NEW EXACT SOLUTIONS OF SOME NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS VIA THE IMPROVED EXP-FUNCTION METHOD

www.arpapress.com/volumes/vol8issue2/ijrras_8_2_04.pdf NEW EXACT SOLUTIONS OF SOME NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS VIA THE IMPROVED EXP-FUNCTION METHOD M. F. El-Sabbagh, R. Zait and R. M. Abdelazeem Mathematics Department, Faculty of Science, MiniaUniversity, Egypt. Corresponding e-mail: rasha_math24@yahoo.com ABSTRACT In this paper, we establish new exact solutions of some nonlinear partial differential equations (PDEs) of interest such as the Kaup Kupershmidt, the generalized shallow water, the Boussinesq equations via the improved Exp function method. Also the method is used to construct periodic and solitary wave solutions for the considered equations as well. Keywords: Nonlinear PDEs, Exact solutions, The improved Exp function method.. INTRODUCTION The nonlinear evolution equations (NLEEs) are widely used as models to describe complex physical phenomena in various field of science, particularly in fluid mechanics, solid state physics, plasma waves and chemical physics. Nonlinear equations covers also the following subjects: surface wave in compressible fluid, hydro magnetic waves in cold plasma, acoustic waves in un harmonic crystal, ect.. The wide applicability of these equations is the main reason for they have attracted so much attention from mathematicians in the last decades. The investigation of the exact solutions of non linear partial differential equations (PDEs) plays an important role in the study of non-linear physical phenomena. When we want to understand the physical mechanism of phenomena in nature, described by non linear PDEs, exact solutions have to be explored. The study of nonlinear PDEs becomes one of the most important topics in mathematical physics. Recently there are many new methods to obtain exact solutions of nonlinear PDEs such as sine-cosine function method [-5], tanh function method [6-8], ( G/ G )-expansion method [9-3 ], extended Jacobi elliptic function method [4, 5]. He and Wu [6], proposed a straightforward and concise method, called Exp-function method [7-24], to obtain generalized solitary wave solutions of nonlinear PDEs. Ali [25] improved this method and obtained new exp-function solutions and periodic solutions as well. In this paper, we use the improved Exp-function method [25, 26 ] to search for new solitary wave solutions, compact like solutions and periodic solutions of some nonlinear PDEs, such as the Kaup Kupershmidt equation [27-35], the generalized shallow water equation [36, 37], and the Boussinesq equation [38-42]. 2. THE IMPROVED EXP-FUNCTION METHOD: We present the improved Exp-function method [25] in the following steps: - Consider the following nonlinear PDE with two independent variables x, t and dependant variable u: N(u, u t, u x, u xx, u xt, u tt, ) = 0 (), where N is in general a polynomial function of its argument and the subscripts denote the partial derivatives. 2- We seek a traveling wave solution of Eq. () in the form u x, t = u ξ, ξ = kx + ωt 2, where k and ω are constants to be determined 3- Using the transformation (2), Eq. () can be reduced to an ordinary differential equation (ODE): G u, u, u, = 0 3, where G is a polynomial of u and its derivatives. 4- Through this method, we express the solution of the nonlinear PDE () in the form: 32

