Exact solutions to complex Ginzburg–Landau equation

In this paper, the trial equation method and the complete discrimination system for polynomial method are applied to retrieve the exact travelling wave solutions of complex Ginzburg–Landau equation. Both the Kerr and power laws of nonlinearity are considered. All the possible exact travelling wave solutions consisting of the rational function-type solutions, solitary wave solutions, triangle function-type periodic solutions and Jacobian elliptic functions solutions are obtained, and some of them are new solutions. In addition, concrete examples are presented to ensure the existence of obtained solutions. Moreover, four types of representative solutions are depicted to present the nature of the obtained solutions.     

Exact solutions to fractional Drinfel’d–Sokolov–Wilson equations

In this paper, the complete discrimination system for polynomial method is applied to retrieve the exact traveling wave solutions of time-fractional coupled Drinfel’d–Sokolov–Wilson equations. All of the possible exact traveling wave solutions which consist of the rational function type solutions, solitary wave solutions, triangle function type periodic solutions and Jacobian elliptic functions solutions are obtained, and some of them are new solutions. Moreover, the concrete examples are presented to ensure the existences of obtained solutions. In addition, four types of representative solutions are depicted to show the nature of 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.     

Some new radiating Kerr-Newman solutions

Three exact non-static solutions of Einstein-Maxwell equations corresponding to a field of flowing null radiation plus an electromagnetic field are presented. These solutions are non-static generalizations of the well known Kerr-Newman solution. The current vector is null in all the three solutions. These solutions are the electromagnetic generalizations of the three generalized radiating Kerr solutions discussed by Vaidya and Patel. The solutions discussed by us describe the exterior gravitational fields of rotating radiating charged bodies. Many known solutions are derived as particular cases.     

(G′/G)展开法和(2+1)维非对称Nizhnik-Novikov-Veselov系统的新精确解

通过引入并扩展 （G′/G）展开法, 构造出（2+1）维非对称Nizhnik-Novikov-Veselov系统的三种形式的新精确通解: 双曲函数通解, 三角函数通解, 有理函数通解. 当双曲函数通解中的常数取特定值时, 通解变为相应孤立波解. 关键词： G′/G）展开法')" href="#">（G′/G）展开法 （2+1）维非对称Nizhnik-Novikov-Veselov系统 精确解 孤立波解     

含高阶非线性效应的薛定谔方程的精确解研究

利用孤子理论,研究了含三次和五次非线性项的非线性薛定谔方程,在参数取不同值时得到了方程的新型亮孤子解、新型暗孤子解和新的三角函数周期解。     

Static,Spherically Symmetric Solutions of Yang-Mills-Dilaton Theory

Static, spherically symmetric solutions of the Yang-Mills-Dilaton theory are studied. It is shown that these solutions fall into three different classes. The generic solutions are singular. Besides there is a discrete set of globally regular solutions further distinguished by the number of nodes of their Yang-Mills potential. The third class consists of oscillating solutions playing the role of limits of regular solutions, when the number of nodes tends to infinity. We show that all three sets of solutions are non-empty. Furthermore we give asymptotic formulae for the parameters of regular solutions and confront them with numerical results.     

One-soliton solutions from Laplace’s seed

One-soliton solutions of axially symmetric vacuum Einstein field equations are presented in this paper. Two sets of Laplace’s solutions are used as seed and it is shown that the derived solutions reduce to some already known solutions when the constants are properly adjusted. An analysis of the solutions in terms of the Ernst potential is also presented. It is found that the solutions do not reduce to the Euclidean form at spatial infinity. However, in the static limit, Weyl solutions are obtained for half integral ∂-values.     

Anisotropic fluid spheres in general relativity

A procedure is developed to find static solutions for anisotropic fluid spheres from known static solutions for perfect fluid spheres. The method is used to obtain four exact analytical solutions of Einstein’s equations for spherically symmetric self-gravitating distribution of anisotropic matter. The solutions are matched to the Schwarzschild exterior metric. The physical features of one of the solutions are briefly discussed. Many previously known perfect fluid solutions are derived as particular cases.     

Bifurcations and new exact travelling wave solutions for the bidirectional wave equations

By using the method of dynamical system, the bidirectional wave equations are considered. Based on this method, all kinds of phase portraits of the reduced travelling wave system in the parametric space are given. All possible bounded travelling wave solutions such as dark soliton solutions, bright soliton solutions and periodic travelling wave solutions are obtained. With the aid of Maple software, numerical simulations are conducted for dark soliton solutions, bright soliton solutions and periodic travelling wave solutions to the bidirectional wave equations. The results presented in this paper improve the related previous studies.     

New Rational Form Solutions to Coupled Nonlinear Wave Equations

The new rational form solutions to the elliptic equation are shown, and then these solutions to the elliptic equation are taken as a transformation and applied to solve nonlinear coupled wave equations. It is shown that more novel kinds of solutions are derived, such as periodic solutions of rational form, solitary wave solutions of rational form,and so on.     

