Modified Ginzburg-Landau Equation Research Articles

Unusual Sequence of the Critical Magnetic Fields Hc1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{c1}$$\\end{document}, Hc2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{c2}$$\\end{document}, and Hc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{c}$$\\end{document} in Multicomponent Superconductors

All superconductors in a magnetic field are characterized by three critical magnetic fields: lower critical Hc1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{c1}$$\\end{document}, upper critical Hc2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{c2}$$\\end{document} and thermodynamic critical field Hc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{c}$$\\end{document}. Only two sets of inequalities Hc2>Hc>Hc1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{c2}>H_c>H_{c1}$$\\end{document} or Hc1>Hc>Hc2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{c1}>H_c>H_{c2}$$\\end{document} are possible in a single-component superconductor. Here, we report our study of the critical fields in multicomponent superconductors with two superconducting components in the framework of the Ginzburg-Landau functional. We derive the relationship between the phases of the components of the superconducting complex order parameter from the charge conservation law in explicit form and insert it into the Ginzburg-Landau functional. Using the modified Ginzburg-Landau equation, we acquire the single vortex state including the analytical expression for asymptotics. Also, we obtain the analytical form for the state in the upper critical field. We find that in some cases an unusual sequence of critical fields Hc1,Hc2>Hc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{c1},H_{c2}>H_c$$\\end{document} can be realized in multicomponent superconductors.

Hybrid Superconducting/Superconducting Mesoscopic Heterostructure Studied by Modified Ginzburg–Landau Equations

Studies involving vortexes in hybrid superconducting devices and their interactions with different components inside samples are important for reaching higher values of critical parameters in superconducting materials. The vortex distribution on each side of a sample with different fundamental parameters, such as temperature T, penetration depth λ, coherence length ξ, electron mass m, and the order parameter Ψ, may help to improve the superconducting properties. Thus, in this work, we used the modified Ginzburg–Landau theory to investigate a hybrid superconductor (HS), as well as to provide a highly tunable and adjustable theoretical tool for theoretically explaining the experimental results involving the HS in order to study the vortex behavior in superconductors of mesoscopic dimensions with extreme differences among their fundamental parameters. Therefore, we evaluated the influence of the HS on the vortex configuration and its effects on field-dependent magnetization. The results show that when the applied magnetic field H was increased, the diamagnetic response of the HS (Meissner effect) included additional jumps in magnetization, while diamagnetism continued to increase in the sample. In addition, the differences among parameters created an interface between both components, and two different magnitudes of supercurrent and vortex sizes caused less degradation of the local superconductivity, which increased the upper critical field. On the other hand, this type of HS with differences in parameters on both sides can be used to control the vortex movement in the selected sample of the superconducting region with more accuracy.

On nonstandard chemotactic dynamics with logistic growth induced by a modified complex Ginzburg–Landau equation

AbstractIn this paper, we derive a variant of the classical Keller–Segel model of chemotaxis incorporating a growth term of logistic type for the cell population , say with , and a nonstandard chemical production–degradation mechanism involving first‐ and second‐order derivatives of the logarithm of the cell density, say with , via the ()‐hydrodynamical system associated with a modified Ginzburg–Landau equation governing the evolution of the complex wavefunction . In a chemotactic context, will play the role of the concentration of chemical substance. Then, after carrying out a detailed analysis of the modulational stability of uniform‐in‐space plane waves, dark soliton‐shaped traveling wave densities of the former system are constructed from solitary wave solutions of the latter.

Study of a dissipative soliton in the distributed mathematical model of ultra-long mode-locked fiber lasers

We propose the new, to the best of our knowledge, distributed mathematical model of dissipative soliton evolution in ultra-long fiber lasers. The model is based on a modified Ginzburg–Landau equation, has a stable analytical solution, and takes into account the saturated gain and saturable absorption. The analytical results show a good accordance with the results of numerical simulation of ultra-long ring cavity fiber lasers. The results may be applied to the analysis of a wide range of fiber laser systems.

