Perturbed sine-Gordon Equation Research Articles

ВЗАИМОДЕЙСТВИЕ И ДИНАМИКА ТОПОЛОГИЧЕСКИХ ДЕФЕКТОВ В ДОМЕННОЙ СТРУКТУРЕ ЗАКРУЧЕННОГО НЕМАТИКА

The dynamics and interaction of topological defects in the domain structure of π/2 twisted nematic liquid crystal are studied. A feature of Williams domains in twisted nematics is that hydrodynamic flows in them, along with the tangential velocity component, also have an axial component, the direction of which is opposite in neighboring domains. The axial velocity component arises as a result of the strong coupling between the initial twisted director orientation n and the hydrodynamic flow velocity. In domain structures of π/2 twisted nematics, as well as in planar oriented ones, edge dislocations with topological charges S =±1 arise. With increasing applied voltage density of dislocations grows. Dislocations move both along the Williams domains ( climb ) and perpendicular to them ( glide ). At a certain rate of the increasing applied voltage U > U c, distortions of the domain structure begin to propagate on both sides of the defect core. This leads to the formation of a special topological defects - dislocations with a “diffused” core, or linear defects, which are oriented perpendicular to the Williams domains and move mainly along them. The arising linear defect has the same topological charge as the initial dislocation. It is shown that their interaction is qualitatively well described by the perturbed sine-Gordon equation.

Some Important Points of the Josephson Effect via Two Superconductors in Complex Bases

In this paper, we study the extraction of some analytical solutions to the nonlinear perturbed sine-Gordon equation with the long Josephson junction properties. The model studied was formed to observe the long Josephson junction properties separated by two superconductors. Moreover, it is also used to explain the Josephson effect arising in the highly nonlinear nature of the Josephson junctions. This provides the shunt inductances to realize a Josephson left-handed transmission line. A powerful scheme is used to extract the complex function solutions. These complex results are used to explain deeper properties of Josephson effects in the frame of impedance. Various simulations of solutions obtained in this paper are also reported.

Invariant manifold of modified solitons for the perturbed sine-Gordon equation

We study the perturbed sine-Gordon equation θ tt − θ xx + sin θ = F(ɛ, x), where we assume that the perturbation F is analytic in ɛ and that its derivatives with respect to ɛ satisfy certain bounds at ɛ = 0. We construct implicitly an, adjusted to the perturbation F, manifold which is invariant in the following sense: the initial value problem for the perturbed sine-Gordon equation with an appropriate initial state on the manifold has a unique solution which follows a trajectory on the manifold. The trajectory is precisely described by two parameters which satisfy specific ODEs. In our approach, the invariant manifold is generated as a limit of a sequence whose elements are created one after another by successive distortion of the classical solitary manifold. For the proof, we firstly modify the iteration scheme introduced in Mashkin (2020 Stability of the solitary manifold of the perturbed sine-Gordon equation J. Math. Anal. Appl. 486 123904). By using the modified iteration scheme we build the mentioned sequence. Thereafter we show convergence of the sequence to establish the result.

Random matrix ensembles in hyperchaotic classical dissipative dynamic systems

We study the statistical fluctuations of Lyapunov exponents in the discrete version of the non-integrable perturbed sine-Gordon equation, the dissipative AC- and DC-driven Frenkel–Kontorova model. Our analysis shows that the fluctuations of the exponent spacings in the strictly overdamped limit, which is nonchaotic, conform to an uncorrelated Poisson distribution. By studying the spatiotemporal dynamics, we relate the emergence of the Poissonian statistics to Middleton’s no-passing rule. Next, by scanning values of the DC drive and the particle mass, we identify several parameter regions for which this one-dimensional model exhibits hyperchaotic behavior. Furthermore, in the hyperchaotic regime where roughly fifty percent of the exponents are positive, the fluctuations exhibit features of the correlated universal statistics of the Gaussian orthogonal ensemble (GOE). Due to the dissipative nature of the dynamics, we find that the match between the Lyapunov spectrum statistics and the universal statistics of GOE is not complete. Finally, we present evidence supporting the existence of the Tracy–Widom distribution in the fluctuation statistics of the largest Lyapunov exponent.

