Perturbed sine-Gordon Equation Research Articles

Numerical simulation of the self-pumped long Josephson junction using a modified sine-Gordon model

We have numerically investigated the dynamics of a long Josephson junction (flux-flow oscillator) biased by a DC current in the presence of magnetic field. The study is performed in the frame of the modified sine-Gordon model, which includes the surface losses, RC-load at both FFO ends and the self-pumping effect. In our model the dumping parameter depends both on the spatial coordinate and the amplitude of the AC voltage. In order to find the DC FFO voltage the damping parameter has to be calculated by successive approximations and time integration of the perturbed sine-Gordon equation. The modified model, which accounts for the presence of the superconducting gap, gives better qualitative agreement with experimental results compare to the conventional sine-Gordon model.

Periodic oscillations, peak-splitting and phase transitions of Josephson vortex flow resistance inBi2Sr2CaCu2O8+δ

We investigate both experimentally and theoretically transport properties of the Josephson vortices (JV) with a finite bias current along the $c$ axis of the intrinsic Josephson junctions in layered high-${T}_{c}$ superconductor of ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8+\ensuremath{\delta}}$. Employing the perturbed sine-Gordon equation and integrating a novel characterization factor of the difference between current distributions on neighboring layers $\ensuremath{\Delta}J$, we are successful in explaining new phenomena such as (1) oscillation with period ${H}_{p}$ corresponding to adding one JV per layer, (2) its peak splitting, and (3) reverse of the peaks and valleys with increase of current in JV flow resistance as a function of magnetic field observed in our experiments. Additionally, our simulation shows concrete evidence for the sample size dependent phase transition between ${H}_{p}∕2$-period and ${H}_{p}$-period oscillations. The novel phenomena are expected to be useful for designing practical devices in experiments.

Green's Function Method for Perturbed sine-Gordon Equation**The project supported by National Natural Science Foundation of China under Grant Nos. 10375041 and 10174054

The usual Green's function method is introduced to show the completeness of the squared Jost solutions of the multi-soliton case in this work. Since sine-Gordon equation contains second derivative in time, the known procedure with generalized Marchenko equation to show completeness relation of the eigenfunctions of linearized equation is unavailable. And the explicit expressions of Jost solutions are not necessary here. Thus a general method of direct perturbation method for the perturbed sine-Gordon equation is developed.

A superconducting phase-locked local oscillator for a submillimetre integratedreceiver

Comprehensive measurements of the flux flow oscillator (FFO) radiation linewidthare performed using an integrated harmonic SIS mixer; the FFO linewidth andspectral line profile are compared to a theory. An essential dependence of the FFOlinewidth on frequency is found; a possible explanation is proposed. The results of thenumerical solution of the perturbed sine–Gordon equation qualitatively confirm thisassumption. To optimize the FFO design, the influence of the FFO parameters on theradiation linewidth is studied. A novel FFO design at a moderate current density hasresulted in a free-running FFO linewidth of about 10 MHz in the flux flow regimeup to 712 GHz, limited only by the gap frequency of Nb. This relatively narrowfree-running linewidth (along with implementation of a wide-band phase locking loopsystem) allows continuous phase locking of the FFO in the wide frequency range of500–710 GHz. These results are the basis for the development of a 550–650 GHz integratedreceiver for the terahertz limb sounder (TELIS) intended for atmosphere study andscheduled to fly on a balloon in 2005. We report here also on the design of the secondgeneration of the phase-locked superconducting integrated receiver chip for TELIS.

Breather Dynamics in the Perturbed sine-Gordon Equation**The project supported in part by National Natural Science Foundation of China under Grant No. 10175058

We show that a direct perturbation theory can be used to give a systematic description of the evolution of breather in perturbed sine-Gordon equation. The error in inverse scattering method is also pointed out and corrected to obtain the accurate results.

Length-scale competition for the sine-Gordon kink in a random environment

This paper deals with the transmission of a kink in a random medium described by a randomly perturbed sine-Gordon equation. Different kinds of perturbations are addressed, with time or spatial random fluctuations, with or without damping. We derive effective evolution equations for the kink velocity and width by applying a perturbation theory of the inverse scattering transform and limit theorems of stochastic calculus. Results are

Homoclinic tubes and chaos in perturbed sine-Gordon equation

Sine-Gordon equation under a quasi-periodic perturbation or a chaotic perturbation is studied. Existence of a homoclinic tube is proved. Established are chaos associated with the homoclinic tube, and “chaos cascade” referring to the embeddings of smaller scale chaos in larger scale chaos.

Travelling waves in a singularly perturbed sine-Gordon equation

We determine the linearised stability of travelling front solutions of a perturbed sine-Gordon equation. This equation models the long Josephson junction using the RCSJ model for currents across the junction and includes surface resistance for currents along the junction. The travelling waves correspond to the so-called fluxons and their linear stability is determined by calculating the Evans function. Surface resistance corresponds to a singular perturbation term in the governing equation, which specifically complicates the computation of the corresponding Evans function. Both the flow of quasi-particles across and along the junction stabilise the waves.

Solitary Waves of a Perturbed sine-Gordon Equation**The project supported by the China Postdoctoral Science Foundation (Grant No. 28) and the Shanghai Scientic and Technological Development Foundation (Grant No. 98JC14032)

Based on the bifurcation and the idea that the solitary waves and shock waves of partial differential equations correspond respectively to the homoclinic and heteroclinic trajectories of nonlinear ordinary differential equations satisfied by the travelling waves, different conditions for the existence of solitary waves of a perturbed sine-Gordon equation are obtained. All of the corresponding approximate solitary wave solutions are given by integrating the derived approximate equations directly.