u η = m j =0 n i=0 A j exp (jξ) B i exp (iξ) 4, where m and n are positive integers that could be freely chosen. 5- To determine u x, t explicitly, one may apply the following producer: (i) Substitute Eq. (4) into Eq. (3), then the lift hand side of Eq. (3) is converted into a polynomial in exp (ξ). Setting all coefficients of exp (ξ) to zero yield a system of algebraic equations for A 0, A, A 2,. A m, B 0, B, B 2,. B n, k and ω. (ii) Solve these algebraic equations to obtain A 0, A, A 2,, A m, B 0, B, B 2,, B n, k and ω. 3. APPLICATIONS In order to illustrate the effectiveness of the above method, examples of mathematical and physical interests are chosen as follows: 3.The Kaup Kupershmidt equation [27-35] The Kaup Kupershmidt equation is the nonlinear fifth-order partial differential equation; It is the first equation in a hierarchy of integrable equations with Lax operator 3 3 + 2u. It has properties similar (but not identical) to x + u x x those of the better known KdV hierarchy in which the Lax operator has order two. In the present paper we introduce new exact solutions of the Kaup Kupershmidt equation via the improved Expfunction method as follows: Consider the Kaup Kupershmidt equation is given as: u t = u xxxxx 20uu xxx 50u x u xx + 80u 2 u x (5) Using the transformation: u = u η, η = x ct 6, where c is a constant to be determined later. Substituting Eq. (6) into Eq. (5) we get cu u + 20uu + 50u u 80u 2 u = 0 7, where primes denote derivatives with respect to η. Now we study the following cases: Case : m = 2, n = 2: According to the improved Exp-function method, the travelling wave solution of Eq. (5) in this case can be written as: u x, t = A 0 + A exp (η) + A 2 exp (2η) B 0 + B exp (η) + B 2 exp (2η) (8) In case B 2 0, Eq. (8) can be simplified as: u x, t = A 0 + A exp (η) + A 2 exp (2η) B 0 + B exp (η) + exp (2η) (9) Substituting Eq. (9) into Eq. (7), and using the Maple, equating to zero the coefficients of all powers of exp (η)yields a set of algebraic equations for A 0, A, A 2, B 0, B and c. Solving these system of algebraic equations, with the aid of Maple, we obtain families of solutions as the following: 33

Family : B 0 =, A = 5, A 0 = 2, c =, A 2 = 2, B = 2. u x, t = 2 5exp (x + t) + exp (2x + 22t) 2 + 2exp (x + t) + exp (2x + 22t) (0) Figure. Traveling wave solution of Eq. 5 for solution 0. Family 2: B 0 =, A = 5 8, A 0 =, c = 6 6, A 2 = 6, B = 2. u 2 x, t = 6 5 t exp (x + 8 6 ) + 6 exp (2x + t 8 ) + 2exp (x + t 6 ) + exp (2x + t () 8 ) Figure 2. Traveling wave solution of Eq. 5 for solution. Case 2: m = 3, n = 3: According to the improved Exp-function method, the travelling wave solution of Eq. (5) in this case can be written as: 34

u x, t = A 0 + A exp (η) + A 2 exp (2η) + A 3 exp (3η) B 0 + B exp (η) + B 2 exp (2η) + B 3 exp (3η) (2) In case B 3 0, Eq. (2) can be simplified as: u x, t = A 0 + A exp (η) + A 2 exp (2η) + A 3 exp (3η) B 0 + B exp (η) + B 2 exp (2η) + exp (3η) 3 Substituting Eq. (3) into Eq. (7), and using the Maple, equating to zero the coefficients of all powers of ex p η yields a set of algebraic equations for A 0, A, A 2, A 3, B 0, B, B 2 and c. Solving the system of algebraic equations given above, with the aid of Maple, we obtain: Family : A 0 = 0, A = B 8 2 2, A 2 = 5 B 2 2, A 3 =, B 2 0 = 0, B = B 4 2 2, B 2 = B 2, c =. u 5 x, t = 4 B 2 2 exp (x + t) 5B 2 exp (2x + 22t) + exp (3x + 33t) 2 B 2 2 exp (x + t) + 2B 2 exp (2x + 22t) + 2exp (3x + 33t) (4) If B 2 =, then exp (x + t) 5exp (2x + 22t) + exp (3x + 33t) u 5 x, t = 4 (5) 2 exp (x + t) + 2exp (2x + 22t) + 2exp (3x + 33t) Figure 3. Traveling wave solution of Eq. 5 for solution 5, B 2 =. Family 2: A 0 = 0, A = B 64 2 2, A 2 = 5 B 6 2, A 3 =, B 6 0 = 0, B = B 4 2 2, B 2 = B 2, c = u 6 x, t = 4 B 2 2 exp (x + t 6 ) 5B 2exp (2x + t 3t ) + exp (3x + 8 6 ) 4(B 2 2 exp (x + t 6 ) + 4B 2exp (2x + t (6) 3t ) + 4exp (3x + 8 6 )) 6. 35

