The homogeneous balance method and its application to the Benjamin-Bona-Mahoney (BBM) equation

We make use of the homogeneous balance method and symbolic computation to construct new exact traveling wave solutions for the Benjamin-Bona-Mahoney (BBM) equation. Many new exact traveling wave solutions are successfully obtained, which contain rational and periodic-like solutions. This method is straightforward and concise, and it can also be applied to other nonlinear evolution equations.     

Zhiber-Shabat方程的孤立波解与周期波解

结合齐次平衡法原理并利用F展开法,再次研究了Zhiber-Shabat方程的各种椭圆函数周期解.当椭圆函数的模m分别趋于1或0时,利用这些椭圆函数周期解,得到了Zhiber-Shabat方程的各种孤子解和三角函数周期解,从而丰富了相关文献中关于Zhiber-Shabat波方程的解的类型.     

New soliton solutions for a Kadomtsev-Petviashvili (KP) like equation coupled to a Schrödinger equation

The repeated homogeneous balance is used to construct a new exact traveling wave solution of the Kadomtsev-Petviashvili (KP) like equation coupled to a Schrödinger equation, in which the homogeneous balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation, respectively. Many new exact traveling wave solutions are successfully obtained, which contain rational and periodic-like solutions. This method is straightforward and concise, and it can be applied to other nonlinear evolution equations.     

二维色散长波方程组的精确解

利用齐次平衡法给出了二维色散长波方程组的定态解、孤立波解与非孤立波解等几种显式精确解。这个方法也可用来寻找其它非线性发展方程的不同类型的精确解。     

Traveling wave solutions of nonlinear evolution equations via the enhanced (G′/G)-expansion method

In this article, an enhanced (G′/G)-expansion method is suggested to find the traveling wave solutions for the modified Korteweg de-Vries (mKDV) equation. Abundant traveling wave solutions are derived, which are expressed by the hyperbolic and trigonometric functions involving several parameters. The efficiency of this method for finding these exact solutions has been demonstrated. It is shown that the proposed method is effective and can be used for many other nonlinear evolution equations (NLEEs) in mathematical physics.     

TRAVELING WAVE SOLUTIONS AND THEIR STABILITY OF NONLINEAR SCHRDINGER EQUATION WITH WEAK DISSIPATION

In this paper,several new constant-amplitude and variable-amplitude wave solutions(namely,traveling wave solutions) of a generalized nonlinear Schrdinger equation are investigated by using the extended homogeneous balance method,where the balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation,respectively.In addition,stability analysis of those solutions are also conducted by regular phase plane technique.     

广度kdv方程的精确解:改进的齐次平衡法

In this paper,we present a solution methodology to obtain exact solutions of some nonlinear evolution equation by modifying the homogeneous balance method.Based on the modified homogeneous balance method,several kinds of exact(new)solutions of the generalized KdV equation are obtained.     

New traveling waves solutions to generalized Kaup-Kupershmidt and Ito equations

In this paper we consider a special fifth-order KdV equation with constant coefficients and we obtain traveling wave solutions for it, using the projective Riccati equation method. By mean of a scaling, exact solutions to general Kaup-Kupershmidt (KK) equation are obtained. As a particular case, exact solutions to standard KK equation can be derived. Using the same method, we obtain exact solutions to standard Ito equation. By mean of scaling, new exact solutions to general Ito equation are formally derived.     

Some new soliton-like solutions and periodic wave solutions with loop or without loop to a generalized KdV equation

In this paper, by using the integral bifurcation method, we study a generalized KdV equation which was first derived by Fokas from physical considerations via a methodology of Fuchssteiner. All kinds of soliton-like or kink-like wave solutions and periodic wave solutions with loop or without loop are obtained. Smooth compacton-like periodic wave solution and non-smooth periodic cusp wave solution are also obtained. Their dynamic properties are investigated and their profiles are given by Mathematical software.     

