The modified ( G ' / G )-expansion method for the ( 1 + 1 ) Hirota-Ramani and ( 2 + 1 ) breaking soliton equation

In this article, we apply the modified (G'/G)-expansion method to construct hyperbolic, trigonometric and rational function solutions of nonlinear evolution equations. This method can be thought of as the generalization of the (G'/G)-expansion method given recently by Wang et al. (2008). To illustrate the validity and advantages of this method, the (1+1)-dimensional Hirota-Ramani equation and the (2+1)dimensional breaking soliton equation are considered and more general traveling wave solutions are obtained. It is shown that the proposed method provides a more general powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.


DESCRIPTION OF THE MODIFIED (G'/G)-EXPANSION METHOD
A given nonlinear evolution equation is in the form ( , , , , , ,...), Where ( , , ) u x y t , we use the wave transformation ( , , ) ( ), Where , ,... 0, Where V  is a new function of  , we further introduce the following anstaz: Where () are constants to be determined later.
To determine () u  explicitly, we take the following four steps (Zayed and Al-Joudi, 2010;Zayed and Al-Joudi, 2009;Zhang et al., 2011): Step 1: Determine the positive integer m in Equation 5 by balancing the highest-order nonlinear terms and the highest-order derivatives in Equation 4.
Step 2: Substitute Equation 5 along with Equation 1 into Equation 4 and collect all terms with the same powers of

 
Step 3: Solve these algebraic equations by the use of Mathematica to find the values of 12 , , , , i kk  (0,1,..., ).im  Step 4: Use the results obtained in above steps to derive a series of fundamental solutions () , since the solutions of Equation have been well known for us as follows: Where c 1 and c 2 are constants, we can obtain exact solutions of Equation 2 by integrating each of the obtained fundamental solutions () V  with respect to  and r times as follows:

Remark 1
It can easily be found that when 0 r  , ( ) ( ) uV   then Equation 5 becomes the anstaz solutions obtained in Wang et al. (2008).When 1 r  , the solution of Equation 9 can be found in Zhang et al. (2011) and cannot be obtained by the methods in Wang et al. (2008).

APPLICATIONS
Here, we used the modified (G'/G)-expansion method to find the exact solutions of the following nonlinear partial differential equations (PDEs): Example 1: Nonlinear Hirota-Ramani equation Here, we used the proposed method previously used in the work, to find the solutions to Hirota-Ramani equation (Hirota and Ramani, 1980;Reza and Rasoul, 2011): (1 ) 0, Where 0   is a constant.To this end, we use the wave transformation ( , ) ( ), Where k, ω are constants, to reduce Equation 10 to the following ODE: , where () According to Step 1, we get m + 2 = 2m, and hence m = 2 .We then suppose that Equation 13 has the formal solution It is easy to see that Substituting Equation 14 .Setting each coefficient of this polynomial to zero, we get the following algebraic equations: On solving the algebraic Equations 17 to 21, we have the results: Where 22 ( 4 ) 1.
k   Consequently, we deduce the following exact solutions of Equation 10: Substituting Equations 8, 10, 12 and 14 obtained in Peng (2008Peng ( , 2009) ) into Equation 23, we have respectively the following kink-type traveling wave solutions: (1) If Where cc is the sign function, while  is given by: In this case, we have We now simplify Equation 28 to get the following periodic solutions: (1) Where Where In this case, we have Remark 2 If we multiply Equation 13 by () and integrate with zero constant of integration, we deduce that Where On solving Equation 32 we have the two cases: Integrating Equations 33 and 34, we have the solutions of Equation 10 in the forms: and 36, we arrive at the same solutions of Equations 24 and 25 or 26, respectively.
Integrating Equations 37 and 38, we have the solutions of Equation 10 in the forms: and 40, we arrive at the same solutions of Equation 29and 30, respectively.

Example 2: Nonlinear breaking soliton equation
Here, we used the proposed method previously used in the work, to find the solutions of the breaking soliton equation (Zayed et al., 2011;Zayed and Al-Joudi, 2010;Zayed and Al-Joudi, 2009): To this end, we use the wave transformation ( , , ) ( ), Where 12 , kkand  are constants, to reduce Equation 41to the following ODE: Integrating Equation 43once with respect to  , with zero constant of integration, we get Setting 1 r  , and uV   , we deduce that () V  satisfies the equation: According to Step 1, we get m = 2. Thus, the formal solution of Equation 45 has the same form of Equation 14.
Substituting Equations 14 to 16 into Equation 45and Setting each coefficient to zero, we get the following algebraic equations: On solving the algebraic Equations 46 to 50, we have the following results: Consequently, we deduce the following exact solutions of Equation 41: Substituting the results of Equations 8, 10, 12 and 14 of Peng (2008Peng ( , 2009) ) into Equation 52, we have respectively, the following Kink-type traveling wave solutions: ( Where In this case, we have We now simplify Equation 57 to get the following periodic solutions: (1) Where Where Integrating Equations 62 and 63, we have the solutions of Equation 41 in the forms: Integrating Equations 66 and 67, we have the solutions of Equation 41 in the forms: