Generalized and Improved (G /G)-Expansion Method Combined with Jacobi Elliptic Equation

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1 Commun. Theor. Phys Vol. 61, No. 6, June 1, 2014 eneralized and Improved /-Expansion Method Combined with Jacobi Elliptic Equation M. Ali Akbar, 1,2, Norhashidah Hj. Mohd. Ali, 1 and E.M.E. Zayed 3 1 School of Mathematical Sciences, Universiti Sains Malaysia, Malaysia 2 Department of Applied Mathematics, University of Rajshahi, Bangladesh 3 Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt Received November 20, 2013; revised manuscript received January 13, 2014 Abstract In this article, we propose an alternative approach of the generalized and improved /-expansion method and build some new exact traveling wave solutions of three nonlinear evolution equations, namely the Boiti Leon Pempinelle equation, the Pochhammer Chree equations and the Painleve integrable Burgers equation with free parameters. When the free parameters receive particular values, solitary wave solutions are constructed from the traveling waves. We use the Jacobi elliptic equation as an auxiliary equation in place of the second order linear equation. It is established that the proposed algorithm offers a further influential mathematical tool for constructing exact solutions of nonlinear evolution equations. PACS numbers: Ik, Jr, Yv Key words: Boiti Leon Pempinelle equation, Painleve integrable burgers equation, Pochhammer Chree equation, /-expansion method, traveling wave solutions 1 Introduction The exact solutions of nonlinear evolution equations NLEEs play an important role in the study of nonlinear physical phenomena. One of the fundamental problems is to obtain their traveling wave solutions. Therefore, development of new methods to find exact traveling wave solutions of NLEEs still draw a lot of interest by a diverse group of researchers. In recent times, there has been remarkable progress in the development of new methods for exact traveling wave and solitary wave solutions and some of the established methods are: the Hirota s bilinear transformation method, [1] the homogeneous balance method, [2] the Backlund transform method, [3] the tanhfunction method, [4] the homotopy analysis method, [5 6] the Jacobi elliptic function method, [7] the F-expansion method, [8 9] the variational iteration method, [10] the Expfunction method, [11 12] the inverse scattering method, [13] and others. [14 16] Recently, Wang et al. [17] established a widely used direct and concise method called the /-expansion method for obtaining the traveling wave solutions of NLEEs, where ξ satisfies the second order linear ordinary differential equation ODE +λ +µ = 0. Applications of the /-expansion method can be found in the articles [18 21] for better comprehension. The main advantage of the /-expansion method over the other methods is that it gives more general solutions with some free parameters which, by suitable choice of the parameters, turn out to be some known solutions obtained by the existing methods. Moreover, i most of the methods provide solutions in a series form and it is essential to examine the convergence of the approximate series. For example, the Adomian decomposition method depends only on the initial conditions and obtains a solution in a series which converges to the exact solution of the problem, but, by the /-expansion method, one may obtain a general solution without approximation and thus does not need to test the convergence, and ii the solution procedure, using a computer algebra system, like Maple, is utter simple. In order to establish the effectiveness and reliability of the /-expansion method and to expand the possibility of its use, further research has been carried out by several researchers. For instance, uo and Zhou [22] presented the solutions in the form u = a 0 + n a i / i + b i / i 1 σ 1 + 1/µ / } i= and obtained new traveling wave solutions of the Whitham Broer Kaup-like equation and couple Hirota Satsuma KdV equations. Zayed and Al-Joudi [23] constructed the traveling wave solutions of some nonlinear evolution equations by making use of the extended /-expansion method. Naher et al. [24] also applied the extended method to the Caudrey Dodd ibbon equation and obtained new solutions. Zhang et al. [25] presented an improved /-expansion method to seek more general traveling wave solutions. Zayed and epreel [26] Corresponding author, ali math74@yahoo.com c 2014 Chinese Physical Society and IOP Publishing Ltd