If B 2 =, then u 6 x, t = t exp (x + 4 6 ) 5exp (2x + t 3t ) + exp (3x + 8 6 ) 4(ex p x + t 6 + 4ex p 2x + t 8 + 4ex p 3x + 3t 6 ) (7) Figure 4. Traveling wave solution of Eq. 5 for solution 7, B 2 =. 3.2 The generalized shallow water equation: [36, 37] The shallow water wave equations describe the evolution of incompressible flow, neglecting density change along the depth. The shallow water wave equations are applicable to cases where the horizontal scale of the flow is much bigger than the depth of fluid. The shallow water equations have been extensively used for a wide variety of coastal phenomena, such as tide-currents, pollutant- dispersion storm-surges, tsunami-wave propagation, etc.. In the present paper we introduce new exact solutions of the generalized shallow water equation via the improved Exp-function Method as follows: Consider the generalized shallow water equation: u xxxt + αu x u xt + βu t u xx u xt u xx = 0 8, where α and β are arbitrary nonzero constants. Using the transformation: u = u η, η = x ct 9, where c is a constant to be determined later. Substituting Eq. (9) into Eq. (8) we get cu cαu u cβu u cu u = 0 (20), where the prime denotes the differential with respect to η. Now we study the following cases: Case : m = 2, n = 2: According to the improved Exp-function method, the travelling wave solution of Eq. (8) in this case can be written as: u x, t = A 0 + A exp (η) + A 2 exp (2η) B 0 + B exp (η) + B 2 exp (2η) (2) 36

In case B 2 0, Eq.(2) can be simplified as: u x, t = A 0 + A exp (η) + A 2 exp (2η) B 0 + B exp (η) + exp (2η) 22 Substituting Eq. (22) into Eq. (20), and using the Maple, equating to zero the coefficients of all powers of exp (η)yields a set of algebraic equations for A 0, A, A 2, B 0, B and c. Solving the system of algebraic equation given above, with the aid of Maple, we obtain: A 0 = B 0( 24+A 2 β +A 2 α), A α+β = 0, A 2 = A 2, B 0 = B 0, B = 0, c =. 3 u x, t = B 0 ( 24 + A 2 β + A 2 α) + A α + β 2 exp (2x + 2t 3 ) B 0 + exp (2x + 2t (23) 3 ) If A 2 = B 0 =, α = β = 2, then u x, t = 5 + exp (2x + 2t 3 ) + exp (2x + 2t 3 ) (24) Figure 5. Traveling wave solution of Eq. 8 for solution Eq. 24, A 2 = B 0 =, α = β = 2. Case 2: m = 3, n = 3: According to the improved Exp-function method, the travelling wave solution of Eq. (8) in this case can be written as: u x, t = A 0 + A exp (η) + A 2 exp (2η) + A 3 exp (3η) B 0 + B exp (η) + B 2 exp (2η) + B 3 exp (3η) (25) In case B 3 0, Eq.(25) can be simplified as: u x, t = A 0 + A exp (η) + A 2 exp (2η) + A 3 exp (3η) B 0 + B exp (η) + B 2 exp (2η) + exp (3η) 26 Substituting Eq. (26) into Eq. (20), and using the Maple, equating to zero the coefficients of all powers of exp (η) yields a set of algebraic equations fora 0, A, A 2, A 3, B 0, B, B 2 and c. Solving the system of algebraic equation given above, with the aid of Maple, we obtain: 37