Inelastic Interactions of Solitons in a Linear Defect of the Electroconvective Structure of a Nematic Liquid Crystal

The inelastic interactions of solitons in a linear defect appearing in a one-dimensional electroconvective structure of a π/2-twisted nematic liquid crystal have been studied for the first time. Linear defects in twisted nematic liquid crystals are characterized by a sufficiently extended strain field and are oriented normally to Williams domains. Hydrodynamic flows in domains have not only the tangential velocity component but also the axial component whose directions in neighboring domains are opposite. In contrast to the case of planar orientation, the flow continuity condition for an anisotropic liquid in twisted nematic liquid crystals prevents the decay of a linear defect into individual dislocations when an applied voltage increases. This behavior results in the formation of domain zigzag oscillations in the defect core. The boundaries between zig and zag regions are dislocations with the topological charges S = +1 and –1. It has been established that the periodic interaction of dislocations (kinks) with opposite topological charges S = +1 and –1 is accompanied by the formation of a breather. It has been found that the collision of a kink and the breather in the linear defect is inelastic because it leads to the decay of the breather into a kink–antikink pair. It has been shown that such interactions of dislocations with topological charges S = +1 and –1 are qualitatively well described by the perturbed sine-Gordon equation.

Stability of the solitary manifold of the perturbed sine-Gordon equation

We study the perturbed sine-Gordon equation θtt−θxx+sin⁡θ=F(ε,x), where F is of differentiability class Cn in ε and the first k derivatives vanish at ε=0, i.e., ∂εlF(0,⋅)=0 for 0≤l≤k. We construct implicitly a virtual solitary manifold by deformation of the classical solitary manifold in n iteration steps. Our main result establishes that the initial value problem with an appropriate initial state εn-close to the virtual solitary manifold has a unique solution, which follows up to time 1/(C˜εk+12) and errors of order εn a trajectory on the virtual solitary manifold. The trajectory on the virtual solitary manifold is described by two parameters, which satisfy a system of ODEs. In contrast to previous works our stability result yields arbitrarily high accuracy as long as the perturbation F is sufficiently often differentiable.

Dynamics of dislocations in the domain structure of the nematic liquid crystal

Dynamics and interaction of classical dislocations in the domain structure of π/2 nematic liquid crystal is studied. A feature of twisted nematics is that hydrodynamic flows in Williams domains, together with the tangential component of velocity, also have an axial component, the direction of which is opposite in neighboring domains. Dislocations can move both perpendicular (glide) to Williams domains, and along (climb) them. It was found that when dislocations collide with opposite topological charges S = ±1 at given voltage, their speed increases. It has been shown that dynamics and interaction of dislocations with topological charges S = ±1 are qualitatively well described by the perturbed sine-Gordon equation.

Traveling electromagnetic waves in annular Josephson tunnel junctions

It is well known that long Josephson tunnel junctions (JTJs) act as active transmission lines for the slow-mode propagation of magnetic flux-quanta (in the form of solitary waves) that is at the base of many superconducting circuits. At the same time, they support the propagation of quasi-TEM dispersive waves with which the magnetic flux non-linearly interact. In this work, we study the properties of the electromagnetic resonances, under different conditions of practical interest, in annular JTJs (AJTJs), in which the wavelengths are limited to the length of the circumference divided by an integer. Our analysis is based on perturbed sine–Gordon equations the (1+1)-dimensional space with periodic boundary conditions. We discuss the discrete modes of the traveling EM waves in circular AJTJs in the presence of an in-plane magnetic field, as well as in the recently introduced confocal AJTJs (in the absence of magnetic field). In both cases, a variable-separation method leads to quantitatively different Mathieu equations characterized by even and odd spatially periodic solutions with different eigenfrequencies. It implicates that a single mode circulating wave is given by the superposition of two standing waves with the same wavelengths but different frequencies, and so has a periodically inverting direction of propagation. The control parameters of this frequency splitting are the in-plane magnetic field amplitude for the circular AJTJ and the aspect ratio for the confocal AJTJs. In the appropriate limits, the previously known solutions are recovered.

Exact Topological Soliton Solutions to the Strongly Perturbed Family of Sine-Gordon Type Equations

This paper uses the Ansatz method to solve for exact topological soliton solutions to sine-Gordon type equations. Single, double, and triple sine-Gordon and sine-cosine-Gordon equations are investigated along with dispersive and highly dispersive variations. After these solutions are found, strong perturbations are added to each equation and the new solutions are found. In solving both the perturbed and unperturbed sine-Gordon type equations, constraints are imposed on the parameters. The novel contributions of the authors are the soliton solutions to the strongly perturbed sine-Gordon equation and its variations. These results are important to the study of Josephson junctions, crystal dislocations, ultra-short optical pulses, relativistic field theory, and elementary particles.

