 American Journal of Computational Mathematics, 2011, 1, 219-225 doi:10.4236/ajcm.2011.14025 Published Online December 2011 (http://www.SciRP.org/journal/ajcm) Copyright © 2011 SciRes. AJCM General Solution of Generalized (2 + 1)-Dimensional Kadomtsev-Petviashvili (KP) Equation by Using the GG-Expansion Method Abdollah Borhanifar, Reza Abazari Department of Mathematics, University of Mohaghegh Ardabili, Ardabil, Iran E-mail: borhani@uma.ac.ir Received May 5, 2011; revised May 28, 2011; accepted June 10, 20 11 Abstract In this work, the GG-expansion method is proposed for constructing more general exact solutions of the (2 + 1)-dimensional Kadomtsev-Petviashvili (KP) equation and its generalized forms. Our work is motivated by the fact that the GG-expansion method provides not only more general forms of solutions but also periodic and solitary waves. If we set the parameters in the obtained wider set of solutions as special values, then some previously known solutions can be recovered. The method appears to be easier and faster by means of a symbolic computation system. Keywords: GG-Expansion Method, Generalized Kadomtsev-Petviashvili (KP) Equation, Hyperbolic Function Solutions, Trigonometric Function Solutions 1. Introduction Nonlinear evolution equations (NLEEs) have been the subject of study in various branches of mathematical- physical sciences such as physics, biology, chemistry, etc. The analytical solutions of such equations are of funda-mental importance since a lot of mathematical-physical models are described by NLEEs. Among the possible solutions to NLEEs, certain special form solutions may depend only on a sing le combination of variables su ch as traveling wave variables. In the literatu re, there is a wide variety of approaches to nonlinear problems for con-structing traveling wave solutions. Some of these ap-proaches are the Jacobi elliptic function method , in-verse scattering method , Hirotas bilinear method , homogeneous balance method , homotopy perturba-tion method , Weierstrass function method , sym-metry method , Adomian decomposition method , sine/cosine method , tanh/coth method , the Exp-function method [11-16] and so on. But, most of the methods may sometimes fail or can only lead to a kind of special solution and the solution procedures become very complex as the degree of nonlinearity increases. Recently, the GGtroduced by Wang et al. , has become widely used to -expansion method, firstly in-search for various exact solutions of NLEEs [17-27]. The value of the GG-expansion method is that one treats nonlinear proby essentially linear methods. The method is based on the explicit linearization of NLEEs for traveling waves with a certain substitution which leads to a second-order differential equation with con-stant coefficients. Moreover, it transforms a nonlinear equation to a simple algebraic computation. The generalized (2 + 1)-dimensional Kablems domtsov-Pe- tviashivilli (gKP) equ ation given by =0, >12ntx xxxyyxuuu uun  The objectives of this work are twofold. First, we de-scribe the GG-expansion method. Second, we aim to implemenresent method to obtain general exact travelling wave solutions of governing equation. t the p2. Description of the GG-Expansion The objective of this section is to outline the use of the Method GG-expansion method for solving certain nonlinear ifferential equations (PDEs). Suppose we have a partial dnonlinear PDE for ,, ,uxyt in the form A. BORHANIFAR ET AL. 220 where is a polynomial in its argumencludes inear terms and the highest ord,,,, ,=0,xyPuuu uu (1) x txxP nonlnsfts, which in-er derivatives. The traormation ,,, =uxyzt U, =kx yt, reduces Equation (4) to the ordinary differential equation (ODE) 2,,, ,,=0,PU kUUkUkU  (2) where =(),UU and prime denotes derivatrespect tive with o . be expresseWe assume that the solution of Equation (2) cand by a polynomial in GG as fol-lows:  0=1=,0niiniUGG . (3) where 