Family : A 0 = 0, A = B ( 24 + A 3 β + A 3 α), A α + β 2 = 0, A 3 = A 3, B 0 = 0, B = B, B 2 = B 2, c = 3 u 2 x, t = B ( 24 + A 3 β + A 3 α) exp (x + t α + β 3 ) + A 3exp (3x + t) B exp (x + t (27) 3 ) + exp (3x + t) If B = A 3 = 2, α = β =, then u 2 x, t = 20exp (x + t ) + 2exp (3x + t) 3 2exp (x + t (28) 3 ) + exp (3x + t) Figure 6. Traveling wave solution of Eq. 8 for solution Eq. 28, A 3 = B = 2, α = β =. Family 2: A 0 = B 0(A 3 β + A 3 α 36), A α + β = 0, A 2 = 0, A 3 = A 3, B 0 = B 0, B = 0, B 2 = 0, c = 8 u 3 x, t = B 0 (A 3 β + A 3 α 36) + A α + β 3 exp (3x + 3t 8 ) B 0 + exp (3x + 3t (29) 8 ) If B 0 = A 3 = 2, α = β =, then u 3 x, t = 32 + 2exp (3x + 3t 8 ) 2 + exp (3x + 3t 8 ) (30) 38

Figure 7. Traveling wave solution of Eq. 8 for solution Eq. 30, A 3 = B 0 = 2, α = β =. 3.3 The Boussinesq equation: [38-42] The Boussinesq-type equations, which include the lowest-order effects of nonlinearity and frequency dispersion as additions to the simplest non-dispersive linear long wave theory, provide a sound and increasingly well-tested basis for the simulation of wave propagation in coastal regions. The standard Boussinesq equations for variable water depth were first derived by Peregrine (967), who used depth-averaged velocity as a dependent variable. In the present paper we introduce new exact solutions of the Boussinesq equation via the improved Exp-function method as follows: Consider the Boussinesq equation: u tt u xx u xxxx 6(u x ) 2 6uu xx = 0 (3) Using the transformation: u = u η, ξ = kx + ωt 32, where k, ω are constants to be determined later. Substituting Eq. (32) into Eq. (3) we get ω 2 u k 2 u k 4 u 6k 2 (u ) 2 6k 2 uu = 0 33, where the prime denotes the differential with respect to ξ. Now we study the following cases: Case : m = 2, n = 3: According to the improved Exp-function method, the travelling wave solution of Eq. (3) in this case can be written as: u x, t = A 0 + A exp (ξ) + A 2 exp (2ξ) B 0 + B exp (ξ) + B 2 exp (2ξ) + B 3 exp (3ξ) (34) In case B 3 0, Eq.(34) can be simplified as: u x, t = A 0 + A exp (ξ) + A 2 exp (2ξ) B 0 + B exp (ξ) + B 2 exp (2ξ) + exp (3ξ) 35 Substituting Eq. (35) into Eq. (33), and using the Maple, equating to zero the coefficients of all powers of ex p ξ yields a set of algebraic equations for A 0, A, A 2, B 0, B, B 2, k and ω. Solving the system of algebraic equation given above, with the aid of Maple, we obtain: 39

Family : A 0 = 0, A = 0, A 2 = k 2 B 2, B 0 = 0, B = B 2 2 4, B 2 = B 2, ω = k + k 2 u x, t = k 2 B 2 exp (2kx + 2kt + k 2 ) B 2 2 4 exp (kx + kt + k2 ) + B 2 exp (2kx + 2kt + k 2 ) + exp (3kx + 3kt + k 2 ) (36) If k =, B 2 = 2, then 2exp (2x + 2t 2) u x, t = exp (x + t 2) + 2exp (2x + 2t 2) + exp (3x + 3t 2) (37) Figure 8. Traveling wave solution of Eq. 3 for solution Eq. 37, B 2 = 2, k =. Family 2: A 0 = 0, A = 0, A 2 = k 2 B 2, B 0 = 0, B = B 2 2 4, B 2 = B 2, ω = k + k 2 u 2 x, t = k 2 B 2 exp (2kx 2kt + k 2 ) B 2 2 4 exp (kx kt + k2 ) + B 2 exp (2kx 2kt + k 2 ) + exp (3kx 3kt + k 2 ) (38) If B 2 = 2, k =, then 2ex p 2x 2t 2 u 2 x, t = exp (x t 2) + 2exp (2x 2t 2) + exp (3x 3t 2) (39) 40

