The Self-similar Solution for Ginzburg-Landau Equation and Its Limit Behavior in Besov Spaces

In this paper, we study the limit behavior of self-similar solutions for the Complex Ginzburg-Landau (CGL) equation in the nonstandard function space E_{s,p}. We prove the uniform existence of the solutions for the CGL equation and its limit equation in E_{s,p}. Moreover we show that the self-similar solutions of CGL equation converge, globally in time, to those of its limit equation as the parameters tend to zero. Key Words Ginzburg-Landau equation; Schrödinger equation; self-similar solution; limit behavior.     

Slow Time-periodic solutions of the cubic-quinitic Ginzburg-Landau equation

In this paper, the Ginzburg-Landau equation with small complex coefficients is considered. A translation is introduced to transform the Ginzburg-Landau equation into a dynamical system. Moreover, the existence and the properties of the equilibria are discussed. The spatial quasiperiodic solutions disappear due to the perturbation are proved. Finally, several types of heteroclinic orbits are proposed and numerical analysis are provided.     

Traveling waves in the complex Ginzburg-Landau equation

Summary In this paper we consider a modulation (or amplitude) equation that appears in the nonlinear stability analysis of reversible or nearly reversible systems. This equation is the complex Ginzburg-Landau equation with coefficients with small imaginary parts. We regard this equation as a perturbation of the real Ginzburg-Landau equation and study the persistence of the properties of the stationary solutions of the real equation under this perturbation. First we show that it is necessary to consider a two-parameter family of traveling solutions with wave speedυ and (temporal) frequencyθ; these solutions are the natural continuations of the stationary solutions of the real equation. We show that there exists a two-parameter family of traveling quasiperiodic solutions that can be regarded as a direct continuation of the two-parameter family of spatially quasi-periodic solutions of the integrable stationary real Ginzburg-Landau equation. We explicitly determine a region in the (wave speedυ, frequencyθ)-parameter space in which the weakly complex Ginzburg-Landau equation has traveling quasi-periodic solutions. There are two different one-parameter families of heteroclinic solutions in the weakly complex case. One of them consists of slowly varying plane waves; the other is directly related to the analytical solutions due to Bekki & Nozaki [3]. These solutions correspond to traveling localized structures that connect two different periodic patterns. The connections correspond to a one-parameter family of heteroclinic cycles in an o.d.e. reduction. This family of cycles is obtained by determining the limit behaviour of the traveling quasi-periodic solutions as the period of the amplitude goes to ∞. Therefore, the heteroclinic cycles merge into the stationary homoclinic solution of the real Ginzburg-Landau equation in the limit in which the imaginary terms disappear.     

Proof of the Absence of Elliptic Solutions of the Cubic Complex Ginzburg-Landau Equation

We consider the cubic complex Ginzburg-Landau equation. Using Hone's method, based on formal Laurent-series solutions and the residue theorem, we prove the absence of elliptic standing-wave solutions of this equation. This result complements a result by Hone, who proved the nonexistence of elliptic traveling-wave solutions. We show that it is more efficient to apply Hone's method to a system of polynomial differential equations rather than to an equivalent differential equation. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 161–171, January, 2006.     

应用多项式完全判别系统方法求解时空分数阶复Ginzburg-Landau方程

研究了时空分数阶复Ginzburg-Landau方程.首先通过分数阶复变换将时空分数阶复Ginzburg-Landau方程转化为一个常微分方程.然后将常微分方程化为初等积分形式.最后用多项式完全判别系统法求得一系列精确解,其中包含有孤立波解、有理函数解、三角函数周期解、Jacobi椭圆函数双周期解.     

Pathfollowing for essentially singular boundary value problems with application to the complex Ginzburg-Landau equation

We present a pathfollowing strategy based on pseudo-arclength parametrization for the solution of parameter-dependent boundary value problems for ordinary differential equations. We formulate criteria which ensure the successful application of this method for the computation of solution branches with turning points for problems with an essential singularity. The advantages of our approach result from the possibility to use efficient mesh selection, and a favorable conditioning even for problems posed on a semi-infinite interval and subsequently transformed to an essentially singular problem. This is demonstrated by a Matlab implementation of the solution method based on an adaptive collocation scheme which is well suited to solve problems of practical relevance. As one example, we compute solution branches for the complex Ginzburg-Landau equation which start from non-monotone ‘multi-bump’ solutions of the nonlinear Schrödinger equation. Following the branches around turning points, real-valued solutions of the nonlinear Schrödinger equation can easily be computed.     

Exact front, soliton and hole solutions for a modified complex Ginzburg-Landau equation from the harmonic balance method

A novel approach of using harmonic balance (HB) method is presented to find front, soliton and hole solutions of a modified complex Ginzburg-Landau equation. Three families of exact solutions are obtained, one of which contains two parameters while the others one parameter. The HB method is an efficient technique in finding limit cycles of dynamical systems. In this paper, the method is extended to obtain homoclinic/heteroclinic orbits and then coherent structures. It provides a systematic approach as various methods may be needed to obtain these families of solutions. As limit cycles with arbitrary value of bifurcation parameter can be found through parametric continuation, this approach can be extended further to find analytic solution of complex quintic Ginzburg-Landau equation in terms of Fourier series.     

Insensitizing controls for a class of nonlinear Ginzburg-Landau equations

This paper shows the existence of insensitizing controls for a class of nonlinear complex Ginzburg-Landau equations with homogeneous Dirichlet boundary conditions and arbitrarily located internal controller. When the nonlinearity in the equation satisfies a suitable superlinear growth condition at infinity, the existence of insensitizing controls for the corresponding semilinear Ginzburg-Landau equation is proved. Meanwhile, if the nonlinearity in the equation is only a smooth function without any additional growth condition, a local result on insensitizing controls is obtained. As usual, the problem of insensitizing controls is transformed into a suitable controllability problem for a coupled system governed by a semilinear complex Ginzburg-Landau equation and a linear one through one control. The key is to establish an observability inequality for a coupled linear Ginzburg-Landau system with one observer.     

