Two-dimensional Ginzburg-Landau Equation Research Articles

Extension of the stability criterion for dissipative vector solitons of a laser coupled two-dimensional Ginzburg–Landau Equation generated from vector asymmetric inputs

Pulses propagating in inhomogeneous nonlinear media with linear/nonlinear gain and loss described by the vector (2+1)−dimensional cubic-quintic Ginzburg−−Landau equations (CQCGLEs) are considered. The evolution and the stability of the vector dissipative optical solitons generated from an elliptic input are studied. Based on the variational approach, we analyze the influence of various physical parameters on the dynamics of the propagating signal and its relevant parameters. Following a suitable choice of the test function, the stability analysis has been fulfilled by the analysis of the total energy. Numerical results have confirmed the analytical prediction of the collision on the stability of the vector dissipative solitons.

Engineering dissipative chirped solitons in the cardiac tissue under electromagnetic induction

This work deals with the engineering dissipative chirped solitary waves in the cardiac tissue under electromagnetic induction. Employing the a two-dimensional Ginzburg-Landau (GL) equation describing the spatiotemporal evolution of transmembrane potential in cardiac cells under magnetic flow effect, we use the phase imprint technique to reduce the model equation into a derivative GL equation (alias chirping model equation) that allow us to investigate the dynamics of dissipative chirp solitary wave propagating through a cardiac tissue. The baseband modulational instability (MI) of the chirping model is investigated and the analytical expression of the MI growth rate is derived. In the baseband MI regime, we use the approximative solitonlike solution of the chirping model to investigate the dynamics of transmembrane potential in cardiac cells. We show that the frequency chirp associated to the solitary wave is directly proportional to the wave amplitude and the nonlinear chirping parameter can be used to modulate the chirping amplitude. Also, our results show that the approximate solution of the chirping model can be used to generate bright (dissipative) solitons, dark solitons, breather solitons and first-order rogue wave in the cardiac cell under electromagnetic induction. Our theoretical results are confirmed by numerical simulations. The results of this work can motivate the investigations of qualitative new phenomena in the presence of large dissipation.

HOC–ADI schemes for two-dimensional Ginzburg–Landau equation in superconductivity

In this paper, we apply the high-order compact scheme coupled with alternating direction implicit (HOC–ADI) method to solve the two-dimensional Ginzburg–Landau (2D GL) equation. Five HOC–ADI schemes with second order accuracy in time and fourth order in space are proposed for 2D GL equation. Scheme I and Scheme II are both nonlinear, which need nonlinear iteration.To overcome this shortcoming, Scheme III and Scheme IV are proposed in order to avoid nonlinear iteration. With the three-level ADI scheme and the method of extrapolation, we obtain a linearized scheme V. Some numerical experiments are shown to testify and compare the superiority of the new numerical schemes.

Bounded solutions for an ordinary differential system from the Ginzburg–Landau theory

In this paper, we look at a linear system of ordinary differential equations as derived from the two-dimensional Ginzburg–Landau equation. In two cases, it is known that this system admits bounded solutions coming from the invariance of the Ginzburg–Landau equation by translations and rotations. The specific contribution of our work is to prove that in the other cases, the system does not admit any bounded solutions. We show that this bounded solution problem is related to an eigenvalue problem.

Effects of a magnetic field on vortex states in superfluid He3−B

Superfluid $^3$He-B possesses three locally stable vortices known as a normal-core vortex ($o$-vortex), an A-phase-core vortex ($v$-vortex), and a double-core vortex ($d$-vortex). In this work, we study the effects of a magnetic field parallel or perpendicular to the vortex axis on these structures by solving the two-dimensional Ginzburg-Landau equation for two different sets of strong coupling correction. The energies of the $v$- and $d$-vortices have nontrivial dependence on the magnetic field. As a longitudinal magnetic field increases, the $v$-vortex is energetically unstable even for high pressures and the $d$-vortex becomes energetically most stable for all possible range of pressure. For a transverse magnetic field the energy of the $v$-vortex becomes lower than that of the $d$-vortex in the high pressure side. In addition, the orientation of the double cores in the $d$-vortex prefers to be parallel to the magnetic field at low pressures, while the $d$-vortex with the double cores perpendicular to the magnetic field is allowed to continuously deform into the $v$-vortex by increasing the pressure.

Efficient numerical schemes for two-dimensional Ginzburg-Landau equation in superconductivity

The objective of this paper is to propose some high-order compact schemes for two-dimensional Ginzburg-Landau equation. The space is approximated by high-order compact methods to improve the computational efficiency. Based on Crank-Nicolson method in time, several temporal approximations are used starting from different viewpoints. The numerical characters of the new schemes, such as the existence and uniqueness, stability, convergence are investigated. Some numerical illustrations are reported to confirm the advantages of the new schemes by comparing with other existing works. In the numerical experiments, the role of some parameters in the model is considered and tested.

