F-Expansion Method and New Exact Solutions of the Schrödinger-KdV Equation

F-expansion method is proposed to seek exact solutions of nonlinear evolution equations. With the aid of symbolic computation, we choose the Schrödinger-KdV equation with a source to illustrate the validity and advantages of the proposed method. A number of Jacobi-elliptic function solutions are obtained including the Weierstrass-elliptic function solutions. When the modulus m of Jacobi-elliptic function approaches to 1 and 0, soliton-like solutions and trigonometric-function solutions are also obtained, respectively. The proposed method is a straightforward, short, promising, and powerful method for the nonlinear evolution equations in mathematical physics.

In the present research, we shall apply the the -expansion method to obtain 52 types of exact solution: six for the Weierstrass-elliptic function solutions and the rest for Jacobian-elliptic function solutions of the Schrdinger-KdV equation: Among the methods mentioned above, the auxiliary equation method [29] is based on the assumption that the travelling wave solutions are in the form where ( ) satisfies the following auxiliary ordinary differential equation: where , , and are real parameters. Although many exact solutions were obtained in [29] via the auxiliary equation (3), all these solutions are expressed only in terms of hyperbolic and trigonometric functions. In this paper, we want to generalize the work in [29]. We propose a new auxiliary 2 The Scientific World Journal equation which has more general exact solutions in terms of Jacobian-elliptic and the Weierstrass-elliptic functions. Moreover, many exact solutions in terms of hyperbolic and trigonometric functions can be also obtained when the modulus of Jacobian-elliptic functions tends to one and zero, respectively. The rest of the paper is arranged as follows. In Section 2, we briefly describe the auxiliary equation method (expansion method) for nonlinear evolution equations. By using the method proposed in Section 2, Jacobian-elliptic and the Weierstrass-elliptic functions solutions are presented in Sections 3 and 4, respectively. Soliton-like solutions and trigonometric-function solutions are listed in Sections 5 and 6, respectively. Some conclusions are given in Section 7. The paper is ended by Appendices A-D which play an important role in obtaining the solutions.
In Appendices A and B, we present 52 types of exact solution for (7) (see [34][35][36][37]43] for details). In fact, these exact solutions can be used to construct more exact solutions for (1).

New Exact Jacobian-Elliptic Function Solutions of the Schrödinger-KdV Equation
The coupled Schrödinger-KdV equation is known to describe various processes in dusty plasma, such as Langmuir, dust-acoustic wave, and electromagnetic waves [44][45][46][47]. Exact solution of (9) was studied by many authors [48][49][50][51]. Here the -expansion method is applied to system (9) and gives some new solutions. Let where , , and are constants. Substituting (10) into (9), we find that = 2 and , satisfy the following coupled nonlinear ordinary differential system: Balancing the highest nonlinear terms and the highest order derivative terms in (11), we find = 2 and = 2. Therefore, we suppose that the solution of (11) can be expressed by where 0 , 1 , 2 , 0 , 1 , and 2 are constants to be determined later and ( ) is a solution of ODE (7). Inserting (12) into (11) with the aid of (7), the left-hand side of (11) becomes polynomials in ( ) if canceling and setting the coefficients of the polynomial to zero yields a set of algebraic equations, 0 , 1 , 2 , 0 , 1 , and 2 . Solving the system of algebraic equations with the aid of Mathematica, we obtain Substituting these results into (12), we have the following formal solution of (11): With the aid of Appendix A and from the formal solution of (14) along with (10), one can deduce more general combined Jacobian-elliptic function solutions of (1). Hence, the following exact solutions are obtained.

The New Weierstrass-Elliptic Function Solutions of the Schrödinger-KdV Equation
On using the solutions given in [43], mentioned in Appendix B, and from the formal solution (14) along with (10), we get then the following exact solutions.

New Soliton-Like Solutions of the Schrödinger-KdV Equation
Some soliton-like solutions of (1) can be obtained in the limited case when the modulus → 1 (see Appendix C), as follows: The Scientific World Journal Here, it should be noted that each exact solution given in (67) can be split into two solutions if one chooses the (+ve) and (−ve) signs, respectively, but they have not been calculated. Also, all the exact solutions given by (67) can be verified by substitution. The main feature for some of these exact solutions is the inclusion of the free parameters , , and .

New Trigonometric-Function Solutions of the Schrödinger-KdV Equation
Some trigonometric-function solutions of (1) can be obtained in the limited case when the modulus → 0. For example, Here, we note also that each trigonometric-function solution obtained in this section can split into two solutions if we choose the (+ve) and (−ve) signs, respectively. Besides, all these solutions can be verified by direct substitution. Also, the main feature for some of these exact solutions is the inclusion of the free parameters , , and .

Conclusion
In this paper, the -expansion method has been applied to construct 52 types of exact solution of the the Schrödinger-KdV equation. The main advantage of this method over other methods is that it possesses all types of exact solution, including those of Jacobian-elliptic and Weierstrasselliptic functions. Moreover, the soliton-like solutions and trigonometric-function solutions have been also obtained as the modulus of Jacobi-elliptic function approaches to 1 and 0. It can be said that the results in this paper provide good supplements to the existing literature and are useful for describing certain nonlinear phenomena. This method can be applied to many other nonlinear evolution equations. Finally, it is worthwhile to mention that the proposed method is also a straightforward, short, promising, and powerful method for other nonlinear evolution equations in mathematical physics.

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