New Exact Analytical Solutions for the General Kdv Equation with Variable Coefficients

In this paper, a general algebraic method based on the generalized Jacobi elliptic functions expansion method, the improved general mapping deformation method and the extended auxiliary function method with computerized symbolic computation is proposed to construct more new exact solutions of a generalized KdV equation with variable coefficients. As a result, eight families of new generalized Jacobi elliptic function wave solutions and Weierstrass elliptic function solutions of the equation are obtained by using this method, some of these solutions are degenerated to soliton-like solutions, trigonometric function solutions in the limit cases when the modulus of the Jacobi elliptic functions 1 m  or 0, which shows that the general method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differential equations arising in mathematical physics.


INTRODUCTION
Nonlinear partial differential equations (NLPDEs) are widely used to describe complex physical phenomena arising in the world around us and various fields of science.The investigation of exact solutions of NLPDEs plays an important role in the study of these phenomena such as the nonlinear dynamics and the mechanism behind the phenomena.With the development of soliton theory, many powerful methods for obtaining exact solutions of NLPDEs have been presented, such as inverse scattering transformation [1], Hirota bilinear method [2], Bäcklund transformation [3], Darboux transformation [4], homotopy perturbation method [5], extended Riccati equation rational expansion method [6], asymptotic methods [7], extended auxiliary function method [8], algebraic method [9], Jacobi elliptic function expansion method [10],and so on [11][12][13].
In [14][15], Hong proposed a generalized Jacobi elliptic functions expansion method to obtain generalized exact solutions of NLPDEs.In [16], Hong et al. proposed an improved general mapping deformation method to obtain generalized exact solutions of the general KdV equation with variable coefficients (GVKDV).Which is more general than many other algebra expansion methods [6,[8][9][10][11][12][13][14][15] etc.The solution procedure of this method, by the help of Matlab or Mathematica, is of the utmost simplicity, and this method can be easily extended to all kinds of NLPDEs.In this work, we will proposed the general algebraic method to obtain several new families of exact solutions for the GVKDV equations.
The rest of this paper is organized as follows.In section 2,we briey describe the new general algebraic method.In section 3, several families of solutions for the GVKdV equation are obtained, some of which are degenerated to new solitary-like solutions and new triangular-like functions solutions in the limit case.In section 4, some conclusions are given.

SUMMARY OF THE GENERAL ALGEBRAIC METHOD
Consider a given nonlinear evolution equation with one physical field ( , ) u x t in two variables x and t ( , , , , ) 0 We seek the following formal solutions of the given system by a new intermediate transformation: Substituting Eqs. ( 3) and (2) into Eq.( 1), and setting the coefficients of  into Eq. ( 2) we can get the solutions of Eq. ( 1).In order to obtain some new general solutions of Eq.( 3),we assume that (3) have the following solutions: Where , , , p q r l are arbitrary constants which ensure denominator unequal to zero, so do the following situations, and () Where " ' " denotes d d , " '' " denotes

And
, , , e f g h satisfy one of the following relations at the same time.
If we let , our method contain the improved general mapping deformation method [16]etc.
In the following, we will use this method to solve the GVKdV equation
By balancing the highest-order linear term Where are functions of t to be determined later.

Remark 4:
4 u are in full agreement with the results in Ref. [16],which contain the results (19) constructed by Zhao in Ref. [17] and 3 u obtained by Zhu in Ref. [18].

4．CONCLUSION
In this paper, we succeed to propose a general algebraic method for finding new exact solutions of the GVKdV equation (8).More importantly, our method is much simple and powerful to find new solutions to various kinds of nonlinear evolution equations, such as KdV equation, Boussinesq equation, zakharov equation, etc. we believe that this method should play an important role for finding exact solutions in the mathematical physics.

,
then return to determine the balance constant n again; (b) when n is a negative integer, we suppose ( ) ( ) n uv   , then return to determine the balance constant n again.

5 k
are arbitrary constants in all above cases.
After solving the ODEs by Mathematica we could determine the following solutions: i at. ,