Ginzburg-Landau Equation Research Articles

Numerical Method for Stochastic Nonlinear Schrödinger Equation Driven by Multivariate Gaussian Measure: Algorithms and Applications

In this paper, we present a novel Galerkin spectral method for numerically solving the stochastic nonlinear Schrödinger (NLS) equation driven by multivariate Gaussian random variables, including the nonlinear term. Our approach involves deriving the tensor product of triple random orthogonal basis and random functions, which enables us to effectively handle the stochasticity and nonlinear term in the equation. We apply this newly proposed method to solve both one- and two-dimensional stochastic NLS equations, providing detailed analysis and comparing the results with Monte Carlo simulation. In addition, the proposed method is applied to the stochastic Ginzburg–Landau equation. Our method exhibits excellent performance in both spatial and random spaces, achieving spectral accuracy in the numerical solutions.

THERMODYNAMIC MAGNETIC FIELD AND SPECIFIC HEAT JUMP OF IRON-BASED SUPERCONDUCTOR BY TWO-BAND GINZBURG-LANDAU APPROACH

In this research, temperature-dependent thermodynamic magnetic field and specific heat jump of Fe-based superconductor were studied by two-band Ginzburg-Landau theory. The analytical form of the thermodynamic magnetic field was investigated by four temperature-dependent models; Chen’s model, Zhu’s model, Shanenko’s model and Changjan & Udomsamuthirun’s model. For the calculation of specific heat jump, we use the Ruthgers expression. Calculation findings indicate an inverse relationship between the critical temperature and the specific heat jump. When compared with various models, the results from Changjan and Udomsamuthirun's models suit experimental data of Ba1-xKxFe2As2 (x = 0.4) and KFe2As2 superconductors better

Nonlinear magnetoconvection of a Maxwell fluid in a porous layer with chemical reaction

ABSTRACT In this paper, the linear and nonlinear instability of magnetoconvection in a Darcy-Benard setup saturated by a Maxwell fluid with chemical reactions is studied. The governing non-dimension equations are solved using the normal modes, and we obtain the expressions for steady and oscillatory thermal Rayleigh numbers. The effects of different physical parameters such as the Damkohler number (), Hartmann number (), Solute Rayleigh number (), relaxation parameter (), Magnetic Prandtl number (), Lewis number () on stationary and oscillatory critical thermal Rayleigh numbers are presented and described. Enhancing the values of the Solute Rayleigh number and Lewis number makes the system unstable. Also, the Hartmann number and Damkohler number have a contrasting effect on stationary and oscillatory convection. Enhancing the value of the relaxation parameter makes the system more stable. In order to study heat transport by convection, the well-known equation, the Landau-Ginzburg equation, is derived.

Hybrid Schrödinger-Ginzburg-Landau (Sch-GL) approach to study of superconducting integrated structures

Hybrid Schrödinger-Ginzburg-Landau approach is based on the fact of the mathematical similarity of Ginzburg-Landau (GL) and Schrödinger formalisms. Starting from Schrödinger approach by replacing the term V (x)−E with the term α(x)+β(x)|ψ(x)|2 we get the Ginzburg-Landau equation. In the proposed relaxation algorithm, we use one- and two-dimensional ground energy solutions of the Schrödinger equation and apply them as a starting trial solution for the GL relaxation method. The obtained numerical results and used methodology form the basis of the simulation platform for studying superconducting integrated structures and modeling various superconducting devices, including their time-dependent geometry.

Cubic-quartic optical solitons of the complex Ginzburg-Landau equation: A novel approach

Cubic-quartic optical solitons of the complex Ginzburg-Landau equation: A novel approach

Fully discrete scheme for a time-dependent Ginzburg-Landau equation in macromolecular microsphere composite hydrogels

In this article, we propose and analyze a fully discrete scheme for the time-dependent Ginzburg-Landau equation, which describes the phase field evolution of macromolecular microsphere composite hydrogels. This fully discrete scheme is a combination of the mass lumped finite element approximation for spatial discretization and the first order backward Euler scheme for temporal discretization. Besides, the convex-concave decomposition is adopted technically in the treatment of energy functional, where the concave part is treated explicitly and the convex part is treated implicitly. Moreover, we obtain the unique solvability and unconditional energy stability of the scheme. Finally, several numerical experiments are presented to illustrate the convergence order and effectiveness of the presented full-discrete scheme.