0, and ,i are constants to be determined later, ()G satisfies a second order linear ordinary dif-ferential equation (LODE): 22dd =0.ddGGG (4) where  and  are arbitrary constants. Using the ge- neral solutions of Equation (4), we have 22122222122212221244sinh cosh224,42244cosh sinh2244sin cos22424cos sin2CCCCGGCCCC 0,   22,40,242 (5) and it follows, from (3) and (4), that   1=121 1222=1,=121(2) 211,iinii iiiiiGGUi iGGiGGiGGiGGiGG      (6) and so on, here the prime denotes the derivative with spective to 1=niiiUiGGGGre. To determine u explicitly, we take the following four steps: Step 1. Determin the integer n by substituting Equa- tion (3) aloeng with Equation (4) into Equation (2), and ba nl Equation (4) into Equtiolancing the highest order noinear term(s) and the highest order partial derivative. Step 2. Substitute Equation (3) give the value of n determined in Step 1, along witha-fn (2) and collect all terms with the same order o GG together, the left-hand side of Equation (2) is converted into a polynomial in GG. Then set each ent of this polynomial to zero to derive a set of algebraic equations for 0,,kcoeffici and i. Step 3. Solve the system of algebraic equations ob-tained in Step 2, for 0,,,,abc a ndi by use of Mobtained e eries of fundamental solutionsaple. Step 4. Use the results in abovsteps to de- rive a s u of Equa- tion (2) depending on GG, since the solutions of Equation (4) have bn for us, then we can obe een well knowtain exact solutions of Equation (1). 3. Application In this section, we will demonstrate thGG-dimension-expan- sion method on the generalized (2 + 1)al Ka- ation given by domtsev-Petviashvili (KP) equ=0,|2ntxxxx yyxuuu uun|>1,  (7) where ,  and  are constants. Using the wave vari-able =,kx yt in (7) and integrating the result-ing equation and neglecting the constant of iwe find ntegration, 22140, 1,21nkUkUnn kU our goal, we use the transformation  (8) To achieveCopyright © 2011 SciRes. AJCM A. BORHANIFAR ET AL.221  ,nUV1 that will carry (8) into the ODE  222222341112nnkVknVnknVVnV    =0, (9) According to Step 1, we get2, hence We then suppose that Equatioe fol-lo 32mmn (9) has th2.m wing formal solutions: 221,0,VGG GG (10)02 where 21,, and 0, are constants whichknown to be determined later. titutingon me order of are un-Subs Equati(10) into Equation (9) and col- lecting all terms with the saGG22 22012222 42 42222222 322 3=,=22 328=,=2knnknnnnknn kknk   ,,nn(11) Substitute the above general case in (10), we get  to-gether, the left-hand sides of Equation (9) are converted into a polynomial in GG. Setting each coefficient of each polynomial to zero, we derive a set of algebraic equations for 01,, ,k,,, and 2, and solving them by use of Maple, we get the following gen- eral result: 22 222 3,knnGGG21, 2.VGnnn (12) then use the transformation  1=,nUV when ,240 the hyperbolic function solutions of Equa-tion (7), becomes: 224cos221122222221221244sinh h2222 3=2442sinh cosh2244sinh cosh2 nCCknnunCCCC     12222142,2442sinh cosh22nCC   (13) and when ,the trigonometric function solutions of Equation (7), will be: 240 224221122222221221244sin cos2222 32442sin cos22444sin cos2 nCCknnunCCCC        1222212,2442sin cos22nCC   (14) here w42 422228=,2kk nkx ytkn  12,,,CC and  are arbitrary constants. In particular, when then the general solutions and 2=0,C Copyright © 2011 SciRes. AJCM A. BORHANIFAR ET AL. 222 (13) and (14) reduces , res pectively, 1242 222282kk nkxytn22 22 42122 42 422222 34422 24428tanh ,22 22nnknnnkkk nkx ytkn tanhu        (15) 12tn222 22 42 4 222 2122 42 422222 34428tan22 24428tan ,22 22nnknnkk nukxynkkk nkx ytkn         (16) and when then we deduce from general solutions (13) and (14) that, 1=0,C 1222 22 42 4222 2122 42 422222 34428coth22 24428coth ,22 22nnknnkk nukxynkkk nkx ytkn 2tn        (17) 1222 22 42 4222 2122 42 422222 34428() cot22 24428cot ,22 22nnknnkk nukxynkkk nkx ytkn 2tn      (18) where ,, ,k and  are arbitrary