2 670 Communications in Theoretical Physics Vol. 61 used the improved /-expansion method to some NLEEs to construct traveling wave solutions. Zayed [27] presented a new approach of the /-expansion method where ξ satisfies the Jacobi elliptic equation [ ξ] 2 = e 2 4 ξ+e 1 2 ξ+e 0 and obtained some new exact solutions for higher dimensional nonlinear evolution equations. Zayed [28] again presented a further alternative approach of this method in which ξ satisfies the Riccati equation ξ = A + B 2 ξ. Still substantial work has to be done in order for the /-expansion method to be well established, since every nonlinear equation has its own physically significant rich structure. Very recently, Akbar et al. [29] presented a generalized and improved /-expansion method and obtained abundant solutions of the KdV equation and the Zakharov Kuznetsov Benjamin Bona Mahony ZKBBM equations and strain waves equation in microstructured solids. In this article, we present an alternative approach of the generalized and improved /-expansion to obtain some new solutions to the Painleve integrable Burgers equation, the Boiti Leon Pempinelle equation and the Pochhammer Chree equations. The key idea of this approach is the traveling wave solutions of the NLEEs can be expressed by a polynomial in d + /, where ξ satisfies the Jacobi elliptic equation instead of the second order linear equation. 2 Description of the Method Let us consider the nonlinear partial differential equation of the form, Pu, u t, u x, u y, u tt, u t x, u t y, u xx, u x y, u y y,... = 0, 1 where u = ux, y, t is an unknown function, P is a polynomial in ux, y, t and its partial derivatives in which the highest order partial derivatives and the nonlinear terms are involved. The main steps of the generalized and improved /-expansion method combined with the Jacobi elliptic equation are as follows: Step 1 The traveling wave variable ux, y, t = uξ, ξ = x + y V t, 2 where V is the speed of the traveling wave, permits us to transform Eq. 1 into an ODE for u = uξ: Qu, u, u, u,... = 0, 3 where prime refers the ordinary derivative and Q is a polynomial in uξ and its derivatives. Step 2 If possible, integrate Eq. 3 term by term one or more times, yields constants of integration. For simplicity, the integration constants may be zero. Step 3 We assume the traveling wave solution of the ODE 3 can be expressed by a polynomial in d + / as follows: m α n uξ = d + / n, 4 n= m where either α m or α m may be zero, but both α m and α m can not be zero together, α n n = 0, ±1, ±2,..., ±m and d are constants to be determined later and = ξ satisfies the Jacobi elliptic equation, [ ] 2 = e e e 0, 5 where e 2, e 1, and e 0 are arbitrary constants. Step 4 To determine the positive integer m, substitute Eq. 4 along with Eq. 5 into Eq. 3 and then consider homogeneous balance between the highest order derivatives and the nonlinear terms appearing in Eq. 3. If the value of m does not become an integer, put u = v m ; this transforms Eq. 3 in which m becomes an integer. Step 5 Substituting Eq. 4 along with Eq. 5 into Eq. 3, we obtain polynomial in i j d + / m, i = 0, 1; j = 0, ±1, ±2, ±3,...; m = 0, 1, 2, 3,.... Collecting all coefficients of the same power and setting them equal to zero, yields a set of algebraic equations for α n, e 2, e 1, e 0, d, and V. Step 6 Suppose the value of the constants α n, e 2, e 1, e 0, d, and V can be obtained by solving the set of algebraic equations obtained in Step 3. Since the general solution of Eq. 5 is well known for us, substituting the values of α n, e 2, e 1, e 0, d, and V into Eq. 4, we obtain more general exact traveling wave solutions of the nonlinear evolution Eq Applications of the Method In this section, we utilize the proposed method to construct exact traveling wave solutions for the Painleve integrable Burgers equation, the Boiti Leon Pempinelle equation and the Pochhammer Chree equations, which are very important in mathematical physics. 3.1 Painleve Integrable Burgers Equation Let us consider the 2+1-dimensional Painleve integrable Burgers equation of the form: [19,30] u t = u u y + α v u x + β u y y + α β u x x, 6 u x = v y, 7 where the values of α and β are non-zero. This system of equations is developed from the generalized Painleve integrability classification. It has applications in various areas of applied mathematics, such as modelling of gas dynamics and traffic flow.

