Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2010, Article ID 498249, 10 pages doi:10.1155/2010/498249 Research Article Exact Solutions for the Generalized BBM Equation with Variable Coef?cients Cesar A. G ? omez1 and Alvaro H. Salas2, 3 1 Department of Mathematics, Universidad Nacional de Colombia, Bogot? a, Colombia 2 Department of Mathematics and Statistics, Universidad Nacional de Colombia, Manizales, Cll 45, Cra 30, Colombia 3 Department of Mathematics, Universidad de Caldas, Manizales, Colombia Correspondence should be addressed to Alvaro H. Salas, asalash2002@yahoo.com Received 23 November 2009; Accepted 21 January 2010 Academic Editor: Jihuan He Copyright q 2010 C. A. G? omez and A. H. Salas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The variational iteration algorithm combined with the exp-function method is suggested to solve the generalized Benjamin-Bona-Mahony equation BBM with variable coe?cients. Periodic and soliton solutions are formally derived in a general form. Some particular cases are considered. 1. Introduction The BBM equation ut uux ux ? μuxxt 0, 1.1 which describes approximately the unidirectional propagation of long waves in certain nonlinear dispersive systems, has been proposed by Benjamin et al. in 1972 1 as a more satisfactory model than the KdV equation 2 ut uux uxxx 0. 1.2 It is easy to see that 1.1 can be derived from the equal width EW-equation 3 : ut uux ? μuxxt 0, 1.3 by means of the change of variable u u 1, that is, by replacing u with u 1. This last equation is considered as an equally valid and accurate model for the same wave phenomena 2 Mathematical Problems in Engineering simulated by 1.1 and 1.2 . On the other hand, some researches analyzed the generalized KdV equation with variable coe?cients ut σ t up ux μ t uxxx 0, 1.4 because this model has important applications in several ?elds of science 4–7 . Motivated by these facts, we will consider here the generalized EW-equation with variable coe?cients ut σ t up ux ? μ t uxxt 0. 1.5 Using the solutions of 1.5 we obtain exact solutions to the generalized BBM equation ut σ t u 1 p ux ? μ t uxxt 0, 1.6 of order p > 0. 2. Exact Solutions to Generalized BBM Equation 2.1. The Variational Iteration Method Consider the following nonlinear equation: Lu x, t Nu x, t g x, t , 2.1 where L and N are linear and nonlinear operators, respectively, and g x, t is an inhomogeneous term. According to the variational iteration method VIM 8–14 , a functional correction to 2.1 is given by un 1 x, t un x, t t 0 θ τ Lun x, τ Nun x, τ ? g x, τ dτ, 2.2 where θ τ is a general Lagrange's multiplier, which can be identi?ed via the variational theory; the subscript n ≥ 0 denotes the nth order approximation and u is a restricted variation which means δu 0. In this method, we ?rst determine the Lagrange multiplier θ τ that will be identi?ed optimally via integration by parts. The successive approximation un 1 of the solution u will be readily obtained upon using the determined Lagrangian multiplier and any selective function u0. One of the advantages of the VIM, is the free choice of the initial