Solitons in the presence of a small, slowly varying perturbation

ABSTRACT We consider the perturbed sine-Gordon equation , where the external perturbation is small and slowly varying. We show that the initial value problem with an appropriate initial state close enough to the solitary manifold has a unique solution, which follows up to time and errors of order a trajectory on the solitary manifold. The trajectory on the solitary manifold is described by ODEs, which agree up to errors of order with Hamilton equations for the restricted to the solitary manifold sine-Gordon Hamiltonian.

Microscopic Tunneling Model of Nb–AlN–NbN Josephson Flux-Flow Oscillator

Since the very first experimental realization of a Josephson flux-flow oscillator (FFO), its theoretical description has been limited by the phenomenological perturbed sine-Gordon equation (PSGE). While PSGE can qualitatively describe the topological excitations in Josephson junctions that are sine-Gordon solitons or fluxons, it is unable to capture essential physical phenomena of a realistic system such as the coupling between tunnel currents and electromagnetic radiation. Furthermore, PSGE neglects any dependence on energy gaps of superconductors and makes no distinction between symmetric and asymmetric junctions: those made of two identical or two different superconducting materials. It was not until recently when it became possible to calculate properties of FFO by taking into account information about energy gaps of superconductors (Gulevich et al. in Phys Rev B 96:024215, 2017). Such approach is based on the microscopic tunneling theory and has been shown to describe essential features of symmetric Nb–AlO\(_\mathrm{x}\)–Nb junctions. Here, we extend this approach to asymmetric Nb–AlN–NbN junctions and compare the calculated current–voltage characteristics to our experimental results.

On the transition from parabolicity to hyperbolicity for a nonlinear equation under Neumann boundary conditions

An integro differential equation which is able to describe the evolution of a large class of dissipative models, is considered. By means of an equivalence, the focus shifts to the perturbed sine-Gordon equation that in superconductivity finds interesting applications in multiple engineering areas. The Neumann boundary problem is considered, and the behaviour of a viscous term, defined by a higher-order derivative with small diffusion coefficient $$\varepsilon ,$$ is investigated. The Green function, expressed by means of Fourier series, is considered, and an estimate is achieved. Furthermore, some classes of solutions of the hyperbolic equation are determined, proving that there exists at least one solution with bounded derivatives. Results obtained prove that diffusion effects are bounded and tend to zero when $$\varepsilon$$ tends to zero.

Nonlinear Wave Simulation on the Xeon Phi Knights Landing Processor

We consider an interesting from computational point of view standing wave simulation by solving coupled 2D perturbed Sine-Gordon equations. We make an OpenMP realization which explores both thread and SIMD levels of parallelism. We test the OpenMP program on two different energy equivalent Intel architectures: 2× Xeon E5-2695 v2 processors, (code-named “Ivy Bridge-EP”) in the Hybrilit cluster, and Xeon Phi 7250 processor (code-named “Knights Landing” (KNL). The results show 2 times better performance on KNL processor.

Josephson flux-flow oscillator: The microscopic tunneling approach

We elaborate a theoretical description of large Josephson junctions which is based on the Werthamer's microscopic tunneling theory. The model naturally incorporates coupling of electromagnetic radiation to the tunnel currents and, therefore, is particularly suitable for description of the self-coupling effect in Josephson junction. In our numerical calculations we treat the arising integro-differential equation, which describes temporal evolution of the superconducting phase difference coupled to the electromagnetic field, by the Odintsov-Semenov-Zorin algorithm. This allows us to avoid evaluation of the time integrals at each time step while taking into account all the memory effects. To validate the obtained microscopic model of large Josephson junction we focus our attention on the Josephson flux flow oscillator. The proposed microscopic model of flux flow oscillator does not involve the phenomenological damping parameter, rather, the damping is taken into account naturally in the tunnel current amplitudes calculated at a given temperature. The theoretically calculated current-voltage characteristics is compared to our experimental results obtained for a set of fabricated flux flow oscillators of different lengths. Our theoretical calculation agrees well with the obtained experimental results, and, to our knowledge, is the first where theoretical description of Josephson flux flow oscillator is brought beyond the perturbed sine-Gordon equation.

