An Extended Auxiliary Function Method and Its Application in mKdV Equation

An extended auxiliary functionmethod is presented for constructing exact traveling wave solutions to nonlinear partial differential equations. The main idea of this method is to take full advantage of the solutions to the elliptic equation to construct exact traveling wave solutions for nonlinear partial differential equations. mKdV equation is chosen to illustrate the application of the extended auxiliary function method. Consequently, more new exact traveling wave solutions are derived that are not obtained by the previously known methods.

With the rapid development of computer algebraic systems like Maple or Mathematical, many powerful algebraic methods have been presented to obtain a great many exact solutions (especially traveling wave solutions) for nonlinear partial differential equation (NLPDE), for example, sinecosine method [18], Fu's method [19], the hyperbolic tangent function expansion method [8,20], Riccati equation expansion method [9,21], the generalized Riccati equation method [22], auxiliary ordinary differential equation method [23], modified extended tanh method [24], Fan's unified algebraic method [25], the extended Fan's subequation method [26], and so on [26][27][28][29][30][31][32][33][34][35][36].Generally speaking, these methods have a common characteristic: the possible solutions are constructed from the known functions or the solutions for some simple and solvable nonlinear ordinary differential equations such as sine-cosine function, tanh function, Jacobi elliptic function or the Riccati equation, the coupled Riccati equation, and sinh-Gordon equation.Recently, Xu presents the elliptic equation expansion method to construct new and more general solutions of NLPDE [37].The main purpose of this paper is to summarize and extend Xu's method to unified basic frame so that it can be used to obtain more exact traveling wave solutions for NLPDE.For illustration, we apply the extended method to the modified Korteweg-de Vries (mKdV) equation and get successfully many new exact solutions.
This paper is organized as follows.In Section 2, we give the main steps of our extended algorithm for computing exact solutions of NLPDE.In Section 3, many families of traveling wave solutions of mKdV equation are obtained.Finally, some conclusions and discussions are provided in Section 4.
Step 1 (reduce the nonlinear PDE to the nonlinear ODE).For a Given nonlinear partial differential equation with the physical field (, ) in two variables ,  Δ (,   ,   ,   ,   ,   , . ..) = 0. (1) We consider its traveling wave solution in the form where  and  are constants to be determined later.Substituting (2) into (1) gives rise to a nonlinear ordinary differential equation with constant coefficients Step 2 (set the series of formal solutions).To seek the traveling wave solutions to (3), in [37] Xu assumes that (3) has the solutions in the form of with the new variable () satisfying the elliptic equation where , , , and  are constants and  is the integer to be determined later.
In order to get more exact solutions for (3), here we assume that (3) has the following formal solutions: Step 3 (determine the truncation expansion terms in (6)).In order to obtain the value of  in (6).We define the degree of () as [()] =  which gives rise to the degrees of other expressions as Therefore we can get the value of  in (6) by balancing the highest-order contributions from such terms in (1) or (3).If  is a nonnegative integer, we first make the transformation  =   .
Step 6 (solve ( 5)).Motivated by [37], we obtain some special solutions for (5) which are now listed in the following.

The Application of the Elliptic Equation Expansion Method
In the following, we will apply the extended elliptic equation method to construct new explicit solutions of mKdV equation [1]: where  and  are constants.The mKdV equation is of important significance in many branches of nonlinear science field.When  = 6, (13) is called the positive mKdV equation, while  = −6, (13) is called the negative mKdV equation.The mKdV equation appears in many fields such as acoustic waves in certain anharmonic lattices, Alfvén waves in a collisionless plasma, transmission lines in Schottky barrier, models of traffic congestion, ion acoustic solitons, elastic media, and so forth.It possesses many remarkable properties such as miura transformation, conservation laws, inverse scattering transformation, bilinear transformation, N-solitons, breather solutions, Bäcklund transformation, Painlevé integrability, Darboux transformation, and doublyperiodic solutions.
According to Step 1 in Section 2, in order to look for the traveling wave solutions for (13), we make the following traveling wave transformation: and substitute ( 14) into (13).Then (13) changes into the form According to Steps 2 and 3 in Section 2, by balancing the highest-order partial derivative term   with the highest nonlinear term  2   in (15), we can obtain  = 1.So we have where  0 ,  1 ,  1 ,  1 and  1 are constants to be determined later and  satisfies (5).According to Step 4 in Section 2, with the aid of Maple, substituting ( 16) into (15) yields a set of algebraic equations for     ( = −1, 0, 1;  = 0, 1).Setting the coefficients of these terms     of the resulting system's numerator to zero yields a set of overdetermined algebraic equations with respect to the unknowns  0 ,  1 ,  1 ,  1 , and  1 : According to Step 5 in Section 2, by use of the Maple software package "Charsets" by Dongming Wang, which is based on the Wu-elimination method [38], solving the previous overdetermined algebraic equations, we can get the following three sets of explicit expressions for  0 ,  1 ,  1 ,  1 , and  1 .
Remark 2. The previous solutions obtained here, to our knowledge, are all new families of rational formal solutions to mKdV equation.

Conclusion and Discussions
In summary, we have presented the extended elliptic equation method and its algorithm.As a result, some new exact traveling wave solutions for the mKdV equation are obtained which may be useful for describing certain nonlinear physical phenomena.The method which we have proposed in this paper is a standard, direct, and computerized method, which allow us to do complicated and tedious algebraic calculation.It is shown that the algorithm can be also applied to other NLPDE in mathematical physics.In addition, a natural problem is that whether the extended elliptic equation method can be further extended to generate more types of solutions of NLPDE in nonlinear science.Hence, further study of the extended elliptic equation method is needed.