Traveling wave solutions for the (3+1)-dimensional breaking soliton equation by ( G  G )-expansion method and modified F-expansion method Mohammad Taghi Darvishi, Maliheh Najafi and Mohammad Najafi Abstract—In this paper, using ( G  G )-expansion method and mod- ified F-expansion method, we give some explicit formulas of exact traveling wave solutions for the (3+1)-dimensional breaking soliton equation. A modified F-expansion method is proposed by taking full advantages of F-expansion method and Riccati equation in seeking exact solutions of the equation. Keywords—Exact solution, The (3+1)-dimensional breaking soli- ton equation, ( G  G )-expansion method, Riccati equation, Modified F- expansion method. I. I NTRODUCTION I N many different fields of science and engineering, it is very important to obtain exact or numerical solutions of nonlinear partial differential equations. It is well known that nonlinear phenomena are very important in a variety of scientific fields, especially in fluid mechanics, solid state physics, plasma physics, plasma waves and chemical physics. Searching for exact and numerical solutions, especially for traveling wave solutions, of nonlinear equations in mathemat- ical physics plays an important role in soliton theory [1], [2]. Recently many new approaches to nonlinear equations were proposed, such as the homotopy perturbation method [3], [4], [5], [6], [7], the variational iteration method [8], [9], [10], parameter expansion method [11], [12], [13], [14], spectral collocation method [15], [16], [17], [18], [19], homotopy analysis method [20], [21], [22], [23], [24], [25], and the Exp- function method [26], [27], [28], [29], [30], [31]. In this paper, we solve a (3+1)-dimensional breaking soliton equation by the ( G  G )-expansion method and modified F-expansion method, and obtain some exact and new solutions for it. The(2+1)-dimensional breaking soliton equation has the following form u xt − 4u xy u x − 2u xx u y − u xxxy =0, (1) this equation describes the (2+1)-dimensional interaction of the Riemann wave propagated along the y-axis with a long wave propagated along the x-axis [32]. Wazwaz [33] presented an extension to equation (1) by adding the last three terms with y replaced by z. By his work, one enables to establish M.T. Darvishi, Maliheh Najafi and Mohammad Najafi: Depart- ment of Mathematics, Faculty of Science, Razi University, Kermanshah 67149, Iran. e-mail: darvishimt@yahoo.com, malihe math87@yahoo.com and m najafi82@yahoo.com. the following (3+1)-dimensional breaking soliton equation u xt − 4 u x (u xy + u xz ) − 2 u xx (u y + u z )− (u xxxy + u xxxz )=0, (2) where u = u(x,y,z,t): R x × R y × R z × R t → R. In this paper, by means of the ( G  G )-expansion method and modified F-expansion method, we obtain some exact traveling wave solutions for equation (2). The outline of this paper is as follows. In the following section we have a brief review on the ( G  G )-expansion. In Section III we apply the ( G  G )-expansion method on equation (2) to obtain some traveling wave solution for the equation. In Section IV a review on the modified F-expansion method is presented. We obtain some traveling wave solutions for equation (2) by the modified F-expansion method in Section V. The paper is concluded is Section VI. II. THE ( G  G )- EXPANSION METHOD Wang et al. [34] proposed the ( G  G )-expansion method to solve nonlinear partial differential equations, where G = G(ξ) satisfies a second order linear ordinary differential equation. In this section we describe the ( G  G )-expansion method to find traveling wave solutions of nonlinear evolution equations. Suppose that a nonlinear equation, say in two independent variables x, t, is given by P (u, u t ,u x ,u tx ,u xx , ··· )=0, (3) where u = u(x, t) and P is a polynomial of u and its derivatives in which the highest order derivatives and nonlinear terms are involved. The main steps of the ( G  G )- expansion method are as follows: • First. Suppose that u(x, t)= u(ξ), ξ = x + wt (4) the traveling wave variable (4) permits us reducing (3) to an ordinary differential equation (shortly ODE) for u = u(ξ ) such as P (u, u  ,u  ,u  , ··· )=0, (5) • Second. Now, we suppose that the solution of (5) can be expressed by a polynomial in ( G  G ) as u(ξ )= α m ( G  G ) m + ... (6) World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:5, No:7, 2011 1100 International Scholarly and Scientific Research & Innovation 5(7) 2011 scholar.waset.org/1307-6892/12188 International Science Index, Mathematical and Computational Sciences Vol:5, No:7, 2011 waset.org/Publication/12188