New Rational Form Solutions to Coupled Nonlinear Wave Equations

The new rational form solutions to the elliptic equation are shown, and then these solutions to the elliptic equation are taken as a transformation and applied to solve nonlinear coupled wave equations. It is shown that more novel kinds of solutions are derived, such as periodic solutions of rational form, solitary wave solutions of rational form,and so on.     

A Pair of Resonance Stripe Solitons and Lump Solutions to a Reduced(3+1)-Dimensional Nonlinear Evolution Equation

With symbolic computation, some lump solutions are presented to a(3+1)-dimensional nonlinear evolution equation by searching the positive quadratic function from the Hirota bilinear form of equation. The quadratic function contains six free parameters, four of which satisfy two determinant conditions guaranteeing analyticity and rational localization of the solutions, while the others are free. Then, by combining positive quadratic function with exponential function, the interaction solutions between lump solutions and the stripe solitons are presented on the basis of some conditions. Furthermore, we extend this method to obtain more general solutions by combining of positive quadratic function and hyperbolic cosine function. Thus the interaction solutions between lump solutions and a pair of resonance stripe solitons are derived and asymptotic property of the interaction solutions are analyzed under some specific conditions. Finally, the dynamic properties of these solutions are shown in figures by choosing the values of the parameters.     

A nonlocal Boussinesq equation: Multiple-soliton solutions and symmetry analysis

A nonlocal Boussinesq equation is deduced from the local one by using consistent correlated bang method. To study various exact solutions of the nonlocal Boussinesq equation, it is converted into two local equations which contain the local Boussinesq equation. From the N-soliton solutions of the local Boussinesq equation, the N-soliton solutions of the nonlocal Boussinesq equation are obtained, among which the (N=2,3,4)-soliton solutions are analyzed with graphs. Some periodic and traveling solutions of the nonlocal Boussinesq equation are derived directly from the known solutions of the local Boussinesq equation. Symmetry reduction solutions of the nonlocal Boussinesq equation are also obtained by using the classical Lie symmetry method.     

Bosonized Supersymmetric Sawada–Kotera Equations: Symmetries and Exact Solutions

The Bosonized Supersymmetric Sawada–Kotera(BSSK) system is constructed by applying bosonization method to a Supersymmetric Sawada–Kotera system in this paper. The symmetries on the BSSK equations are researched and the calculation shows that the BSSK equations are invariant under the scaling transformations, the space-time translations and Galilean boosts. The one-parameter invariant subgroups and the corresponding invariant solutions are researched for the BSSK equations. Four types of reduction equations and similarity solutions are proposed. Period Cnoidal wave solutions, dark solitary wave solutions and bright solitary wave solutions of the BSSK equations are demonstrated and some evolution curves of the exact solutions are figured out.     

Radial asymptotically periodic solutions of the Kuramoto-Sivashinsky equation

Radial steady solutions of the Kuramoto-Sivashinsky equation are studied. It is shown that there exist solutions that approach at infinity the one-dimensional periodic solutions. Both hyperbolic and elliptic periodic solutions are considered.     

Symbolic Computation and Construction of New Exact Travelling Wave Solutions to （3＋1）-Dimensional KP Equation

With the aid of symbolic computation system Maple, many exact solutions for the （3＋1）-dimensional KP equation are constructed by introducing an auxiliary equation and using its new Jacobi elliptic function solutions, where the new solutions are also constructed. When the modulus m → 1 and m →0, these solutions reduce to the corresponding solitary evolution solutions and trigonometric function solutions.     

Pipe Shape Solution of Faddeev Model

Utilizing the results that the Faddeev model is equivalent to the mesonic sector of the SU（2） Skyrme model, where the baryon number current vanishes everywhere, some exact solutions including the vortex solutions of the Faddeev model axe discussed. The solutions are classified by the 2, the new multisoliton solutions are obtained and the pipe shape found. first Chern number. When the Chern number equals distribution of the energy density of the solutions are     

Exact solutions to the generalized lienard equation and its applications

Some new exact solutions of the generalized Lienard equation are obtained, and the solutions of the equation are applied to solve nonlinear wave equations with nonlinear terms of any order directly. The generalized one-dimensional Klein-Gordon equation, the generalized Ablowitz (A) equation and the generalized Gerdjikov-Ivanov (GI) equation are investigated and abundant new exact travelling wave solutions are obtained that include solitary wave solutions and triangular periodic wave solutions.      

Variable-Coefficient Mapping Method Based on Elliptical Equation and Exact Solutions to Nonlinear SchrSdinger Equations with Variable Coefficient

In this paper, by means of the variable-coefficient mapping method based on elliptical equation, we obtain explicit solutions of nonlinear Schrodinger equation with variable-coefficient. These solutions include Jacobian elliptic function solutions, solitary wave solutions, soliton-like solutions, and trigonometric function solutions, among which some are found for the first time. Six figures are given to illustrate some features of these solutions. The method can be applied to other nonlinear evolution equations in mathematical physics.