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A nonuniform magnetic field has been applied to a long Josephson junction to create two separated vortex and antivortex regions. Such a configuration is the Josephson version of the so‐called fluxonic diode, an electrical dual to the semiconductor p‐n junction diode. Volt‐ampere relations of a junction subjected to this magnetic field are compared with those obtained using a unidirectional magnetic field. The study, supported by simulation, shows a number of unique characteristics including rectification and fluxon injection across the junction. Both are necessary phenomena for realization of a bipolar transistor dual.

The phase dynamics of a long Josephson junction with spatially inhomogeneously distributed bias current is considered for the case of a dense soliton chain (regime of the Flux Flow oscillator). To derive the analytical solution of the corresponding sine-Gordon equation the Poincare method has been used. In the range of the validity of the theory good coincidence between analytically derived and numerically computed current-voltage characteristics have been demonstrated for the simplest example of unitstep function distribution of bias current (unbiased tail). It is shown, that for the considered example of bias current distribution, there is an optimal length of unbiased tail that maximizes the amplitude of the main harmonic and minimizes the dynamical resistance (thus leading to reduction of a linewidth)., 7 pages, 5 figures

We report the effect of quantum and thermal fluctuations on stability of mutual phase locking in one-dimensional long Josephson junction devices, involving layered superconductors. Accounting for both the induction coupling and the charging effect, we determined the zero-temperature $(T=0)$ phase diagram, using renormalization-group analysis, and found that the in-phase mode is stable, but some out-of-phase modes are unstable against quantum fluctuations. At finite T, all stable phase-locking modes (at $T=0)$ are unstable, but stability is still maintained within a finite length, which decreases inversely with T. In ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8+y},$ this length for the in-phase mode is roughly 1000 \ensuremath{\mu}m at 1 K.

We investigate the collective phase dynamics in conventional long Josephson junction (LJJ) stacks and in layered superconductors, exhibiting intrinsic LJJ behaviors. Using a theoretical model which accounts for both the magnetic induction effect and the breakdown of local charge neutrality (i.e., charging effect), we show that the collective motion of Josephson vortices, including the dispersion of Josephson plasma mode and the Swihart-type velocity, in an intrinsic LJJ stack such as Bi$_2$Sr$_2$CaCu$_2$O$_{8+y}$ (BSCCO) is significantly modified from those in a conventional LJJ stack. In BSCCO, the strength of the charging effect $\alpha$ is small (i.e., $\alpha \sim 0.1-0.4$), but it leads to notable changes in collective phase dynamics, including changes to the stability condition. Also, we show that splitting of the supercurrent branch in the resistive state is due to collective motion of Josephson vortices. The width of spread of these sub-branches in the linear current-voltage regime depends on $\alpha$, suggesting another way to measure the charging effect in BSCCO., Comment: 27 pages, 6 figures, to appear in Physica C (2002)

We investigate numerically a long Josephson junction with several phase pi-discontinuity points. Such junctions are usually fabricated as a ramp between an anisotropic cuprate superconductor like YBCO and an isotropic metal superconductor like Nb. From the top, they look like zigzags with pi-jumps of the Josephson phase at the corners. These pi-jumps, at certain conditions, lead to the formation of half-integer flux quanta, which we call semifluxons (SF), pinned at the corners. We show (a) that the spontaneous formation of SFs depends on the junction length, (b) that the ground state without SFs can be converted to a state with SFs by applying a bias current, (c) that the SF configuration can be rearranged by the bias current. All these effects can be observed using a SQUID microscope., Comment: ~8 pages, 6 figures, submitted to PRB

The discovery of the Josephson effect stimulated the development of superconductor microelectronics with a huge number of both theoretical and experimental papers devoted to it. Such interest is caused by its great technological potential and the value of the Josephson effect in information processing, generation of ultra-high frequency oscillators, fast switching devices, mixers, detectors, amplifiers and so on (see, e.g., [3.1–3,3.7]).

The resonance instability of breathers and the dynamics vortex production for weak nonstationary action on a long Josephson junction are examined. A combined fluctuation-dynamic mechanism is proposed for disruption of the spatially homogeneous state in a long Josephson junction. The resulting resistive state is analyzed.

