Exact traveling wave solutions for the generalized Hirota-Satsuma couple KdV system using the exp(−φ(ξ))-expansion method

In this research, we find the exact traveling wave solutions involving parameters of the generalized Hirota-Satsuma couple KdV system according to the exp(− ( ))-expansion method and when these parameters are taken to be special values we can obtain the solitary wave solutions which is derived from the exact traveling wave solutions. It is shown that the proposed method provides a more powerful mathematical tool for constructing exact traveling wave solutions for many other nonlinear evolution equations. Subjects: Mathematics & Statistics; Physical Sciences; Science


PUBLIC INTEREST STATEMENT
In this paper, we use the exp(− ( ))-expansion method to find the exact and solitary wave solutions of the generalized Hirota-Satsuma couple KdV system. The exact traveling wave solutions are obtained from the explicit solutions by choosing the particular value of the physical parameters. So, we can choose appropriate value of the physical parameters to obtain exact solutions we need in varied instances. There are various types of traveling wave solutions that are of particular interest in solitary wave theory.
The objective of this article was to apply The exp(− ( ))-expansion method for finding the exact traveling wave solution of the generalized Hirota-Satsuma couple KdV system which play an important role in mathematical physics.
The rest of this paper is organized as follows: In Section 2, we give the description of The exp(− ( ))-expansion method. In Section 3, we use this method to find the exact solutions of the nonlinear evolution equations pointed out above. In Section 5, conclusions are given.

Description of method
Consider the following nonlinear evolution equation where F is a polynomial in u(x, t) and its partial derivatives in which the highest order derivatives and nonlinear terms are involved. In the following, we give the main steps of this method Step 1. We use the wave transformation where c is a positive constant, to reduce Equation (2.1) to the following ODE: where P is a polynomial in u( ) and its total derivatives, while � = d d � .
Step 2. Suppose that the solution of ODE (Equation 2.3) can be expressed by a polynomial in exp(− ( )) as follows Since a m 0 ≤ m ≤ n are constants to be determined, such that a m ≠ 0.
The positive integer m can be determined by considering the homogenous balance between the highest order derivatives and nonlinear terms appearing in Equation (2.3). Moreover precisely, we define the degree of u( ) as D(u( )) = m, which gives rise to degree of other expression as follows: Therefore, we can find the value of m in Equation (2.5) and where a m , … … , , are constants to be determined later.
Step 3. After we determine the index parameter m, we substitute Equation (2.4) along Equation (2.5) into Equation (2.3) and collecting all the terms of the same power exp(−m ( )), m = 0, 1, 2, 3, … and equating them to zero, we obtain a system of algebraic equations, which can be solved by Maple or Mathematica to get the values of a i .
Step 4 It is to be noted here that the construction of the exp (− ( )) is similar to the construction of the G ′ G -expansion. For better understanding of the duality of both methods we cite Alquran and Qawasmeh (2014), Alquran (2014a, 2014b).

Application
Here, we will apply the exp(− ( ))-expansion method described in Section 2 to find the exact traveling wave solutions and the solitary wave solutions of the generalized Hirota-Satsuma couple KdV system (Yan, 2003). We consider the generalized Hirota-Satsuma couple KdV system when w = 0, Equation (3.1) reduce to be the well-known Hirota-Satsuma couple KdV equation.
Balancing between the highest order derivatives and nonlinear terms appearing in P ′′ and P 3 ⇒ N + 2 = 3N ⇒ (N = 1). So that, by using Equation (2.4) we get the formal solution of Equation (3.5) substituting Equation (3.6) and its derivative into Equation (3.5) and collecting all term with the same power of exp(−3 ), exp(−2 ), exp(− ), exp(0 ) we get: Solving above system by using maple 16, we get:

Thus the solution is
Let us now discuss the following cases: When (3.12) (3.14) ).

Physical interpretations of the solutions
In this section, we depict the graph and signify the obtained solutions to the generalized Hirota-Satsuma couple KdV system. Now, we will discuss all possible physical significances for parameter.

Conclusion
The exp(− ( ))-expansion method has been applied in this paper to find the exact traveling wave solutions and then the solitary wave solutions of the generalized Hirota-Satsuma couple KdV system . Let us compare between our results obtained in the present article with the well-known results obtained by other authors using different methods as follows: our results of nonlinear dynamics of the generalized Hirota-Satsuma couple KdV system are new and different from those obtained in Yan (2003), and Figures 1-16, show the solitary traveling wave solution of the generalized Hirota-Satsuma couple KdV system . We can conclude that the exp(− ( ))-expansion method is a very powerful and efficient technique in finding exact solutions for wide classes of nonlinear problems and can be applied to many other nonlinear evolution equations in mathematical physics. Another possible merit is that the reliability of the method and the reduction in the size of computational domain give this method a wider applicability.