solution u0 x, t . If we consider a special form to u0 with arbitrary parameters, using the relations un x, t un 1 x, t , ?k ?tk un x, t ?k ?tk un 1 x, t , 2.3 Mathematical Problems in Engineering 3 we can obtain a set of algebraic equations in the unknowns given by the parameters that appear in u0. Solving this system, we have exact solutions to 2.1 . To solve 1.5 , we construct the following functional equation un 1 x, t un x, t t 0 θ τ Lun x, τ Nun x, τ dτ, 2.4 where Lun x, τ un τ x, τ , Nun x, τ σ τ u 1 p ux x, τ ? μ τ uxxτ x, τ . 2.5 Taking in 2.4 variation with respect to the independent variable un, and noticing that δNun 0 we have δun 1 x, t δun x, t δ t 0 θ τ Lun x, τ Nun x, τ dτ δun x, t θ t δun x, t ? t 0 θ τ δun x, τ dτ 0. 2.6 This yields the stationary conditions 1 θ t 0, θ t 0. 2.7 Therefore, θ t ?1. 2.8 Substituting this value into 2.4 we obtain the formula un 1 x, t un x, t ? t 0 Lun x, τ Nun x, τ dτ. 2.9 Using the wave transformation ξ x λt ξ0, 2.10 4 Mathematical Problems in Engineering setting ? ?t u1 ξ ? ?t u0 ξ , 2.11 and performing one integration, 2.9 reduces to λu0 ξ σ t p 1 u p 1 0 ξ ? λμ t u0 ξ 0, 2.12 where for sake of simplicity we set the constant of integration equal to zero. With the change of variable u0 ξ v2/p ξ , 2.13 equation 2.12 converts to λv2 ξ ? 2μ t 2 ? p p2 λ v 2 ? 2μ t p λv ξ v ξ σ t p 1 v ξ 4 0. 2.14 Observe that if v ξ is a solution to 2.14 , then ?v ξ is also a solution to this equation. 2.2. The Exp-Function Method Recently, He and Wu 15 have introduced the Exp-function method to solve nonlinear di?erential equations. In particular, the Exp-function method is an e?ective method for solving nonlinear equations with high nonlinearity. The method has been used in a satisfactory way by other authors to solve a great variety of nonlinear wave equations 15– 21 . The Exp-function method is very simple and straightforward, and can be brie?y revised as follows: Given the nonlinear partial di?erential equation F u, ux, ut, uxx, uxt, utt, . . . 0, 2.15 it is transformed to ordinary di?erential equation F u, u , u , u , uxt, . . . 0, 2.16 by mean of wave transformation ξ x λt ξ0. Solutions to 2.16 can then be found using the expression u ξ d n ?c an exp nξ q n ?p bn exp nξ , 2.17 Mathematical Problems in Engineering 5 where c, d, p, and q are positive integers which are unknown to be determined later, an and bn are unknown constants. After balancing, we substitute 2.17 into 2.16 to obtain an algebraic systems in the variable ζ exp nξ . Solving the algebraic system we can obtain exact solutions to 2.16 and reversing, solutions to 2.15 in the original variables. 3. Solutions to 2.14 by the Exp-Function Method Using the Exp-function method, we suppose that solutions to 2.14 can be expressed in the form v ξ 1 n ?1 an exp nrξ 1 m ?1 bm exp mrξ a?1 exp ?rξ a0 a1 exp rξ b?1 exp ?rξ b0 b1 exp rξ . 3.1 We obtain following solutions to 2.14 : v1 ± 2λk p 1 p 2 2 p 1 p 2 λ exp p/2 μ t ξ ? k2σ t exp ? p/2 μ t ξ , λ λ t , v2 ± 2λk p 1 p 2 σ t k2 exp p/2 μ t ξ ? 2λ p 1 p 2 exp ? p/2 μ t ξ , λ λ t . 3.2 Some special solutions are obtained if λ λ t ± k2 2 p2 3p 2 σ t . 