Exact Solutions of Discrete Complex Cubic Ginzburg-LandauEquation and Their Linear Stability

The discrete complex cubic Ginzburg-Landau equation is an important model to describe a number of physical systems such as Taylor and frustrated vortices in hydrodynamics and semiconductor laser arrays in optics. In this paper, the exact solutions of the discrete complex cubic Ginzburg-Landau equation are derived using homogeneous balance principle and the G' / G-expansion method, and the linear stability of exact solutions is discussed.     

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.     

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.     

Propagation of nonlinear waves in bi-inductance nonlinear transmission lines

We consider a one-dimensional modified complex Ginzburg-Landau equation, which governsthe dynamics of matter waves propagating in a discrete bi-inductance nonlineartransmission line containing a finite number of cells. Employing an extended Jacobielliptic functions expansion method, we present new exact analytical solutions whichdescribe the propagation of periodic and solitary waves in the considered network.     

Exact periodic and blow up solutions for 2D Ginzburg-Landau equation

New exact periodic wave solutions for the 2D Ginzburg-Landau equation are obtained using the homogeneous balance principle and general Jacobi elliptic-function method. Furthermore, a blow up solution is provided. At the end, some properties about these solutions are showed by the graphs.     

复变系数Ginzburg-Landau方程的啁啾组合孤波解

基于描述非均匀光纤系统的复系数Ginzburg-Landau方程,通过拟解法得到了该方程的精确啁啾组合孤波解,并分析了该解的特性.通过大量的数值模拟,发现在有限的初始扰动下这些组合孤波解是在非均匀光纤系统中稳定的.最后,为了进一步研究组合孤波解的稳定性,我们还探讨了组合孤波的相互作用.     

Modulation instability analysis,optical and other solutions to the modified nonlinear Schrödinger equation

This paper studies the new families of exact traveling wave solutions with the modified nonlinear Schrödinger equation, which models the propagation of rogue waves in ocean engineering. The extended Fan sub-equation method with five parameters is used to find exact traveling wave solutions. It has been observed that the equation exhibits a collection of traveling wave solutions for limiting values of parameters. This method is beneficial for solving nonlinear partial differential equations, because it is not only useful for finding the new exact traveling wave solutions, but also gives us the solutions obtained previously by the usage of other techniques (Riccati equation, or first-kind elliptic equation, or the generalized Riccati equation as mapping equation, or auxiliary ordinary differential equation method) in a combined approach. Moreover, by means of the concept of linear stability, we prove that the governing model is stable. 3D figures are plotted for showing the physical behavior of the obtained solutions for the different values of unknown parameters with constraint conditions.     

Exact solutions of discrete complex cubic-quintic Ginzburg-Landau equation with non-local quintic term

In this paper, via the extended tanh-function approach, the abundant exact solutions for discrete complex cubic-quintic Ginzburg-Landau equation, including chirpless bright soliton, chirpless dark soliton, constant magnitude solution (plane wave solution), triangular function solutions and some solutions with alternating phases, etc. are obtained. Meanwhile, the range of parameters where some exact solution exist are given. Among these solutions, solutions with alternating phases do not have continuous analogs. Moreover, in the lattice, the points of singularity of tan-type and sec-type solutions can be ‘between sites’ and thus the singularities can be avoided.     

Stability Analysis of Solitary Wave Solutions for Coupled and(2+1)-Dimensional Cubic Klein-Gordon Equations and Their Applications

The searching exact solutions in the solitary wave form of non-linear partial differential equations(PDEs play a significant role to understand the internal mechanism of complex physical phenomena. In this paper, we employ the proposed modified extended mapping method for constructing the exact solitary wave and soliton solutions of coupled Klein-Gordon equations and the(2+1)-dimensional cubic Klein-Gordon(K-G) equation. The Klein-Gordon equation are relativistic version of Schr¨odinger equations, which describe the relation of relativistic energy-momentum in the form of quantized version. We productively achieve exact solutions involving parameters such as dark and bright solitary waves, Kink solitary wave, anti-Kink solitary wave, periodic solitary waves, and hyperbolic functions in which severa solutions are novel. We plot the three-dimensional surface of some obtained solutions in this study. It is recognized that the modified mapping technique presents a more prestigious mathematical tool for acquiring analytical solutions o PDEs arise in mathematical physics.     

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.     

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.     