Phase Transition of Vortex States in Two-Dimensional Superconductors under a Oscillating Magnetic Field from the Chiral Helimagnet

We have investigated vortex states in two-dimensional superconductors under a oscillating magnetic field from a chiral helimagnet. We have solved the two-dimensional Ginzburg-Landau equations with finite element method. We have found that when the magnetic field from the chiral helimagnet increases, vortices appear all at once in all periodic regions. This transition is different from that under the uniform magnetic field. Under the composite magnetic field with the oscillating and uniform fields (down-vortices), vortices antiparallel to the uniform magnetic field disappear. Then, the small uniform magnetic field easily remove down-vortices.

The solution of the two-dimensional Ginzburg-Landau equations using a higher-order moment technique with Lattice-Boltzmann and He's semi-inverse method

The solution of the two-dimensional Ginzburg-Landau equations using a higher-order moment technique with Lattice-Boltzmann and He's semi-inverse method

Accessible solitons in complex Ginzburg-Landau media

We construct dissipative spatial solitons in one- and two-dimensional (1D and 2D) complex Ginzburg-Landau (CGL) equations with spatially uniform linear gain; fully nonlocal complex nonlinearity, which is proportional to the integral power of the field times the harmonic-oscillator (HO) potential, similar to the model of "accessible solitons;" and a diffusion term. This CGL equation is a truly nonlinear one, unlike its actually linear counterpart for the accessible solitons. It supports dissipative spatial solitons, which are found in a semiexplicit analytical form, and their stability is studied semianalytically, too, by means of the Routh-Hurwitz criterion. The stability requires the presence of both the nonlocal nonlinear loss and diffusion. The results are verified by direct simulations of the nonlocal CGL equation. Unstable solitons spontaneously spread out into fuzzy modes, which remain loosely localized in the effective complex HO potential. In a narrow zone close to the instability boundary, both 1D and 2D solitons may split into robust fragmented structures, which correspond to excited modes of the 1D and 2D HOs in the complex potentials. The 1D solitons, if shifted off the center or kicked, feature persistent swinging motion.

Numerical Study of Quantized Vortex Interaction in the Ginzburg-Landau Equation on Bounded Domains

AbstractIn this paper, we study numerically quantized vortex dynamics and their interaction in the two-dimensional (2D) Ginzburg-Landau equation (GLE) with a dimensionless parameterε> 0 on bounded domains under either Dirichlet or homogeneous Neumann boundary condition. We begin with a review of the reduced dynamical laws for time evolution of quantized vortex centers in GLE and show how to solve these nonlinear ordinary differential equations numerically. Then we present efficient and accurate numerical methods for discretizing the GLE on either a rectangular or a disk domain under either Dirichlet or homogeneous Neumann boundary condition. Based on these efficient and accurate numerical methods for GLE and the reduced dynamical laws, we simulate quantized vortex interaction of GLE with differentεand under different initial setups including single vortex, vortex pair, vortex dipole and vortex lattice, compare them with those obtained from the corresponding reduced dynamical laws, and identify the cases where the reduced dynamical laws agree qualitatively and/or quantitatively as well as fail to agree with those from GLE on vortex interaction. Finally, we also obtain numerically different patterns of the steady states for quantized vortex lattices under the GLE dynamics on bounded domains.

Nonlinear Magnetoconvection in a Sparsely Packed Porous Medium

Linear and weakly nonlinear properties of magnetoconvection in a sparsely packed porous medium are investigated. We have obtained the values of Takens-Bogdanov bifurcation points and codimension two bifurcation points by plotting graphs of neutral curves corresponding to stationary and oscillatory convection for different values of physical parameters relevant to magnetoconvection in a sparsely packed porous medium near a supercritical pitchfork bifurcation. We have derived a nonlinear two-dimensional Ginzburg-Landau equation with real coefficients by using Newell-Whitehead (1969) method. The effect of the parameter values on the stability mode is investigated and shown the occurrence of secondary instabilities namely, Eckhaus and Zigzag instabilities. We have studied Nessult number contribution at the onset of stationary convection. We have also derived two nonlinear one-dimensional coupled Ginzburg-Landau-type equations with complex coefficients near the onset of oscillatory convection at a supercritical Hopf bifurcation and discussed the stability regions of standing and travelling waves.

Analysis of some finite difference schemes for two-dimensional Ginzburg-Landau equation

Analysis of some finite difference schemes for two-dimensional Ginzburg-Landau equation

General soliton solutions for nonlinear dispersive waves in convective type instabilities

The two-dimensional Ginzburg–Landau equation (GLE) is obtained from basic equations by a linear stability analysis. This equation governs the evolution of slowly varying envelopes of periodic spatio-temporal patterns related to Rayleigh–Bénard convective instabilities. In addition, the phase instabilities of the complex GLE (CGLE) with quintic and space-dependent cubic terms modelling the Eckhaus and zigzag convective instabilities are reported. We find soliton solution classes to the elliptic and hyperbolic CGLE, by applying the function transformation method. The two-dimensional CGLE is transformed to a sine-Gordon equation, a sinh-Gordon equation and other equations, which depend only on one function χ. The general solution of the equation in χ leads to a general soliton solution of the two-dimensional CGLE. The obtained solutions contain some interesting specific solutions such as plane solitons, N multiple solitons and propagating breathers. We also discuss the soliton stability of the CGLE.

