Ginzburg-Landau System Research Articles

Boundedness of Solutions of the Ginzburg–Landau System Involving a Subelliptic Operator

Boundedness of Solutions of the Ginzburg–Landau System Involving a Subelliptic Operator

Strong convergence of the tamed Euler-Maruyama method for stochastic singular initial value problems with non-globally Lipschitz continuous coefficients

In our previous works [1] and [2], we delved into numerical methods for solving stochastic singular initial value problems (SSIVPs) that involve coefficients satisfying the global Lipschitz condition. The paper addresses the limitations of our previous work by introducing an explicit method, called the tamed Euler-Maruyama method, for numerically solving SSIVPs with non-globally Lipschitz continuous coefficients, which is both easy-to-implement and exceptionally efficient. We prove the existence and uniqueness theorem and the boundedness of the moments of the solution to SSIVPs under the non-globally Lipschitz condition. Moreover, we provide a sharp analysis of the strong convergence of the proposed method, along with the uniform boundedness of numerical solutions. We also apply our results to the stochastic singular Ginzburg-Landau system and provide numerical simulations to illustrate our theoretical findings.

The Existence of Meissner Solutions to the Full Ginzburg–Landau System in Three Dimensions

The Existence of Meissner Solutions to the Full Ginzburg–Landau System in Three Dimensions

Vortex sheet solutions for the Ginzburg–Landau system in cylinders: symmetry and global minimality

Vortex sheet solutions for the Ginzburg–Landau system in cylinders: symmetry and global minimality

Evolution of periodic wave and dromion-like structure solutions in the variable coefficients coupled high-order complex Ginzburg–Landau system

Evolution of periodic wave and dromion-like structure solutions in the variable coefficients coupled high-order complex Ginzburg–Landau system

On the existence of Meissner solutions to the Ginzburg-Landau system in two dimensions

In this paper we establish the existence of the stable Meissner solutions of the Ginzburg-Landau equations for superconductivity in R2.

Bifurcation analysis and optical soliton solutions for the fractional complex Ginzburg–Landau equation in communication systems

The nonlinear fractional complex Ginzburg–Landau system is a classical equation which has been developed vigorously in the fields of the combustion theory, nonlinear optics, signal processing, statistical mechanics and so on. This paper mainly concentrates on the bifurcations and single dispersive optical solitons for the fractional complex Ginzburg–Landau equation in communication systems. Starting with the traveling wave transformations and some other suitable transformations, the fractional complex Ginzburg–Landau system is converted into an equivalent ordinary differential system. Secondly, the bifurcation phase diagram and Hamiltonian of fractional complex Ginzburg–Landau system are also given. In addition to, the dispersive optical solitons and traveling wave solutions are derived via the dynamical theory method, the polynomial complete discriminant system technique and symbolic computation. It is notable that the obtained optical soliton solutions may improve or complement the corresponding results in the literature (Akram et al., 2022, Ouahid et al., 2021). Finally, in order to further understand the propagation of the fractional complex Ginzburg–Landau equation in nonlinear optics, three-dimensional and two-dimensional diagrams are given.

Validity of Whitham's modulation equations for dissipative systems with a conservation law: Phase dynamics in a generalized Ginzburg-Landau system

It is well-established that Whitham's modulation equations approximate the dynamics of slowly varying periodic wave trains in dispersive systems. We are interested in its validity in dissipative systems with a conservation law. The prototype example for such a system is the generalized Ginzburg-Landau system that arises as a universal amplitude system for the description of a Turing-Hopf bifurcation in spatially extended pattern-forming systems with neutrally stable long modes. In this paper we prove rigorous error estimates between the approximation obtained through Whitham's modulation equations and true solutions to this Ginzburg-Landau system. Our proof relies on analytic smoothing, Cauchy-Kovalevskaya theory, energy estimates in Gevrey spaces, and a local decomposition in Fourier space, which separates center from stable modes and uncovers a (semi)derivative in front of the relevant nonlinear terms.

Iterative learning control of complex Ginzburg–Landau system

Different from the existing research on iterative learning control (ILC) real-valued systems, complex-valued partial differential system is studied in this paper. All variables and parameters of the system are complex, and distributed control is imposed on the system. First, the original complex system is transformed into two coupled real systems, which represent the real and imaginary parts of the original complex system, respectively. Second, a complex [Formula: see text]-type ILC algorithm is designed that the real and imaginary parts of the learning law are coupled to each other. Then, the contraction mapping principle and analytical techniques are used to ensure that the tracking error converges to zero in the sense of [Formula: see text]-norm. Finally, a numerical simulation is presented to show the effectiveness of the proposed algorithm.