The paper reports on experimental observation of sub-terahertz sound wave generation and detection by a long Josephson junction. This effect was discovered at spectral measurements of sub-terahertz electromagnetic emission from a flux-flow oscillator (FFO) deposited on optically polished Si substrate. The back action of the acoustic waves generated by the FFO and reflected by the bottom surface of the Si substrate results in the appearance of resonant steps in the FFO IVCs with spacing as small as 29 nV for 0.3 mm substrate thickness; these steps manifest themselves in a pronounced resonant structure in the emission spectra with spacing of about 14 MHz precisely according to the Josephson relation. The mechanism of acoustic wave generation and detection by the FFO is discussed; a possibility for employing the discovered effect for FFO frequency stabilization has been demonstrated. A simple and reliable way to suppress the superfine resonant structure has been developed and proven; this invention allows for continuous frequency tuning and FFO phase locking at any desired frequency, all of which is vitally important for most applications., 6 pages, 7 figures

We present a theoretical and experimental study of Josephson vortex (fluxon) moving in the presence of spatially homogeneous dc and ac bias currents. By mapping this problem to the problem of calculating the current-voltage characteristic of a small Josephson junction, we derive the dependence of the average fluxon velocity on the dc bias current. In particular we find that the low frequency ac bias current results in an additional nonlinear damping of fluxon motion. Such ac induced damping crucially depends on the intrinsic damping parameter and increases drastically as this parameter is reduced. We find a good agreement of the analysis with both the direct numerical simulations and the experimentally measured current-voltage characteristics of a long annular Josephson junction with one trapped fluxon., Physical Review B, in press

We studied thermally activated and macroscopically quantum-tunneled switching from the zero-voltage state to a first resistive state in stacks of short and long Josephson junctions (JJs) naturally formed in Bi 2 Sr 2 CaCu 2 O 8+ δ single crystals. In the short (long) JJs the switching corresponds to an escape of a phase particle (a flexible phase line) from a potential well. The effective barrier height in the long JJs is reduced in proportion to the length of the junctions.

A report of numerical experiment results on long Josephson junction with one and two rectangular inhomogeneities in the barrier layer is presented. In case of one inhomogeneity we demonstrate the existence of the asymmetric fluxon states. The disappearance of mixed fluxon-antifluxon states when the position of inhomogeneity shifted to the end of the junction is shown. In case with two inhomogeneities the change of the amplitude of Josephson current through the inhomogeneity at the end of junction makes strong effect on the stability of the fluxon states and smoothes the maximums on the dependence ``critical current - magnetic field''., Presented for M2S, Dresden, July 9-14, 2006

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.

A half-fluxon on a long Josephson junction with a chiral p-wave superconductor is studied. The half-fluxon is similar to the half quantum vortex (half-fluxoid) proposed by Volovik and Mineev in that the half quantum vortex results from the cooperation between two kinds of phases provided by a superconducting order parameter: one is a global phase related to the symmetry \(U(1)\) and the other is the rotation phase of a \(\hat{\mathbf{d}}\) vector related to the symmetry \(\hat{D}^{(+)}_{(1)}\) in spin space. Each phase for the half-fluxon varies with \(\pi\) along a long Josephson junction. In contrast, each phase varies with \(2\pi(n+1/2)\) along a half-fluxoid. A half-fluxon and a half-fluxoid are accompanied by a spin current, which is induced by the spatial modulation of a \(\hat{\mathbf{d}}\) vector. This half-fluxon is different from a half-fluxon on a Josephson 0–\(\pi\)-junction in terms of the spin current. A spin current is parallel or antiparallel to a supercurrent. Therefore, the spin current ...

Fluxon collisions have recently been successfully imaged by a focused electron beam [A. Laub {ital et} {ital al}., Phys. Rev. Lett. {bold 75}, 1372 (1995)]. This report presents numerical and analytical calculations of the effect of a beam-induced dissipative spot on a fluxon-fluxon collision in a long Josephson junction. In contrast to the fluxon-antifluxon collision, the energy loss for the colliding fluxons of the same polarity is minimal at the collision point. Good accord between analytical and numerical results is found. Presented calculations are relevant for interpretation of the low temperature scanning electron microscopy experiments on fluxon imaging in long Josephson junctions. {copyright} {ital 1996 The American Physical Society.}