Weakly nonlinear instability analysis of triple diffusive convection under internal heat generator and modulated boundaries

A nonlinear instability analysis of triple diffusive convection under the time-dependent heat and mass transfer boundary conditions in the presence of internal heat source is evaluated in this study. On various physical parameters, the momentary behavior of both Sherwood and Nusselt number profiles is examined. In the geometry, we have considered two parallel infinite horizontal plates acting gravity vertically downward z-direction. By using the weakly nonlinear analysis, the Ginzburg–Landau equation is generated for the rate of heat and mass transport. Here, we have considered the temperature and concentration of two solutes. The temperature and first concentration of the solute at the lower plate are higher than the upper plate, while the second concentration of the solute at the upper plate is higher compared to that of the lower plate. According to the different modulation, we have considered four cases based on the phase angle of the modulations. The convective heat and mass transports are measured as a function of the Nusselt number (Nu) and Sherwood number (Sh1 and Sh2) for both the concentration. From the results, it is found that the first Lewis number increases all the considered profiles, while Ri increases the Nusselt number profile only. The principal discovery elucidated by this article resides in the observation that the internal heat source, subject to modulated boundaries, maintains the convective instability if different solutes are used from both ends.

Interlayer Interaction and Coercivity of Three-Layer Films Оbtained bу Chemical Deposition

Abstract—The results of experimental and theoretical studies of the coercivity and the displacement field of the hysteresis loop on the thickness of the nonmagnetic interlayer in magnetic films obtained by electroless plating are presented. With the help of model calculations based on the Landau–Ginzburg equations, the exchange interactions between magnetic layers with the participation of atoms of the nonmagnetic layer are studied. The resulting expression for the displacement field describes well the exponential changes in the displacement field as a function of the interlayer thickness in structures with both magnetically soft layers and layers with significantly different values of the coercivity.

Combined FEM and phase field method for reliability design of forward degradation in SiC bipolar device

The reliability of 4H-SiC bipolar devices is compromised by the expansion of single Shockley stacking faults (SSFs) during forward-current operation. Because SSF expansion is governed by multiphysical aspects, including electrical, thermal, and stress states, analysis of the mounted structure is important for improving power module design. We propose a practical design method that analyzes the critical condition due to SSF expansion using a combined method with a multiphysical finite element method (FEM) and phase field model based on the time-dependent Ginzburg–Landau equation. In preliminary studies, the thermal deformation of the demonstration module and the variation of threshold current of a bar-shaped SSF were verified from experimental and reference data. Estimating the SSF expansion rate on the constructed response surface under the mutiphysical inputs from FEM, the proposed design method can be used effectively in the design process by changing the various design variables.

On Attractors of Ginzburg–Landau Equations in Domain with Locally Periodic Microstructure: Subcritical, Critical, and Supercritical Cases

On Attractors of Ginzburg–Landau Equations in Domain with Locally Periodic Microstructure: Subcritical, Critical, and Supercritical Cases

Homogenization of Attractors to Ginzburg-Landau Equations in Media with Locally Periodic Obstacles: Critical Case

In this paper the Ginzburg-Landau equation is considered in locally periodic porous medium, with rapidly oscillating terms in the equation and boundary conditions. It is proved that the trajectory attractors of this equation converge in a weak sense to the trajectory attractors of the limit Ginzburg-Landau equation with an additional potential term. For this aim we use an approach from the papers and monographs of V.V. Chepyzhov and M.I. Vishik concerning trajectory attractors of evolution equations. Also we apply homogenization methods appeared at the end of the XX-th century. First, we apply the asymptotic methods for formal construction of asymptotics, then, we verify the leading terms of asymptotic series by means of the methods of functional analysis and integral estimates. Defining the appropriate axillary functional spaces with weak topology, we derive the limit (homogenized) equation and prove the existence of trajectory attractors for this equation. Then we formulate the main theorem and prove it with the help of axillary lemmas.

