色散渐减光纤中Ginzburg-Landau方程的自相似脉冲演化的解析解

采用对称约简的分析方法，得出了变系数Ginzburg-Landau方程的抛物渐近自相似脉冲解析解的一般表达式.给出了二阶色散系数纵向双曲型变化和纵向指数型变化的色散渐减光纤中自相似脉冲的振幅、啁啾以及脉冲宽度的具体形式，并与数值解进行了对比，其结果符合得很好.从而证实了稀土元素掺杂的色散渐减光纤中，在增益色散因子的影响下，脉冲的演化具有抛物型自相似特性.     

Time Dependent Ginzburg-Landau方程的Weierstrass椭圆函数解

将文[22]中提出的求解非线性演化方程的Weierstrass椭圆函数解的一个新方法应用于Time Dependent Ginzburg-Landau方程,获得了该方程的一些新的双周期解,并在退化情形下得到了一些新的精确孤波解.     

高阶色散效应常系数Ginzburg-Landau方程自相似脉冲演化的解析分析

采用自相似分析方法,基于常系数高阶色散的Ginzburg-Landau方程,通过分离变量法得出了高阶色散效应自相似脉冲演化的解析解,给出了自相似脉冲的振幅、相位、啁啾以及脉冲宽度的一般表达式.研究表明,在增益光纤的二阶正常色散区域,同时考虑高阶色散和增益色散双重效应影响下演化的自相似孤子脉冲仍然保持线性啁啾;振幅解析解的三阶色散效应显著.这与数值计算的结果非常一致. 关键词： 三阶色散 Ginzburg-Landau方程 自相似脉冲 二阶正常色散     

Cascade replication of soliton solutions in the one-dimensional complex cubic-quintic Ginzburg-Landau equation

In the framework of the one-dimensional cubic-quintic complex Ginzburg-Landau equation, we demonstrate two methods to achieve cascade replication of dissipative solitons from a single one by using external forces whose positions and magnitudes are being precisely controlled. The first method combines high amplitude positive currents and negative currents to increase the efficiency of the process significantly and to make the process repeatable. By using two repulsive forces whose locations are time dependent, the second method enables one to obtain multiple dissipative solitons in only one cascade replication process.     

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 Hölder continuous generalized synchronization exist in the coupling system. Numerical result analysis shows the correctness of theory.     

Soliton Solutions of Discrete Complex Ginzburg-Landau Equation via Extended Hyperbolic Function Approach

In this paper, we extend the hyperbolic function approach for constructing the exact solutions of nonlinear differential-difference equation (NDDE) in a unified way. Applying the extended approach and with the aid of Maple,we have studied the discrete complex Ginzburg-Landau equation (dCGLE). As a result, we find a set of exact solutions which include bright and dark soliton solutions.     

耦合复金兹堡-朗道(Ginzburg-Landau)方程中的模螺旋波

以双层耦合复金兹堡-朗道(Ginzburg-Landau)方程系统为时空模型, 研究了其中的模螺旋波, 讨论了这种特殊波动现象的稳定条件和相关影响因素. 模螺旋波与该类时空系统中常见的相螺旋波相比, 其中心不存在缺陷点, 同时仅在其变量的振幅部分(而非相位部分) 表现为螺旋结构. 本文通过数值方法研究了耦合复金兹堡-朗道方程中产生模螺旋波所需要的初始和参数条件.研究表明, 当双层耦合系统的初始斑图之间的差距较大时, 才能够产生模螺旋波; 同时观察到系统在参数不匹配的条件下会发生相螺旋波向模螺旋波的转变.通过对同步函数的计算, 发现该转变过程具有非连续性.     

Exact homoclinic wave and soliton solutions for the 2D Ginzburg-Landau equation

New exact wave solutions including homoclinic wave, kink wave and soliton solutions for the 2D Ginzburg-Landau equation are obtained using the auxiliary function method, generalized Hirota method and the ansatz function technique under the certain constraint conditions of coefficients in equation, respectively. The result shows that there exists a kink-wave solution which tends to one and the same periodic wave solution as time tends to infinite.     

复Ginzburg-Landau方程中模螺旋波的稳定性研究

研究了复Ginzburg-Landau方程系统中模螺旋波与其他斑图在同一平面内的竞争行为,发现演化结果在系统参数平面内可分为四个主要区域:在I区和III区中,模螺旋波与相螺旋波相比稳定性较差,模螺旋波的空间被相螺旋波所入侵.在II区中,模螺旋波具有较强的稳定性,相螺旋波的空间被模螺旋波所入侵.在IV区内,由于时空混沌所导致的频率不稳定性,演化的结果较为复杂.我们通过对模螺旋波、相螺旋波以及时空混沌的频率分析,发现当模螺旋波的系统参数为α1=-1.34,β1=0.35时,较高频率的模螺旋波具有较好的稳定性,高频模螺旋波可以入侵低频斑图空间.竞争结果主要受系统变量实部的频率影响,频率分析所得到的理论结果与数值实验结果符合得非常好.     

Analytical self-similar solutions of the Ginzburg-Landau equation with three-order dispersion effect

Based on the technique of the symmetry reduction, we find the asymptotic self-similarity analytical resolutions from the constant coefficient Ginzburg-Landau equation considering both influences of the thirdorder dispersion and gain dispersion on the evolution of pulses. We have obtained the self-similar pulse amplitude function, phase function, strict linear chirp function, and the effective temporal pulse width. Numerical simulations show qualitative agreement with these theoretical results.     

