Exact Solutions of Equation Generated by the Jaulent-Miodek Hierarchy by ( G / G )-Expansion Method

The (G 󸀠 /G)-expansion method is proposed for constructing more general exact solutions of the nonlinear (2 + 1)-dimensional equation generated by the Jaulent-Miodek Hierarchy. As a result, when the parameters are taken at special values, some new traveling wave solutions are obtained which include solitary wave solutions which are based from the hyperbolic functions, trigonometric functions, and rational functions. We find in this work that the (G 󸀠 /G)-expansion method give some new results which are easier and faster to compute by the help of a symbolic computation system. The results obtained were compared with tanh method.


Introduction
In the present paper, we obtained exact solutions of the nonlinear (2 + 1)-dimensional equation (1) generated by the Jaulent-Miodek Hierarchy [1] in the form where  is a spatial variable, and (, , ) is the analytic function with respect to , , and .In addition,  −1  is the inverse of   with    −1  =  −1     = 1 and ( −1  )() = ∫  −∞ (), under the decaying condition at infinity.This equation was studied by Liu and Yan by using qualitative analysis method [2].It is clear that searching for explicit solutions of nonlinear evolution equation (NLEEs) by using various method has become the main aim for many authors.Many powerful methods have been created and developed to obtain analytic solutions of (NLEEs), such as the the tanhcoth method [3][4][5], sine-cosine method [6], homogeneous balance method [7], exp-function method [8,9], Hirota bilinear transformation [10,11], Jacobi elliptic function method [12], and solitary wave ansatz method [13].One of the most effective direct method to build traveling wave solution of (NLEEs) is the (  /)-expansion method, which was first proposed by Wang et al. [14].It is assumed that the traveling wave solutions can be expressed by a polynomial in (  /), where  = () satisfies the following second-order linear ordinary differential equation   () +   () + () = 0, where  =  − , and , , and  are constants.Next, Bekir applied the method to some nonlinear evolution equations earning some traveling wave solutions [15].After that, Zhang et al. have improved and generalized the (  /)-expansion method to seek exact solutions of other nonlinear evolution equations [16].Moreover, Zhang et al. have further extended the method to deal with evolution equations with variable coefficients [17].In fact the (  /)-expansion method has been successfully applied to obtain exact solution for a variety of (NLEEs) [12,[18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35].In this paper, the (  /)expansion method is proposed for constructing more general exact solutions of the above nonlinear (2 + 1)-dimensional equation generated by the Jaulent-Miodek Hierarchy.Our paper is organized as follows: in Section 2, the description of the (  /)-expansion method is provided; in Section 3, we spell out the applications of this method to the nonlinear (2 + 1)-dimensional equation; in Section 4, application of the tanh method on the nonlinear (2 + 1)-dimensional equation and finally in Section 5, some conclusions are given.

Description the (𝐺 󸀠 /𝐺)-Expansion Method
In this section we describe the (  /)-expansion method for finding traveling wave solutions of (NLEEs).Suppose that a nonlinear evolution equation, say in three independent variables , , and , is given by  (,   ,   ,   ,   ,   ,   ,   ,   , . ..) = 0, where  = (, , ) is an unknown function, and  is a polynomial in  = (, , ) and its various partial derivatives, in which highest-order derivatives and nonlinear terms are involved.To determine the (  /)-expansion method, we take the following six steps.
Step 1.To find the traveling wave solutions of (2), we introduce the wave variable  (, , ) =  () ,  =  +  + , where  is constant.Substituting (3) into (2), we obtain the following ordinary differential equations (ODE): Step 2. If necessary we integrate (4) as many times as possible and set the constants of integration to be zero for simplicity.
Step 3. We suppose the solution of nonlinear partial differential equation can be expressed by a polynomial in (  /) as where  = () satisfies the second-order linear ordinary differential equation where   = /,   =  2 / 2 , and   , , and  are real constants with   ̸ = 0.
Step 4. The positive integer  can be determined by considering the homogeneous balance between the highest-order derivatives and nonlinear terms appearing in (4) as follows: if we define the degree of () as [()] = , then the degree of other expressions is defined by Therefore, we can get the value of  in (5).
Step 5. Substituting ( 5) into (4), using ( 6), collecting all terms with the same order of (  /) together, and then setting each coefficient of this polynomial to zero yield a set of algebraic equations for   , , , and .
Step 6. Substituting   , , , and  obtained in Step 5 and the general solutions of ( 6) into (5), we obtain traveling wave solutions of the nonlinear (PDE).