Gauge-invariant microscopic kinetic theory of superconductivity in response to electromagnetic fields

Within a gauge-invariant microscopic kinetic theory, we study the electromagnetic response in the superconducting states. Both superfluid and normal-fluid dynamics are involved. We predict that the normal fluid is present only when the excited superconducting velocity $v_s$ is larger than a threshold $v_L=|\Delta|/k_F$. Interestingly, with the normal fluid, we find that there exists friction between the normal-fluid and superfluid currents. Due to this friction, part of the superfluid becomes viscous. Therefore, a three-fluid model: normal fluid, non-viscous and viscous superfluids, is proposed. For the stationary magnetic response, at $v_s<v_L$ with only the non-viscous superfluid, the Meissner supercurrent is excited and the gap equation can reduce to Ginzburg-Landau equation. At $v_s{\ge}v_L$, with the normal fluid, non-viscous and viscous superfluids, in addition to the directly excited Meissner supercurrent in the superfluid, normal-fluid current is also induced through the friction drag with the viscous superfluid current. Due to the normal-fluid and viscous superfluid currents, the penetration depth is influenced by the scattering effect. In addition, a modified Ginzburg-Landau equation is proposed. We predict an exotic phase in which both the resistivity and superconducting gap are {\em finite}. As for the optical response, the excited ${v_s}$ oscillates with time. When $v_s<v_L$, only the non-viscous superfluid is present whereas at $v_s{\ge}v_L$, normal fluid, non-viscous and viscous superfluids are present. We show that the excited normal-fluid current exhibits the Drude-model behavior while the superfluid current consists of the Meissner supercurrent and Bogoliubov quasiparticle current. Due to the friction between the superfluid and normal-fluid currents, the optical conductivity is captured by the three-fluid model. ......

The Magnetic Field Distribution of Type II Superconductors Based on the Modified GL Equations

The standard Ginzburg-Landau (GL) equations are only valid in the vicinity of the critical temperature. Based on the Eilenberger equations for a single band and s-wave superconductor, we derive a modified version of the standard GL equations to improve the applicability of the standard formalism at temperature away from the critical temperature. It is shown that in comparison with previous studies, our method is more convenient to calculate and our modified equations are also compatible with a dirty superconductor. To illustrate the usefulness of our formalism, we solve the modified equations numerically and give the magnetic field distribution in the mixed state at any temperature. The results show that the vortex lattice could be still observed even away from the critical temperature (e.g., T/Tc = 0.3).

Dissipative Phase Solitons in Semiconductor Lasers.

We experimentally demonstrate the existence of nondispersive solitary waves associated with a 2π phase rotation in a strongly multimode ring semiconductor laser with coherent forcing. Similarly to Bloch domain walls, such structures host a chiral charge. The numerical simulations based on a set of effective Maxwell-Bloch equations support the experimental evidence that only one sign of chiral charge is stable, which strongly affects the motion of the phase solitons. Furthermore, the reduction of the model to a modified Ginzburg-Landau equation with forcing demonstrates the generality of these phenomena and exposes the impact of the lack of parity symmetry in propagative optical systems.

Modulational stability of solitary states in a lossy nonlinear electrical line

The modified Ginzburg–Landau equation that describes the pulse propagation in a lossy electrical transmission line is used to derive an eigenvalue problem that allows a detailed investigation of the modulational stability of the solitary states in the line. It is found that the growth rates of the perturbation are complex functions of the spatial variable and that, in general, the solitary states in the network can be either modulationally stable, unstable, or destabilized under a given perturbation function.

