一般加性噪声扰动的分数阶随机复Ginzburg-Landau方程解的适定性

证明带有一般加性噪声的分数阶复Ginzburg-Landau方程解的存在性与唯一性.首先通过一族正则函数逼近原方程,然后再建立逼近解的一致估计,最后再通过极限过程证明方程的存在性和唯一性.

耦合慢扩散模式的五次实Ginzburg-Landau方程的脉冲解存在性

利用Fenichel的几何奇异摄动理论,将一个耦合慢扩散模式的次临界五次实Ginzburg-Landau方程脉冲解的存在性问题转化为几何摄动问题,展示了临界流形之间的横截相交性,并通过计算临界流形上Melnikov函数的零点进一步验证了同宿轨道的存在性.在一定的参数条件下,证明了慢扩散的次临界五次实Ginzburg-Landau方程有脉冲解.

Interactions between optical bullets with different velocities in the three-dimensional cubic-quintic complex Ginzburg-Landau equation

By using the three-dimensional complex Ginzburg--Landau equation with cubic--quintic nonlinearity, this paper numerically investigates the interactions between optical bullets with different velocities in a dissipative system. The results reveal an abundance of interesting behaviours relating to the velocities of bullets： merging of the optical bullets into a single one at small velocities; periodic collisions at large velocities and disappearance of two bullets after several collisions in an intermediate region of velocity. Finally, it also reports that an extra bullet derives from the collision of optical bullets when optical bullets are at small velocities but with high energies.

Amplitude wave in one-dimensional complex Ginzburg-Landau equation

The wave propagation in the one-dimensional complex Ginzbur-Landau equation （CGLE） is studied by considering a wave source at the system boundary. A special propagation region, which is an island-shaped zone surrounded by the defect turbulence in the system parameter space, is observed in our numerical experiment. The wave signal spreads in the whole space with a novel amplitude wave pattern in the area. The relevant factors of the pattern formation, such as the wave speed, the maximum propagating distance and the oscillatory frequency, are studied in detail. The stability and the generality of the region are testified by adopting various initial conditions. This finding of the amplitude pattern extends the wave propagation region in the parameter space and presents a new signal transmission mode, and is therefore expected to be of much importance.

VORTEX DYNAMICS OF THE ANISOTROPIC GINZBURG-LANDAU EQUATION

In this article, using coordinate transformation and Gronwall inequality, we study the vortex motion law of the anisotropic Cinzburg-Landau equation in a smooth bounded domain Ω （R^2,that is ,Эtuε=j,k=1∑2（ajkЭxkuε）xj＋ε^2^-b（x）（1-｜uε｜^2）uε,x∈Ω,and conclude that each vortex,bj（t）（j=1,2,…,N）satisfies dt^-dbj（t）=-（a（bj（t））^-a1k（bj（t））Эxka（bj（t））,a（aj（t））^-a2k（bj（t））Эxka（bj（t）））,where a（x）=√a11a22-a12^2. We prove that all the vortices are pinned together to the critical points of a（x）. Furthermore, we prove that these critical points can not be the maximum points.

Multi-symplectic Runge-Kutta methods for Landau-Ginzburg-Higgs equation

Nonlinear wave equations have been extensively investigated in the last sev- eral decades. The Landau-Ginzburg-Higgs equation, a typical nonlinear wave equation, is studied in this paper based on the multi-symplectic theory in the Hamilton space. The multi-symplectic Runge-Kutta method is reviewed, and a semi-implicit scheme with certain discrete conservation laws is constructed to solve the first-order partial differential equations （PDEs） derived from the Landau-Ginzburg-Higgs equation. The numerical re- sults for the soliton solution of the Landau-Ginzburg-Higgs equation are reported, showing that the multi-symplectic Runge-Kutta method is an efficient algorithm with excellent long-time numerical behaviors.