Macroscopic quantum tunneling (MQT) for a single fluxon moving along a long Josephson junction is studied theoretically. To introduce a fluxon-pinning force, we consider inhomogeneities made by modifying thickness of an insulating layer locally. Two different situations are studied: one is the quantum tunneling from a metastable state caused by a single inhomogeneity, and the other is the quantum tunneling in a two-state system made by two inhomogeneities. In the quantum tunneling from a metastable state, the decay rate is estimated within the WKB approximation. Dissipation effects on a fluxon dynamics are taken into account by the Caldeira-Leggett theory. We propose a device to observe quantum tunneling of a fluxon experimentally. Required experimental resolutions to observe MQT of a fluxon seem attainable within the presently available micro-fabrication technique. For the two-state system, we study quantum resonance between two stable states, i.e., macroscopic quantum coherence (MQC). From the estimate for dissipation coefficients due to quasiparticle tunneling, the observation of MQC appears to be possible within the Caldeira-Leggett theory., 30 pages LaTeX including 11 PS figures, using jpsj.sty. To be published on J. Phys. Soc. Jpn. Overestimates for damping amplitude is corrected

Theoretical model for the radiation linewidth in a multi-fluxon state of a long Josephson junction is presented. Starting from the perturbed sine-Gordon model with the temperature dependent noise term, we develop a collective coordinate approach which allows to calculate the finite radiation linewidth due to excitation of the internal degrees of freedom in the moving fluxon chain. At low fluxon density, the radiation linewidth is expected to be substantially larger than that of a lumped Josephson oscillator. With increasing the fluxon density, a crossover to a much smaller linewidth corresponding to the lumped oscillator limit is predicted., Comment: 11 pages LaTeX, to appear in Phys Rev B

Static and dynamical states in one‐dimensional parallel arrays of small, all refractory Josephson tunnel junctions, are experimentally investigated at 4.2 K. The dependence of the maximum Josephson current on the applied magnetic field in parallel arrays shows asymmetric behavior around secondary maxima which are discussed in terms of a nonuniform applied magnetic field. Current steps and linear branches induced by the applied magnetic field are observed together in the current‐voltage characteristics for the first time in the strongly underdamped regime. The resonant structures appearing in the current‐voltage characteristics of the arrays show features similar to high‐order Fiske modes in a long Josephson junction.

We study the phase locking of a long Josephson junction operating in a fluxon oscillator regime to external rf signals whose frequency is an even harmonic of the oscillator frequency. The phenomenon is investigated for different values of the dc bias current in the junction, and around each value the current intervals that allow phase locking are evaluated. These intervals can be well identified even when the drive frequency is sixteen times the fluxon oscillator frequency. The explanation of the numerical data that we obtain is given in terms of analytical approximations for a long Josephson junction model. An equation containing relevant experimental parameters is derived for the current-locking ranges generated by the different harmonics. This equation establishes that the amplitude of these intervals decreases exponentially with the harmonic number and we show how this result can be related to a property of the Fourier spectrum of fluxon oscillations. In all the cases that we analyze very good agreement is found between the numerical evidence and theoretical analysis.

The dynamic behavior of fluxons in a current biased long Josephson junction provides a system for studying quantum tunneling of a macroscopic variable, the total magnetic flux through the junction. Adjacent Fiske steps on the I-V curve provide potential wells where, at appropriate bias conditions and temperatures, tunneling can be observed. The junction parameters determine the tunneling behavior of the variable; it is observed to be incoherent. However, for certain conditions the system may exhibit coherent behavior.

We report first systematic measurements of zero-field steps in I-V characteristics of long Nb/Al-AlO/sub x//Nb Josephson junctions made in a window of a stripline structure. We compared the data for junction series of the same dimensions but with different ratio of the junction width W to the width of the idle window part W' surrounding them. Zero-field steps were found to be stable only in structures with W'/W >

The second harmonic contribution to the current-phase relation changes the properties of the static magnetic flux distributions in the long Josephson junction (LJJ) and inspires new homogenious and fluxon static states. We study stability properties and bifurcations of these static regimes within the frame of a model described by the double sine-Gordon equation. The critical curves behavior and the interconnection between different types of magnetic flux distributions are analyzed.

For creation of a noisy non-stationary spectrometer operating in the sub-THz frequency range, the emitting power and spectral characteristics of a flux-flow oscillator based on a long Josephson junction are considered. The effect of bias current profiles on the spectral linewidth of a long overlap Josephson junction is investigated and compared with the inline junction geometry. It has been demonstrated that the spectral linewidth can be increased by a factor of three with a moderate reduction of the emitted power.