New analytical approximation forms for non-linear instability of electric porous media

We study the Kelvin–Helmholtz instability for parallel flow between two dielectric fluids in porous media. A normal electric field stresses the system. A non-linear perturbation technique is introduced. This technique is based on the Fourier transform and the multiple scales method. We discussed the stability conditions for non-Darcian and Darcian flows. A linear analytical dispersion relation and several coupled non-linear equations contain non-linear correction terms in uniform analytical descriptions derived. For non-Darcian, a complex non-linear Ginzburg–Landau equation is used to analyse the stability of the problem obtained. We introduce a simple analytical technique for calculating the non-linear cutoff wavenumber, and a new analytical solution form is derived. The newly derived analytical solution is compared with several previously obtained approximation solutions. The analytical solution is more realistic than the previously proposed descriptions, which were widely used in the past. For some special physical parameters, a non-linear modified diffusion equation for low Reynolds number is obtained. Also, for high Reynolds number we obtain a new analytical non-linear dispersion relation (complex non-linear modified Ginzburg–Landau equation) in terms of the linear analytical dispersion relation. Numerical results and stability diagrams support the analytical proofs. Regions of stability and instability are identified. The non-linear numerical results showed that the linear model is inadequate.

Soliton-like solutions for the modified variable-coefficient Ginzburg–Landau equation

This paper is devoted to studying the solutions of the modified Ginzburg–Landau equation with variable coefficient. Based on computerized symbolic computation, several families of new exact dark- and bright-soliton-like solutions are presented. Moreover, we derive some similarity solutions, which can be illustrated in terms of the elliptic and the second kind of Painlevé transcendent equation.

Generalized chaotic synchronization in coupled Ginzburg-Landau equations

Generalized synchronization is analyzed in unidirectionally coupled oscillatory systems exhibiting spatiotemporal chaotic behavior described by Ginzburg-Landau equations. Several types of coupling between the systems are analyzed. The largest spatial Lyapunov exponent is proposed as a new characteristic of the state of a distributed system, and its calculation is described for a distributed oscillatory system. Partial generalized synchronization is introduced as a new type of chaotic synchronization in spatially nonuniform distributed systems. The physical mechanisms responsible for the onset of generalized chaotic synchronization in spatially distributed oscillatory systems are elucidated. It is shown that the onset of generalized chaotic synchronization is described by a modified Ginzburg-Landau equation with additional dissipation irrespective of the type of coupling. The effect of noise on the onset of a generalized synchronization regime in coupled distributed systems is analyzed.

Traveling ion channel density waves affected by a conservation law

A model of mobile, charged ion channels embedded in a biomembrane is investigated. The ion channels fluctuate between an opened and a closed state according to a simple two-state reaction scheme whereas the total number of ion channels is a conserved quantity. Local transport mechanisms suggest that the ion channel densities are governed by electrodiffusionlike equations that have to be supplemented by a cable-type equation describing the dynamics of the transmembrane voltage. It is shown that the homogeneous distribution of ion channels may become unstable to either a stationary or an oscillatory instability. The nonlinear behavior immediately above threshold of an oscillatory bifurcation occurring at finite wave number is analyzed in terms of amplitude equations. Due to the conservation law imposed on ion channels, large-scale modes couple to the finite-wave-number instability and have thus to be included in the asymptotic analysis near the onset of pattern formation. A modified Ginzburg-Landau equation extended by long-wavelength stationary excitations is established, and it is highlighted how the global conservation law affects the stability of traveling ion channel density waves.

Modulational instability of partially coherent signals in electrical transmission lines

We present an investigation of the modulational instability of partially coherent signals in electrical transmission lines. Starting from the modified Ginzburg-Landau equations and the Wigner-Moyal representation, we derive a nonlinear dispersion relation for the modulational instability. It is found that the effect of signal broadbandness reduces the growth rate of the modulational instability.