Figure 9. Traveling wave solution of Eq. 3 for solution Eq. 39, B 2 = 2, k =. Case 2: m = 2, n = 4: According to the improved Exp-function method, the travelling wave solution of Eq. (3) in this case can be written as: u x, t = A 0 + A ex p ξ + A 2 ex p 2ξ B 0 + B ex p ξ + B 2 ex p 2ξ + B 3 ex p 3ξ + B 4 ex p 4ξ (40) In case B 4 0, Eq.(40) can be simplified as: u x, t = A 0 + A ex p ξ + A 2 ex p 2ξ B 0 + B ex p ξ + B 2 ex p 2ξ + B 3 ex p 3ξ + ex p 4ξ 4 Substituting Eq. (4) into Eq. (33), and using the Maple, equating to zero the coefficients of all powers of ex p ξ yields a set of algebraic equations for A 0, A, A 2, B 0, B, B 2, B 3, k and ω. Solving the system of algebraic equation given above, with the aid of Maple, we obtain: Family : A 0 = 0, A = 0, A 2 = A 2, B 0 = A 2 2 64k 4, B = 0, B 2 = A 2 4k 2, B 3 = B 3, ω = k + 4k 2 u 3 x, t = A 2 exp (2kx + 2kt + 4k 2 ) A 2 2 64k 4 + A 2 4k 2 exp (2kx + 2kt + 4k2 ) + exp (4kx + 4kt + 4k 2 ) (42) If A 2 = 2, k =, then 2exp (2x + 2t 5) u 3 x, t = 6 + (43) 2 exp (2x + 2t 5) + exp (4x + 4t 5) 4

Figure 0. Traveling wave solution of Eq. 3 for solution Eq. 43, A 2 = 2, k =. Family 2: A 0 == 0, A = 0, A 2 = A 2, B 0 = A 2 2 64k 4, B = 0, B 2 = A 2 4k 2, B 3 = B 3, ω = k + 4k 2 u 4 x, t = A 2 exp (2kx 2kt + 4k 2 ) A 2 2 64k 4 + A 2 4k 2 exp (2kx 2kt + 4k2 ) + exp (4kx 4kt + 4k 2 ) (44) If A 2 = 2, k =, then 2exp (2x 2t 5) u 4 x, t = 6 + (45) 2 exp (2x 2t 5) + exp (4x 4t 5) Figure. Traveling wave solution of Eq. 33 for solution Eq. 45, A 2 = 2, k =. 42

4. CONCULSION In this paper, the improved Exp-function method has been successfully applied to obtain new solutions of three nonlinear partial differential equations. Thus, the improved Exp-function method can be extended to solve the problems of nonlinear partial differential equations which arising in the theory of solitons and other areas. 5. REFERENCES []. M.T. Alquran, Solitons And Periodic Solutions To Nonlinear Partial Differential Equations By The Sine- Cosine Method, Appl. Math. Inf. Sci. 6, No., pp. 85-88, 202. [2]. S. A. Mohammad-Abadi, Analytic Solutions Of The Kadomtsev-Petviashvili Equation With Power Law Nonlinearity Using The Sine-Cosine Method, American Journal of Computational and Applied Mathematics, Vol.,No. 2, pp. 63-66, 20. [3]. A. J. M. Jawad, The Sine-Cosine Function Method For The Exact Solutions Of Nonlinear Partial Differential Equations, IJRRAS, Vol. 3, No., 202. [4]. M. Hosseini, H. Abdollahzadeh, M.Abdollahzadeh, Exact Travelling Solutions For The Sixth-Order Boussinesq Equation, The Journal of Mathematics and Computer Science Vol. 2, No.2, pp. 376-387, 20. [5]. S. Arbabi, M. Najafi, M. Najafi, New Periodic And Soliton Solutions Of (2 + )-Dimensional Soliton Equation, Journal of Advanced Computer Science and Technology, Vol., No. 4, pp. 232-239, 202. [6]. A. J. M. Jawad, M. D. Petkovic and A. Biswas, Soliton Solutions To A few Coupled Nonlinear Wave Equations By Tanh Method, IJST, 37A2,pp. 09-5, 203. [7]. A. J. Muhammad-Jawad, Tanh Method For Solutions Of Non-linear Evolution Equations, Journal of Basrah Researches (Sciences), Vol. 37. No. 4, 20. [8]. W. Malfliet, W. Hereman, The Tanh Method:. Exact Solutions Of Nonlinear Evolution And Wave Equations, Physica Scripta. Vol. 54, pp. 563-568, (996). [9]. J. F.Alzaidy, The (G'/G) - Expansion Method For Finding Traveling Wave Solutions Of Some Nonlinear Pdes In Mathematical Physics, IJMER, Vol.3, Issue., pp. 369-376, 203. [0]. G. Khaled A, AGeneralized (G /G)-Expansion Method To Find The Traveling Wave Solutions Of Nonlinear Evolution Equations, J. Part. Diff. Eq., Vol. 24, No., pp.55-69, 20. []. J. Manafianheris, Exact Solutions Of The BBM And MBBM Equations By The Generalized (G'/G )- Expansion Method Equations, International Journal of Genetic Engineering, Vol. 2, No. 3, pp. 28-32, 202. [2]. Y. Qiu and B. Tian, Generalized G'/G-Expansion Method And Its Applications, International Mathematical Forum, Vol. 6, No. 3, pp. 47 57, 20. [3]. R. K. Gupta, S. Kumar, and B. Lal, New Exact Travelling Wave Solutions Of Generalized Sinh-Gordon And (2 + )-Dimensional ZK-BBM Equations, Maejo Int. J. Sci. Technol.,Vol. 6, pp. 344-355, 202. [4]. A. S. Alofi, Extended Jacobi Elliptic Function Expansion Method For Nonlinear Benjamin-Bona-Mahony Equations, International Mathematical Forum, Vol. 7, No.53, pp. 2639 2649, 202. [5]. B. Hong, D. Lu2, and F. Sun, The Extended Jacobi Elliptic Functions Expansion Method And New Exact Solutions For The Zakharov Equations, World Journal of Modelling and Simulation, Vol. 5, No. 3, pp. 26-224, 2009. [6]. J. H. He and X. H. Wu, Exp-function Method For Nonlinear Wave Equations, Chaos, Solitons & Fractals, Vol. 30, pp.700-708, 2006. [7]. A. Ebaid, Exact Solitary Wave Solutions For Some Nonlinear Evolution Equations Via Exp-function Method, Physics Letters A 365, pp. 23 29, 2007. [8]. A. Boz, A. Bekir, Application Of Exp-function Method For (2 + )-Dimensional Nonlinear Evolution Equations, Chaos, Solitons and Fractals, Vol. 40, issue, pp. 458-465, 2009. [9]. A. Boz, A. Bekir, Application Of Exp-function Method For (3 + )-Dimensional Nonlinear Evolution Equations, Computers and Mathematics with Applications, Vol. 56, pp.45 456, 2008. [20]. Z.Sheng, Application Of Exp-function Method To High-dimensional Nonlinear Evolution Equation, Chaos, Solitons and Fractals, Vol. 38, pp. 270 276, 2008. [2]. Z. Sheng, Exp-function Method Exactly Solving A KdV Equation With Forcing Term, Applied Mathematics and Computation, Vol. 97, pp. 28 34, 2008. [22]. Yusufo glu. E, New Solitonary Solutions For Modified Forms Of DP And CH Equations Using Exp-function Method, Chaos, Solitons and Fractals xxx (2007) xxx xxx. [23]. Yusufo glu. E, New Solitonary Solutions For The MBBM Equations Using Exp-function Method, Physics Letters A 372, pp. 442 446, 2008. [24]. E. Misirli, Y. Gurefe, EXP-Function Method For Solving Nonlinear Evolution Equations, Mathematical and Computational Applications, Vol. 6, No., pp. 258-266, 20. 43

