Nonlinear Ginzburg-Landau Equations Research Articles

Dissipative Soliton Resonance: Adiabatic Theory and Thermodynamics

We present the adiabatic theory of dissipative solitons (DS) of complex cubic-quintic nonlinear Ginzburg–Landau equation (CQGLE). Solutions in the closed analytical form in the spectral domain have the shape of Rayleigh–Jeans distribution for a positive (normal) dispersion. The DS parametric space forms a two-dimensional (or three-dimensional for the complex quintic nonlinearity) master diagram connecting the DS energy and a universal parameter formed by the ratio of four real and imaginary coefficients for dissipative and non-dissipative terms in CQGLE. The concept of dissipative soliton resonance (DSR) is formulated in terms of the master diagram, and the main signatures of transition to DSR are demonstrated and experimentally verified. We show a close analogy between DS and incoherent (semicoherent) solitons with an ensemble of quasi-particles confined by a collective potential. It allows applying the thermodynamical approach to DS and deriving the conditions for the DS energy scalability.

Unconditionally convergent and superconvergent analysis of second-order weighted IMEX FEMs for nonlinear Ginzburg-Landau equation

Unconditionally convergent and superconvergent analysis of second-order weighted IMEX FEMs for nonlinear Ginzburg-Landau equation

Convergence and superconvergence analysis of finite element methods for nonlinear Ginzburg–Landau equation with Caputo derivative

Convergence and superconvergence analysis of finite element methods for nonlinear Ginzburg–Landau equation with Caputo derivative

Unconditional error analysis of a linearized BDF2 virtual element method for nonlinear Ginzburg–Landau equation with variable time step

We consider a virtual element method in space for the nonlinear Ginzburg–Landau equation, while a linearized time-variable-step second order backward differentiation formula (BDF2) is adopted in time. The error splitting approach is used to prove the unconditional optimal error estimate of the derived scheme under the mild restriction on the ratio of adjacent time-steps ratios (similarly proposed in Liao and Zhang 2021, Zhang and Zhao 2021). By using the techniques of the discrete complementary convolution (DOC) kernels and the discrete complementary convolution (DCC) kernels, we obtain the boundedness and error estimates of the solution of time-discrete system. Moreover, the optimal convergence in L2-norm for the fully discrete scheme is finally derived. Numerical examples on a set of polygonal meshes are given to validate our theoretical results.

Tanh Jacobi spectral collocation method for the numerical simulation of nonlinear Schrödinger equations on unbounded domain

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.

Conservative Finite-Difference Scheme for 1D Ginzburg–Landau Equation

In this study, our attention is focused on deriving integrals of motion (conservation laws; invariants) for the problem of an optical pulse propagation in an optical fiber containing an optical amplifier or attenuator because, to date, such invariants are absent in the literature. The knowledge of a problem’s invariants allows us develop finite-difference schemes possessing the conservativeness property, which is crucial for solving nonlinear problems. Laser pulse propagation is governed by the nonlinear Ginzburg–Landau equation. Firstly, the problem’s conservation laws are developed for the various parameters’ relations: for a linear case, for a nonlinear case without considering the linear absorption, and for a nonlinear case accounting for the linear absorption and homogeneous shift of the pulse’s phase. Hereafter, the Crank–Nicolson-type scheme is constructed for the problem difference approximation. To demonstrate the conservativeness of the constructed implicit finite-difference scheme in the sense of preserving difference analogs of the problem’s invariants, the corresponding theorems are formulated and proved. The problem of the finite-difference scheme’s nonlinearity is solved by means of an iterative process. Finally, several numerical examples are presented to support the theoretical results.

Stochastic Ginzburg–Landau equation on a half-line driven by the multiplicative noise

We consider the stochastic nonlinear Ginzburg–Landau equations driven by the multiplicative noise on the half-line with Neumann brown-noise boundary conditions. We establish the existence and uniqueness of a solution to an initial-boundary value problem with values in Sobolev space H1. We are also interested in the regularity behavior of the solution, especially near the origin, where the boundary data are highly irregular.