3 No. 6 Communications in Theoretical Physics 671 Using the traveling wave transformation Eq. 2; Eqs. 6 and 7 are converted into the following ODE: V u + u u + α v u + β 1 + αu = 0, 8 C + u v = 0, 9 where C is an integral constant. According to step 3, the solution of Eqs. 8 and 9 can be expressed by polynomial in d + / as follows: α m uξ = d + / m + α m 1 d + / m α α m 1 d + β n d + / n + β n 1 m 1 + αm d + m, 10 vξ = d + / n β β n 1 d + n 1 n, + βn d + 11 where α i i = 0, ±1, ±2,...,±m and β j j = 0, ±1, ±2,..., ±n are constants. Now substituting Eqs. 10 and 11 into Eqs. 8 and 9 respectively and considering the homogeneous balance between the highest order derivatives and the nonlinear terms, we obtain m = 1 and n = 1. Therefore, we can write the solution Eqs. 10 and 11 in the form: uξ = vξ = α 1 d + / + α 0 + α 1 β 1 d + / + β 0 + β 1 d + d +, Substituting Eqs. 12 and 13 along with Eq. 5 into Eqs. 8 and 9, the left hand sides are converted into polynomial in i j d + / m, i = 0, 1; j = 0, ±1, ±2,...; m = 0, 1, 2,.... Setting each coefficient of this polynomial to zero, we obtain a set of algebraic equations we will omit to display them for simplicity for α 1, α 0, α 1, β 1, β 0, β 1, d, e 2, e 1, e 0, C, and V. Solving this over-determined set of algebraic equations by using the symbolic computation systems, such as Maple, we obtain α 1 = 2 β, α 0 = α 0, α 1 = 0, β 1 = 2 β, β 0 = β 0, β 1 = 0, d = d, e 0 = 0, V = 2 β d1 + α α 0 α β 0, C = β 0 α 0, 14 where α 0, β 0, and d are free parameters. Substituting Eq. 14 into Eqs. 12 and 13 yields, uξ = α 0 2 β d +, 15 vξ = β 0 2 β d +, 16 where ξ = x + y 2 β d1 + α α 0 α β 0 }t. According to the Appendix see Zayed [27] for detail, we have the following clusters of exact solutions. Cluster 1: If e 0 = 0, e 1 = 1, and e 2 = 1, we obtain uξ = α 0 2 βd tanhξ, 17 vξ = β 0 2 βd tanhξ. 18 Cluster 2: If e 0 = 0, e 1 = 1, and e 2 = 1, we obtain uξ = α 0 2 βd cothξ, 19 vξ = β 0 2 βd cothξ. 20 Cluster 3: If e 0 = 0, e 1 = 1, and e 2 = 1, we obtain uξ = α 0 2 βd + tanξ, 21 vξ = β 0 2 βd + tanξ. 22 Cluster 4: If e 0 = 0, e 1 = 0, and e 2 = 1, we obtain uξ = α 0 2 β d 1, 23 ξ vξ = β 0 2 β d ξ Cluster 5: If e 0 = 0, e 1 = 1 + m 2, and e 2 = m 2, we obtain uξ = α 0 2 βd + cnξdsξ, 25 vξ = β 0 2 βd + cnξdsξ. 26 Further solutions for u and v can be found for other values of e 0, e 1, and e 2. The Painleve integreble Burgers Eqs. 6 and 7 were studied by Hong et al. [31] via variable separation scheme. Zayed [19] solved these equations by using the basic /-expansion method and obtained three types of solutions, like, the hyperbolic, the trigonometric and the rational function solutions, but in this article we obtain these three types of solutions and in addition, we obtain the elliptic function solution. More apparently, if we set λ 2 = µ in Zayed s [19] solution and d = λ/2 in our solution then Zayed s solutions 3.15 and 3.16 coincide with our solutions 17 and 18 when A = 0 and with 19 and 20 when B = 0. Again, if we set λ 2 = 4 µ 1 in Zayed s solution and d = λ/2 in our solution then Zayed s solutions 3.17 and 3.18 coincide with our solutions 21 and 22. On the other hand, if we set A = 0 and B = 1 in Zayed s solution and d = λ/2 in our solution then Zayed s solutions 3.19 and 3.20 are identical to our solutions 23 and 24. Zayed [19] did not obtain any further solution, but apart from these solutions we obtain elliptic function solutions 25 and Boiti Leon Pempinelle BLP Equation In this subsection, we construct the traveling wave solutions of the BLP equation by the alternative approach. Let us consider the BLP equation: u y t u 2 x y + u x x y 2 v x x x = v t v x x 2 u v x = 0. 28