Application the (𝐺 󸀠 /𝐺)-Expansion Method
3.1.Equation Generated by the Jaulent-Miodek Hierarchy.In this section, the (  /)-expansion method will be applied to the equation generated by the Jaulent-Miodek Hierarchy in the form We first remove the integral term in ( 8) by introducing the potential to carry (8) to the equation Let (, , ) = (),  =  +  + , where  is the wave speed.Substituting the above traveling wave variable  =  +  +  in to (10) yields Integrating (11) above with respect to  once yields For simplicity, the integration constant is taken as zero.Assume that Substituting ( 13) into (12), we obtain ordinary differential equation as follows: We suppose that the solution of ( 14) has the following formal solution: where  = () satisfies the second-order linear ordinary differential equation   () +   () + () = 0.According to Step 4  = 1, and therefore (15) become On substituting (16) along with ( 6) into ( 14), collecting all terms with the same powers of (  /), and setting each cofficient to zero yield a set of simultaneous algebraic equations for  0 ,  1 , , , and  as follows: Solving this set of algebraic equations by Maple or Mathematica, we can obtain the following results: first solution set: second solution set: where  is arbitrary constant.By using (18), expression ( 16) can be written as where  =  +  − (1/4).Equation ( 19) is the formula of a solution of (8).

First Traveling Wave Solution
Set. Substituting the general solutions of ( 6) into (19) we have three types of traveling wave solutions of the equation generated by the Jaulent-Miodek Hierarchy as follows.
In particular, if  1 and  2 are taken as special values, the various known results in the literature can be rediscovered; for instance, when setting  1 = 0,  2 = 1 and  = −1/16 + (1/4) 2 ,  becomes which is the solitary wave solution of ( 8).
On the other hand, when setting  1 = 0 and  2 = 1, the solution of ( 23) can be written as Setting again  2 = 0 and  1 = 1, the solution of ( 23) can be written as Case c.When  2 − 4 = 0, the rational function solutions of ( 8) are as follows: Then (26) becomes By  = / →  = ∫  and  = /, we find the all traveling wave solutions of (8) as follows: then we found Similarly a second traveling wave solution set can be obtained by following the above procedure.

Tanh Method
In this paper, we used tanh method in its standard form presented by Malfliet [36,37], and the outline of tanh method is known.For more details, refer to [38].
For computation the exact traveling wave solutions of equation generated by the Jaulent-Miodek Hierarchy.We used the ansatz where  = tanh{}.Substituting (33) in the ordinary differential equation of the equation generated by the Jaulent-Miodek Hierarchy, we found To determine the parameter , we usually balance the linear term of highest order in the resulting equation (34) with the highest order nonlinear terms, and we find  = 1.This means that Substituting (),   (),   (),  2 (), and  3 () from ( 35) into (34) yields the system of algebraic equations for  0 ,  1 , , and  as follows: Solving the system, we obtain In view of (37), we obtain the solitary wave solutions where  =  +  − 0.25.Equation ( 38) is the same as the result obtained by the (  /)-expansion method.
We obtain the behaviour of exact traveling wave solution of (8).

Conclusion
In this work, we have established the traveling wave solutions of the nonlinear equation generated by the Jaulent-Miodek Hierarchy.It appears that three types of traveling wave solutions have been successfully found by using the (  /)expansion method.The general traveling wave solutions are hyperbolic function solutions, trigonometric function solutions, and rational function solutions.The solitary wave solutions are derived from these functions when the parameters are taken as special values.The solutions of the equation via (  /)-expansion method are exactly the same as those obtained by tanh method if the condition  2 = 0 is satisfied, this means that the tanh method is a special case of (  /)expansion method.The (  /)-expansion method is quite efficient and practically well suited for use in finding the exact solutions for the equation generated by the Jaulent-Miodek Hierarchy.Some solutions given in this paper are new solutions which have not been reported yet.Threedimensional plots (Figure 1) of some of the investigated solutions are also drawn to visualize the underlying dynamics of such results.The method which we have proposed in this work is also a standard, direct, and computerizable method.