Synchronization in nonidentical complex Ginzburg-Landau equations

A cross-correlation coefficient of complex fields has been investigated for diagnosing spatiotemporal synchronization behavior of coupled complex fields. We have also generalized the subsystem synchronization way established in low-dimensional systems to one- and two-dimensional Ginzburg-Landau equations. By applying the indicator to examine the synchronization behavior of coupled Ginzburg-Landau equations, it is shown that our subsystem approach may be of better synchronization performance than the linear feedback method. For the linear feedback Ginzburg-Landau equation, the nonidentical system exhibits generalized synchronization characteristics in both amplitude and phase. However, the nonidentical subsystem may exhibit complete-like synchronization properties. The difference between complex fields for driven and response systems gives a linear scaling with the change of their parameter difference.

A Note on the Two-Dimensional Ginzburg–Landau Equation for Directional, Nearly Monochromatic Waves

In this paper we deal with the two-dimensional Ginzburg–Landauequation. First we simply expand the one-dimensional Ginzburg–Landauequation to the two-dimensional one. Then the concept of thedirectionality is imported into the two-dimensional Ginzburg–Landauequation. Directional, nearly monochromatic waves have a fixedwavenumber but spread over some propagation area in propagatingdirections. Moreover, most of the energy of waves is concentrated in asingle propagating direction. In slightly unstable, directional, nearlymonochromatic waves, the fact that the envelope surface created by theamplitude modulation is presented by the product of the solution of theSchrodinger–Nohara equation and the time function is shown. In thenonlinear case, the time function depends on space.

Erupting, flat-top, and composite spiral solitons in the two-dimensional Ginzburg–Landau equation

Abstract We present three novel varieties of spiraling and nonspiraling axisymmetric solitons in the complex cubic–quintic Ginzburg–Landau equation. These are irregularly “erupting” pulses and two different types of very broad stationary ones found near a border between ordinary pulses and expanding fronts. The region of existence of each pulse is identified numerically. We test their stability and compare their features with those of their counterparts in the one-dimensional and conservative two-dimensional models.

Dual point description of mesoscopic superconductors

We present an analysis of the magnetic response of a mesoscopic superconductor, i.e., a superconducting sample of dimensions comparable to the coherence length and to the London penetration depth. Our approach is based on special properties of the two-dimensional Ginzburg-Landau equations, satisfied at the dual point $(\ensuremath{\kappa}=1/\sqrt{2}).$ Closed expressions for the free energy and the magnetization of the superconductor are derived. A perturbative analysis in the vicinity of the dual point allows us to take into account vortex interactions, using a scaling result for the free energy. In order to characterize the vortex/current interactions, we study vortex configurations that are out of thermodynamical equilibrium. Our predictions agree with the results of recent experiments performed on mesoscopic aluminum disks.

Evaporation of droplets in the two-dimensional Ginzburg–Landau equation

We consider the problem of coarsening in two dimensions for the real (scalar) Ginzburg–Landau equation. This equation has exactly two stable stationary solutions, the constant functions +1 and −1. We assume most of the initial condition is in the “−1” phase with islands of “+1” phase. We use invariant manifold techniques to prove that the boundary of a circular island moves according to Allen–Cahn curvature motion law. We give a criterion for non-interaction of two arbitrary interfaces and a criterion for merging of two nearby interfaces.

Weak solutions to the two-dimensional derivative Ginzburg-Landau equation

In this paper, we deal with the generalized derivative Ginzburg-Landau equation in two spatial dimensions, and obtain the existence of global weak solutions for this equation subject to periodic boundary conditions.

Numerical calculation of the vortex-columnar-defect interaction and critical currents in extreme type-II superconductors - a two-dimensional model based on the Ginzburg-Landau approximation

We extend our previous one-dimensional Ginzburg-Landau calculations of the pinning energy of vortices to two dimensions, in order to achieve an understanding of the pinning forces exerted on vortices by defects. By minimizing the free energy using a relaxation scheme, we obtain the spatial variation of the order parameter and supercurrents for a vortex in the vicinity of a cylindrical defect in an extreme type-II superconductor. The resulting two-dimensional field distributions provide a direct mapping of the spatial dependence of the vortex-defect pinning potential, thereby yielding the pinning force and depinning current as a function of the defect size and magnetic field. We also use periodic boundary conditions in the two-dimensional Ginzburg-Landau equations to solve for the known vortex-vortex interaction, in order to verify the resolution and accuracy of our approach for extreme type-II superconductors. Our direct numerical derivation of the pinning force per vortex is shown to be applicable to a wide range of magnetic fields and columnar-defect densities, and the calculated results are consistent with experimental observation.