the continuum approximation. Hence, the square of the pinning frequency is doubled. We also find that for short kinks (small ${l}_{0}$) the static dressing develops some spatial structure which we believe will have important consequences in the dynamics, namely in the spontaneous radiation of phonons by the moving kink. We show that our analytic results agree closely with results obtained from molecular-dynamics simulations. The agreement with the simulations provides strong evidence for the reliability of our formalism to predict manifestations of discreteness and their ramifications.

Period doubling and chaos in large area Josephson junctions induced by rf signals

The influence of an applied rf signal on the emitted radiation from a large area Josephson junction is examined. A model of the system is presented in the framework of the one-dimensional sine-Gordon equation. The model linearizes for small and large values of the amplitude of the applied signal. In the intermediate regime lower and upper threshold values (for the amplitude) for transition to chaos are found. Feigenbaum period doubling (period-doubling bifurcation cascade) appears when chaos is approached for increasing amplitude. The findings are supported experimentally.

Complex classical paths and the one-dimensional sine-Gordon system

The semiclassical limit of the Green function for a particle in the one-dimensional sine-Gordon potential is obtained by summing over complex classical paths. The results are the same as those obtained in the less physically intuitive WKB approach. In addition to being of practical utility for solving quantum mechanical problems involving tunnelling, the classical path method may show how to deal with dense configurations of instantons.

Wigner-like expansion for the quantum-statistical mechanics of solids: Application to the sine-Gordon chain.

A semiclassical expansion for the quantum partition function of solids is presented. With respect to the usual Wigner method, the temperature range where the expansion is significant is broadened owing to the correct quantum-statistical treatment of the harmonic modes, while a Wigner-like expansion is retained only for the anharmonic part. As an application, the lowest quantum correction to the specific heat of a one-dimensional sine-Gordon model is calculated analytically.

Path-integral method for soliton-bearing systems. Higher-order corrections in the sine-Gordon model in the classical limit

We investigate systematically the higher-order corrections in $t=\frac{T}{{E}_{s}}$ in the one-dimensional sine-Gordon system in the classical limit, where $T$ is the temperature and ${E}_{s}$ is the soliton energy. Making use of the collective-coordinate technique, we obtain the expression of the momentum-dependent soliton density ${n}_{s}(P)$, which includes the higher-order corrections. This ${n}_{s}(P)$ gives rise to the $t$-dependent corrections in the soliton free energy, which coincide exactly with the corrections found recently by the use of the transfer-integral technique.

Defects in the Sine-Gordon model: Statics

The one-dimensional Sine-Gordon model where the field couples linearly to localized defects is studied. This model describes the phase fluctuations of one-dimensional charge density waves if amplitude fluctuations are neglected. For isolated defects the nonlinear stationarity conditions can be solved analytically. Given these stationary solutions the fluctuation spectrum consisting of a bound state and the scattering states and the relevant correlation functions can be computed exactly. The stationary states for a system of two defects are presented. The free energies as a function of the separation of the defects is computed for the absolutely stable states and for the local minima of the free energy function. This allows us to consider the interaction of the defects induced by the Sine-Gordon field. Finally we compute the order parameter correlation function for a random distribution of defects for small concentrationn. Devising a cumulant expansion the correlation function is found exactly in the first order inn. Our result contains the combined effect of the defects on the stationary states and on the phonon Green's function. It has applicability beyond the present context.

Statistical Mechanics of the 1D Sine-Gordon System. II. Transfer Integral Analysis in the Intermediate Temperature Region

We study thermodynamical properties of the classical one-dimensional sine-Gordon system in finite temperatures comparable with and less than the soliton energy E S by means of the transfer integral method. Comparing its analytical solutions with its exact results obtained numerically, we show that the ideal soliton gas phenomenology breaks down at the temperature region around \(k_{\text{B}}T{\gtrsim}0.2E_{\text{S}}\), above which the interaction between solitons plays an appreciable role even quantitatively.