High-dimensional nonlinear Ginzburg–Landau equation with fractional Laplacian: Discretization and simulations

In this paper, we propose a three-level linearized finite difference scheme for the high-dimensional nonlinear Ginzburg–Landau equation with fractional Laplacian. The Crank–Nicolson scheme is used for time discretization, and the fractional Laplacian is discretized by the fractional centered difference scheme. The proposed difference scheme (i.e. a linear system) can be solved efficiently by fast Fourier transform (FFT) and complex conjugate gradient method, since the coefficient matrix is a multi-level block Toeplitz matrix. Furthermore, we analyze the unique solvability and boundedness of solution of the difference scheme by the discrete energy method. It is also proved that the difference scheme is unconditionally stable and second-order accurate in time and space with respect to l∞-norm. Finally, several numerical examples are provided to validate the theoretical results.

Vieta-Lucas polynomials for the coupled nonlinear variable-order fractional Ginzburg-Landau equations

In this article, the non-singular variable-order fractional derivative in the Heydari-Hosseininia concept is used to formulate the variable-order fractional form of the coupled nonlinear Ginzburg-Landau equations. To solve this system, a numerical scheme is constructed based upon the shifted Vieta-Lucas polynomials. In this method, with the help of classical and fractional derivative matrices of the shifted Vieta-Lucas polynomials (which are extracted in this study), solving the studied problem is transformed into solving a system of nonlinear algebraic equations. The convergence analysis and the truncation error of the shifted Vieta-Lucas polynomials in two dimensions are investigated. Numerical problems are demonstrated to confirm the convergence rate of the presented algorithm.

Linear and Weakly Nonlinear Analyses of Magneto-Convection in a Sparsely Packed Porous Medium under Gravity Modulation

This paper deals with the linear stability analysis and weakly non-linear analysis of Magneto-convection in a sparsely packed porous medium with constant vertical Magnetic field and gravity modulation. A linear stability analysis reported here and shows that the gravity modulation has significant effect on the stability limits of the system. The gravity modulation is know to have effect and is treated by a perturbation expansion in powers of the amplitude of modulation. The shift in the critical Rayleigh number is evaluated and depends on the prandtl number and frequency of modulation, using the Venezian method. It is also shown that the onset of convection can be advance or delay by the regulation of various parameters. Weakly nonlinear analysis is performed based on the method of power series, where the disturbance is expressed in terms of power series. A nonlinear Ginzburg-Landau equation to investigate the three different types of gravity modulation on heat transfer is derived as part of this work. Heat transfer have been shown to depend on Nusselt number, further the effect of different types of parameter on heat transport have been studied graphically. Nusselt number graph is also shown for different parameter and explain in detail. The effect of magnetic prandtl number and Chandrasekhar number are stabilize the system. The control of convection is a major issue in systems with fluids as a working media. This is all the more difficult if the fluid system is housed in a porous medium. The paper presents three mechanisms of controlling onset of convection and thereby the heat transfer in such fluid systems. In order the modulation effect is effective in its role, we have considered the system to be a fluid-saturated porous media.

Linearized ADI schemes for two-dimensional space-fractional nonlinear Ginzburg–Landau equation

Space and time approximations for two-dimensional space fractional complex Ginzburg–Landau equation are examined. The schemes under consideration are discreted by the second-order backward differential formula (BDF2) in time and two classes of the fractional centered finite difference methods in space. A linearized technique is employed by the extrapolation. We prove the unique solvability and stability for both numerical methods. The convergence of both numerical methods is analyzed at length utilizing the energy argument, and the convergence orders under the optimal step size ratio are O(τ2+h2) and O(τ2+h4) in the sense of the discrete L2-norm, where τ is the time step size, h=max{hx,hy}, and hx,hy are spatial grid sizes in the x-direction and y-direction, respectively. In addition, we construct a multistep alternating direction implicit (ADI) scheme and a multistep compact ADI scheme based on BDF2 for the efficiently numerical implementation. Finally, numerical examples are carried out to verify our theoretical results.

