A new rational auxiliary equation method and exact solutions of a generalized Zakharov system

A new rational auxiliary equation method for obtaining exact traveling wave solutions of constant coefficient nonlinear partial differential equations of evolution is proposed. Its effectiveness is evinced by obtaining exact solutions of a generalized Zakharov system, some of which are new. It is shown that the G^′/G and the generalized projective Ricatti expansion methods are special cases of the auxiliary equation method. Further, due the solutions obtained, four other new and practicable rational methods are deduced.     

构造非线性差分方程精确解的一种方法

在齐次平衡法、试探函数法的基础上,给出指数函数所组成的两种试探函数法,并借助符号计算系统Mathematica构造了Hybrid-Lattice系统、mKdV差分微分方程、Ablowitz-Ladik.Lattice6系统等非线性离散系统的新的精确孤波解.     

Application of rational expansion method for stochastic differential equations

By means of the Hermite transformation, a new general ansätz and the symbolic computation system Maple, we apply the Riccati equation rational expansion method [24] to uniformly construct a series of stochastic non-traveling wave solutions for stochastic differential equations. To illustrate the effectiveness of our method, we take the stochastic mKdV equation as an example, and successfully construct some new and more general solutions. The method can also be applied to solve other nonlinear stochastic differential equation or equations.     

An extended subequation rational expansion method with symbolic computation and solutions of the nonlinear Schrödinger equation model

To construct exact analytical solutions of nonlinear evolution equations, an extended subequation rational expansion method is presented and used to construct solutions of the nonlinear Schrödinger equation with varing dispersion, nonlinearity, and gain or absorption. As a result, many previous known results of the nonlinear Schrödinger equation can be recovered by means of some suitable selections of the arbitrary functions and arbitrary constants. With computer simulation, the properties of a new non-travelling wave soliton-like solutions with coefficient functions and some elliptic function solutions are shown by some figures.     

Discrete exact solutions of modified Volterra and Volterra lattice equations via the new discrete sine-Gordon expansion algorithm

Recently, we have presented a sine-Gordon expansion method to construct new exact solutions of a wide of continuous nonlinear evolution equations. In this paper we further develop the method to be the discrete sine-Gordon expansion method in nonlinear differential-difference equations, in particular, discrete soliton equations. We choose the modified Volterra lattice and Volterra lattice equation to illustrate the new method such as many types of new exact solutions are obtained. Moreover some figures display the profiles of the obtained solutions. Our method can be also applied to other discrete soliton equations.     

Application of the exp‐function method to nonlinear lattice differential equations for multi‐wave and rational solutions

In this paper, we extend the basic Exp‐function method to nonlinear lattice differential equations for constructing multi‐wave and rational solutions for the first time. We consider a differential‐difference analogue of the Korteweg–de Vries equation to elucidate the solution procedure. Our approach is direct and unifying in the sense that the bilinear formalism of the equation studied becomes redundant. Copyright © 2011 John Wiley & Sons, Ltd.     

Traveling wave solutions of the generalized nonlinear evolution equations

Solitary wave solutions for a family of nonlinear evolution equations with an arbitrary parameter in the exponents are constructed. Some of the obtained solutions seem to be new.     

Extended Jacobi elliptic function expansion solutions of variant Boussinesq equations

With the aid of symbolic computation Maple, an extended Jacobi elliptic function expansion method is presented and successfully applied to variant Boussinesq equations. As a result, abundant periodic wave solutions in terms of the Jacobi elliptic functions are obtained. When the modulus m → 1 or m → 0, exact solitary wave solutions and trigonometric function solutions are also derived. The properties of four new solutions are graphically studied.     

NEW EXACT TRAVELLING WAVE SOLUTIONS TO THREE NONLINEAR EVOLUTION EQUATIONS

Based on the computerized symbolic computation, some new exact travelling wave solutions to three nonlinear evolution equations are explicitly obtained by replacing the tanhξ in tanh-function method with the solutions of a new auxiliary ordinary differential equation.     