Brownian Motion of a Kink in Sine-Gordon System and Diffusion Constant

In a one-dimensional sine-Gordon system, a propagating vibration with small amplitude collides with a kink to produce a translation of the kink, giving rise to its Brownian-like motion, since incoming vibrations are excited thermally. The diffusion constant of the kink is obtained, at low temperatures, using the fluctuation-dissipation theorem and the thermal average over the vibrations. It turns out to be one eighth as large as that obtained previously by Fesser. This reduction would be due to the fact that wave packets of the incident vibrations overlap each other. Assuming the validity of the Einstein relation, we use the result to calculate the low field conductivity of quasi-one-dimensional substances at low temperatures. The magnitudes of the observed conductivity prefactors are reproduced for TaS 3 , but not for TTF–TCNQ and TSeF–TCNQ.

Lock-in transitions and charge transfer in one-dimensional fermion systems

The one-dimensional quantum sine-Gordon Hamiltonian with a density rho s of solitons is studied. rho s( mu ) is the order parameter for the lock-in transition, which happens when the chemical potential mu equals the single soliton energy and temperature is T=0. The soliton density rho s( mu ,T) and the critical behaviour at T=0 are studied in the classical limit (quantum coupling beta to 0) and in the quantum system with beta 2=4 pi ; the critical exponent for rho s is 0 and 1/2 respectively. An important application of these results is the temperature dependence of an incommensurate charge transfer rho e in one-dimensional conductors due to Umklapp scattering. rho e(T) can be determined by the function rho s( mu ,T).

A theory of the bion in a one-dimensional sine-Gordon system: II. AC conductivity

A linear response theory is developed for the ac conductivity due to one-dimensional sine-Gordon bions within a semiclassical approximation. As a damping mechanism we consider the diffusive motion of the phase of the internal motion of the bion due to collisions with other bions. The line shape is either Lorentzian or asymmetric Lorentzian with a lower frequency tail. The lower frequency tail has its origin in the bion to bion transition accompanying one-photon annihilation; a new process other than one-bion creation from the vacuum. The effect of an extrinsic damping is also examined, which, in some cases, results in an asymmetric Lorentzian with a higher frequency tail. The results of the far infrared reflection experiments on quasi-one-dimensional conductors TTF-TCNQ and KCP are reanalyzed in the light of the present theory.

A theory of the bion in the one-dimensional sine-Gordon system: I. Quantum statistical mechanics

Introducing an appropriate renormalization to the phase space of the bion, we study the quantum statistical mechanics of the bion in the one-dimensional sine-Gordon system. A bion representation is found for the partition function of the system in the limit of weak coupling, i.e., the free field. From the partition function, the bion distribution is calculated and an approximate analytic formula is obtained both at low and intermediate temperatures. The obtained bion distribution is neither the Bose distribution nor the Fermi distribution, nor the Boltzmann distribution at low temperatures. The self-energy of the bion at finite temperature is also discussed.

Quantum-statistical mechanics of extended objects. I. Kinks in the one-dimensional sine-Gordon system

Making use of thermal-Green's-function technique, we study the quantum-statistical mechanics of a sine-Gordon system in 1 + 1 dimensions. In the weak-coupling limit, the temperature dependences of the soliton energy, ${E}_{s}$, the soliton inertial mass, and the soliton density are determined. At high temperatures ($Tgm$, where $m$ is the mass of the fundamental field), ${E}_{s}$ decreases monotonically as the temperature increases, and ${E}_{s}$ jumps to zero around $T={T}_{\mathrm{cr}}$ ($\ensuremath{\equiv}{e}^{\ensuremath{-}1}{{E}_{s}}^{0}$), where ${{E}_{s}}^{0}$ is the soliton energy at $T=0$ K. The soliton density agrees with the classical statistical-mechanics results for ${T}_{\mathrm{cr}}gT\ensuremath{\gg}m$, if ${E}_{s}$ in the classical theory is replaced by the temperature-dependent one of the present theory.