Homoclinic intersections and Mel'nikov method for perturbed sine—Gordon equation

We describe and characterize rigorously the homoclinic structure of the perturbed sine-Gordon equation under periodic boundary conditions. The existence of invariant manifolds for a perturbed sine-Gordon equation is established. The Mel'nikov method, together with geometric analysis are used to assess the persistence of the homoclinic orbits under bounded and time-periodic perturbations.

Homoclinic intersections and Mel'nikov method for perturbed sine–Gordon equation

Homoclinic intersections and Mel'nikov method for perturbed sine–Gordon equation

Boundedness of voltages and currents in Josephson junctions represented by the perturbed sine-Gordon equation

In this note, a superconducting Josephson junction represented by the perturbed sine-Gordon equation is considered. The boundedness of the voltage and the current of such a nonlinear junction is established by showing that an energy-like function corresponding to the junction is bounded for bounded inputs.

Homoclinic intersections and Mel'nikov method for perturbed sine-Gordon equation

We describe and characterize rigorously the homoclinic structure of the perturbed sine-Gordon equation under periodic boundary conditions. The existence of invariant manifolds for a perturbed sine-Gordon equation is established. The Mel'nikov method, together with geometric analysis are used to assess the persistence of the homoclinic orbits under bounded and time-periodic perturbations.

Analytical description of the flux-flow mode in a long Josephson junction

The flux-flow mode in a long one-dimensional Josephson junction is studied analytically. An approximate steady-state solution of the perturbed sine-Gordon equation is derived in the form of a dense fluxon chain accompanied by small-amplitude plasma waves. Next, some time-averaged quantities are calculated, making it possible to evaluate the constant bias current and the constant voltage between the junction electrodes. The analytical results are compared with numerical simulations both for the field distribution within the junction and for the $I\ensuremath{-}V$ characteristics.

Backbending current-voltage characteristic for an annular Josephson junction in a magnetic field

Excitation of the Josephson plasma radiation by a fluxon moving in an annular Josephson junction is studied experimentally, numerically, and using an analytical approach. An externally applied magnetic field H forms a cosinelike potential relief for the fluxon in a ring-shaped junction. The motion of the fluxon in the junction leads to an emission of plasma waves, which give rise to a resonance at a certain fluxon velocity. The experimental data agree well with numerical simulations which indicate a locking of the fluxon to the radiation frequency. The peculiar feature indicated by both experiment and numerical simulations is the shape of the resonance in the current-voltage $(I\ensuremath{-}V)$ characteristic which shows a clear backbending, with a negative differential resistance. The analytical approach developed in this work is based on the perturbation theory for radiation emission generated by a kink in the perturbed sine-Gordon equation. To explain the observed effect, we introduce an addition to the perturbation theory, which proves to be crucial for explanation of the backbending $I\ensuremath{-}V$ curves: We take into account the fact that the background radiation field, supported by a balance between emission from the moving kink and dissipative absorption, narrows the junction's plasma frequency gap. In the case when the emission has a resonant character, even a small change of the gap produces a strong reciprocal effect on the emission power. Following this idea, we develop a fully analytical self-consistent approximation that readily allows us to obtain the backbending $I\ensuremath{-}V$ curves.

First-Order Correction of the Perturbed sine-Gordon Equation

A new direct approach based on the Fourier transformation is developed and applied to study the perturbed sine-Gordon equation. Since a new basis for perturbation expansion is introduced, the first-order correction is expressed in terms of Bessell functions.

Initial condition dependence of the residence time for scattering soliton in a perturbed sine-Gordon equation system

Behaviors of scattering soliton in a perturbed sine-Gordon equation system are numerically studied. We measure the residence time which is defined as the time that the soliton is trapped by an impurity. The initial condition dependence of the residence time has a self-similar fractal structure and the distribution function of the residence time can be expressed as an exponential function e−βτ, where τ is the residence time. The value of β depends on the amplitude of external periodic force G. Such a dependence is expressed as β ~ (G − Gc)η, where Gc is a critical value defined as the value that the soliton cannot pass over the impurity. The value of the critical exponent η is determined from this relation and we obtain η = 0.328. This relation can be derived from an analysis based on the consideration of critical phenomenological behavior.

Direct Approach Based on the Separation of Variables for the Soliton Study of Perturbed Sine-Gordon Equation

A new direct approach based on the separation of variables for the study of soliton perturbations is developed. As an example, the effects of perturbation on a soliton (kink), i.e., both the time-dependence of soliton parameters and the first-order correction are derived for perturbed sine-Gordon equation. Our approach is very concise and easy to understand, and it can be extended directly to deal with other nonlinear evolution equations.

Chaotic fluxon-motion in a Josephson junction transmission line

Chaotic motion of one-fluxon pinned at impurity in a system of a Josephson junction transmission line is studied by numerically solving a perturbed sine-Gordon equation. Time series of voltage at various positions of this line are shown. Intensity of chaotic oscillation spatially decays and feature of chaos changes as the position changes.

An Intermittency Caused by Kink-Antikink Interaction in a Perturbed Sine-Gordon Equation System

Kink-antikink interaction is numerically studied in a system of a perturbed sine-Gordon equation. We study a system in which a kink and an antikink are localized by placing two impurities between them. The kink and the antikink oscillate chaotically near the impurities by applying the external force. We observe kink-antikink motion with changing the strength of coupling, where the strength of coupling can be changed by changing the distance between two impurities. Time series of the distance between two kinks shows similar characteristics to that in the coupled chaotic oscillator system. An intermittency caused by chaos-chaos interaction is also observed. Power spectrum of this intermittency is a type of 1/ f .