We analyze motion of a fluxon in a weakly damped ac-driven long Josephson junction with a periodically modulated maximum Josephson current density. We demonstrate both analytically and numerically that a pure {\it ac} bias current can drive the fluxon at a {\it resonant} mean velocity determined by the driving frequency and the spatial period of the modulation, provided that the drive amplitude exceeds a certain threshold value. In the range of strongly ``relativistic'' mean velocities, the agreement between results of a numerical solution of the effective (ODE) fluxon equation of motion and analytical results obtained by means of the harmonic-balance analysis is fairly good; morever, a preliminary PDE result tends to confirm the validity of the collective-coordinate (PDE-ODE) reduction. At nonrelativistic mean velocities, the basin of attraction, in position-velocity space, for phase-locked solutions becomes progressively smaller as the mean velocity is decreased., 15 pages, 26 kbytes, of text in plain LaTeX. A uuencoded, Z-compressed tar archive, 21 kbytes, containing 3 PostScript, filatrella@pit.physik.uni-tuebingen.de, malomed@math.tau.ac.il, parment@salerno.infn.it

The superconducting Josephson junctions (Barone and Paterno 1982; Parmentier 1978; Lomdahl 1985; Likharev 1986; Pedersen 1986) have proven to be one of the most successful testing grounds for nonlinear wave theory; their use for information processing and storage is quite attractive. In the long Josephson junction or transmission line, the physical quantity of interest is a quantum of magnetic flux, or a fluxon, which has a soliton behavior. It is a remarkably robust and stable object, which can be easily manipulated at high speed and stored electronically. Consequently it should be used as a basic bit in information processing systems.

We consider the damped and parametrically driven sine-Gordon equation $$ {u_{tt}} - {u_{xx}} + \sin u = \alpha {u_t} + \Gamma \sin (\omega t)\sin \left( {\frac{u}{n}} \right), $$ (1) with periodic boundary conditions u (x − L/2, t) = u(x + L/2, t). Equation (1) with n = 2 may be derived, for example, as an effective equation of motion for the magnetization vector in several magnetic models, u being an angle describing the orientation of the magnetic vector in a selected (easy anisotropy) plane. The perturbation from the r. h. s. of Eq. (1) appears if one considers a variable magnetic field1, 2. Another physically relevant example of Eq. (1) with n = 1 is a long Josephson junction with parametrically varying critical current3.

The current-voltage and spectral characteristics of a flux flow oscillator (FFO) based on a long Josephson junction are studied. The investigations are performed in the range of small bias currents and magnetic fields where the FFO radiates a quasi-chaotic signal with extremely large radiation linewidth, and the displaced linear slope (DLS) is observed at the current-voltage characteristic. Using direct numerical simulation of the sine-Gordon equation, it is shown that for large lengths of the Josephson junction or in the case of partial matching of the FFO with external waveguide system, the DLS with extremely large linewidth is transformed into Fiske steps with very narrow linewidth. As for Fiske steps, the appearance of regime of chaotic oscillations can be explained by multiple reflections of the traveling waves from junction ends rather than simply by excitation of the internal oscillation modes in the “soft” fluxon chain at weak magnetic fields.

Phase locking of fluxon motion in a long Josephson junction is investigated theoretically for microwave coupling through the boundaries of the system. We demonstrate that the wave properties of the fluxon motion during reflection are crucial for predicting the size of the locking range. Analytical results are compared to direct numerical simulations of the sine-Gordon model and excellent agreement is found.

The authors report on results of phase-locking experiments in a system composed of two long Josephson junctions oscillating at millimeter-wave frequencies. The effects of the coherent behavior of the two junctions are visible on the current-voltage characteristic of the long junctions and in the power enhancement effects observed on the current-voltage characteristic of a detector junction. It is shown that is possible to couple two long junctions in such a way that the enhanced radiation can be transmitted to other devices. Results from another mode of operation of the long Josephson junction as an oscillator are also presented. This mode is based on magnetic-field-induced singularities in long junctions (Fiske steps) generated by the interaction of the cavity modes of the junction seen as an open-ended transmission line and the Josephson oscillations. This mode has given very promising results above 200 GHz, where a power of 20 nW was coupled in a 30- Omega normal state resistance tunnel junction. >

Numerical simulations of the multiple fluxon dynamics stimulated by an external oscillating force applied at a boundary of a long Josephson junction are presented. The calculated I-V characteristics agree well with a recent experimental observation of rf-induced satellite flux-flow steps. The voltage spacing between the satellites and the main flux-flow step corresponds to the second harmonic of the external frequency. We find that the satellite flux-flow modes are characterized by collective excitations propagating through the fluxon chain. These dynamical states can be interpreted as fluxon density waves