Traveling fronts of a real supercritical quintic Ginzburg-Landau equation coupled by a slow diffusion mode

In this paper, we investigate the existence of traveling front solutions for a class of quintic Ginzburg-Landau equations coupled with a slow diffusion mode. By employing the theory of geometric singular perturbations, we turn the problem into a geometric perturbation problem. We demonstrate the intersection property of the critical manifold and further validate the existence of heteroclinic orbits by computing the zeros of the Melnikov function on the critical manifold. The results demonstrate that under certain parameters, there is 1 or 2 heteroclinic solutions, confirming the existence of traveling front solutions for the considered quintic Ginzburg-Landau equation coupled with a slow diffusion mode.     

三维全空间上Ginzburg Landau方程的整体吸引子

作者在三维全空间中考虑研究复Ginzburg Landau方程(CGL)的解的长时间行为。通过引入权空间，应用内插不等式和在权空间的先验估计，获得复 Ginzburg Landau方程整体解的存在性，进一步建立了整体吸引子的存在性。     

非自治Ginzburg-Landau方程的周期解和全局周期吸引子

研究受周期外力影响的非自治Ginzburg-Landau方程的解的长时间行为.首先证明系统在空间H上存在周期解,而且周期解包含在空间V中的一个有界吸收集内.然后证明了当耗散系数λ满足一定条件时,该系统在空间H上具有唯一的周期解,该周期解指数吸引H中的任意有界集.     

On the Global Attractor of Generalized Ginzburg-Landau Equation in One Spatial Dimension

The target of this paper is the long time behaviour of solutions for a generalized Ginzburg-Landau equation on IR. The authors establish the existence of a global attractor of finite Hausdorff and fractal dimension in a weighted Hilbert space for the equation.     

Multi-hump pulses in systems with reflection and phase invariance

We investigate the existence and stability of standing and travelling multi-hump waves in partial differential equations with reflection and phase symmetries. We focus on 2- and 3-pulse solutions that arise near bi-foci and apply our results to the complex cubic-quintic Ginzburg-Landau equation.     

Meromorphic Traveling Wave Solutions of the Complex Cubic-Quintic Ginzburg-Landau Equation

We look for singlevalued solutions of the squared modulus M of the traveling wave reduction of the complex cubic-quintic Ginzburg-Landau equation. Using Clunie??s lemma, we first prove that any meromorphic solution M is necessarily elliptic or degenerate elliptic. We then give the two canonical decompositions of the new elliptic solution recently obtained by the subequation method.     

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 various Lpspaces, at different rates along different time sequences going to infinity, for complex Ginzburg-Landau equation on RN, of various parameters θ and γ.     

Stabilization by Slow Diffusion in a Real Ginzburg-Landau System

The Ginzburg-Landau equation is essential for understanding the dynamics of patterns in a wide variety of physical contexts. It governs the evolution of small amplitude instabilities near criticality. It is well known that the (cubic) Ginzburg-Landau equation has various unstable solitary pulse solutions. However, such localized patterns have been observed in systems in which there are two competing instability mechanisms. In such systems, the evolution of instabilities is described by a Ginzburg-Landau equation coupled to a diffusion equation. In this article we study the influence of this additional diffusion equation on the pulse solutions of the Ginzburg-Landau equation in light of recently developed insights into the effects of slow diffusion on the stability of pulses. Therefore, we consider the limit case of slow diffusion, i.e., the situation in which the additional diffusion equation acts on a long spatial scale. We show that the solitary pulse solution of the Ginzburg-Landau equation persists under this coupling. We use the Evans function method to analyze the effect of the slow diffusion and to show that it acts as a control mechanism that influences the (in)stability of the pulse. We establish that this control mechanism can indeed stabilize a pulse when higher order nonlinearities are taken into account.     

Soliton solutions and Bäcklund transformation for the complex Ginzburg-Landau equation

Symbolically investigated in this paper is the complex Ginzburg-Landau (CGL) equation. With the Hirota method, both bright and dark soliton solutions for the CGL equation are obtained simultaneously. New Bäcklund transformation in the bilinear form is derived. Relevant properties and features are discussed. Solitons can be compressed (amplified) when the nonlinear (linear) dispersion effect is enhanced. Meanwhile, central frequency of the soliton can be affected by the nonlinear and linear dispersion effects. Besides, directions of the movement for the soliton central frequency can be adjusted. Results of this paper would be of certain value to the studies on the soliton compression and amplification.     

The effect of nonlocal interactions on the dynamics of the Ginzburg-Landau equation

Nonlocal amplitude equations of the complex Ginzburg-Landau type arise in a few physical contexts, such as in ferromagnetic systems. In this paper, we study the effect of the nonlocal term on the global dynamics by considering a model nonlocal complex amplitude equation. First, we discuss the global existence, uniqueness and regularity of solutions to this equation. Then we prove the existence of the global attractor, and of a finite dimensional inertial manifold. We provide upper and lower bounds to their dimensions, and compare them with those of the cubic complex Ginzburg-Landau equation. It is observed that the nonlocal term plays a stabilizing or destabilizing role depending on the sing of the real part of its coefficient. Moreover, the nonlocal term affects not only the diameter of the attractor but also its dimension.     

R~3中各向异性Landau-Lifshitz方程的部分正则解

证明了当三维空间中各向异性Landau-Lifshitz方程的弱解满足稳定性条件时,其解具有部分正则性,并且对易面类型的方程,利用Ginzburg-Landau逼近构造了一个整体的部分正则解。