[25]. A.T.Ali, A note On The Exp-function Method And Its Application To Nonlinear Equations, Phys. Scr., Vol. 79, Issue 2, id. 025006, 2009. [26]. M. F. El-Sabbagh, M. M. Hassan and E Hamed, New Solutions For Some Non-linear Evolution Equations Using Improved Exp-function Method, El-Minia Science Bulletin,Vol. 2, pp. -25, 200. [27]. A. Khajeh, A. Yousefi-Koma, V. Maryam and M.M. Kabir, Exact Travelling Wave Solutions For Some Nonlinear Equations Arising In Biology And Engineering, World Applied Sciences Journal, Vol. 9, No. 2, pp. 433-442, 200. [28]. F. Dahe, L. Kezan, On Exact Traveling Wave Solutions For ( + ) Dimensional Kaup Kupershmidt Equation, Applied Mathematics, Vol. 2, pp. 752-756, 20. [29]. F. Qinghua, New Analytical Solutions For (3+) Dimensional Kaup-Kupershmidt Equation, International Conference on Computer Technology and Science, Vol. 47, No. 59, 202. [30]. F. XIE and X.GAO, A Computational Approach To The New Type Solutions Of Whitham Broer Kaup Equation In Shallow Water, Commun. Theor. Phys. (Beijing, China), Vol. 4, No. 2, pp. 79 82, 2004. [3]. H. L U and X. LIU, Exact Solutions To (2+)-Dimensional Kaup Kupershmidt Equation, Commun. Theor. Phys.(Beijing, China), Vol. 52, No. 5, pp. 795 800, 2009. [32]. H. Salas Alvaro, Exact Solutions To Kaup-Kupershmidt Equation By Projective Riccati Equations Method, International Mathematical Forum, Vol. 7, No. 8, pp. 39 396, 202. [33]. L. Jibin and Z. Qiao, Explicit Soliton Solutions Of The Kaup-Kupershmidt Equation Through The Dynamical System Approach, Journal of Applied Analysis and Computation, Vol., No. 2, pp. 243 250, 20. [34]. P. Saucez, A.Vande Wouwer, W.E. Schiesserc, P.Zegeling, Method Of Lines Study Of Nonlinear Dispersive Waves, Journal of Computational and Applied Mathematics, Vol. 68, pp. 43 423, 2004. [35]. V. D. Maria and Nikolai A. Kudryashov, Point Vortices And Polynomials Of The Sawada Kotera And Kaup Kupershmidt Equations, Regular and Chaotic Dynamics, Vol. 6, No. 6, pp. 562 576, 20. [36]. S. A. El-Wakil, M.A. Abdou, A. Hendi, New Periodic Wave Solutions Via Exp-function Method, Physics Letters A 372, pp. 830 840, 2008. [37]. V. Anjali, J. Ram and K. Jitender, Traveling Wave Solutions For Shallow Water Wave Equation By Expansion Method, International Journal Of Mathematical Sciences, Vol. 7, No., 203. [38]. A. R. Seadawy, The Solutions Of The Boussinesq And Generalized Fifth-Order KdV Equations By Using The Direct Algebraic Method, Applied Mathematical Sciences, Vol. 6, No. 82, pp.408 4090, 202. [39]. E. Yusufo glu, New Solitonary Solutions For The MBBM Equations Using Exp-function Method, Physics Letters A 372, pp. 442 446, 2008. [40]. L.V. Bogdanov, V.E. Zakharov, The Boussinesq Equation Revisited, Physica D, Vol. 65, pp. 37 62, 2002. [4]. L.Wei, Z. Jinhua, Z. Yunmei, Exact Solutions For A family Of Boussinesq Equation With Nonlinear Dispersion, International Journal of Pure and Applied Mathematics, Vol. 76, No., pp. 49-68, 202. [42]. M. A. Abdou, A. A. Soliman, S.T. El-Basyony, New Application Of Exp-function Method For Improved Boussinesq Equation, Physics Letters A 369, pp. 469 475, 2007. G \ G - 44