Optical solitons of the complex Ginzburg-Landau equation having dual power nonlinear form using $\varphi^{6}$-model expansion approach

This paper employs a novel $\varphi ^{6}$-model expansion approach to get dark, bright, periodic, dark-bright, and singular soliton solutions to the complex Ginzburg-Landau equation with dual power-law non-linearity. The dual-power law found in photovoltaic materials is used to explain nonlinearity in the refractive index. The results of this paper may assist in comprehending some of the physical effects of various nonlinear physics models. For example, the hyperbolic sine arises in the calculation of the Roche limit and the gravitational potential of a cylinder, the hyperbolic tangent arises in the calculation of the magnetic moment and the rapidity of special relativity, and the hyperbolic cotangent arises in the Langevin function for magnetic polarization. Frequency values, one of the soliton's internal dynamics, are used to examine the behavior of the traveling wave. Finally, some of the obtained solitons' three-, two-dimensional, and contour graphs are plotted.

Achieving optimal magnetic flux expulsion of a Nb3Sn superconducting radio-frequency cavity via spatial temperature gradient

The trapped magnetic flux leads to residual resistance, thereby reducing the quality factor of the superconducting cavity and affecting its power consumption. It has been experimentally shown that the trapped magnetic flux of the superconducting radio-frequency cavity can be expelled during cooldown under a spatial temperature gradient. In the present paper, we simulate the magnetic flux expulsion behavior during the cooling process of a Nb3Sn superconducting cavity using time-dependent Ginzburg-Landau (TDGL) theory. In order to achieve optimal magnetic flux expulsion, the effect of grain size, grain boundary strength |ψ|GB, and cooling time on flux expulsion ratio ε for different temperature gradients are studied theoretically. The results show that ε first increases rapidly and then varies slightly to a steady value with increasing temperature gradient, which is in good agreement with the experimental results. In general, for larger |ψ|GB, ε decreases with increasing grain size. While for smaller |ψ|GB, ε first increases to the maximum then decreases with increasing grain size. Moreover, It is found that a larger cooling time is more beneficial for the expulsion of the flux vortices from the sample. The results of this paper The vortex dynamics behavior can give an insight into the mechanism of magnetic flux expulsion, which can provide a theoretical basis for further enhancing the performance of superconducting cavities.

Complex Ginzburg–Landau equation in the modified Peyrard–Bishop–Dauxois model

Complex Ginzburg–Landau equation in the modified Peyrard–Bishop–Dauxois model

The exact solutions of the variable-order fractional stochastic Ginzburg–Landau equation along with analysis of bifurcation and chaotic behaviors

This article explores the exact solutions of the variable order fractional derivative of the stochastic Ginzburg–Landau equation (GLE) using the G′G2-expansion method with the assistance of Matlab R2021a software. The paper presents three key aspects that contribute to its novelty: (1) Our study introduces and examines the variable order fractional derivative of the stochastic Ginzburg–Landau equation (VOFDSGLE) for the first time, demonstrating our best cognitive effort. We successfully obtain a significant number of exact solutions and provide illustrative examples and visual representations of the VOFDSGLE. These obtained solutions have the potential to offer enhanced availability and practicality in understanding the mechanisms behind the complex physical phenomena observed in various fields. (2) Additionally, we investigate the phase portraits of the variable-order fractional stochastic Ginzburg–Landau equation under a specific condition. We analyze the associated sensitivity and chaotic behaviors, which have not been examined in prior studies on stochastic Ginzburg–Landau equation (Mohammed et al., 2021; Alhojilan and Ahmed, 2023). This exploration adds a unique contribution to the existing literature and offers valuable insights into the dynamics of the system. (3) Our numerical observations underscore the significant impact of multiplicative noise on the solutions. The progressive degradation of patterns and the tendency towards planar surfaces indicate the disruption caused by increasing noise intensity. Moreover, the convergence of solution magnitudes to zero underlines the stabilizing influence of the multiplicative noise. Furthermore, we advance the existing knowledge regarding the response of the stochastic GLE model to small noise, offering valuable insights into the system’s transition behaviors in the energy landscape.

