Exact Solutions for String Cosmology

Some more general cosmological solutions of Bianchi types Ⅱ，Ⅷ，and IX for a cloud string are presented.The physical implications of the solutions are briefly discussed.Our solutions include some of results previously given in the literature as special cases.     

New Explicit Solitary Wave Solutions and Periodic Wave Solutions for the Generalized Coupled Hirota－Satsuma KdV system

In this paper,we study the generalized coupled Hirota-Satsuma KdV system by using the new generalized transformation in homogeneous balance method.As a result,many explicit exact solutions,which contain new solitary wave solutions,periodic wave solutions,and the combined formal solitary wave solutions,and periodic wave solutions ,are obtained.     

New Exact Solutions of the N-Coupled Nonlinear Schroedinger System

In this letter, abundant families of Jacobi elliptic function envelope solutions of the N-coupled nonlinear Schroedinger （NLS） system are obtained directly. When the modulus m → 1, those periodic solutions degenerate as the corresponding envelope soliton solutions, envelope shock wave solutions. Especially, for the 3-coupled NLS system, five types of Jacobi elliptic function envelope solutions are illustrated both analytically and graphically. Two types of those degenerate as envelope soliton solutions.     

Fractal Solutions of the Nizhnik—Novikov—Veselov Equation

Considering that some types of fractal solutions may appear in many (2 1)-dimensional soliton equations because some arbitrary functions can be included in the exact solutions,we use some special types of lower dimensional fractal functions to construct higher dimensional fractal solutions of the Nizhnik-Novikov-Veselov equation.The static eagle-shape fractal solutions,fractal dromion solutions and the fractal lump solutions are given in detail.     

Rational Solutions in a Coupled Burgers System

Three types of the rational solutions for a new coupled Burgers system are studied in detail in terms of the reduction and decoupled procedures. The first two types of rational solutions are singular and valid for one type of model parameter c 〉 0, and another type of rational solutions is nonsingular at any type and valid for another type of model parameter c 〈 0.     

Exact Solutions to （2＋1）-Dimensional Kaup-Kupershmidt Equation

In this paper, by using the symmetry method, the relationships between new explicit solutions and old ones of the （2＋1）-dimensional Kaup-Kupershmidt （KK） equation are presented. We successfully obtain more general exact travelling wave solutions for （2＋ 1）-dimensional KK equation by the symmetry method and the （G1/G）-expansion method. Consequently, we find some new solutions of （2＋1）-dimensional KK equation, including similarity solutions, solitary wave solutions, and periodic solutions.     

Searching for Infinitely Many Symmetries and Exact Solutions via Repeated Similarity Reductions

A simple symmetry reduction procedure is repeatedly used to obtain infinitely many symmetries and then the exact solutions of the Burgers equation. Some sets of exact solutions such as the rational solutions, rationalkink solutions and error function solutions are explicitly given. As a byproduct the recursion operators can be re-obtained at the same time.     

Darboux Transformation and Explicit Solutions for Drinfel＇d-Sokolov-Wilson Equation

A generalized Drinfel＇d Sokolov-Wilson （DSW） equation and its Lax pair are proposed. A Daorboux transformation for the generalized DSW equation is constructed with the help of the gauge transformation between spectral problems, from which a Darboux transformation for the DSW equation is obtained through a reduction technique. As an application of the Darboux transformations, we give some explicit solutions of the generalized DSW equation and DEW equation such as rational solutions, soliton solutions, periodic solutions.     

Solving the AKNS Hierarchy by Its Bilinear Form： Generalized Double Wronskian Solutions

Through the Wronskian technique, a simple and direct proof is presented that the AKNS hierarchy in the bilinear form has generalized double Wronskian solutions. Moreover, by using a unified way, soliton solutions, rational solutions, Matveev solutions and complexitons in double Wronskian form for it are constructed.     

New Jacobian Elliptic Function Solutions to Modified KdV Equation: Ⅰ

An extended Jacobian elliptic function expansion method presented recently by us is applied to the mKdVequation such that thirteen families of Jacobian elliptic function solutions including both new solutions and Fu‘s allresults are obtained. When the modulus m → 1 or 0, we can find the corresponding six solitary wave solutions and sixtrigonometric function solutions. This shows that our method is more powerful to construct more exact Jacobian ellipticfunction solutions and can be applied to other nonlinear differential equations.     

