A connection between the(G'/G)-expansion method and the truncated Painlevé expansion method and its application to the mKdV equation

Recently the （G′/G）-expansion method was proposed to find the traveling wave solutions of nonlinear evolution equations. This paper shows that the （G′/G）-expansion method is a special form of the truncated Painlev＇e expansion method by introducing an intermediate expansion method. Then the generalized （G′/G）-（G/G′） expansion method is naturally derived from the standpoint of the nonstandard truncated Painlev＇e expansion. The application of the generalized method to the mKdV equation shows that it extends the range of exact solutions obtained by using the （ G′/ G）-expansion method.

Konopelchenko-Dubrovsky方程的Bcklund变换和多孤子解

利用标准Painleve截断分析法,将Konopelehenko-Dubrovsky(KD)方程约化为两个线性偏微分方程和一个双线性偏微分方程,建立起相应的Backlund变换,进而获得该(2+1)维非线性系统的多孤子解.

Analytic Solutions of a Predator-Prey System

A predator prey interaction of both populations was treated by the Painlevé analysis method. The analytic solutions for the related equations was obtained using the truncated Painlevé expansion.

修正的Korteweg de-Vries方程的孤立波和周期波相互作用解

修正的Korteweg de—Vries方程有许多形式的解，例如孤立波解、周期波解等，是一个十分重要的物理模型．利用截断的Painlevfi分析和tanh函数展开法，成功解决了修正的Kortewegde—Vries方程的孤立波与其他形式波的相互作用形式．尤其是通过雅克比椭圆积分和第三类的不完全积分得到了修正的Korteweg de—Vries方程的椭圆一正弦波解．

偏微分方程的非线性变换和精确解

利用截属的Painlev'e展开式、非线性变换和可积的微分方程，可以求出一类非线性偏微分方程的自Backlund变换和它的精确解，孤立波解。波动方程、Hirota-Satsuma方程组和非线性色散与耗散方程作为例子来说明这一方法。

Painleve分析产生的Tu系统的精确解

分析了非线性Ｔｕ系统的截断Ｐａｉｎｌｅｖｅ级数，得到了Ｔｕ系统与Ｒｉｃｃａｔｉ型方程Φｉ一Φｘ＝αΦ２＋βΦ＋γ的关系，并揭示了Ｒｉｃｃａｔｉ型方程本身的一些性质．最后借助于Ｒｉｃｃａｔｉ型方程给出了Ｔｕ系统的三类显式的精确解．

The residual symmetry,Bäcklund transformations,CRE integrability and interaction solutions:(2+1)-dimensional Chaffee–Infante equation

In this paper,we consider the(2+1)-dimensional Chaffee-Infante equation,which occurs in the fields of fluid dynamics,high-energy physics,electronic science etc.We build Bäcklund transformations and residual symmetries in nonlocal structure using the Painlevétruncated expansion approach.We use a prolonged system to localize these symmetries and establish the associated one-parameter Lie transformation group.In this transformation group,we deliver new exact solution profiles via the combination of various simple(seed and tangent hyperbolic form)exact solution structures.In this manner,we acquire an infinite amount of exact solution forms methodically.Furthermore,we demonstrate that the model may be integrated in terms of consistent Riccati expansion.Using the Maple symbolic program,we derive the exact solution forms of solitary-wave and soliton-cnoidal interaction.Through 3D and 2D illustrations,we observe the dynamic analysis of the acquired solution forms.

Nonlinear Self-Adjointness, Conservation Laws and Soliton-Cnoidal Wave Interaction Solutions of(2+1)-Dimensional Modified Dispersive Water-Wave System

This paper mainly discusses the(2+1)-dimensional modified dispersive water-wave(MDWW) system which will be proved nonlinear self-adjointness. This property is applied to construct conservation laws corresponding to the symmetries of the system. Moreover, via the truncated Painlev′e analysis and consistent tanh-function expansion(CTE)method, the soliton-cnoidal periodic wave interaction solutions and corresponding images will be eventually achieved.

Construction of Soliton-Cnoidal Wave Interaction Solution for the (2+1)-Dimensional Breaking Soliton Equation

In this paper, the truncated Painleve analysis and the consistent tanh expansion （CTE） method are developed for the （2＋1）-dimensional breaking soliton equation. As a result, the soliton-cnoidal wave interaction solution of the equation is explicitly given, which is dimcult to be found by other traditional methods. When the value of the Jacobi elliptic function modulus rn = 1, the soliton-cnoidal wave interaction solution reduces back to the two-soliton solution. The method can also be extended to other types of nonlinear evolution equations in mathematical physics.