Exact solutions and dynamic properties of the complex Ginzburg-Landau equation with law of four powers of nonlinearity

In this paper, we analyze the complex Ginzburg-Landau equation with law of four powers of nonlinearity by using the complete discrimination system for polynomial method and establish the exact solutions of this equation. As the model is visualised under specific parameters, we give two-dimensional diagrams of the exact solutions to demonstrate the existence of the solutions. In addition, we also establish the dynamic properties of the equation and give the Hamiltonian and topological properties of the equation.

Analytical solution for ginzburg–landau equation in discrete solitons laser arrays lattices via non-perturbation methods

Analytical solution for ginzburg–landau equation in discrete solitons laser arrays lattices via non-perturbation methods

Interpolatory input and output projections for flow control

Eigenvectors of the observability and controllability Gramians represent responsive and receptive flow structures that enjoy a well-established connection to resolvent forcing and response modes. However, whereas resolvent modes have demonstrated great potential to guide sensor and actuator placement, observability and controllability modes have been leveraged exclusively in the context of model reduction via input and output projections. In this work, we introduce interpolatory, rather than orthogonal, input and output projections, that can be leveraged for sensor and actuator placement and open-loop control design. An interpolatory projector is an oblique projector with the property of preserving certain entries in the vector being projected. We review the connection between the resolvent operator and the Gramians, and present several numerical examples where we perform both orthogonal and interpolatory input and output projections onto the dominant forcing and response subspaces. Input projections are used to identify dynamically relevant disturbances, place sensors to measure disturbances, and place actuators for feedforward control in the linearized Ginzburg–Landau equation. Output projections are used to identify coherent structures and place sensors aiming at state reconstruction in the turbulent flow in a minimal channel at$Re_{\tau }=185$. The framework does not require data snapshots and relies only on knowledge of the steady or mean flow.

Helicity-controlled switching of superconducting states by radiation pulse

We provide a theoretical analysis of the possibility of using circularly polarized radiation to switch between two different quantum states of a superconducting nanoring subjected to the half quantum flux. Numerical modeling, performed in the framework of the time-dependent Ginzburg–Landau equation, reveals the condition for on-demand switching between current-carrying states with different helicities with 100% probability. An experimental demonstration of this effect will be unambiguous evidence of the inverse Faraday effect in superconductors and pave the way for the design of light-operated superconducting devices.

Analysis of lump solutions and modulation instability to fractional complex Ginzburg–Landau equation arise in optical fibers

In this paper, the fractional complex Ginzburg–Landau equation (CGLE) with Kerr law in nonlinear optics, which simulates soliton propagation in various waveguides in the presence of a detuning component which comes from the nonlinear Schrödinger equation (NLSE) with the inclusion of the growth and damping terms, is examined. The Hirota bilinear method is exercised to retrieve lump solitons such as the 1-kink wave solution, 2-kink wave solution, double exponential wave solution, and homoclinic breather wave solution to the model. We also scrutinize some M-shaped solutions in the forms of M-shaped rational solutions and the M-shaped interaction with rogue and kink waves. In addition, the instability modulation and gain spectra of the CGLE are examined. The originality of the study lies in the secured outcomes, which were never before produced and effectively balance the nonlinear physical aspects. To illustrate the dynamic of these waves, some of the solutions are sketched in three-dimensional, two-dimensional, contour, and density plots. The produced results are encouraging which can be used to describe the phenomena occurring in nonlinear optical or plasma physics. The computed solutions demonstrate that the suggested approaches are skillful, categorical, consistent, and effective in identifying exact solutions to a variety of complicated nonlinear problems that have recently arisen in nonlinear optics, applied sciences, and engineering.