A Series of Exact Solutions of （2＋l）-Dimensional CDGKS Equation

An algebraic method with symbolic computation is devised to uniformly construct a series of exact solutions of the （2＋1）-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawda equation. The solutions obtained in this paper include solitary wave solutions, rational solutions, triangular periodic solutions, Jacobi and Weierstrass doubly periodic solutions. Among them, the Jacobi periodic solutions exactly degenerate to the solutions at a certain limit condition. Compared with most existing tanh method, the method used here can give new and more general solutions. More importantly, this method provides a guldeline to classifj, the various types of the solution according to some parameters.     

Application of Modified G′/G-Expansion Method to Traveling Wave Solutions for Whitham-Broer-Kaup-Like Equations

A modified G′/G-expansion method is presented to derive traveling wave solutions for a class of nonlinear partial differential equations called Whitham -Broer- Kaup-Like equations. As a result, the hyperbolic function solutions, trigonometric function solutions, and rational solutions with parameters to the equations are obtained. When the parameters are taken as special values the solitary wave solutions can be obtained.     

Auto-Backlund Transformation and Analytic Solutions of （2＋1）-Dimensional Boussinesq Equation

Using the truncated Painleve expansion, symbolic computation, and direct integration technique, we study analytic solutions of （2＋1）-dimensional Boussinesq equation. An auto-Backlund transformation and a number of exact solutions of this equation have been found. The set of solutions include solitary wave solutions, solitoff solutions, and periodic solutions in terms of elliptic Jacobi functions and Weierstrass ＆ function. Some of them are novel.     

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.     

Solitary Wave Solutions of Triple Sine-Gordon Equation

Through a mapping relation between the triple sine-Gordon and Ф5 model, we discuss the equation in detail. A series of solitary wave solutions for the triple sine-Gordon equation is obtained by the existing ones of the Ф5 model. Some detailed solitary wave solutions are given for some special coefficients which have significance in physics.     

An Automated Algebraic Method for Finding a Series of Exact Travelling Wave Solutions of Nonlinear Evolution Equations

Based on a type of elliptic equation,a new algebraic method to construct a series of exact solutions for nonlinear evolution equations is proposed,meanwhile,its complete implementation TRWS in Maple is presented.The TRWS can output a series of travelling wave solutions entirely automatically,which include polynomial solutions,exponential function solutions,triangular function solutions,hyperbolic function solutions,rational function solutions,Jacobi elliptic function solutions,and Weierstrass elliptic function solutions.The effectiveness of the package is illustrated by applying it to a variety of equations.Not only are previously known solutions recovered but also new solutions and more general form of solutions are obtained.     

New Exact Solutions to （2＋1）-Dimensional Variable Coefficients Broer-Kaup Equations

In this paper, using the variable coefficient generalized projected Rieatti equation expansion method, we present explicit solutions of the （2＋1）-dimensional variable coefficients Broer-Kaup （VCBK） equations. These solutions include Weierstrass function solution, solitary wave solutions, soliton-like solutions and trigonometric function solutions. Among these solutions, some are found for the first time. Because of the three or four arbitrary functions, rich localized excitations can be found.     

Approximate Solutions of Perturbed Nonlinear Schroedinger Equations

By applying Lou＇s direct perturbation method to perturbed nonlinear Schroedinger equation and the critical nonlinear SchrSdinger equation with a small dispersion, their approximate analytical solutions including the zero-order and the first-order solutions are obtained. Based on these approximate solutions, the analytical forms of parameters of solitons are expressed and the effects of perturbations on solitons are briefly analyzed at the same time. In addition, the perturbed nonlinear Schroedinger equations is directly simulated by split-step Fourier method to check the validity of the direct perturbation method. It turns out that the analytical results given by the direct perturbation method are well supported by numerical calculations.     

New Exact Solutions to （2＋1）-Dimensional Dispersive Long Wave Equations

Based upon a further extended tanh method [Phys. Lett. A307 (2003) 269; Chaos, Solitons and Fractals 17 (2003) 669] and the symbolic computation system, Maple, we consider the (2 1)-dimensional dispersive long waveequations. We obtain many new solutions of the equation. These solutions contain solitomlike solutions, periodic form solutions, and some rational solutions.