Periodic spatio-temporal oscillations governed by a globally controlled Ginzburg-Landau equation

A new type of periodic oscillations in a globally controlled subcritical cubic complex Ginzburg-Landau equation, formerly observed in numerical simulations, is explained and investigated analytically by means of a multiscale perturbation theory. Using an appropriate class of solutions of the nonlinear Schrödinger equation as a starting point, we construct a new class of asymptotic solutions of the cubic complex Ginzburg-Landau equation in the limit of large dispersion and nonlinear frequency shift.     

Envelope self-similar solutions for the nonautonomous and inhomogeneous nonlinear Schrödinger equation

By means of the similarity transformation connecting with the solvable stationary equation, the self-similar combined Jacobian elliptic function solutions and fractional form solutions of the generalized nonlinear Schrödinger equation (NLSE) are obtained when the dispersion, nonlinearity, and gain or absorption are varied. The propagation dynamics in a periodic distributed amplification system is investigated. Self-similar cnoidal waves and corresponding localized waves including bright and dark similaritons (or solitons) for NLSE and arch and kink similaritons (or solitons) for cubic-quintic NLSE are analyzed. The results show that the intensity and the width of chirped cnoidal waves (or similaritons) change more distinctly than that of chirp-free counterparts (or solitons).     

Spatiotemporal self-similar solutions for the nonautonomous (3+1)-dimensional cubicben quintic Grossben Pitaevskii equation

With the help of similarity transformation, we obtain analytical spatiotemporal self-similar solutions of the nonautonomous (3+1)-dimensional cubic-quintic Gross-Pitaevskii equation with time-dependent diffraction, nonlinearity, harmonic potential and gain or loss when two constraints are satisfied. These constraints between the system parameters hint that self-similar solutions form and transmit stably without the distortion of shape based on the exact balance between the diffraction, nonlinearity and the gain/loss. Based on these analytical results, we investigate the dynamic behaviours in a periodic distributed amplification system.     

基于复Ginzburg-Landau方程的双核光纤中调制不稳定性的仿真研究

在双核光纤光学系统中,应用复Ginzburg-Landau方程,研究了连续波的不稳定性问题.双核光纤光学系统是由一个非线性离散主核和一个线性附核构成的.研究发现,在线性微扰下存在调制不稳定性.系统仿真结果表明：如果充分考虑调制不稳定性,则该系统将产生规则或者不规则的脉冲序列.反之,如果不考虑调制不稳定性它将产生一连串具有连续增长振幅的离散峰.这表明在反常群速度色散情况下,一串归零脉冲的峰值或者单一归零脉冲峰值仍然是增强的.在光纤中产生归零序列脉冲源,这一研究结果对全光纤通信有一定的价值,对光纤光学及物理学 关键词： 光孤子 复Ginzburg-Landau方程 双核光纤 调制不稳定性     

Time-dependent Ginzburg-Landau equation in a car-following model considering the driver’s physical delay

In this paper, an extended car-following model considering the delay of the driver’s response in sensing headway is proposed to describe the traffic jam. It is shown that the stability region decreases when the driver’s physical delay in sensing headway increases. The phase transition among the freely moving phase, the coexisting phase, and the uniformly congested phase occurs below the critical point. By applying the reductive perturbation method, we get the time-dependent Ginzburg-Landau (TDGL) equation from the car-following model to describe the transition and critical phenomenon in traffic flow. We show the connection between the TDGL equation and the mKdV equation describing the traffic jam.     

Analytic solutions of self-similar pulse based on Ginzburg-Landau equation with constant coefficients

Based on the constant coefficients of Ginzburg-Landau equation that considers the influence of the doped fiber retarded time on the evolution of self-similar pulse, the parabolic asymptotic self-similar solutions were obtained by the symmetry reduction algorithm. The parabolic asymptotic amplitude function, phase function, strict linear chirp function and the effective temporal pulse width of self-similar pulse are given in this paper. And these theoretical results are consistent with the numerical simulations. Supported by the Natural Science Foundation of Guangdong Province of China (Grant No. 04010397)     

Exact solutions of the generalized Bretherton equation

The generalized Bretherton equation is studied. The Bäcklund transformations between traveling wave solutions of the generalized Bretherton equation and solutions of polynomial ordinary differential equation are constructed. The classification problem for meromorphic solutions of the latter equation is discussed. Several new families of exact solutions for the generalized Brethenton equation are given.     

Spiral Solutions of the Two-Dimensional Complex Ginzburg-Landau Equation

The multi-order exact solutions of the two-dimensional complex Ginzburg-Landau equation are obtained by making use of the wave-packet theory. In these solutions, the zeroth-order exact solution is a plane wave, the first-order exact solutions are shock waves for the amplitude and spiral waves both between the amplitude and the shift of phase and between the shift of phase and the distance.     

Noise-sustained structure,intermittency, and the Ginzburg-Landau equation

The time-dependent generalized Ginzburg-Landau equation is an equation that is related to many physical systems. Solutions of this equation in the presence of low-level external noise are studied. Numerical solutions of this equation in thestationary frame of reference and with anonzero group velocity that is greater than a critical velocity exhibit a selective spatial amplification of noise resulting in spatially growing waves. These waves in turn result in the formation of a dynamic structure. It is found that themicroscopic noise plays an important role in themacroscopic dynamics of the system. For certain parameter values the system exhibits intermittent turbulent behavior in which the random nature of the external noise plays a crucial role. A mechanism which may be responsible for the intermittent turbulence occurring in some fluid systems is suggested.     

Existence of self-similar blow-up solutions for Zakhrov equation in dimension two. Part I

We consider the Zakharov equation in space dimension two