Multiple pulse passive mode locking in lasers with spectrally inhomogeneous gain

A new mechanism of formation of multiple pulse operation, multistability, and hysteresis in lasers with spectrally inhomogeneous gain media such as ion-doped glasses has been discovered. The multistability manifests itself as a dependence on the initial conditions of the number of pulses in the laser operation that is established after the transient process. It has been found that the dependence of the number of pulses in the established operation on the pumping power is hysteretic in character. The mechanism revealed is connected with a change in the frequency dispersion of the active medium caused by the spectrally inhomogeneous saturation of the gain. Analysis and numerical simulation have been carried out on the basis of the modified Ginzburg-Landau equation, which takes into account the change in the spectral profile of the gain during laser operation.

Generalized synchronization in coupled Ginzburg-Landau equations and mechanisms of its arising

Generalized chaotic synchronization regime is observed in the unidirectionally coupled one-dimensional Ginzburg-Landau equations. The mechanism resulting in the generalized synchronization regime arising in the coupled spatially extended chaotic systems demonstrating spatiotemporal chaotic oscillations has been described. The cause of the generalized synchronization occurrence is studied with the help of the modified Ginzburg-Landau equation with additional dissipation.

Use of the modified Ginzburg–Landau equations in high temperature superconductors

AbstractThe modified Ginzburg–Landau equations are used to study some selected problems for the high temperature superconductors. The derivations of modified Ginzburg–Landau equations are briefly demonstrated by taking into consideration the layered features of the sample according to the relation between the coherence length and the separation distance of the layers. The domain wall energy, the maximum supercurrent and the parallel critical field of high temperature superconducting thin film were evaluated. Comparison with previous works is given. (© 2005 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Generalized Synchronization in Autooscillatory Media

The phenomenon of generalized chaotic synchronization has been found in autooscillatory media described by the Ginzburg-Landau equations. A mechanism responsible for the establishment of generalized synchronization regimes in one-way coupled autooscillatory media exhibiting chaotic spatiotemporal behavior is proposed. The mechanism of generalized synchronization is described based on an analysis of a modified Ginzburg-Landau equation with an additional dissipative term.

Mesoscopic samples: the superconducting condensate via the Gross–Pitaevskii scenario

Mesoscopic superconductors are routinely investigated with the Ginzburg–Landau equations, whereby the confinement is taken into account by imposing that the normal component of the superconducting current vanishes through the sample boundary. We argue that this approach gives misleading results when applied to micron- and submicron-sized devices, and we introduce modified Ginzburg–Landau equations that take the confinement potential into account on the level of the free energy functional. For devices much larger than the Ginzburg–Landau coherence length, both approaches agree, but deviations appear for samples of the scale of the coherence length. In the absence of a magnetic field, the modified Ginzburg–Landau equation for the order parameter reduces to the Gross–Pitaevskii equation.

Modified Ginzburg–Landau Equation and Benjamin–Feir Instability

In this paper, a modulated wave train in a nonlinear monoinductance LC circuit is studied. Using the method of multiple scales in the general form, we establish that the evolution of nonlinear excitations is governed by what we called the Modified Ginzburg–Landau Equation (MGLE). Benjamin–Feir instability for the MGLE is analyzed.

Exact solutions of the one-dimensional generalized modified complex Ginzburg–Landau equation

The one-dimensional (1D) generalized modified complex Ginzburg–Landau (MCGL) equation for the traveling wave systems is analytically studied. Exact solutions of this equation are obtained using a method which combines the Painlevé test for integrability in the formalism of Weiss–Tabor–Carnevale and Hirota technique of bilinearization. We show that pulses, fronts, periodic unbounded waves, sources, sinks and solution as collision between two fronts are the important coherent structures that organize much of the dynamical properties of these traveling wave systems. The degeneracies of the 1D generalized MCGL equation are examined as well as several of their solutions. These degeneracies include two important equations: the 1D generalized modified Schrödinger equation and the 1D generalized real modified Ginzburg–Landau equation. We obtain that the one parameter family of traveling localized source solutions called “Nozaki–Bekki holes” become a subfamily of the dark soliton solutions in the 1D generalized modified Schrödinger limit.