ATTRACTORS FOR THE GINZBURG-LANDAU-BBM EQUATIONS IN AN UNBOUNDED DOMAIN

By the interpolating inequality and a priori estimates in the weighted space,the existence of the global solutions for the Ginzburg-Landau equation coupled with the BBM equation in an unbounded domain is considered, and the existence of the maximal attractor is obtained.

Asymptotic behavior of 2D generalized stochastic Ginzburg-Landau equation with additive noise

The 2D generalized stochastic Ginzburg-Landau equation with additive noise is considered. The compactness of the random dynamical system is established with a priori estimate method, showing that the random dynamical system possesses a random attractor in H^1 0.

EQUI-ATTRACTION AND BACKWARD COMPACTNESS OF PULLBACK ATTRACTORS FOR POINT-DISSIPATIVE GINZBURG-LANDAU EQUATIONS

A new concept of an equi-attractor is introduced, and defined by the minimal compact set that attracts bounded sets uniformly in the past, for a non-autonomous dynam- ical system. It is shown that the compact equi-attraction implies the backward compactness of a pullback attractor. Also, an eventually equi-continuous and strongly bounded process has an equi-attractor if and only if it is strongly point dissipative and strongly asymptotically compact. Those results primely strengthen the known existence result of a backward bounded pullback attractor in the literature. Finally, the theoretical criteria are applied to prove the existence of both equi-attractor and backward compact attractor for a Ginzburg-Landau equation with some varying coefficients and a backward tempered external force.

Generalized spatiotemporal chaos synchronization of the Ginzburg-Landau equation

In this paper, the generalized synchronization of two unidirectionally coupled Ginzburg Landau equations is studied theoretically. It is demonstrated that the drive-response system has bounded attraction domain and compact attractors. It is derived that the correction equation has asymptotically stable zero solutions under certain conditions and that the sufficient conditions for smooth generalized synchronization and Holder continuous generalized synchronization exist in the coupling system. Numerical result analysis shows the correctness of theory.

Time-dependent Ginzburg-Landau equation for lattice hydrodynamic model describing pedestrian flow

A thermodynamic theory is formulated to describe the phase transition and critical phenomena in pedestrian flow. Based on the extended lattice hydrodynamic pedestrian model taking the interaction of the next-nearest-neighbor persons into account, the time-dependent Ginzburg-Landau （TDGL） equation is derived to describe the pedestrian flow near the critical point through the nonlinear analysis method. The corresponding two solutions, the uniform and the kink solutions, are given. The coexisting curve, spinodal line, and critical point are obtained by the first and second derivatives of the thermodynamic potential.

Intermittencies in complex Ginzburg-Landau equation by varying system size

The dynamical behaviour of the one-dimensional complex Ginzburg-Landau equation （CGLE） with finite system size L is investigated, based on numerical simulations. By varying the system size and keeping other system parameters in the defect turbulence region （defect turbulence in large L limit）, a number of intermittencies new for the CGLE system are observed in the processes of pattern formations and transitions while the system dynamics varies from a homogeneous periodic oscillation to strong defect turbulence.

Soliton structures in the (1+1)-dimensional Ginzburg–Landau equation with a parity-time-symmetric potential in ultrafast optics

In this paper, the （l＋l）-dimensional variable-coefficient complex Ginzburg-Landau （CGL） equation with a parity- time （PT） symmetric potential U（x） is investigated. Although the CGL equations with a PT-symmetric potential are less reported analytically, the analytic solutions for the CGL equation are obtained with the bilinear method in this paper. Via the derived solutions, some soliton structures are presented with corresponding parameters, and the influences of them are analyzed and studied. The single-soliton structure is numerically verified, and its stability is analyzed against additive and multiplicative noises. In particular, we study the soliton dynamics under the impact of the PT-symmetric potential. Results show that the PT-symmetric potential plays an important role for obtaining soliton structures in ultrafast optics, and we can design fiber lasers and all-optical switches depending on the different amplitudes of soliton-like structures.