We have measured the Josephson radiation emitted at millimeter‐wave frequencies by a long Josephson junction dc‐current biased on Fiske steps of the current‐voltage characteristic. Our measurements demonstrate that differences between dynamic excitation generating Fiske steps and fluxon oscillations giving rise to zero‐field steps in long junctions may exist. The radiation, whose maximum measured frequency is 240 GHz, is generated by an in‐line junction and detected on a chip by a small planar tunnel junction. From measurements of Shapiro steps amplitude and photon‐assisted tunneling steps we find that the coupled radiation has a power tunability of 350 nW and the maximum coupled power is 20 nW in a 30‐Ω normal‐state resistance tunnel junction.

We have used low-temperature scanning electron microscopy for the investigation of the interaction of fluxons (Josephson vortices) with a superconducting (Abrikosov) vortex trapped in one of the electrodes of a long quasi-one-dimensional Josephson junction. The Abrikosov vortex penetrated a top superconducting film and spread out in the tunnel barrier area between the films. In addition, we were able to move the Abrikosov vortex across the junction by the electron beam and to measure at the same time the value of the critical current of the junction. A model explaining a critical-current dependence on the external magnetic field and on the Abrikosov vortex position is proposed. We have also observed the bound states of a fluxon trapped by the Abrikosov vortex in the Josephson junction.

We study the long Josephson junction (LJJ) model which takes into account the second harmonic of the Fourier expansion of Josephson current. The dependence of the static magnetic flux distributions on parameters of the model are investigated numerically. Stability of the static solutions is checked by the sign of the smallest eigenvalue of the associated Sturm-Liouville problem. New solutions which do not exist in the traditional model, have been found. Investigation of the influence of second harmonic on the stability of magnetic flux distributions for main solutions is performed., Comment: 4 pages, 6 figures, to be published in Proc. of Dubna-Nano2010, July 5-10, 2010, Russia

A Flux-flow oscillator based on a long Josephson junction is a local oscillator in fully superconducting integrated receivers (SIRs). The low frequency voltage fluctuations define the free-running FFO radiation linewidth, an important parameter for employing the SIR as a submm spectrometer. The DC voltage of the oscillator is independently controlled by both the dissipative DC bias current IB and the loss-free external magnetic field at the two FFO ends. It is mainly the natural wideband current noise of the Josephson barrier, which is converted into the fluctuations of voltage by the differential resistance Rd on the current IB. We have developed and experimentally demonstrated the method for suppression of the voltage fluctuations by convey of the bias current noise into the anticorrelated noise of the magnetic field. The experiments were made with Nb/AlN/Nb FFOs, having a specially designed shunting structure at the end, where flux quanta enter the junction.

Collision of a free fluxon in a long dc-driven damped Josephson junction with a fluxon pinned by the microshort or microresistor is analyzed by means of perturbation theory. It is demonstrated that, in the case of like polarities of the fluxons, three different outcomes of the collision are possible: depinning, capture, and exchange. If the polarities are opposite, the possible outcomes are depinning, annihilation, and a quasielastic collision. The boundaries between the different regimes in the parametric space of the model are found.

We present a numerical scheme for solving a nonlinear partial integro-differential equation with nonlocal time dependence. The equation describes the dynamics in a long Josephson junction modeled by use of the microscopic theory for tunneling between superconductors. We demonstrate that the detailed behavior of a solitonic mode (fluxon dynamics) in the junction is different from the results of the conventional perturbed sine-Gordon model.

The influence of thermal noise on fluxon motion in a long Josephson junction is investigated when the motion is phase locked to an external microwave signal. It is demonstrated that the thermal noise can be treated theoretically within the context of a two-dimensional map that models the dynamics of a single fluxon.

Experiments are reported on a new geometry designed to couple long Josephson junction fluxon oscillators to a resonant cavity. The junctions were made with a niobium-aluminum oxide-niobium trilayer process with a critical-current density of around 1000 A/cm2. Various numbers of such junctions were incorporated directly as part of a coplanar strip half-wave resonator, with fundamental mode of about 34 GHz. Both the current density and oscillation frequency are higher than in previous experiments. Evidence for phase locking of multiple junctions is presented. Applied Physics Letters is copyrighted by The American Institute of Physics.