4 672 Communications in Theoretical Physics Vol. 61 This system of equations has application in water waves. Boiti et al. [32] investigated solutions of this system by using Backlund transformation. The traveling wave transformation 2 permits us to reduce Eqs. 27 and 28 into the following ODE: 2v u + u 2 + V u = 0, 29 v + v + 2 u v = According to step 3, the solution structure of Eqs. 29 and 30 are similar to Eqs. 10 and 11. By homogeneous balance method, we get m = 1 and n = 1. Therefore, the form of solutions of Eqs. 29 and 30 are also the same as Eqs. 12 and 13. Substituting Eqs. 12 and 13 along with Eq. 5 into Eqs. 29 and 30, and executing the same procedure as described in Subsec. 3.1, we obtain an over-determined set of algebraic equations we will omit to display them for simplicity for α 1, α 0, α 1, β 1, β 0, β 1, d, e 2, e 1, e 0, C, V and solving this set of algebraic equations by using the symbolic computation systems, such as Maple, we obtain α 1 = 1, α 0 = α 0, α 1 = 0, β 1 = 1, β 0 = β 0, β 1 = 0, d = α 0 ± e 1, e 0 = 0, V = ±2 e 1, 31 where α 0 and β 0 are free parameters. Substituting Eq. 31 into Eqs. 12 and 13, yields uξ = α 0 d +, 32 vξ = β 0 d +, 33 where ξ = x + y 2 e 1 t and d = α 0 ± e 1. According to the Appendix, we write down only three clusters of exact solutions for Eqs. 27 and 28 as follows: Cluster 1: If e 0 = 0, e 1 = 1, and e 2 = 1, we obtain uξ = 1 + tanhξ, 34 vξ = β 0 α tanhξ, 35 where ξ = x + y 2 t. Cluster 2: If e 0 = 0, e 1 = 1, and e 2 = 1, we obtain uξ = i tanξ, 36 vξ = β 0 α 0 i tanξ, 37 where ξ = x + y 2it. Cluster 3: If e 0 = 0, e 1 = 1 + m 2, and e 2 = m 2, we obtain uξ = i 1 + m 2 cnξdsξ, 38 vξ = β 0 α 0 i 1 + m 2 cnξdsξ, 39 where ξ = x + y 2 i 1 + m 2 t. Similarly, we can write down the other clusters of exact solutions of Eqs. 27 and 28 which are omitted for convenience. The exact traveling wave solutions of BLP equation were investigated in [33 35]. In [33], Xiong and Xia constructed the exact solutions of the BLP equation by using the /-expansion method. If we set λ 2 = µ, k = 1, g t = 2, f y = 1, and f 1 y = β 0 α 0 1, then their solutions coincide with our solutions respectively. Xiong and Xia [33] obtained another solution, namely the rational function solution. In this article, we also obtain this solution but for simplicity we do not write down these solutions here. They did not find further solution, but in this article we obtain additional solution, namely the elliptic function solution Eqs. 38 and Pochhammer Chree PC Equation In this subsection, we construct the traveling wave solutions of the generalized PC equation by the proposed new approach. Let us consider the generalized PC equation: u t t u t t x x α u + β u 3 + γ u 5 x x = 0, 40 which describes the nonlinear model for longitudinal wave propagation of elastic rods. [36 37] The traveling wave variable, ξ = x V t allows us to reduce Eq. 40 into the following ODE: V 2 u V 2 u iv α u + β u 3 + γ u 5 = 0, 41 where u iv denotes the fourth derivative and u denotes the second derivative of u with respect to ξ. Equation 41 