constants. For important case 3=,2n 32 2=0txxxxyy ,xuuuuu  where (19) and  ,are constants, then according to re-sults in (11), the general hyperbolic and trigonometric function solution of (19) will be the KP Equation (7) re-duce to 23222135kC2224cosh 4CC (20) 222214=,1118 sinh422uC2322 22 212222222212 1235 44,444182 sincoscos 222kC CuCCC CC  2          (21) Copyright © 2011 SciRes. AJCM A. BORHANIFAR ET AL.223 where and ,42 4241692,9kkkx ytk    12,,,,CC k=0, then the ge and μ are arbitrary constants. When neral hyperbolic and trigonometric function solution (20) and (21) reduce to 2C232242 42,94162kk  (22) 235 4418cos 29kukx ytk 232242 422235 4,94164218cos 29kukkkx ytk (23) and when then the general solution (20)-(21) reduce to 1=0,C 232242 42244kk235 4=,916218sinh 29kukx ytk   (24) 232242 422235 4.94164218sin 29kukkkx ytk  (25) We would like to note that the obtained solutions with an explicit linear function in  have been checked with Maple by putting them back into the original Equations (7). 4. Conclusions and Future Work This study shows that the GG-expansion method is the equations considered, they might serve as seeding quite efficient and practically well suited for use in find-ing exact solutions for the gesional Kadomtsev-Petviashvili (gKP) equation. The reli-ability of the method and the reduction in the size of computational domain give this method a wider applica-bility. Though the obtained solutions represent only a small part of the large variety of possible solutions for neralized (2 + 1)-dimen- Copyright © 2011 SciRes. AJCM A. BORHANIFAR ET AL. 224 ical systems. Furthermore, our solutions are in mgeneral forms, and many known solutions to these equa-he aid of Ma-ple, we have assured the correctness of the obtained so-k into solutions for a class of localized structures existing in the physore tions are only special cases of them. With tlutions by putting them bacthe original equation. We hope that they will be useful for further studies in applied sciences. According to Case 5, present method failed to obtain the general solution of gKP for =1,n and =2,n therefore the authors hope to extend the GG-expansion method to solve these especial type of gKP. 5. Acknowledgments This work is partially supported by Grant-in-Aid from the University of Mohaghegh Ardabili, Ardabil, Iran. 6. References  G. T. Liu and T. Y. Fan, “New Applications of Devel- oped Jacobi Elliptic Function Expansion Methods,” Phy- sics Letters A, Vol. 345, No. 1-3, 2005, pp. 161-166. doi:10.1016/j.physleta.2005.07.034  . J. Ablowitz and H. Segur, “Solitons and Inverse Scat-tering Transform,” SIAM, Philadelphia, 1981. doi:1 M0.1137/1.9781611970883 ethod in Soliton Theory,” Cam-ambridge, 2004.  R. Hirota, “The Direct Mbridge University Press, C M. L. Wang, “Exact Solutions for a Compound KdV-Burg- er s Equation,” Physics Letters A, Vol. 213, No. 5-6, 1996, pp. 279-287. doi:10.1016/0375-9601(96)00103-X  J. H. He, “Thelinear Oscillat Homotopy Perturbation Method for Non-ors with Discontinuities,” Applied Mathe-matics and Computation, Vol. 151, No. 1, 2004, pp. 287- 292. doi:10.1016/S0096-3003(03)00341-2  Z. Y. Yan, “An Improved Algebra Method and Its Ap-plications in Nonlinear Wave Equations,” Chaos Solito& Fractals, Vol. 21, No. 4, 2004, ppns . 1013-1021. doi:10.1016/j.chaos.2003.12.042  G. W. Bluman and S. Kumei, “Symmetries and tial Equations,” Springer-Verlag Differen-, New York, 1989. 994. car Partial Dif-ional Nizhnik-Novikov-.064 G. Adomian, “Solving Frontier Problems of Physics: The Decomposition Method,” Kluwer, Boston, 1 A. Borhanifar, H. Jafari and S. A. Karimi, “New Solitons and Periodic Solutions for the Kadomtsev-Petviashvili Equa- tion,” The Journal of Nonlinear Science and Applitions, Vol. 1, No. 4, 2008, pp. 224-229.  H. Jafari, A. Borhanifar and S. A. Karimi, “New Solitary Wave Solutions for the Bad Boussinesq and Good Bous- sinesq Equations,” Numerical Methods fo ferential Equations, Vol. 25, No. 5, 2000, pp. 1231-1237.  