Spectrum of oscillations in the dynamic state of long Josephson junction (JJ) in parallel magnetic field is studied analytically and numerically. In the flux-flow state, interaction of the moving fluxon train with the edges of the JJ results in resonant generation of an acoustic fluxon density wave. A new type of resonance occurs at higher voltages, above the flux-flow branch. Here the spectrum consists of two correlated frequencies. Resonant mixing with the fundamental Josephson frequency results in large amplitude of oscillations. Possible applications of such resonant mixing as a frequency multiplier and adjustable narrow line-width generator are discussed.

The motion of magnetic flux quanta in long Josephson junctions has been studied intensively in the last decade and became a very interesting area of both theoretical and experimental soliton physics. In a long Josephson junction (LJJ) a magnetic flux quantum (fluxon) corresponds to a soliton solution of the perturbed sine-Gordon equation

A flux-flow type Josephson oscillator (FFO), which utilizes the unidirectional flux flow (a moving vortex array) in a long Josephson junction has been consided for use as a mm and submm wave oscillator in integrated superconducting receiver systems mainly because it gives high output power and wide bandwidth.1–3

The coupling of millimeter-wave radiation from a long Josephson junction to a small-area junction is discussed. The coupling is obtained by means of thin-film technology and the design parameters allowing radiation coupling are discussed. Since the frequency of the oscillator radiation is smaller than the plasma frequency of the detector junction, the latter shows evidence of chaotic dynamics in the current-voltage characteristic. However, very stable zero-crossing bias steps appear for high values of the applied radiation power. The tunability of the oscillator power allows measuring the dependence of the detector critical current on the external radiation to be measured; from this measured dependence and from the observed amplitude of the photon-assisted tunneling steps it is estimated that maximum coupled power is the range of tens of nanowatts.

When a long Josephson junction oscillator is phase locked to an external microwave field, vertical steps appear in the IV-curve at voltages corresponding to fluxon frequencies ω F which are rationally related to the external driving frequency ω by

Fluxon dynamics in a variety of Long Josephson Junction CLJJ) systems is studied using the collective coordinate analysis of McLaughlin and Scott, and the results of the ODE predictions are compared with the numerical solution of the full PDE for each problem. The LJJ systems we have studied include: Single LJJ*s with several microresistor Cor microshort) inhomogeneities, an LJJ model proposed by Reinisch and Fernandez with added inhomogeneities and a system of two coupled LJJ*s introduced by Kivshar and Malomed. In all cases, the effects of forcing and damping are taken into account and interesting phenomena of fluxon trapping and bifluxon formation are observed and discussed.

In the paper, we consider the problem of numerical simulation of long Josephson Junction (JJ) devices. Our interest to the problem is motivated by need of erective parameters calculation of Flux Flow Oscillator (FFO). Practical FFOs have complicated structures and traditional model in form of simple perturbed sine-Gordon equation is inadequate because it doesn't capture important physical details. In this work, we evaluate variable coeficient 1D sine-Gordon equation and boundary conditions for this 1D equation starting from full 2D model of junction and 3D magnetic field problem for bias and control line currents. Then this 1D problem is solved numerically using implicit time stepping numerical procedure. As result program FFOSIM was developed. It is a general tool that allow calculations of critical current, I-V curves and other values. We present results of calculations of junctions inside microstrips where boundary conditions were evaluated using 3D magnetic field calculation program 3DMLSI.

Numerical experiment results on long Josephson junction with one and two rectangular inhomogeneities in the barrier layer are presented. We demonstrate the efiect of the shifting of the inhomogeneity and the value of the Josephson current on the vortex structure. The disappearance of mixed fluxon-antifluxon states is shown when the position of inhomogeneity shifted to the end of the junction. A change of the amplitude of Josephson current at the end makes a strong efiect on the stability of the fluxon states and smoothes the maximums of the dependence 'critical current-magnetic field'.

The flux-flow dynamics in a long Josephson junction is investigated for a spatially nonuniform bias current density. Starting from the linearization about a rotating background, a simple approximate solution of the modified sine–Gordon equation is presented, making possible to derive the current–voltage characteristic of the junction. As an example of nonuniform driving, an overlap structure with an unbiased tail is discussed. In particular, it is shown how a nonuniform distribution of the bias current density may affect the dynamical resistance of a flux-flow oscillator. Analytical results are compared with direct numerical simulations and good agreement is found for a wide range of junction dimensions.