Mixed function method for obtaining exact solution of nonlinear differential-difference equations and coupled NDDEs

In this paper, we construct a new mixed function method for the first time. By using this new method, we study the two nonlinear differential-difference equations named the generalized Hybrid lattice and two-component Volterra lattice equations. Some new exact solutions of mixed function type such as discrete solitary wave solutions, discrete kink and anti-kink wave solutions and discrete breather solutions with kink and anti-kink character are obtained and their dynamic properties are also discussed. By using software Mathematica, we show their profiles.     

Traveling solitary wave solutions to evolution equations with nonlinear terms of any order

Many physical phenomena in one- or higher-dimensional space can be described by nonlinear evolution equations, which can be reduced to ordinary differential equations such as the Lienard equation. Thus, to study those ordinary differential equations is of significance not only in mathematics itself, but also in physics. In this paper, a kind of explicit exact solutions to the Lienard equation is obtained. The applications of the solutions to the nonlinear RR-equation and the compound KdV-type equation are presented, which extend the results obtained in the previous literature.     

New generalized Jacobi elliptic function rational expansion method

In this work, a new generalized Jacobi elliptic function rational expansion method is based upon twenty-four Jacobi elliptic functions and eight double periodic Weierstrass elliptic functions, which solve the elliptic equation ?^′2=r+p?²+q?⁴, is described. As a consequence abundant new Jacobi-Weierstrass double periodic elliptic functions solutions for (3+1)-dimensional Kadmtsev-Petviashvili (KP) equation are obtained by using this method. We show that the new method can be also used to solve other nonlinear partial differential equations (NPDEs) in mathematical physics.     

New exact solutions for some coupled nonlinear partial differential equations using extended coupled sub-equations expansion method

An extended coupled sub-equations expansion method is proposed to seek new traveling wave solutions of the coupled nonlinear partial differential equations. The generalized Schrödinger-Boussinesq and coupled nonlinear Klein-Gordon-Schrödinger equations are used to illustrate the validity and the advantages of this method.     

Constructing rational and multi‐wave solutions to higher order NEEs via the Exp‐function method

In this paper, we present an application of some known generalizations of the Exp‐function method to the fifth‐order Burgers and to the seventh‐order Korteweg de Vries equations for the first time. The two examples show that the Exp‐function method can be an effective alternative tool for explicitly constructing rational and multi‐wave solutions with arbitrary parameters to higher order nonlinear evolution equations. Being straightforward and concise, as pointed out previously, this procedure does not require the bilinear representation of the equation considered. Copyright © 2011 John Wiley & Sons, Ltd.     

mKdV方程和mKP方程组的新的精确孤立波解

用三角函数假设法和一种新辅助方程的解构造mK dV方程和mKP方程组的精确孤立波解.这种方法也可用于寻找其它非线性发展方程的新的孤立波解.     

New explicit traveling wave solutions for three nonlinear evolution equations

In this paper, using the extended tanh-function method, new explicit traveling wave solutions including rational solutions for three nonlinear evolution equations are obtained with the aid of Mathematica.     

Some exact solutions for Toda type lattice differential equations using the improved (G′/G)‐expansion method

Nonlinear lattice differential equations (also known as differential‐difference equations) appear in many applications. They can be thought of as hybrid systems for the inclusion of both discrete and continuous variables. On the basis of an improved version of the basic (G′/G)‐expansion method, we focus our attention towards some Toda type lattice differential systems for constructing further exact traveling wave solutions. Our method provides not only solitary and periodic wave profiles but also rational solutions with more arbitrary parameters. Copyright © 2012 John Wiley & Sons, Ltd.     

Note on exact solutions for a class of nonlinear diffusion equations

An effective characterization is given for a class of generalized nonlinear diffusion equations with power law dependent terms. Further, a new auxiliary equation ansatz is derived. Consequently, new exact traveling wave trigonometric function, solitary-like and Weierstrass elliptic solutions to a subclass are obtained by means of an auxiliary equation method and a generalized Riccati equation expansion method.     

Exact and explicit solutions to nonlinear evolution equations using the division theorem

In this paper, we show the applicability of the first integral method, which is based on the ring theory of commutative algebra, to the regularized long-wave Burgers equation and the Gilson-Pickering equation under a parameter condition. Our method provides polynomial first integrals for autonomous planar systems. Through the established first integrals, exact traveling wave solutions are derived in a concise manner.