Fluxon Dynamics in Elliptic Annular Josephson Junctions

We analyze the dynamics of a magnetic flux quantum (current vortex) trapped in a current-biased long planar elliptic annular Josephson tunnel junction. The system is modeled by a perturbed sine-Gordon equation that determines the spatial and temporal behavior of the phase difference across the tunnel barrier separating the two superconducting electrodes. In the absence of an external magnetic field the fluxon dynamics in an elliptic annulus does not differ from that of a circular annulus where the stationary fluxon speed merely is determined by the system losses. The interaction between the vortex magnetic moment and a spatially homogeneous in-plane magnetic field gives rise to a tunable periodic non-sinusoidal potential which is strongly dependent on the annulus aspect ratio. We study the escape of the vortex from a well in the tilted potential when the bias current exceeds the depinning current. The smallest depinning current as well as the lowest sensitivity of the annulus to the external field is achieved when the eccentricity is equal to -1. The presented extensive numerical results are in good agreement with the findings of the perturbative approach. We also probe the rectifying properties of an asymmetric potential implemented with an egg-shaped annulus formed by two semi-elliptic arcs.

Elliptic annular Josephson tunnel junctions in an external magnetic field: the statics

We have investigated the static properties of one-dimensional planar Josephson tunnel junctions (JTJs) in the most general case of elliptic annuli. We have analyzed the dependence of the critical current in the presence of an external magnetic field applied either in the junction plane or in the perpendicular direction. We report a detailed study of both short and long elliptic annular junctions having different eccentricities. For junctions having a normalized perimeter less than one the threshold curves are derived and computed even in the case with one trapped Josephson vortex. For longer junctions a numerical analysis is carried out after the derivation of the appropriate perturbed sine-Gordon equation. For a given applied field we find that a number of different phase profiles exist which differ according to the number of fluxon-antifluxon pairs. We demonstrate that in samples made by specularly symmetric electrodes a transverse magnetic field is equivalent to an in-plane field applied in the direction of the current flow. Varying the ellipse eccentricity we reproduce all known results for linear and ring-shaped JTJs. Experimental data on high-quality – elliptic annular junctions support the theoretical analysis provided self-field effects are taken into account.

Semifluxons in 0–π–0 infinitely long Josephson junctions

We investigate both analytically and numerically an infinitely long Josephson junction with two π-discontinuities in the phase characterized by a jump of π in the phase difference of the junction, i.e. a 0–π–0 long Josephson junction on an infinite domain. The dynamics of the system is described by a modified one-dimensional perturbed sine-Gordon equation. We investigate an instability region of the trivial zero solution in which semifluxons are spontaneously generated. A perturbation technique is used to show that the existence of static semifluxons depends on the length of the junction, the facet length, and the applied bias current. Numerical simulations are presented accompanying our analytical results.

Nonlinear waves and solitons in models of fault block geological media

This is an overview of theoretical research on solitons and other nonlinear waves induced by deformation in fault block (fragmented) geological media, including a brief description of available models with their geophysical and geodynamic applications. The cited models correspond to the classical and perturbed sine-Gordon equations and use the rotation angle and displacement of fault blocks as dynamic variables. The models simulate kinematics and dynamics of traveling deformation fronts (kinks and solitons) generated in faulted rocks. The sine-Gordon equation is presented in its seismological and geomechanic applications for description of earthquake data and modeling of strain waves, faulting patterns, and subduction processes, including slow earthquakes and periodic episodic tremor and slow slip (ETS) effects.

On kinks and other travelling-wave solutions of a modified sine-Gordon equation

We give an exhaustive, non-perturbative classification of exact travelling-wave solutions of a perturbed sine-Gordon equation (on the real line or on the circle) which is used to describe the Josephson effect in the theory of superconductors and other remarkable physical phenomena. The perturbation of the equation consists of a constant forcing term and a linear dissipative term. On the real line candidate orbitally stable solutions with bounded energy density are either the constant one, or of kink (i.e. soliton) type, or of array-of-kinks type, or of “half-array-of-kinks” type. While the first three have unperturbed analogs, the last type is essentially new. We also propose a convergent method of successive approximations of the (anti)kink solution based on a careful application of the fixed point theorem.

Propagation of kink—antikink pair along microtubules as a control mechanism for polymerization and depolymerization processes

Among many types of proteinaceous filaments, microtubules (MTs) constitute the most rigid components of the cellular cytoskeleton. Microtubule dynamics is essential for many vital cellular processes such as intracellular transport, metabolism, and cell division. We investigate the nonlinear dynamics of inhomogeneous microtubulin systems and the MT dynamics is found to be governed by a perturbed sine-Gordon equation. In the presence of various competing nonlinear inhomogeneities, it is shown that this nonlinear model can lead to the existence of kink and antikink solitons moving along MTs. We demonstrate kink—antikink pair collision in the framework of Hirota's bilinearization method. We conjecture that the collisions of the quanta of energy propagating in the form of kinks and antikinks may offer a new view of the mechanism of the retrograde and anterograde transport direction regulation of motor proteins in microtubulin systems.