Abstract
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.
Highlights
Nonlinear evolution equations are widely used to describe complex phenomena in many scientific and engineering fields, such as fluid dynamics, plasma physics, hydrodynamics, solid state physics, optical fibers, and acoustics
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
The soliton-like solutions and trigonometric-function solutions have been obtained as the modulus m of Jacobi-elliptic function approaches to 1 and 0
Summary
Nonlinear evolution equations are widely used to describe complex phenomena in many scientific and engineering fields, such as fluid dynamics, plasma physics, hydrodynamics, solid state physics, optical fibers, and acoustics. Finding solutions of such nonlinear evolution equations is important. Determining solutions of nonlinear evolution equations is a very difficult task and only in certain cases one can obtain exact solutions. 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. The Scientific World Journal equation which has more general exact solutions in terms of Jacobian-elliptic and the Weierstrass-elliptic functions. Many exact solutions in terms of hyperbolic and trigonometric functions can be obtained when the modulus of Jacobian-elliptic functions tends to one and zero, respectively. By using the method proposed, Jacobian-elliptic and the Weierstrass-elliptic functions solutions are presented in Sections 3 and 4, respectively. The paper is ended by Appendices A–D which play an important role in obtaining the solutions
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