3.3 This choice gives solutions v3 ± k 2 csch p 2 μ t ξ , λ k2 2 p2 3p 2 σ t , 3.4 v4 k 2 sech p 2 μ t ξ , λ ? k2 2 p2 3p 2 σ t , 3.5 v5 ± k 2 csc p 2 μ t ξ , λ ? k2 2 p2 3p 2 σ t . 3.6 6 Mathematical Problems in Engineering Solution 3.6 follows from 3.4 with the identi?cations μ t → ?μ t and k → ?k √ ?1. v6 ? k 2 sec p 2 μ t ξ , λ ? k2 2 p2 3p 2 σ t . 3.7 Solution 3.7 follows from 3.4 with the identi?cations μ t → ?μ t and k → ?k. 4. Particular Cases 4.1. Case 1: Solutions to 2.14 When p 2 Equation 2.14 takes the form λv2 ξ ? λμ t v ξ v ξ 1 3 σ t v ξ 4 0. 4.1 From 3.2 with p 2: v7 ± 24λk 24λ exp 1/ μ t ξ ? k2σ t exp ? 1/ μ t ξ , v8 ± 24λk k2σ t exp 1/ μ t ξ ? 24λ exp ? 1/ μ t ξ . 4.2 From 3.3 – 3.7 with p 2: v9 ± k 2 csch 1 μ t ξ , λ k2 24 σ t , v10 ± k 2 sech 1 μ t ξ , λ ? k2 24 σ t , v11 ± k 2 csc 1 ?μ t ξ , λ ? k2 24 σ t , v12 ± k 2 sec 1 μ t ξ , λ ? k2 24 σ t . 4.3 Mathematical Problems in Engineering 7 Other exact solutions are: v13 ± 3a2 exp 2 ?2/μ t ξ 2 √ 55a exp ?2/μ t ξ ? 22 k 3a2 exp 2 ?2/μ t ξ 22a exp ?2/μ t ξ 22 , λ ? 1 3 k2 σ t , v14 ± 3a2 ± 2 √ 55a exp ?2/μ t ξ ? 22 exp 2 ?2/μ t ξ k 3a2 22a exp ?2/μ t ξ 22 exp 2 ?2/μ t ξ , λ ? 1 3 k2 σ t , v15 ±k ? ? ?1 ? 44 8 √ 55 3a 11 √ 55 exp ?2/μ t ξ 22 8 √ 55 ? ? ?, λ ? 1 3 k2 σ t , v16 ± k a ± sinh ?2/μ t ξ √ a2 1 ± cosh ?2/μ t ξ , λ ? 1 3 k2 σ t , v17 ± k a ± cosh ?2/μ t ξ √ a2 1 ± sinh ?2/μ t ξ , λ ? 1 3 k2 σ t , v18 ± k cos 2/μ t ξ 1 ± sin 2/μ t ξ , λ k2 3 σ t . 4.4 4.2. Case 2: Solutions to 2.14 When p 4 Equation 2.14 takes the form λv2 ξ μ t λ v 2 ? λ 2 μ t v ξ v ξ 1 5 σ t v ξ 4 0. 4.5 From 3.2 with p 4: v19 ± 60λk 60λ exp 2/ μ t ξ ? k2σ t exp ? 2/ μ t ξ , λ λ t , v20 ± 60λk σ t k2 exp 2/ μ t ξ ? 60λ exp ? 2/ μ t ξ , λ λ t . 4.6 8 Mathematical Problems in Engineering From 3.3 – 3.7 with p 4: v21 ± k 2 csch 2 μ t ξ , λ k2 60 σ t , v22 ± k 2 sech 2 μ t ξ , λ ? k2 60 σ t , v23 ± k 2 csc 2 ?μ t ξ , λ ? k2 60 σ t , v24 ± k 2 sec 2 ?μ t ξ , λ ? k2 60 σ t . 4.7 Other exact solutions are: v25 ± k a exp 2/ ?μ t ξ ? 4 2 a2 exp 4/ ?μ t ξ 16a exp 2/ ?μ t ξ 16 , λ ? 1 5 k2 σ t , v26 ± k 4 exp 2/ ?μ t ξ ? a 2 a2 16a exp 2/ ?μ t ξ 16 exp 4/ ?μ t ξ , λ ? 1 5 k2 σ t , v27 ±2k ? ? ?1 ? 3 2 ± cos 2/ μ t ξ ? ? ?, λ ? 4 5 k2 σ t , v28 ±2k ? ? ?1 ? 3 2 ± sin 2/ μ t ξ ? ? ?, λ ? 4 5 k2 σ t . 4.8 It is clear that using 2.13 we obtain solutions to 1.5 . Finally, observe that if u0 x, t is a solution of 1.5 , then the solutions u x, t to the generalized BBM equation 1.6 are obtained as follows: u x, t u0 x, t ? 1. 4.9 5. Conclusions We have considered the generalized EW-equation with variable coe?cients and the generalized BBM-equation with variable coe?cients. We obtained analytic solutions by using the variational iteration method combined with the exp-function method. With the aid of Mathematica we have derived a lot of di?erent types of solutions for these two models. Combined formal soliton-like solutions as well as kink solutions have been formally derived. Mathematical Problems in Engineering 9 The results obtained show that the technique used here can be considered as a powerful method to analyze other types of nonlinear wave equations. According to 22 , there are alternative iteration alorithms, which might be useful for future work. Furthermore, various modi?cations of the exp-function method have been appeared in open literature, for example, the double exp-function method 23, 24 . Other methods for solving nonlinear di?erential equations may be found in 25–35 . We think that the results presented in this paper are new in the literature. References 1 T. B. Benjamin, J. L. Bona, and J. J. Mahony, "Model equations for long waves in nonlinear dispersive systems," Philosophical Transactions of the Royal Society of London. Series A, vol. 272, no. 1220, pp. 47–78, 1972. 2 D. J. Korteweg and G. de Vries, "On the change of long waves advancing in a rectangular canal and a new type of long stationary wave," Philosophical Magazine, vol. 39, pp. 422–443, 1835. 3 P. J. Morrison, J. D. Meiss, and J. R. Cary, "Scattering of regularized-long-wave solitary waves," Physica D, vol. 11, no. 3, pp. 324–336, 1984. 4 R. M. Miura, "Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation," Journal of Mathematical Physics, vol. 9, pp. 1202–1204, 1968. 5 N. Nirmala, M. J. Vedan, and B. V. Baby, "Auto-B¨ acklund transformation, Lax pairs, and Painlev? e property of a variable coe?cient Korteweg-de Vries equation. I," Journal of Mathematical Physics, vol. 27, no. 11, pp. 2640–2643, 1986. 6 Z. Liu and C. Yang, "The application of bifurcation method to a higher-order KdV equation," Journal of Mathematical Analysis and Applications, vol. 275, no. 1, pp. 1–12, 2002. 7 Y. Zhang, S. Lai, J. Yin, and Y. Wu, "The application of the auxiliary equation technique to a generalized mKdV equation with variable coe?cients," Journal of Computational and Applied Mathematics, vol. 223, no. 1, pp. 75–85, 2009. 8 J.-H. He, "Variational iteration method for autonomous ordinary di?erential systems," Applied Mathematics and Computation, vol. 114, no. 2-3, pp. 115–123, 2000. 9 J.-H. He, "Homotopy perturbation method: a new nonlinear analytical technique," Applied Mathematics and Computation, vol. 135, no. 1, pp. 73–79, 2003. 10 J.-H. He, "The homotopy perturbation method nonlinear oscillators with discontinuities," Applied Mathematics and Computation, vol. 151, no. 1, pp. 287–292, 2004. 11 A.-M. Wazwaz, "The variational iteration method for rational solutions for KdV, K 2, 2 , Burgers, and cubic Boussinesq equations," Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 18–23, 2007. 12 E. Yusufoglu and A. Bekir, "The variational iteration method for solitary patterns solutions of gBBM equation," Physics Letters. A, vol. 367, no. 6, pp. 461–464, 2007. 13 J.-M. Zhu, Z.-M. Lu, and Y.-L. Liu, "Doubly periodic wave solutions of Jaulent-Miodek equations using variational iteration method combined with Jacobian-function method," Communications in Theoretical Physics, vol. 49, no. 6, pp. 1403–1406, 2008. 14 C. A. G? omez and A. H. Salas, "The variational iteration method combined with improved generalized tanh-coth method applied to Sawada-Kotera equation," Applied Mathematics and Computation, 2009. In press. 15 J.-H. He and X.-H. