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Background: Gold ormus is a type of superconductor that can exhibit superconductivity at temperatures below 1 Kelvin, allowing it to conduct electricity without resistance. While not as widely used as other materials like niobium or lead, gold ormus is valuable for research in superconductivity. Limited studies have been conducted on gold ormus. Numerical simulations of the Ginzburg–Landau theory have yielded important results for both gold ormus. Results: Class-I and class-II superconducting gold ormus, have been successfully simulated using the Runge–Kutta fourth-order method. Our analysis shows the convergence of our simulation outcomes and emphasizes the importance of considering truncation error and selecting appropriate step sizes for accurate results. The periodic factor of penetration (PFP) for each superconductor has been determined, with class-I superconducting gold ormus having a PFP of 250 nm, class-II superconducting gold ormus having a PFP of 566.2 nm. The relationship between the PFP and the length of the penetration depth has also been revealed. Conclusions: Our study confirms the accuracy of the Runge–Kutta fourth-order method in simulating superconductors. By analyzing the PFP for different superconducting materials, we have identified trends in penetration depth that contribute to understanding superconductivity. Our simulations provide valuable insights for further research in the field of superconductivity. Adjusting parameters carefully ensures reliable simulations and advances progress in superconductivity research. [ABSTRACT FROM AUTHOR]

The problem of density wave propagation is considered for a logistic equation with delay and diffusion. This equation, called the Kolmogorov–Petrovsky–Piscounov–Fisher equation with delay, is investigated by asymptotic and numerical methods. Local properties of solutions corresponding to this equation with periodic boundary conditions are studied. It is shown that an increase in the period leads to the emergence of stable solutions with a more complex spatial structure. The process of wave propagation from one and from two initial perturbations is analyzed numerically, which allows tracing the process of wave interaction in the second case. The complex spatially inhomogeneous structure arising during the wave propagation and interaction can be explained by the properties of the corresponding solutions of a periodic boundary value problem with an increasing range of the spatial variable. [ABSTRACT FROM AUTHOR]

This article depicts the heat and mass transport of the double-diffusive convective flow of Walter-B viscoelastic fluid in highly permeable porous media with an internal heat source. We used weakly nonlinear analysis to quantify the nature of heat and mass transport using the Ginzburg–Landau equation. The Ginzburg–Landau equation has been derived in terms of the amplitude of the stream function. The effect of physical parameters has been examined on Nusselt and Sherwood numbers, which has represented graphically. According to the boundary condition, we have discussed the four scenarios based on the phase angles. Our study has demonstrated that internal heat plays a significant role in heat transfer processes. Furthermore, the elastic parameter leads to a transient augmentation in the heat and mass transfer rate. The main output of the current study is that the highest transport was found when both the modulations were put in out-phase condition (Scenario 1).

This paper leverages the Ginzburg–Landau equation, a fundamental model for elucidating the dynamics of dissipative optical solitons, in conjunction with the Markov Chain Monte Carlo method and the Lilliefors normality test, to precisely identify key parameters such as dispersion coefficient, nonlinear coefficient, and gain coefficient. Our comprehensive analysis yields not only posterior mean estimates and confidence intervals for these parameters but also their corresponding posterior probability density functions. These findings offer valuable statistical insights and theoretical underpinnings for the design and optimization of fiber lasers, with implications for enhancing the performance of fiber-optic communication systems. [ABSTRACT FROM AUTHOR]

Thermotaxis, the movement in response to temperature gradients, is a widely observed phenomenon prevalent in various natural occurrences, spanning from biological processes to the migration of colloidal particles. Results from studies of thermotaxis may inspire the development of new technologies, such as sensors that mimic the navigation strategies of organisms that respond to temperature gradients. By adjusting temperature gradients through temperature modulation, microorganisms can be manipulated. Therefore, the main aim of this study is to investigate the effect of thermal modulation on biothermal convection in a layer of liquid that contains thermotactic microorganisms. In this study, we explore the impact of non-uniform, time-varying boundary conditions on the system. Our study involves an analysis of weak nonlinear stability based on the Ginzburg-Landau model, leading to the derivation of heat (Nusselt number Nu) and mass (Sherwood number Sh) transfer coefficients that are dependent on various system parameters. In-phase modulation (IPM) has a relatively weak effect on both heat transfer and mass transfer, which closely resembles the unmodulated scenario. On the contrary, in the cases of out-of-phase modulation (OPM) and lower bound modulation (LBM), significant changes in heat and mass transfer are observed. It has been determined that the Peclet number, a parameter characterizing thermotaxis, can either induce stabilization or destabilization within the system. TM instability. [ABSTRACT FROM AUTHOR]

The main novelty of this paper lies in five aspects. (1) Our study on the fractional-order derivative stochastic Ginzburg–Landau equation (FODSGLE) has resulted in numerous exact solutions using Jacobian elliptic functions (JEFs). These solutions offer valuable insights into complex physical phenomena and have practical implications across various fields. (2) By examining the effect of noise on FODSGLE solutions, we found consistent behavior regardless of the form of fractional derivatives. As noise intensity increases, the solutions deteriorate and tend toward zero. This study contributes to the existing literature by providing new insights and surpassing previous efforts in understanding the dynamics of FODSGLE. (3) Our research investigates how fractional order affects noise in solutions of the FODSGLE. Surprisingly, we found that changes in the fractional order (α) have minimal influence on the system when the noise intensity (σ) is fixed. This indicates an aspect that has not been addressed extensively in existing literatures. (4) We compare different types of fractional derivatives within the context of the FODSGLE, presenting a novel contribution, while keeping the noise intensity or fractional order fixed. Our results demonstrate that the CFOD closely aligns with the MFOD, while displaying more differences with the solutions obtained using the BFOD. This comparison sheds light on an aspect that has not been thoroughly investigated in previous literatures. (5) This paper delves into the phase portraits of the FODSGLE and investigates the associated sensitivity and chaotic behaviors. Notably, previous studies on the stochastic Ginzburg–Landau equation (SGLE) have not extensively examined this aspect. By analyzing the phase portraits, we gain valuable insights into the dynamics and stability of FODSGLE solutions, uncovering intricate behaviors within the system. Our exploration of sensitivity and chaotic behaviors adds another dimension to understanding the (FODSGLE), laying a foundation for further research in this domain. [ABSTRACT FROM AUTHOR]

This paper is on the transition scenario of the class of spiral Poiseuille flows that results from the onset, propagation and evolution of disturbances according to mechanisms of Tollmien-Schlichting , and Taylor , acting simultaneously. The problem is approached from the fundamental point of view of following the growth of initially infinitesimally small disturbances into their nonlinear stage when the effect of Reynolds stresses makes itself felt. To this end a set of Generalised Nonlinear Orr–Sommerfeld, Squire and Continuity Equations is set up that enables accounting for effects of growth of initially infinitesimally small disturbances into nonlinearities through a rational iteration scheme. The present proposal closely follows the method put forth for this pupose in 1971 by Stuart and Stewartson in their seminal papers on the influence of nonlinear effects during transition in the bench-mark flows of the class of spiral Poiseuille flows; which are the plane-walled channel flow and the flow in the gap between concentric circular cylinders (Taylor instability). The basic feature of the proposed method is the introduction of an Amplitude Parameter and of a slow/long- scale variable through which the effects of growing disturbances are accounted for within the framework of a rational iteration scheme. It is shown that the effect of amplified disturbances is capturable, as in the bench-mark flows, by a Ginzburg–Landau type differential equation for an Amplitude Function in terms of suitably defined slow/long-scale variables. However, the coefficients in this equation are numbers that depend upon the flow parameters of the spiral Poiseuille flow, which are a suitably defined Reynolds Number, the Swirl Number, and the geometric parameter of transverse curvature inherent in the flow geometry. The Ginzburg–Landau equation derived hints at the drastic changes in flow pattern that the spiral Poiseuille flow in transition may undergo, as its Swirl Number is taken from very small to very large values. [ABSTRACT FROM AUTHOR]

The frequency of the applied electric field and the epitaxial strain can affect the microstructure and the overall properties of ferroelectric thin films. In this study, the ferroelectric characteristics of BaTiO3 nano single crystal thin films were investigated via phase field simulation under the epitaxial strain of -0. 1% and -0. 7%, respectively, in the frequency range of 0. 1 ~100 kHz. It is found that, with the increase in frequency, the square-shaped hysteresis loop gradually changes to an elliptic loop and the butterfly loop changes to a kidney-shaped loop. When the applied frequency is below 50 kHz, the coercive field increases rapidly with the increase in frequency, but the change in remnant polarization is not significant. When the frequency is over 50 kHz, the coercive field only increases slightly, and the remnant polarization exhibits a downward trend. It is observed that the frequency dependence of ferroelectric properties is more significantly affected by the epitaxial strain while the frequency is low. The frequency dependence is not very sensitive to the epitaxial strain while the frequency is high. Meanwhile, the compressive epitaxial strain causes significant rise in both the remnant polarization and coercive field, and the tensile epitaxial strain leads to opposite effects. By analysis, it is found that the underlying mechanism of the frequency-dependent hysteresis is due to the competition between the speed of microstructure evolution and the frequency of the applied electric field. The findings of this study serve as theoretical basis for the experiment and design of ferroelectric functional thin films, and they provide knowledge-based support for the application of high-frequency electronic devices. [ABSTRACT FROM AUTHOR]

For the logistic equation with diffusion two generalizations are considered: one with time delay and the other with deviation in the spatial variable. For these equations, the problem of density wave propagation is considered. These equations, known as Kolmogorov–Petrovsky–Piskunov–Fisher equations with delay or deviation (KPPF with delay or deviation), are investigated using both asymptotic and numerical methods.To investigate the qualitative behavior of solutions of such equations, the wave profile equation is considered and the conditions for the emergence of its oscillatory modes are identified.The local properties of solutions with periodic boundary conditions are examines, revealing that an increase in the period leads to the emergence of stable solutions with a more complex spatial structures.Numerical analysis of the wave propagation process indicates that at sufficiently small delay values, this equation has solutions that are close to those of the standard KPPF equation. Increasing the delay parameter leads first to the appearance of a damped oscillatory component in the spatial distribution of the solution. With further increases of this parameter, a complex spatial profile of the wave appears.In the final part of the chapter, which focuses on numerical calculation, the numerical analysis of the wave propagation process from two initial perturbations is performed. This analysis tracks the process of wave interaction. The complex spatially inhomogeneous structure that arises during wave propagation and interaction can be explained by the properties of the corresponding solutions of the periodic boundary value problem, especially as the interval of change in the spatial variable increases.

There are many types of superconductors, including gold ormus and some fullerene derivatives. Gold can become a superconductor at extremely low temperatures (

AbstractThe study investigates the linear and weakly nonlinear stability of Jeffrey fluids in an anisotropic porous medium with throughflow. In the linear stability analysis, the study examines how convective motion changes with variations in the Jeffrey parameter, Peclet number, and anisotropic parameters. The study finds that thermal anisotropy parameter, positive Peclet number, and mechanical anisotropy parameter delay the onset of fluid convection, whereas the Jeffrey parameter enhances the onset of convection in the presence of throughflow. In the nonlinear analysis, heat transfer is calculated using the Ginzburg–Landau Equation (GLE). The study observes that heat transfer increases with the Vadasz number and positive Peclet number, making the system unstable, whereas heat transfer decreases with an increased value of the Jeffrey parameter, thermal anisotropy parameter, mechanical anisotropic parameter, and negative Peclet number, making the system stable. [ABSTRACT FROM AUTHOR]

We revisit the nonlinear stability of the critical invasion front in the Ginzburg–Landau equation. Our main result shows that the amplitude of localized perturbations decays with rate t - 3 / 2 , while the phase decays diffusively. We thereby refine earlier work of Bricmont and Kupiainen as well as Eckmann and Wayne, who separately established nonlinear stability but with slower decay rates. On a technical level, we rely on sharp linear estimates obtained through analysis of the resolvent near the essential spectrum via a far-field/core decomposition which is well suited to accurately describing the dynamics of separate neutrally stable modes arising from far-field behavior on the left and right. [ABSTRACT FROM AUTHOR]

We consider the Ginzburg–Landau equation in the perforated domain, with rapidly oscillating coefficients. We derive the homogenized Ginzburg–Landau equation with a 'strange term' (potential) and prove that the trajectory attractors of the given equation tend in a weak sense to the trajectory attractors of the homogenized one. Assuming additional conditions to be satisfied for the coefficients, we provide also a convergence of the global attractor. [ABSTRACT FROM AUTHOR]

In the recent decade, several numerical procedures are developed based on the radial basis functions (RBFs) in the strong and the weak form of the mathematical model. These numerical algorithms have been used to solve a wide range of differential equations. However, the accuracy of RBFs collocation method for some partial differential equations (PDEs) is low thus a modification of RBFs collocation plan is required. The main aim of the current paper is to employ an improvement of RBFs collocation algorithm i.e. rational RBFs (RRBFs) collocation method based on the partition of unity (PU) idea to get the numerical solution of the multi-dimensional Ginzburg–Landau equation. It is clear that the RBFs collocation approach is an important numerical procedure for solving PDEs in non-rectangular physical regions. For differential equations with sufficiently smooth solutions, the RBF collocation technique generates acceptable and efficient accuracy. The RBF collocation method may produce solutions with non-physical oscillations for the underlying functions which have steep gradients or discontinuities. Using a fourth-order time-split approach and rational RBFs (RRBFs) collocation method, a new numerical procedure is proposed. First, a fourth-order time-split approach is used to discrete the time variable. Then, a combination of RBFs-PU collocation technique has been developed to get a full-discrete scheme. At the end, several examples have been studied to show the stability, convergence, and accuracy of the proposed numerical algorithm. [ABSTRACT FROM AUTHOR]

The general characteristics of an optical signal as a result of generation in a resonator can be described using a dynamic model based on the complex cubic Ginzburg–Landau equation, which takes into account the saturated gain and dissipative terms responsible for the distributed action of various intracavity devices. The paper proposes two new effective modifications of the split-step Fourier method for a numerical solution to equations of this type. The first algorithm is based on the application of a new way of separating physical processes affecting the optical signal during propagation in a fiber, which made it possible to express the action of both nonlinear and dispersive spatial steps by explicit analytical expressions. The second proposed method enabled significant improvement in the accuracy of calculations due to including energy evolution in the coefficients of the equation. Numerical experiments have shown that the new schemes can produce the second order of approximation with respect to the evolutionary variable in contrast to the classical scheme that provides only the first order of approximation. [ABSTRACT FROM AUTHOR]

In this paper, we research initial boundary value problems for time-space fractional Ginzburg–Landau equations with Caputo time fractional derivatives and fractional Laplacian operators. As far as we know, there have been few studies concerning stability and optimal control for the corresponding system. First, by Galërkin method, we establish global existence and a few decay estimates of weak solutions. What's more, we study the dependence of weak solutions on these given data, then we get stability of the system. Further, we obtain an optimal control of the time-space fractional Ginzburg–Landau system. [ABSTRACT FROM AUTHOR]

The Ginzburg–Landau equation (GLE) plays an important role in discrete solitons laser array lattices. In this manuscript, the analytical solution for GLE which is a discrete analogue of laser array using non-perturbation methods is represented. The laser array for stability analysis of time-dependent solution in terms of damping and coupling is investigated. Further, GLE equation for periodic travelling wave solution with homotopy perturbation method and variational iteration method is derived. These methods provide a straightforwardly computable, readily demonstrable and rapidly convergent solution. For the sake of the consistency, validation and effectiveness of these methods, a convergence of GLE has also been revealed. [ABSTRACT FROM AUTHOR]

Superconductor Nb open nanotubes are analyzed under a strong transport current in an external magnetic field using the time‐dependent Ginzburg–Landau equations. The geometric properties of the hysteresis loop in current–voltage characteristics are described for nanotubes of different radii from 190 to 440 nm as a function of the magnetic field. The hysteresis effect appears at a constant temperature as a result of the order parameter dynamics. In nanotubes of radii equal to 240 nm and larger, the area of the hysteresis loop manifests a non‐monotonic behavior as a function of the external magnetic field with a maximum ranging from 15 to 20 mT. Normal conductivity is shown to have a crucial impact on the hysteresis effect in superconductor open nanotubes. Namely, a higher normal conductivity leads to the disappearance of the hysteresis effect. As a rule, the upper and lower branches of the hysteresis loop are provided by the phase‐slip and vortex regimes, correspondingly. Just before the transition of the system from the upper branch to the lower one, the way of nucleation and annihilation of vortices in the region with suppressed superconductivity experiences readjustment. [ABSTRACT FROM AUTHOR]

This paper presents a method for extracting cubic–quartic optical soliton solutions for the complex Ginzburg–Landau equation with five distinct forms of nonlinear refractive index. By utilizing the proposed algorithm, we can obtain a diverse range of optical solitons, including hybrid types, that satisfy the specified parameter restrictions. [ABSTRACT FROM AUTHOR]

This paper deals with the well-posedness of the solutions of the fractional complex Ginzburg-Landauequation driven by locally Lipschitz nonlinear diffusion terms defined on $ R^n $. We first give the pathwise uniform estimates and uniform estimates on average. Then we prove the existence, uniqueness and measurability of solutions for the equation. [ABSTRACT FROM AUTHOR]

This paper is concerned with numerical analysis of a finite element method for the time-dependent Ginzburg–Landau equations under the Coulomb gauge. The main challenge is that the magnetic potential A is divergence-free and satisfies a Stokes-like structure under the Coulomb gauge. The proposed method uses linear Lagrange element P 1 to solve for the order parameter ψ , the lowest order Nédélec edge element N D 1 and linear Lagrange element P 1 to approximate the magnetic potential A and the electric potential ϕ , respectively. In particular, the proposed method preserves a weakly divergence-free property for A in the discrete level. The main aim of this work is to establish the second order spatial convergence of the most important variable ψ , though the numerical solutions of A are only O(h) in space. Our analysis is based on a nonstandard quasi-projection for ψ and the corresponding H - 1 -norm estimates for Maxwell projection. With the quasi-projection, we prove that the lower-order approximation to A does not pollute the accuracy of ψ h . An effective one step recovery is also proposed to obtain second order numerical solution for A . Our numerical experiments confirm the optimal second order convergence of ψ h and the efficiency of the recovery step. [ABSTRACT FROM AUTHOR]

In this paper, we prove a global Carleman estimate for the complex Ginzburg-Landau operator with a cubic nonlinear term in a bounded domain of Rn, n = 2, 3. As applications, we study state observation problems for the Ginzburg-Landau equation. [ABSTRACT FROM AUTHOR]

This paper investigates the effect of gravity modulation on Rayleigh–Bénard convection using the rigid isothermal boundary conditions. We calculate heat transfer results using the Nusselt and mean Nusselt numbers through the finite-amplitude of convection, which we got from the Ginzburg–Landau equation (GLE). The Ginzburg–Landau equation is derived analytically from the Fredholm solvability condition at third order. The finite amplitude equation (GLE) is a function of system parameters and solved numerically. The gravity modulation considered in terms of steady and sinusoidal parts. The sinusoidal part defines gravity modulation in terms of amplitude and frequency. Our study shows that gravity modulation controls the heat transfer results. The amplitude of modulation enhances heat transfer for low frequencies and diminishes for high frequencies. Further, we found that rigid isothermal boundary conditions are diminishing heat transfer than free and isothermal boundaries. Finally, we concluded that rigid isothermal boundary conditions and gravity modulation controls heat transfer results. [ABSTRACT FROM AUTHOR]

Linear and weakly nonlinear stability analyses of thermosolutal convection in a couple‐stress fluid with effects of helical force and rotation are performed. The governing nondimensional equations are solved using the normal modes. We have shown the effect of the helical force parameter, solutal Rayleigh number, Couple stress parameter, Lewis number, Taylor number, and Prandtl number on stationary and oscillatory convection regions and presented graphically. Solutal Rayleigh number, Couple stress parameter, Lewis number, and Taylor number have a stabilizing effect on the system whereas the helical force parameter has a destabilizing effect on the system. To study heat transport by convection we have derived the Ginzburg–Landau equation. [ABSTRACT FROM AUTHOR]

The chaotic behavior of thermal convection in a nonuniformly rotating electrically conductive fluid under the action of a constant vertical magnetic field B0 is studied. In the presence of vertical temperature gradients ∇T0 and the thermo-electromotive force coefficient ∇Tα, thermomagnetic (TM) instability arises, leading to the generation of magnetic fields. The magnetic field B1 excited by the effect of the Biermann battery is directed perpendicular to the plane of the vectors ∇T0, ∇Tα and the gradient of temperature disturbances ∇T1. This magnetic field changes the heat transfer regime, and due to the effects of convective heat transfer and Righi-Leduc, a positive feedback is established, which leads to an increase in magnetic field disturbances. Using the truncated Galerkin method, a nonlinear dynamic system of equations is obtained, which describes the processes of generation and regeneration of the magnetic field. Numerical analysis of these equations showed the existence of a regular, quasi-periodic, and chaotic behavior of magnetic field disturbances, accompanied by its inversions. Applying the method of perturbation theory to the nonlinear dynamic system of equations, we obtained the Ginzburg-Landau equation for the weakly nonlinear stage of nonuniformly rotating magnetic convection, taking into account TM effects. The solution of this equation showed that the stationary level of the generated magnetic fields increases with allowance for the influence of the TM instability. [ABSTRACT FROM AUTHOR]

The present analysis mathematically investigates the thermohaline convection problem in viscoelastic fluid layer saturating porous media by utilizing the modified Boussinesq approximation. By performing linear stability analysis, the Darcy–Rayleigh numbers for stationary and oscillatory modes of convection are derived. The effects of different parameters describing the problem are studied numerically. In nonlinear stability analysis, the heat and mass transfer rates in the form of Nusselt and Sherwood numbers, respectively, are obtained for oscillatory convection using the derived Ginzburg–Landau equation. From the results, it is observed that overstability is the preferred mode of instability in linear stability. It is found that in linear double‐diffusive convection problems, the stress relaxation imparts a destabilizing effect whereas the strain retardation time, the coefficient of specific heat variation due to temperature, and the concentration gradient have a stabilizing effect on the system's stability. The numerical values of heat and mass transfer rates varied with the coefficient of specific heat showing that the heat transport decreases while the mass transport increases. Also, the stress relaxation time, the concentration gradient, and the gravity modulation's amplitude increase while the strain retardation time decreases the heat and mass transfer rates. The wavelength of oscillations remains unaltered with the variation of specific heat variation due to temperature. The modulation frequency does not affect the heat/mass transfer rate; though, the wavelength of oscillations decreases with increasing frequency. [ABSTRACT FROM AUTHOR]

This paper is concerned with new error analysis of a lowest-order backward Euler Galerkin-mixed finite element method for the time-dependent Ginzburg-Landau equations. The method is based on a commonly-used nonuniform approximation, in which a linear Lagrange element, the lowest-order Nédélec edge element, and the Raviart-Thomas face element are used for the order parameter ψ, the magnetic field curlA, and the magnetic potential A, respectively. This mixed method has been widely used in practical simulations due to its low cost and ease of implementation. In the Ginzburg-Landau model, the order parameter ψ is the most important variable, which indicates the state of the superconductor. An important feature of the method is the inconsistency of the approximation orders. A crucial question is how the first-order approximation of (curlA, A) influences the accuracy of ψh. The main purpose of this paper is to establish the second-order accuracy for the order parameter in a spatial direction, although the accuracy for (curlA, A) is first order only. Previous analysis only gave the first-order convergence for all three variables due to certain artificial pollution involved in the analysis. Our analysis is based on a nonstandard quasi-projection for ψ and the corresponding more precise estimates, including the H-1 -norm. With the quasi-projection, we prove that the lower-order approximation to (curlA, A) does not pollute the accuracy of ψh. Our numerical experiments confirm the optimal convergence of ψh. The approach can be extended to many other multiphysics models.. [ABSTRACT FROM AUTHOR]

The Ginzburg–Landau equations were proposed in the superconductivity theory to describe mathematically the intermediate state of superconductors in which the normal conductivity is mixed with the superconductivity. It turned out that these equations have interesting and non-trivial generalizations. First of all, they can be extended to arbitrary compact Riemann surfaces. Next, they can be generalized to dimension 3 as dynamical (or hyperbolic) Ginzburg–Landau equations. They also have a 4-dimensional extension provided by Seiberg–Witten equations. In this review we describe all these interesting topics together with some unsolved problems. [ABSTRACT FROM AUTHOR]

In this paper, the fractional complex Ginzburg–Landau equation with time-dependent coefficients, which is used to depict diverse physical phenomena like superfluidity, superconductivity, Bose–Einstein condensation, second-order phase transitions, strings and liquid crystals, is investigated by three different methods under the circumstance of taking two forms of nonlinearity into account. A number of exact solutions are derived and fallen into different categories for the sake of discussing the dynamic behaviors of this equation further. More narrowly, we acquire the solitary, soliton and elliptic wave solutions through the unified method as well as the trigonometric, hyperbolic trigonometric and rational solutions through the improved F-expansion method in the sense of quadratic–cubic nonlinearity. And as another achievement, we obtain the bright, dark, combined bright–dark, singular soliton, mixed singular soliton and singular periodic wave solutions by employing the extended Sinh-Gordon equation expansion method under power law nonlinearity. Moreover, we draw the 2D, 3D and contour images for some of these solutions with fit choices of parameters to realize the propagations of waves more visually and deeply, thereby uncovering more physical phenomena in connection with the governing model. [ABSTRACT FROM AUTHOR]

An efficient difference method is constructed for solving one-dimensional nonlinear time-space fractional Ginzburg-Landau equation. The discrete method is developed by adopting the $ L2 $-$ 1_{\sigma} $ scheme to handle Caputo fractional derivative, while a fourth-order difference method is invoked for space discretization. The well-posedness and a priori bound of the numerical solution are rigorously studied, and we prove that the difference scheme is unconditionally convergent in pointwise sense with the rate of $ O(\tau^2+h^4) $, where $ \tau $ and $ h $ are the time and space steps respectively. In addition, the proposed method is extended to solve two-dimensional problem, and corresponding theoretical analysis is established. Several numerical tests are also provided to validate our theoretical analysis.

Abstract This paper serves as a sequel to previously reported results on cubic‐quartic solitons with complex Ginzburg–Landau equation. This work is with various forms of power law nonlinearity that structures the self‐phase modulation. The enhanced Kudryashov's approach gives the soliton solution. The conservation laws are also identified for the models.

We present a class of orthogonal functions on infinite domain based on Jacobi polynomials. These functions are generated by applying a tanh transformation to Jacobi polynomials. We construct interpolation and projection error estimates using weighted pseudo‐derivatives tailored to the involved mapping. Then, using the nodes of the newly introduced tanh Jacobi functions, we develop an efficient spectral tanh Jacobi collocation method for the numerical simulation of nonlinear Schrödinger equations on the infinite domain without using artificial boundary conditions. The applicability and accuracy of the solution method are demonstrated by two numerical examples for solving the nonlinear Schrödinger equation and the nonlinear Ginzburg–Landau equation. [ABSTRACT FROM AUTHOR]

A dissipative nonlinear transmission network is studied in the regular regime using both analytical approach and numerical technique that allow us to obtain the stable and unstable periodic orbits of this network in a systematic way. Going from a one-dimensional Ginzburg-Landau equation that governs the dynamics of modulated waves in the network system under consideration and assuming variations in both the amplitude and phase of the network voltage signals, we have proposed an amplitude-phase model consisting of a simple three-dimensional continuous quadratic dynamical system that describes, in the situations of weak dissipation, the dynamics of voltage signals (filters) through the network. We investigate, through the use of dynamic analysis tools such as the perturbation theory, the phase portrait, and Hopf bifurcation theory, the rich dynamics of the proposed, which have some interesting characteristics for different network parameters and initial conditions. Explicit analytical results are obtained for the post-bifurcation periodic orbits and their stability, and the conditions under which the constructed 3D dynamical system undergoes supercritical and/or subcritical Hopf bifurcations are presented. Computer simulations and numerical results are carried out to provide numerical verification of the theoretical studies; our numerical analysis confirms the existence of various type of periodic orbits, stable node-foci, unstable saddle-foci, as well as stable limit cycles. Basins of attraction of equilibrium positions for the derived 3D dynamical system are built by means of the Lyapunov's direct method. [ABSTRACT FROM AUTHOR]

An efficient difference method is constructed for solving one-dimensional nonlinear time-space fractional Ginzburg-Landau equation. The discrete method is developed by adopting the - scheme to handle Caputo fractional derivative, while a fourth-order difference method is invoked for space discretization. The well-posedness and a priori bound of the numerical solution are rigorously studied, and we prove that the difference scheme is unconditionally convergent in pointwise sense with the rate of , where and are the time and space steps respectively. In addition, the proposed method is extended to solve two-dimensional problem, and corresponding theoretical analysis is established. Several numerical tests are also provided to validate our theoretical analysis. [ABSTRACT FROM AUTHOR]

For each given n = 2, we construct a family of entire solutions uε(z,t), \varepsilon > 0ε>0, with helical symmetry to the three-dimensional complex-valued Ginzburg-Landau equation... [ABSTRACT FROM AUTHOR]

We propose and analyze residual-based a posteriori error estimator for a new finite element method for the time-dependent Ginzburg–Landau equations with the temporal gauge of superconductivity. The magnetic potential variable is approximated by H (∇ × ; Ω) -conforming element and the scalar order parameter is approximated by the H 1 (Ω) -conforming element. Using the dual problem of a linearization of the original problem, we prove the reliability of the a posteriori error estimator, and an adaptive algorithm with the temporal and spatial refining and coarsening steps is then proposed. Numerical results are presented for illustrating the a posteriori error estimator and the adaptive algorithm in convex and nonconvex domains. [ABSTRACT FROM AUTHOR]

Meshless method is an effective method to solve many kinds of equations on arbitrary domains. However, previous meshless techniques, such as moving least square method, radious basis function method, and element free Galerkin method, often have complex shape functions, which make the ill-condition of the equation increase along with the adding of control points. In order to overcome above shortcomings, in this paper, we construct a new meshless method for solving nonlinear variable order fractional Ginzburg–Landau equation. The method uses the bases of bicubic spline space as shape functions and applies continuation technique to make the operation simple and the calculation accelerated. Eventually, numerical examples demonstrate that our method can obtain higher accuracy and efficiency even though the step size is larger than other methods. [ABSTRACT FROM AUTHOR]

The asymmetric method is employed to derive the soliton solution for the Ginzburg–Landau complex equation with higher-order dispersion and nonlinear gradient terms. The bright soliton solutions have been obtained and the propagation of solitons has been analyzed. Additionally, the stable transport for the resulting soliton is also provided. And the influence of the initial phase, third-order group velocity dispersion coefficient, and the corresponding nonlinear effect on soliton transmission have been investigated. The method for adjusting the location, propagation direction, and amplitude of soliton pulses also have been presented. The research findings could carry theoretical significance for soliton control and offer insights for enhancing fiber laser and communication systems. [ABSTRACT FROM AUTHOR]

Rotational modulation and the third diffusive component are investigated in relation to transports in the Rayleigh Bènard convection in the x − direction of an infinitely extended fluid layer or fluid-saturated porous layer using a weakly non-linear theory. The direction of travel for the z -axis is upward. The z -axis is interpreted as pointing upward. Temperature and concentration of both the solutes at lower plate are higher than the upper plate. The time dependent rotational speed of fluid is assumed to be modulated by small parameter ϵ 2 with amplitude (δ). A linear differential matrix approach is used to examine the system's behaviour. Ginzburg–Landau's equation, a non-autonomous differential equation, has been derived using the Fredholm-solvability condition. The impact of dimensionless parameters on stationary convection is also discussed. The expression of the critical Rayleigh number is obtained for three different cases: clear fluid layer, Darcy porous layer, and sparsely packed porous layer (Darcy–Brinkman model). The Nusselt number (N u) and Sherwood numbers (S h 1 , S h 2 ) for each solute are used to quantify the convective transports. The impact of modulation and other dimensionless parameters on these convective transports has been discussed using graphical representation Furthermore, among the three models under consideration, it is discovered that the heat and mass transports are highest in the fluid layer and minimum in the densely packed porous layer. • The rate of transports under modulated rotation flow are analysed. • Two different solute concentrations are mixed from opposite boundaries. • Asymptotic expansion method is used to determine the solution of system. • The heat and mass transports are highest in the fluid layer. • The parameters δ , R C 1 , M and D a enhance the rate transports. [ABSTRACT FROM AUTHOR]

We consider a time semidiscretization of the Ginzburg-Landau equation by the backward Euler scheme. For each time step τ, we build an exponential attractor of the dynamical system associated to the scheme. We prove that, as τ tends to 0, this attractor converges for the symmetric Hausdorff distance to an exponential attractor of the dynamical system associated to the Allen-Cahn equation. We also prove that the fractal dimension of the exponential attractor and of the global attractor is bounded by a constant independent of τ.

In this paper, a numerical method to solve a class of stochastic nonlinear differential equations is introduced. The proposed method is based on the Temimi-Ansari method. The special states of the four systems are studied to show that the proposed method is efficient and applicable. These systems are stochastic Langevin's equation, Ginzburg-Landau equation, Davis-Skodje, and Brusselator systems. The results clarify the accuracy and efficacy of the presented new method with no need for any restrictive assumptions for nonlinear terms.

The purpose of the paper is to implement the collective variable method to investigate the generalized complex Ginzburg–Landau equation, which characterizes the kinetics of solitons in respect of pulse parameters for fiber optics. The statistical simulations of the interacting system of ordinary differential equations that reflect all the collective variables included in the pulse ansatz have been successfully carried out using a well-known numerical methodology, the fourth-order Runge–Kutta technique. The collective variable method is employed to plot the pulse variation characteristics as a function of propagation distance. The amplitude, temporal position, width, chirp, frequency, and phase of the pulse are all depicted against the propagated coordinate, where the width, phase of soliton, amplitude, and chirp all show a strong periodicity. The numerical dynamics of solitons have also been exhibited against varying values of pulse parameters to highlight differences in collective variables. Other key bits of the current investigation are also determined. [ABSTRACT FROM AUTHOR]

In this paper, the optical solitons for the complex Ginzburg–Landau equation with Kudryashov's law of refractive index are established. An improved modified extended tanh–function technique is used to extract numerous solutions. Bright and dark solitons, as well as singular soliton solutions, are achieved. In addition, as the modulus of ellipticity approaches unity or zero, solutions are formulated in terms of Jacobi's elliptic functions, which provide solitons and periodic wave solutions. [ABSTRACT FROM AUTHOR]

In this paper, we will construct the solution of the Landau-Ginzburg equation by the Adomian decomposition method. This method avoids linearization of space and discretization of time, it often gives a good approximation of the exact solution. [ABSTRACT FROM AUTHOR]

The major goal of the present paper is to construct optical solitons of the Ginzburg–Landau equation including the parabolic nonlinearity. Such an ultimate goal is formally achieved with the aid of symbolic computation, a complex transformation, and Kudryashov and exponential methods. Several numerical simulations are given to explore the influence of the coefficients of nonlinear terms on the dynamical features of the obtained optical solitons. To the best of the authors' knowledge, the results reported in the current study, classified as bright and kink solitons, have a significant role in completing studies on the Ginzburg–Landau equation including the parabolic nonlinearity. [ABSTRACT FROM AUTHOR]