Improved convergence of the spectral proper orthogonal decomposition through time shifting

Spectral proper orthogonal decomposition (SPOD) is an increasingly popular modal analysis method in the field of fluid dynamics due to its specific properties: a linear system forced with white noise should have SPOD modes identical to response modes from resolvent analysis. The SPOD, coupled with the Welch method for spectral estimation, may require long time-resolved datasets. In this work, a linearised Ginzburg–Landau model is considered in order to study the method's convergence. Spectral proper orthogonal decomposition modes of the white-noise forced equation are computed and compared with corresponding response resolvent modes. The quantified error is shown to be related to the time length of Welch blocks (spectral window size) normalised by a convective time. Subsequently, an algorithm based on a temporal data shift is devised to further improve SPOD convergence and is applied to the Ginzburg–Landau system. Next, its efficacy is demonstrated in a numerical database of a boundary layer subject to bypass transition. The proposed approach achieves substantial improvement in mode convergence with smaller spectral window sizes with respect to the standard method. Furthermore, SPOD modes display growing wall-normal and spanwise velocity components along the streamwise direction, a feature which had not yet been observed and is also predicted by a global resolvent calculation. The shifting algorithm for the SPOD opens the possibility for using the method on datasets with time series of moderate duration, often produced by large simulations.

On the shape of Meissner solutions to the 2-dimensional Ginzburg–Landau system

On the shape of Meissner solutions to the 2-dimensional Ginzburg–Landau system

Entire Vortex Solutions of Negative Degree for the Anisotropic Ginzburg–Landau System

The anisotropic Ginzburg-Landau system \[ \Delta u+\delta\, \nabla (\mathrm{div}\: u) +\delta\, \mathrm{curl}^*(\mathrm{curl}\: u)=(|u|^2-1) u, \] for $u\colon\mathbb R^2\to\mathbb R^2$ and $\delta\in (-1,1)$, models the formation of vortices in liquid crystals. We prove the existence of entire solutions such that $|u(x)|\to 1$ and $u$ has a prescribed topological degree $d\leq -1$ as $|x|\to\infty$, for small values of the anisotropy parameter $|\delta| < \delta_0(d)$. Unlike the isotropic case $\delta=0$, this cannot be reduced to a one-dimensional radial equation. We obtain these solutions by minimizing the anisotropic Ginzburg-Landau energy in an appropriate class of equivariant maps, with respect to a finite symmetry subgroup.

Optimal Sensor and Actuator Selection Using Balanced Model Reduction

Optimal sensor and actuator selection is a central challenge in high-dimensional estimation and control. Nearly all subsequent control decisions are affected by these sensor and actuator locations. In this article, we exploit balanced model reduction and greedy optimization to efficiently determine sensor and actuator selections that optimize observability and controllability. In particular, we determine locations that optimize scalar measures of observability and controllability using greedy matrix QR pivoting on the dominant modes of the direct and adjoint balancing transformations. Pivoting runtime scales linearly with the state dimension, making this method tractable for high-dimensional systems. The results are demonstrated on the linearized Ginzburg–Landau system, for which our algorithm approximates known optimal placements computed using costly gradient descent methods.

Alikhanov Legendre—Galerkin Spectral Method for the Coupled Nonlinear Time-Space Fractional Ginzburg–Landau Complex System

A finite difference/Galerkin spectral discretization for the temporal and spatial fractional coupled Ginzburg–Landau system is proposed and analyzed. The Alikhanov L2-1σ difference formula is utilized to discretize the time Caputo fractional derivative, while the Legendre-Galerkin spectral approximation is used to approximate the Riesz spatial fractional operator. The scheme is shown efficiently applicable with spectral accuracy in space and second-order in time. A discrete form of the fractional Grönwall inequality is applied to establish the error estimates of the approximate solution based on the discrete energy estimates technique. The key aspects of the implementation of the numerical continuation are complemented with some numerical experiments to confirm the theoretical claims.

In–Out Intermittency with Nested Subspaces in a System of Globally Coupled, Complex Ginzburg–Landau Equations