Stochastic Ginzburg-Landau equation on a half-line with Neumann type white-noise boundary conditions

We consider the stochastic nonlinear Ginzburg-Landau equations on the half-line with Neumann white-noise boundary conditions. We establish the existence and uniqueness of a solution to initial-boundary value problem with values in an appropriate space. We are also interested in the regularity behavior of the solution, especially near the origin, where the boundary data is highly irregular.

Vortex Configurations of a Nano-sized Superconducting 3D Pyramidal Confinement

By using the state-of-the-art three-dimensional simulations within the phenomenological two coupled nonlinear Ginzburg-Landau equations, we observe the effect of a superconducting pyramidal nano-sized confinement on the vortex configurations for three cases, where magnetic fields are applied to three different directions $\vec {H}_{z}$, $\vec {H}_{x}$, and $\vec {H}_{t}$ and square, triangle, and trapezoid arrangement of vortex configurations are observed respectively. The Cooper pair and screening current density have been calculated on the orthogonal plane to the applied field. Furthermore, we analyze the magnetic behavior of different vortex states.

Weakly Nonlinear Magnetic Convection in a Nonuniformly Rotating Electrically Conductive Medium Under the Action of Modulation of External Fields

In this paper we studied the weakly nonlinear stage of stationary convective instability in a nonuniformly rotating layer of an electrically conductive fluid in an axial uniform magnetic field under the influence of: a) temperature modulation of the layer boundaries; b) gravitational modulation; c) modulation of the magnetic field; d) modulation of the angular velocity of rotation. As a result of applying the method of perturbation theory for the small parameter of supercriticality of the stationary Rayleigh number nonlinear non-autonomous Ginzburg-Landau equations for the above types of modulation were obtaned. By utilizing the solution of the Ginzburg-Landau equation, we determined the dynamics of unsteady heat transfer for various types of modulation of external fields and for different profiles of the angular velocity of the rotation of electrically conductive fluid.

Intensity and magnetization angle reconfigurable lateral spin-wave coupling and transport

Magnonic directional coupler is important unit of the integrated functional magnonic networks. Here we report on the mechanism of the variation of the spin-wave coupling length which is equal to the longitudinal spatial scale of wave intensity modulation along the magnonic stripes placed in the proximity. The array of yttrium-iron garnet films was proposed to be the main unit of the magnonic directional coupler system. The simultaneous variation of the magnetization orientation and spin-wave intensity leads to the variation of spin-wave coupling length. Using the finite element numerical simulation and numerical integration of the phenomenological equations similar to the nonlinear Ginzburg-Landau equations the dispersion and frequency dependences of the spin wave coupling length were obtained. We also demonstate experimentally the simultaneous control of spin-wave transport by the variation of the signal intensity and magnetization angle. It was shown that the proposed dual-mode control can enhance the functionality of directional coupler, power splitter and multiplexer based on coupled stripes in planar magnonic networks.

Chebyshev polynomials for the numerical solution of fractal–fractional model of nonlinear Ginzburg–Landau equation

This paper introduces a new version for the nonlinear Ginzburg–Landau equation derived from fractal–fractional derivatives and proposes a computational scheme for their numerical solutions. The fractal–fractional derivative is defined in the Atangana–Riemann–Liouville sense with Mittage–Leffler kernel. The proposed approach is based on the shifted Chebyshev polynomials (S-CPs) and the collocation scheme. Through the way, a new operational matrix (OM) of fractal–fractional derivative is derived for the S-CPs and used in the presented method. More precisely, the unknown solution is separated into their real and imaginary parts, and then, these parts are expanded in terms of the S-CPs with undetermined coefficients. These expansions are substituted into the main equation and the generated operational matrix is utilized to extract a system of nonlinear algebraic equations. Thereafter, the yielded system is solved to obtain the approximate solution of the problem. The accuracy of the proposed approach is examined through some numerical examples. Numerical results confirm the suggested approach is very accurate to provide satisfactory results.