Exact Solitary Wave And Periodic Wave Solutions Of The Kaup-Kuperschmidt Equation

In this paper we investigate the exact traveling wave solutions of the fifth-order Kaup-Kuperschmidt equation. The bifurcation and exact solutions of a general first-order nonlinear equation are investigated firstly. With the help of Maple and by using the bifurcation and exact solutions of two derived subequations, we obtain two families of solitary wave solutions and two families of periodic wave solutions of the KK equation. The relationship of the two subequations and the two known rst integrals are analyzed.     

新的辅助方程法构造KdV方程的行波解

应用一种新的辅助方程法成功地获得了(1+1)维KdV方程的多个含有参数的精确行波解,所得的解涵盖了已有结果．与其它方法相比,所给出的方法具有简单高效、计算量小、速度快、易于求解等特点．另外,所给的方法还可以用来求解其它的一大类非线性发展方程的精确行波解．     

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.     

RLW-Burgers方程的一类解析解

本文给出了 RLW-Burgers方程及 Kd V-Burgers方程的一类解析解 ,且可得到 RLW-Burgers方程的振荡激波解 .这些解可以表示为 Burgers方程和 Kd V方程解的线性组合 ,文末还对文 [8]作了讨论 .     

New generalized and improved (G′/G)-expansion method for nonlinear evolution equations in mathematical physics

In this article, new extension of the generalized and improved (G′/G)-expansion method is proposed for constructing more general and a rich class of new exact traveling wave solutions of nonlinear evolution equations. To demonstrate the novelty and motivation of the proposed method, we implement it to the Korteweg-de Vries (KdV) equation. The new method is oriented toward the ease of utilize and capability of computer algebraic system and provides a more systematic, convenient handling of the solution process of nonlinear equations. Further, obtained solutions disclose a wider range of applicability for handling a large variety of nonlinear partial differential equations.     

Analytic investigation of the (2 + 1)-dimensional Schwarzian Korteweg-de Vries equation for traveling wave solutions

By means of the two distinct methods, the Exp-function method and the extended (G′/G)-expansion method, we successfully performed an analytic study on the (2 + 1)-dimensional Schwarzian Korteweg-de Vries equation. We exhibited its further closed form traveling wave solutions which reduce to solitary and periodic waves. New rational solutions are also revealed.     

Bäcklund transformations and abundant exact explicit solutions of the Sharma-Tasso-Olver equation

By using an extension of the homogeneous balance method and Maple, the Bäcklund transformations for the Sharma-Tasso-Olver equation are derived. The connections between the Sharma-Tasso-Olver equation and some linear partial differential equations are found. With the aid of the transformations given here and the computer program Maple 12, abundant exact explicit special solutions to the Sharma-Tasso-Olver equation are constructed. In addition to all known solutions re-deriving in a systematic way, several entirely new and more general exact explicit solitary wave solutions can also be obtained. These solutions include (a) the algebraic solitary wave solution of rational function, (b) single-soliton solutions, (c) double-soliton solutions, (d) N-soliton solutions, (e) singular traveling solutions, (f) the periodic wave solutions of trigonometric function type, and (g) many non-traveling solutions. By using the Airy’s function and the Bäcklund transformations obtained here, the exact explicit solution of the initial value problem for the STO equation is presented. The variety of the structure of the solutions for the Sharma-Tasso-Olver equation is illustrated.     

扩展的(G'/G)展开法求量子Zakharov-Kuznetsov方程的精确解

为得到量子Zakharov-Kuznetsov方程的一些新精确解,借助行波解的思想,结合齐次平衡原理和一类非线性常微分方程解的结构,利用扩展的(G'/G)展开方法,研究了其相应的更加丰富的精确解表达形式.新精确解的表达式主要由双曲函数、三角函数和有理数函数构成,出现了某些怪波解的情形.通过对比不同情况下解的形式,利用M...     

On the ZK equation with a directional dissipation

This paper studies the generalized Zakharov-Kuznetsov-Burgers equation. The initial value problem associated to this equation will be investigated in the nonhomogeneous Sobolev spaces and some suitable weighted spaces, under appropriate conditions. Moreover, an ill-posedness result (in some sense) will be proved in the anisotropic Sobolev spaces. Furthermore some exact traveling wave solutions of this equation will be obtained.