A solution of the complex Ginzburg-Landau equation with a continuum of decay rates

We prove the existence of a uniform initial datum whose solution decays, in var- ious Lp spaces, at different rates along different time sequences going to infinity, for complex Ginzburg-Landau equation on RN, of various parameters θ and γ.

A Series Solution for the Ginzburg-Landau Equation with a Time-Periodic Coefficient

The solution of the real Ginzburg-Landau (GL) equation with a time-periodic coefficient is obtained in the form of a series, with assured convergence, using the computer-assisted ‘Homotopy Analysis Method’ (HAM) propounded by Liao [1]. The formulation has been kept quite general to keep open the possibility of obtaining the solution of the GL equation for different continua as limiting cases of the present study. New ideas have been added and clear explanations are provided in the paper to the existing concepts in HAM. The method can easily be extended to solve complex GL equation, system of GL equations or even the GL equations with a diffusion term, each having a time-periodic coefficient. The necessary code in Mathematica that implements the HAM for the current problem is appended to the paper for use by the readers.

Relation between the complex Ginzburg-Landau equation and reaction-diffusion system

The complex Ginzburg-Landau equation （CGLE） has been used to describe the travelling wave behaviour in reaction-diffusion （RD） systems. We argue that this description is valid only when the RD system is close to the Hopf bifurcation, and is not valid when a RD system is away from the onset. To test this, we study spirals and anti-spirals in the chlorite-iodide-malonic acid （CIMA） reaction and the corresponding OGLE. Numerical simulations confirm that the OGLE can only be applied to the CIMA reaction when it is very near the Hopf onset.

INHOMOGENEOUS INITIAL-BOUNDARY VALUE PROBLEM FOR GINZBURG-LANDAU EQUATIONS

Some integral identities of smooth solution of inhomogeneous initial boundary value problem of Ginzburg-Landau equations were deduced, by which a priori estimates of the square norm on boundary of normal derivative and the square norm of partial derivatives were obtained. Then the existence of global weak solution of inhomogeneous initial-boundary value problem of Ginzburg-Landau equations was proved by the method of approximation technique and a priori estimates and making limit.

SOLUTIONS OF GINZBURG-LANDAU EQUATIONS WITH WEIGHT AND MINIMIZERS OF THE RENORMALIZED ENERGY

In this paper, it is proved that for any given d non-degenerate local minimum points of the renormalized energy of weighted Ginzburg-Landau eqautions, one can find solutions to the Ginzburg-Landau equations whose vortices tend to these d points. This provides the connections between solutions of a class of Ginzburg-Landau equations with weight and minimizers of the renormalized energy.

Liouville Type Theorems for Lichnerowicz Equations and Ginzburg-Landau Equation: Survey

In this survey paper, we firstly review some existence aspects of Lichnerowicz equation and Ginzburg-Landau equations. We then discuss the uniform bounds for both equations in Rn. In the last part of this report, we consider the Liouville type theorems for Lichnerowicz equation and Ginzburg-Landau equations in Rn via two approaches from the use of maximum principle and the monotonicity

Accurate Symbolic Solution of Ginzburg-Landau Equations in the Circular Cell Approximation by Variational Method: Magnetization of Ideal Type II Superconductor

In this paper I access the degree of approximation of known symbolic approach to solving of Ginzburg-Landau (GL) equations using variational method and a concept of vortex lattice with circular unit cells, refine it in a clear and concise way, identify and eliminate the errors. Also, I will improve its accuracy by providing for the first time precise dependencies of the variational parameters;correct and calculate magnetisation, compare it with the one calculated numerically and conclude they agree within 98.5% or better for any value of the GL parameter k and at magnetic field , which is good basis for many engineering applications. As a result, a theoretical tool is developed using known symbolic solutions of GL equations with accuracy surpassing that of any other known symbolic solution and approaching that of numerical one.