A. Borhanifar, M. M. Kabir and L. M. Vahdat, “New Pe- riodic and Soliton Wave Solutions for the Generalized Zak- harov System and (2 + 1)-Dimens Veselov Sy stem,” Chaos Solitons & Fractals, Vol. 42, No. 3, 2009, pp. 1646-1654. doi:10.1016/j.chaos.2009.03 thod for  A. Borhanifar and M. M. Kabir, “New Periodic and Soli- ton Solutions by Application of Exp-Function MeNonlinear Evolution Equations,” Journal of Computa- tional and Applied Mathematics, Vol. 229, No. 1, 2009, pp. 158-167. doi:10.1016/j.cam.2008.10.052  S. A. El Wakil, M. A. Abdou and A. Hendi, “New Peri- odic Wave Solutions via Exp-Function Method,” Physics Letters A, Vol. 372, No. 6, 2008, pp. 830-840. doi:10.1016/j.physleta.2007.08.033  A. Boz and A. Bekir, “Application of Exp-Function Me- thod for (3 + 1)-Dimensional Nonlinear Evolution Equa- tions,” Computers & Mathematics with Applications, Vol. 56, No. 5, 2000, pp. 1451-1456.  H. Zhao and C. Bai, “New Doubly Periodic and Multiple Soliton Solutions of the Generalized (3 + 1)-Dimensional Kadomtsev-Petviashvilli Equation with Variable Coeffi- cients,” Chaos Solitons & Fractals, Vol. 30, No. 1, 2006, pp. 217-226. doi:10.1016/j.chaos.2005.08.148  M. A. Abdou, “Further Improved F-Expansion and New Exact Solutions for Nonlinear Evolution Equations,” Non- linear Dynamics, Vol. 52, No. 3, 2008, pp. 277-288. doi:10.1007/s11071-007-9277-3  M. Wang, X. Li and J. Zhang, “The (G'/G)-EMethod and Traveling Wave Solutioxpansion ns of Nonlinear Evo- lution Equations in Mathematical Physics,” Physics Let- ters A, Vol. 372, No. 4, 2008, pp. 417-423. doi:10.1016/j.physleta.2007.07.051  J. Zhang, X. Wei and Y. J. Lu, “A Generalized (G'/G)- Expansion Method and Its Applications,” Physics Letters A, Vol. 372, No. , 2008, pp. 36-53. doi:10.1016/j.physleta.2008.01.057  A. Bekir, “Application of the (G'/G)-Expansion Method for Nonlinear Evolution Equations,” Physics Letters A, ons Vol. 372, No. 19, 2008, pp. 3400-3406.  A. Bekir and A. C. Cevikel, “New Exact Travelling Wave Solutions of Nonlinear Physical Models,” Chaos Solit& Fractals, Vol. 41, No. 4, 2008, pp. 1733-1739.  E. M. E. Zayed and K. A. Gepreel, “Some Applications of the (G'/G)-Expansion Met hod to Non -Linear Pa rtial Dif- ferential Equations,” Applied Mathematics and Computa- tion, Vol. 212, No. 1, 2009, pp. 1-13. doi:10.1016/j.amc.2009.02.009  D. D. Ganji and M. Abdollahzadeh, “Exact Traveling So- lutions of Some Nonlinear Evolution Equation by (G'/G)- Expansion Method,” Journal of MaVol. 50, No. 1, 2009, Article ID: 013thematical Physics, 519. doi:10.1063/1.3052847  M. Wang, J. Zhang and X. Li, “Application of the (G'/G)- Expansion to Travelling Wave Solutions of the Broerkaup and the Approximate Long Water Wave Equations,” Ap- plied Mathematics and Computation, Vol. 206, No. 1, 2008, pp. 321-326. doi:10.1016/j.amc.2008.08.045  L.-X. Li and M.-L.Wand, “The (G'/G)-Expansion Method and Travelling Wave Solutions for a Higher-Order Non- linear Schrdinger Equation,” Applied Mathematics and Computation, Vol. 208, No. 2, 2009, pp. 440-445. doi:10.1016/j.amc.2008.12.005 Copyright © 2011 SciRes. AJCM A. BORHANIFAR ET AL. Copyright © 2011 SciRes. AJCM 225reel, “The (G'/G)-Expan- E. M. E. Zayed and K. A. Gep sion Method for Finding Traveling Wave Solutions of Nonlinear Partial Differential Equations in Mathematical Physics,” Journal of Mathematical Physics, Vol. 50, No. 1, 2008, Article ID: 013502. doi:10.1063/1.3033750  I. Aslan and T. Ozis, “Analytic Study on Two Nonlinear, Vol. 209, No. 2Evolution Equations by Using the (G'/G)-Expansion Me- thod,” Applied Mathematics and Computation, 2009, pp. 425-429. doi:10.1016/j.amc.2008.12.064  I. Aslan and T. Ozis, “On the Validity and Reliability of the (G'/G)-Expansion Method by Using Higher-Order Non- linear Equations,” Applied Mathematics and Computation, Vol. 211, No. 2, 2009, pp. 531-536. doi:10.1016/j.amc.2009.01.075