is integrable, therefore, integrating twice, we obtain V 2 αu V 2 u β u 3 γ u 5 = Balancing the highest order derivative u with the highest order nonlinear term u 5, we get m = 1/2. To get the closed form analytical solution, m should be an integer. According to Step 2, the transformation u = v 1/2 will fulfil our goal. Therefore, under this transformation, Eq. 42 becomes 4 V 2 αv 2 2 V 2 v v + V 2 v 2 4 β v 3 4 γ v 4 = Now balancing v v and v 4 gives m = 1. Therefore, the solution structure of Eq. 43 is the same as Eq. 13. Now substituting Eq. 13 along with Eq. 5 into Eq. 43, and completing the same process as described in Subsec. 3.1, we obtain an over-determined set of algebraic equations we will omit to display them for simplicity for β 1, β 0, β 1, d, e 2, e 1, e 0, V and solving this over-determined set of algebraic equations by using the symbolic computation system, such as Maple, we obtain β 1 = ± e1 9 β 2 32 αγ γ 2 + e 1, β 0 = β 0, β 1 = 0, d = 8 β 0γ + 3 β2 + e e1 9 β 2 32 αγ,

5 No. 6 Communications in Theoretical Physics 673 e 0 = 18 β4 e β αγ β 2 e β 4 e αγ β2 e α2 γ 2 e β 2 32 αγ 2, e 2 γ 2 + e1 32 αγ 9 β V = ± 2, 44 4 γ 2 + e 1 where β 0 is a free parameter. Substituting Eq. 44 into Eq. 13 yields vξ = 3 β γ ± e1 9 β 2 32 αγ 8 γ 2 + e 1, 45 where γ 2 + e1 32 αγ 9 β ξ = x 2 t. 4 γ 2 + e 1 Recall that u = v 1/2, therefore, we obtain uξ = 3 β γ ± e1 9 β 2 32 αγ } 1/ γ 2 + e 1 According to the Appendix, we have the subsequent clusters of exact solutions for Eq. 40 as follows. Cluster 1: If e 0 = 0, e 1 = 1, and e 2 = 1, then from Eq. 46 we obtain uξ = 3 β } 1/2 8 γ 1 ± tanhξ, 47 where ξ = x β 3 γ t, 4 γ γ < 0 and α = 0. Cluster 2: If e 0 = 0, e 1 = 1, and e 2 = 1, then from Eq. 46 we obtain uξ = 3 β } 1/2 8 γ 1 ± cothξ, 48 where ξ = x β 3 γ/4 γt, γ < 0 and α = 0. Cluster 3: If e 0 = 0, e 1 = 1, and e 2 = 1, from Eq. 46, we obtain uξ = 3 β γ ± β2 32 α γ 1/2, tanξ} 49 8 γ where ξ = x γ 32 α γ 9 β 2 /4 γ t, γ < 0, and 3 β 2 = 8 αγ. Cluster 4: If e 0 = 0, e 1 = 0, and e 2 = 1, from Eq. 46, we obtain uξ = 3 αγ ± 4 γ 1 } 1/2, 50 ξ where ξ = x α t, γ < 0, and β = 0. Cluster 5: If e 0 = 0, e 1 = 1 + m 2, and e 2 = m 2, we obtain uξ = 3 β γ ± m2 9 β 2 32 αγ 8 γ 1 m 2 where cnξdsξ} 1/2, β β 4 96 αγ β m β m αγ β m α 2 γ m 2 2 = 0, γ 1 m2 32 αγ 9 β ξ = x 2 4 γ 1 m 2 t and γ < 0. Other clusters of exact solutions of Eq. 40 are omitted here for convenience. It is interesting to point out that the real solutions are valid if γ < 0. However, if γ > 0, we obtain complex solutions. Li and Zhang [38] studied the bifurcation behaviour of space portraits for the corresponding traveling wave equation using the bifurcation theory of planar dynamical systems. Shawagfeh and Kaya [39] investigated solutions of the generalized PC equation by Adomian decomposition method. Hosseini et al. [40] obtained the analytical solutions of the PC equation by homotopy perturbation method. Zhang and Ma [41] gave some explicit solitary wave solutions of the PC equation by using the method of solving algebraic equation. Via PC equation Zayed [42] demonstrated that, the tanh-coth method is equivalent to the /-expansion method. Zuo [43] investigated solutions of the PC equation by using the extended /-expansion method and obtained the hyperbolic, the trigonometric and the rational function solutions. But in this article, we obtain the above mentioned