Some exact solutions of the (2+1)-dimensional breaking soliton equation using the three-wave method Mohammad Taghi Darvishi and Mohammad Najafi Abstract—This paper considers the (2+1)-dimensional breaking soliton equation in its bilinear form. Some exact solutions to this equation are explicitly derived by the idea of three-wave solution method with the assistance of Maple. We can see that the new idea is very simple and straightforward. Keywords—Soliton solution, Computerized symbolic computation, Painlev´ e analysis, (2+1)-dimensional breaking soliton equation, Hi- rota’s bilinear form. I. I NTRODUCTION W namic processes in physics, mechanics, chemistry and biology which can be represented by nonlinear partial dif- ferential equations. The study of exact solutions of nonlinear evolution equations plays an important role in soliton theory and explicit formulas of nonlinear partial differential equations play an essential role in the nonlinear science. Also, the explicit formulas may provide physical information and help us to understand the mechanism of related physical models. In recent years, many kinds of powerful methods have been proposed to find solutions of nonlinear partial differential equations, numerically and/or analytically, e.g., the variational iteration method [1], [2], [3], the Adomian decomposition method [4], [5], the homotopy perturbation method [6], [7], [8], [9], [10], parameter expansion method [11], [12], [13], spectral collocation method [14], [15], [16], [17], [18], homo- topy analysis method [19], [20], [21], [22], [23], [24], [25], and the Exp-function method [26], [27], [28], [29], [30], [31]. In this paper, by means of the idea of the three-wave method, we will obtain some exact solutions for the (2+1)- dimensional breaking soliton equation in its bilinear form. The paper is organized as follows: in the following section we have a brief review on the three-wave method. In Section III we obtain some exact solutions for the (2+1)-dimensional breaking soliton equation. In Section IV we obtain some soliton solutions for the (2+1)-dimensional Bogoyavlenskii’s breaking soliton equation. Finally the paper is concluded in Section V. II. METHODOLOGY Dai et al. [32], suggested the three-wave method for nonlin- ear evolution equations. The basic idea of this method applies the Painlev` e analysis to make a transformation as u = T (f ) (1) M.T. Darvishi and M. Najafi are with Department of Mathematics, Faculty of Science, Razi University, Kermanshah 67149, Iran. for some new and unknown function f . Then we use this transformation in a high dimensional nonlinear equation of the general form F (u, u t ,u x ,u y ,u z ,u xx ,u yy ,u zz , ···)=0, (2) where u = u(x,y,z,t) and F is a polynomial of u and its derivatives. By substituting (1) in (2), the first one converts into the Hirota’s bilinear form, which it will solve by taking a special form for f and assuming that the obtained Hirota’s bilinear form has three-wave solutions, then we can specify the unknown function f , (for more details see [32], [33]). III. (2+1)- DIMENSIONAL BREAKING SOLITON EQUATION In this section, we investigate explicit soliton solutions of the following (2+1)-dimensional breaking soliton equation given in [34] u xxxy − 2 u y u xx − 4 u x u xy + u xt =0. (3) Equation (3) is used to describe the (2+1)-dimensional inter- action of a Riemann wave propagating along the y-axis with a long wave along the x-axis, which was first described by Calogero and Degasperis in 1977. To solve eq. (3) authors in [34] used of N-soliton solution. In this paper, we use the idea of three-wave method [32], [33], to solve equation (3). By this idea we obtain some analytic solutions for the problem. The process of the method is very easy and more simple than the method of Zheng et al. [34]. To solve eq. (3), we introduce a new dependent variable w by w = −2(ln f ) x (4) where f (x, y, t) is an unknown real function which will be determined. Substituting eq. (4) into eq. (3), we have 2(ln f ) xxt + 2(ln f ) xxxxy + 16(ln f ) xx (ln f ) xxy + 8(ln f ) xxx (ln f ) xy =0, (5) which can be integrated once with respect to x to give 2(ln f ) xt + 2(ln f ) xxxy + 12(ln f ) xx (ln f ) xy +4∂ −1 x ((ln f ) xx (ln f ) xxy − (ln f ) xxx (ln f ) xy )=0. (6) Thus, eq. (6) can be written as (D x D t + D y D 3 x )f · f +4 f 2 ∂ −1 x (D x (ln f ) xx · (ln f ) xy )=0, (7) E can find many important phenomena and dy- World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:5, No:7, 2011 1031 International Scholarly and Scientific Research & Innovation 5(7) 2011 scholar.waset.org/1307-6892/12525 International Science Index, Mathematical and Computational Sciences Vol:5, No:7, 2011 waset.org/Publication/12525