Some complexiton type solutions of the (3+1)-dimensional Jimbo-Miwa equation Mohammad Taghi Darvishi and Mohammad Najafi Abstract—By means of the extended homoclinic test approach (shortly EHTA) with the aid of a symbolic computation system such as Maple, some complexiton type solutions for the (3+1)-dimensional Jimbo-Miwa equation are presented. Keywords—Jimbo-Miwa equation, Painlev´ e analysis, Hirota’s bi- linear form, Computerized symbolic computation. I. I NTRODUCTION T HE Jimbo-Miwa equation is used to describe certain interesting (3+1)-dimensional waves in physics but not pass any of the conventional integrability tests. This equation is the second equation in the well known Painlev` e hierarchy of integrable systems. The (3+1)-dimensional Jimbo-Miwa equation is u xxxy +3 u y u xx +3 u x u xy +2 u yt − 3 u xz =0, (1) where u : R x × R y × R z × R + t → R. There are many efforts to solve equation (1). Tang and Liang [1] applied the multi-linear variable separation scheme to (1). [2] by applying the Painlev´ e test showed that (1) is not inte- grable and through the obtained truncated Painlev´ e expansions constructed two bilinear equations. Starting from these bilinear equations, one soliton, two soliton and dromin solutions are also obtained. [3] obtained a new class of cross kink-wave and periodic solitary-wave solution for (1) by using two-soliton methoid, bilinear method and transforming parameters into complex ones. [4] obtained some exact solutions of (1) by an extended rational expansion method and symbolic compu- tation. [5] obtained two new types of exact periodic solitary- wave and kinky periodic-wave solutions to (1) by applying EHTA. [6] presented exact and explicit generalized solitary solutions for the equation by the Exp-function method. [7] derived multiple front solutions by employing Hirota’s bilinear method for (1). [8] by using rational function transformations approached to exact solution of the equation. [9] obtained new exact solutions, including solitary wave solutions, periodic wave solutions and variable separations solutions of (1) by a kind of classic, efficient and well-developed method, the mapping approach. [10] presented the traveling wave solutions for the equation by the ( G  G )-expansion method. [11] used the generalized three-wave method to obtain exact three-wave solutions including periodic cross-kink wave solutions, doubly periodic solitary wave solutions and breather type of two- solitary wave solutions for (1). [12] presented abundant new M.T. Darvishi and M. Najafi are with Department of Mathematics, Razi University, Kermanshah 67149, Iran, e-mail:darvishimt@yahoo.com and m najafi82@yahoo.com. exact solutions for the Jimbo-Miwa equation (1) by using the generalized Riccati equation mapping method. In that work, authors presented twenty seven solutions for the equation. However, Kudryashov and Sinelshchikov [13] showed that eight from those twenty seven solutions are wrong and do not satisfy the equation. Also, the other nineteen exact solutions are not new and can be found from the well-known solution. One can find another schemes to solve (1) in Refs. [14], [15], [16]. In this paper we present some complexiton type solutions of the equation involve two kinds of transcendental functions. We use EHTA to obtain these solutions. II. EXTENDED HOMOCLINIC TEST APPROACH The basic idea of this method applies the Painlev´ e analysis to make a transformation as u = T (f ) (2) 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, (3) where u = u(x,y,z,t) and F is a polynomial of u and its derivatives. By substituting (2) in (3), 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 solutions in EHTA, then we can specify the unknown function f , (for more details see [17]). III. APPLICATION In this section, we investigate explicit formula of solutions of equation (1). To solve (1), we use the EHTA [17]. By this idea we obtain some analytic solutions for the problem. By using Painlev´ e analysis we set u = 2(ln f ) x (4) where f (x,y,z,t) is an unknown real function which will be determined. Substituting Eq. (4) into Eq. (1), we obtain the following Hirota’s bilinear form (D 3 x D y +2 D t D y − 3 D x D z )f · f =0, (5) where the D-operator, is defined by D m x D k y D p z D n t f (x,y,z,t) · g(x,y,z,t)= ( ∂ ∂x1 − ∂ ∂x2 ) m ( ∂ ∂y1 − ∂ ∂y2 ) k ( ∂ ∂z1 − ∂ ∂z2 ) p ( ∂ ∂t1 − ∂ ∂t2 ) n [f (x 1 ,y 1 ,z 1 ,t 1 )g(x 2 ,y 2 ,z 2 ,t 2 )], World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:5, No:7, 2011 1097 International Scholarly and Scientific Research & Innovation 5(7) 2011 scholar.waset.org/1307-6892/503 International Science Index, Mathematical and Computational Sciences Vol:5, No:7, 2011 waset.org/Publication/503