three types of solutions as well as the elliptic function solutions n addition. For comparison, if we put n = 2 in Zuo s [43] solution then Zuo s solutions 17 and 18 are identical to our solutions 47 and 48 respectively. Again if we fix A 1 = 0 and A 2 = 1 in Zuo s [43] solution then our solution 50 coincides with Zuo s solutions 23 when β = 0. It is lucid that, our trigonometric solution Eq. 49 is distinct from the trigonometric solution obtained by Zuo [43] in Eq. 22. Zuo did not obtain any further solution, but apart from these solutions we obtain the elliptic function solutions 51. To the best of our knowledge the elliptic function solutions 51 is new and has not reported in the past literature. 4 Physical Explanations of Some Obtained Solutions Solitary waves can be obtained from each traveling wave solution by setting particular values to its unknown parameters. By adjusting these parameters, one can get an internal localized mode. In this section, we have presented some graphs of solitary waves constructed by taking suitable values of involved unknown parameters to visualize the underlying mechanism of the original equation. Using mathematical software Maple, three-dimensional

6 674 Communications in Theoretical Physics Vol. 61 plots of some obtained exact traveling wave solutions have been shown in Figs Painleve Integrable Burgers PIB Equations The obtained solutions for the PIB equations incorporate four types of explicit solutions namely hyperbolic, trigonometric, rational and elliptic function solutions. From these explicit results it is easy to say that Eqs. 17, 18 are kink solitons; Eqs. 19, 20 are singular kink solitons; Eqs. 21, 22 are plane periodic solutions; Eqs. 23, 24 are rational solutions and Eqs. 25, 26 are Jacobi elliptic function solutions. For particular values of the parameters α = 1, β = 1, α 0 = 0, β 0 = 1, d = 1, y = 0 within the interval 3 x, t 3, Eq. 17 gives kink soliton which rises or descends from one asymptotic state to another and approaches a constant at infinity. Therefore, Eq. 17 provides kink soliton which is localized traveling wave i.e., it depends on the auxiliary conditions that the dependent variable and its first, second, and higher spatial derivatives tend to zero as ξ ±. On the other hand, for particular values of the parameters α = 1, β = 1, α 0 = 2, β 0 = 1, d = 1, y = 0 within the interval 3 x, t 3, Eq. 21 gives periodic solution which is also traveling. For more convenience the graphical representations of Eq. 17 and Eq. 21 of PIB equations are shown in Figs. 1 and 2 respectively. 4.2 Boiti Leon Pempinelle BLP Equation From the obtained solutions of BPL equations we observe that Eqs. 34, 35 are kink solitons; Eqs. 36, 37 are complex soliton wave solutions and Eqs. 38, 39 are jacobi elliptic solutions. For particular values of the parameters, α 0 = I, β 0 = 0, y = 0 within the interval 1 x, t 1, Eq. 37 provides soliton profile of BLP equations which is represented in Fig. 3. Fig. 3 Soliton profile of BLP equations only shows the modulus plot of Eq. 37 for α 0 = I, β 0 = 0, y = 0 within the interval 1 x, t 1. Fig. 1 Kink soliton profile of PIB equations only shows the shape of Eq. 17 for α = 1, β = 1, α 0 = 0, β 0 = 1, d = 1, y = 0 within the interval 3 x, t 3. Fig. 4 Bell-shaped soliton profile of BLP equations only shows the modulus plot of Eq. 38 for m = 2I, y = 0 within the interval 1 x, t 1. Fig. 2 Periodic wave profile of PIB equations only shows the shape of Eq. 21 for α = 1, β = 1, α 0 = 2, β 0 = 1, d = 1, y = 0 within the interval 3 x, t 3. On the other hand the solution mode of Eqs. 38 and 39 varies for the different values of the modulus 0 k 1 of the corresponding elliptic functions. For

7 No. 6 Communications in Theoretical Physics 675 k = 1, the corresponding elliptic function Eqs. 38 and 39 give hyperbolic solutions and for k = 0 these give plane periodic solutions. Setting modulus of the related elliptic function k = 1, Fig. 4 corresponds bell-shaped soliton of Eq. 38 for m = 2I, y = 0 within the interval 1 x, t Pochhammer Chree PC Equation Fig. 5 Kink soliton profile of PC equations only shows the shape of Eq. 47 for β = 9, γ = 1, y = 0 within the interval 3 x, t 3. to periodic solution, Eq. 50 corresponds to rational solution and Eq. 51 corresponds to jacobi elliptic function solution. The graphical representations of Eqs. 47 and 49 of PC equations are shown in Figs. 5 and 6 respectively. Figure 5 represents kink soliton of PC equation for β = 9, γ = 1, y = 0 within the interval 3 x, t 3 only shows the shape of Eq. 47 and Fig. 6 represents plane periodic traveling wave solution of PC equation for α = 1, β = 1, γ = 1, y = 0 within the interval 5 x, t 5 only shows the shape of Eq Conclusion An alternative approach of the generalized and improved /-expansion method has been proposed in this article for finding new exact traveling wave solutions of NLEEs, where ξ satisfies the Jacobi elliptic equation instead of the second order linear ordinary differential equation. We have employed this method to the Painleve integrable Burgers equation, the Boiti Leon Pempinelle equation and the Pochhammer Chree equation and achieve exact traveling wave solutions in terms of hyperbolic, trigonometric, rational and elliptic functions which might be significant to describe physical phenomena. The results show that the proposed method is effective and a powerful new technique for nonlinear partial differential equations having wide applications in mathematical physics and engineering. The applied method will be used in further works to establish entirely new solutions for other kinds of nonlinear wave equations. Appendix The general solutions of the Jacobi elliptic Eq. 5 and its derivatives are listed in Table 1: Fig. 6 Periodic wave profile of PC equations only shows the modulus plot of Eq. 49 for α = 1, β = 1, γ = 1, y = 0 within the interval 5 x, t 5. Equations are the exact traveling wave solutions of PC equation with various unknown parameters. For particular values of these parameters the traveling wave solutions are instigated into solitary waves. From the obtained solutions of PC equation we can easily conclude that Eq. 47 corresponds to kink solution, Eq. 48 corresponds to singular kink solution, Eq. 49 corresponds Table 1 eneral solutions of the Jacobi elliptic equation see Zayed [27] are as follows. e 0 e 1 e 2 ξ ξ sech ξ sech ξ tanhξ cschξ cschξ cothξ secξ secξ tanξ /ξ 1/ξ m 2 m 2 snξ cnξ dnξ Further solutions for ξ can be found for other values of e 0, e 1 and e 2 which are omitted to display here for simplicity. References [1] R. Hirota, Phys. Rev. Lett [2] M.L. Wang, Phys. Lett. A [3] C. Rogers and W.F. Shadwick, Backlund Transformations, Academic Press, New York 1982.