Chaos and intermittency are studied for the system of globally coupled, complex Ginzburg–Landau equations governing the dynamics of extended, two-dimensional anisotropic systems near an oscillatory (Hopf) instability of a basic state with two pairs of counterpropagating, oblique traveling waves. Parameters are chosen such that the underlying normal form, which governs the dynamics of the spatially constant modes, has two symmetry-conjugated chaotic attractors. Two main states residing in nested invariant subspaces are identified, a state referred to as Spatial Intermittency ([Formula: see text]) and a state referred to as Spatial Persistence ([Formula: see text]). The [Formula: see text]-state consists of laminar phases where the dynamics is close to a normal form attractor, without spatial variation, and switching phases with spatiotemporal bursts during which the system switches from one normal form attractor to the conjugated normal form attractor. The [Formula: see text]-state also consists of two symmetry-conjugated states, with complex spatiotemporal dynamics, that reside in higher dimensional invariant subspaces whose intersection forms the 8D space of the spatially constant modes. We characterize the repeated appearance of these states as (generalized) in–out intermittency. The statistics of the lengths of the laminar phases is studied using an appropriate Poincaré map. Since the Ginzburg–Landau system studied in this paper can be derived from the governing equations for electroconvection in nematic liquid crystals, the occurrence of in–out intermittency may be of interest in understanding spatiotemporally complex dynamics in nematic electroconvection.

Symmetry and asymmetry of components for elliptic Gross-Pitaevskii system

This article examines nonnegative solutions of the two-component elliptic Gross-Pitaevskii system{ϵ1Δu1=u1(u12+λu22−1)inRn,ϵ2Δu2=u2(u22+λu12−1)inRn, for parameters 1<λ<∞ and ϵ1,ϵ2∈R+. We provide a natural counterpart of the De Giorgi's conjecture, regarding the Allen-Cahn equation, for the above system and we settle it for n≤3. We prove and apply a geometric Poincaré inequality to show that∇u1⋅∇u2=−|∇u1||∇u2|. We study symmetry of components when ϵ1=ϵ2 and asymmetry of components when ϵ1≠ϵ2. In one dimension, we explore explicit profiles of solutions for specific parameters λ,ϵ1,ϵ2. In addition, we provide a gradient estimate, various energy estimates, Liouville theorems, monotonicity formulae and a formalization of surface tensions of the energy for solutions of the above system.Note that when λ=0, the above system becomes decoupled and recovers the Allen-Cahn equation. The De Giorgi conjecture (1978) for the Allen-Cahn equation states that bounded monotone solutions of the Allen-Cahn equation are one-dimensional when n≤8. This conjecture is already settled under certain assumptions. When 0<λ≤1, the system refers to the real-valued Ginzburg-Landau system and nonnegative solutions are trivial.

Lie symmetry properties of the two-component complex Ginzburg–Landau system with variable coefficients

We classify the Lie point symmetries of a system of nonlinearly coupled complex Ginzburg–Landau equations with time and space dependent potentials and nonlinearities. New families of systems with exact solutions and conservation laws are identified. When there is only space dependence, the system can be reduced to a set of ordinary differential equations for the amplitude and phase components, for which integrability and first integrals are discussed.

Stable soliton propagation in a coupled (2 + 1) dimensional Ginzburg–Landau system**Project supported by the National Natural Science Foundation of China (Grant Nos. 11674036 and 11875008), Beijing Youth Top Notch Talent Support Program, China (Grant No. 2017000026833ZK08), Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications, Grant No. IPOC2019ZZ01), Fundamental Research Funds

A coupled (2 + 1)-dimensional variable coefficient Ginzburg–Landau equation is studied. By virtue of the modified Hirota bilinear method, the bright one-soliton solution of the equation is derived. Some phenomena of soliton propagation are analyzed by setting different dispersion terms. The influences of the corresponding parameters on the solitons are also discussed. The results can enrich the soliton theory, and may be helpful in the manufacture of optical devices.

Dynamics of non-autonomous fractional Ginzburg-Landau equations driven by colored noise

In this work, the existence and uniqueness of random attractors of a class of non-autonomous non-local fractional stochastic Ginzburg-Landau equation driven by colored noise with a nonlinear diffusion term is established. We comment that compared to white noise, the colored noise is much easier to handle in examining the pathwise dynamics of stochastic systems. Additionally, we prove the attractors of the random fractional Ginzburg-Landau system driven by a linear multiplicative colored noise converge to those of the corresponding stochastic system driven by a linear multiplicative white noise.

Global Uniform Estimate for the Modulus of Two-Dimensional Ginzburg-Landau Vortexless Solutions with Asymptotically Infinite Boundary Energy

For varepsilon>0, let u_varepsilon:Omega to R^2 be a solution of the Ginzburg-Landau system -Delta u_varepsilon=\frac1varepsilon^2 u_varepsilon (1-|u_varepsilon|^2) in a Lipschitz bounded domain Omega\subset\R^2. In an energy regime that excludes interior vortices, we prove that 1-|u_varepsilon| is uniformly estimated by a positive power of varepsilon globally in Omega provided that the energy of u_varepsilon at the boundary \partial Omega does not grow faster than varepsilon^-alpha with alpha in (0,1).