An Insensitizing Control Problem for the Ginzburg–Landau Equation

In this paper, we prove the existence of insensitizing controls for the nonlinear Ginzburg–Landau equation. Here, we have a partially unknown initial data, and the problem consists in finding controls such that a specific functional is insensitive for small perturbations of the initial data. In general, the problem of finding controls with this property is equivalent to prove a partial null controllability result for an optimality system of cascade type. The novelty here is that we consider functionals depending on the gradient of the state, which leads to a null controllability problem for a system with second-order coupling terms. To manage coupling terms of this order, we need a new Carleman estimate for the solutions of the corresponding adjoint system.

The critical state of superconducting multilayered structures

We propose a new method for calculating the dependence of the critical current on an external magnetic field, and the distribution of this current over the layers in superconducting multilayered structures. The method is based on a numerical solution of a system of nonlinear Ginzburg–Landau equations that describe the behavior of a superconducting plate carrying a transport current in a magnetic field, provided that it does not contain Abrikosov vortices. The way in which boundary conditions in the Ginzburg–Landau theory influence the critical state of superconducting layered structures is also considered. From a mathematical point of view, application of a general boundary condition to the system of Ginzburg–Landau equations leads to a change in the order parameter along the thickness of the thin superconducting plates. The physical nature of this phenomenon is explained by the proximity effect at the superconductor–normal metal (SN) interface, which leads to the suppression of the order parameter near the SN interface. The resulting calculated dependences of the plates’ critical current on the magnetic field strength applied parallel to the layers are used to determine the critical current of multilayer structures. It is assumed that the mutual influence of superconducting layers occurs only via the magnetic field that they create.

Instabilities in a Non-Uniformly Rotating Medium with Stratification of the Temperature in an External Uniform Magnetic Field

In this paper the stability of the non-uniformly rotating cylindrical plasma in the axial uniform magnetic field with the vertical temperature gradient is investigated. In the approximation of geometrical optics a dispersion equation for small axisymmetric perturbations is obtained with the effects of viscosity, ohmic and heat conductive dissipation taken into account. The stability criteria for azimuthal plasma flows are obtained in the presence of the vertical temperature gradient and the constant magnetic field. The Rayleigh-Benard problem for stationary convection in the non-uniformly rotating layer of the electrically conducting fluid in the axial uniform magnetic field is considered. In the linear theory of stationary convection the critical value of the Rayleigh number subject to the profile of the inhomogeneous rotation (Rossby number) is obtained. It is shown that the negative values of the Rossby number have a destabilizing effect, since the critical Rayleigh number becomes smaller, than in the case of the uniform rotation , or of the rotation with positive Rossby numbers . To describe the nonlinear convective phenomena the local Cartesian coordinate system was used, where the inhomogeneous rotation of the fluid layer was represented as the rotation with a constant angular velocity and azimuthal shear with linear dependence on the coordinate. As a result of applying the method of perturbation theory for the small parameter of supercriticality of the stationary Rayleigh number a nonlinear Ginzburg-Landau equation was obtaned. This equation describes the evolution of the finite amplitude of perturbations by utilizing the solution of the Ginzburg-Landau equation. It is shown that the weakly nonlinear convection based on the equations of the six-mode Lorentz model transforms into the identical Ginzburg-Landau equation. By utilizing the solution of the Ginzburg-Landau equation, we determined the dynamics of unsteady heat transfer for various profiles of the angular velocity of the rotation of electrically conductive fluid.

Frozen magnetic response in mesoscopics superconductors

For a bulk type II superconducting sample at low temperature, the magnetic field can penetrate in the form of a single quantum fluxoid, for bulk samples this fluxoides are arranged in a hexagonal lattice, this so-called Shubnikov-Abrikosov state or vortex state and takes place between the first and the second critical thermodynamics magnetic fields. Under the first magnetic critical field is present the Meissner-Oschenfeld state. For mesoscopic samples, the magnetic response can present very interesting properties due the proximity effect of in-homogeneous boundary conditions and the presence of dots, anti-dots and/or impurities. The superconducting state in a mesoscopic sample with dot/anti-dot/dot/pillar/trench is calculated within the nonlinear Ginzburg-Landau equations. We predict that the critical magnetic fields and magnetization depends on strongly of the nature, geometry, size of the defects and the boundary conditions used.