8 676 Communications in Theoretical Physics Vol. 61 [4] W. Malfliet, American J. Phys [5] S.J. Liao, The Homotopy Analysis Method and Its Applications in Mechanics, Ph. D. Dissertation in English, Shanghai Jiao Tong University, Shanghai [6] J. Liao, A Kind of Linear Invariance under Homotopy and some Simple Applications of It in Mechanics, Bericht Nr. 520, Institut fuer Schifl bau der Universitaet Hamburg [7] S. Liu, Z. Fu, S. Liu, and Q. Zhao, Phys. Lett. A [8] Y.B. Zhou, M.L. Wang, and Y.M. Wang, Phys. Lett. A [9] M.L. Wang, Y.M. Wang, and J.L. Zhang, Chin. Phys [10] J.H. He, Commun. in Nonlinear Science and Numerical Simulation [11] H.J. He and X.H. Wu, Chaos, Solitons & Fractals [12] M.A. Akbar and N.H.M. Ali, Progress in Applied Mathematics [13] M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform, Cambridge University Press, Cambridge [14] J. Li and Y. Zhang, Mathematical Methods in the Applied Sciences 2010 DOI: /mma.1341 [15] A.R. Adem, C.M. Khalique, and A. Biswas, Mathematical Methods in the Applied Sciences 2010 DOI: /mma.1378 [16] I. Aslan, Mathematical Methods in the Applied Sciences 2011 DOI: /mma [17] M.L. Wang, X. Li, and J. Zhang, Phys. Lett. A [18] S. Mesloub and F. Mesloub, Mathematical Methods in the Applied Sciences 2010 DOI: /mma [19] E.M.E. Zayed, J. Applied Mathematics and Computing [20] J. Li and Y. Zhang, Mathematical Methods in the Applied Sciences 2010 DOI: /mma [21] J. Zhang, X. Wei, and Y. Lu, Phys. Lett. A [22] S. uo and Y. Zhou, Applied Mathematics Computations [23] E.M.E. Zayed and S. Al-Joudi, Mathematical Problems in Engineering, 2010 Article ID , 19 pages, DOI: /2010/ [24] H. Naher, F.A. Abdullah, and M.A. Akbar, Mathematical Problems in Engineering, Vol. 2011, 11 pages. DOI: /2011/ [25] J. Zhang, F. Jiang, and X. Zhao, International Journal of Computational Mathematics [26] E.M.E. Zayed and K.A. epreel, International Journal of Nonlinear Science & Numerical Simulation [27] E.M.E. Zayed, Journal of Physics A: Mathematical and Theoretical [28] E.M.E. Zayed, Journal of Applied Mathematics & Informatics [29] M.A. Akbar, N.H.M. Ali, and E.M.E. Zayed, Mathematical Problems in Engineering, Vol. 2012, 22 pages. doi: /2012/ [30] D.S. Wang and H. Li, Chaos, Solitons & Fractals [31] K.Z. Hong, B. Wu, and X.F. Chen, Commun. Theor. Phys [32] M. Boiti, J. Leon, and P. Pempinelle, Inverse Problems [33] S.Q. Xiong and T.C. Xia, Commun. Theor. Phys [34] D.J. Huang and H.Q. Zhang, Chaos, Solitons & Fractals [35] W.. Feng, K.M. Li, Y.Z. Li, and C. Lin, Communications in Nonlinear Science and Numerical Simulations [36] I.L. Bagolubasky, Computer Physics Communications [37] P.A. Clarkson, R.J. Leveque, and R. Saxton, Studies in Applied Mathematics [38] J. Li and L. Zhang, Chaos, Solitons & Fractals [39] N. Shawagfeh and D. Kaya, Computer and Mathematics with Applications [40] M.M. Hosseini, M.H. Heydari, and M. Moshmand, Middle-East Journal of Scientific Research [41] W. Zhang and M. Ma, Applied Mathematics and Mechanics [42] E.M.E. Zayed, AIP Conference Proceedings [43] M.J. Zuo, Applied Mathematics and Computations