利用蜂拥控制算法的反馈方法控制时空混沌

以一维复数Ginzburg-Landau方程系统为模型,研究时空混沌系统的可控性.基于蜂拥控制算法思想,提出了一种反馈控制方法.数值模拟结果表明,无论是选择空间均匀周期解,还是选择行波解为控制目标态,只要选择合适的控制强度,被研究的时空混沌系统都能被控制到有序的状态.最后,利用空间的关联函数解析其控制机制. 关键词： Ginzburg-Landau方程 时空混沌 关联函数     

New travelling wave solutions for combined KdV-mKdV equation and （2＋1）-dimensional Broer- Kaup- Kupershmidt system

Some new exact solutions of an auxiliary ordinary differential equation are obtained, which were neglected by Sirendaoreji {\it et al in their auxiliary equation method. By using this method and these new solutions the combined Korteweg--de Vries (KdV) and modified KdV (mKdV) equation and (2+1)-dimensional Broer--Kaup--Kupershmidt system are investigated and abundant exact travelling wave solutions are obtained that include new solitary wave solutions and triangular periodic wave solutions.     

New Exact Solutions of Two Nonlinear Physical Models

Abundant new exact solutions of the Schamel-Korteweg-de Vries （S-KdV） equation and modified Zakharov- Kuznetsov equation arising in plasma and dust plasma are presented by using the extended mapping method and the availability of symbolic computation. These solutions include the Jacobi elliptic function solutions, hyperbolic function solutions, rational solutions, and periodic wave solutions. In the limiting cases, the solitary wave solutions are obtained and some known solutions are also recovered.     

A Computational Approach to the New Type Solutions of Whitham-Broer-Kaup Equation in Shallow Water

Based on computerized symbolic computation, a new method and its algorithm are proposed for searching for exact travelling wave solutions of the nonlinear partial differential equations. Making use of our approach, we investigate the Whitham-Broer-Kaup equation in shallow water and obtain new families of exact solutions, which include soliton-like solutions and periodic solutions. As its special cases, the solutions of classical long wave equations and modified Boussinesq equations can also be found.     

Traveling wave solutions of conformable time-fractional Zakharov–Kuznetsov and Zoomeron equations

To model physical phenomena more accurately, fractional order differential equations have been widely used. Investigating exact solutions of the fractional differential equations have become more important because of the applications in applied mathematics, mathematical physics, and other areas. In this work, by means of the trial solution method and complete discrimination system, exact traveling wave solutions of the conformable time-fractional Zakharov–Kuznetsov equation and conformable time-fractional Zoomeron equation have been obtained and also solutions have been illustrated. Finding exact solutions of these equations that are encountered in plasma physics, nonlinear optics, fluid mechanics, and laser physics can help to understand nature of the complex phenomena.     

New exact travelling wave solutions of the generalized zakharov equations

A direct algebraic method is introduced for constructing exact travelling wave solutions of nonlinear partial differential equations with complex phases. The scheme is implemented for obtaining multiple soliton solutions of the generalized Zakharov equations, and then new exact travelling wave solutions with complex phases are obtained. In addition, by using new exact solutions of an auxiliary ordinary differential equation, new exact travelling wave solutions for the generalized Zakharov equations are obtained.     

Dynamics of NLS solitons described by the cubic-quintic Ginzburg-Landau equation

The generalized moment method is applied to average the Ginzburg-Landau equation with quintic nonlinearity in the neighborhood of a soliton solution to the nonlinear Schrödinger equation. A qualitative analysis of the resulting dynamical system is presented. New soliton solutions bifurcating from a known exact soliton solution are obtained. The results of the qualitative analysis are compared with those obtained by direct numerical solution of the Ginzburg-Landau equation.     

Soliton solutions for the space-time nonlinear partial differential equations with fractional-orders

Many practical models in interdisciplinary fields can be described with the help of fractional-order nonlinear partial differential equations(NPDEs). Fractional-order NPDEs such as the space-time fractional Fokas equation, the space-time Kaup–Kupershmidt equation and the space-time fractional (2+1)-dimensional breaking soliton equation have been widely applied in many branches of science and engineering. So, finding exact traveling wave solutions are very helpful in the theories and numerical studies of such equations. More precisely, fractional sub-equation method together with the proposed technique is implemented to obtain exact traveling wave solutions of such physical models involving Jumarie’s modified Riemann–Liouville derivative. As a result, some new exact traveling wave solutions for them are successfully established. Also, (1+1)-dimensional plots and 1-dimensional plots of some of the derived solutions are given to visualize the dynamics of the considered NPDEs. The obtained results reveal that the proposed technique is quite effective and convenient for obtaining exact solutions of NPDEs with fractional-order.     

The relation among the hyperbolic-function-type exact solutions of nonlinear evolution equations

First, we investigate the solitary wave solutions of the Burgers equation and the KdV equation, which are obtained by using the hyperbolic function method. Then we present a theorem which will not only give us a clear relation among the hyperbolic-function-type exact solutions of nonlinear evolution equations, but also provide us an approach to construct new exact solutions in complex scalar field. Finally, we apply the theorem to the KdV–Burgers equation and obtain its new exact solutions.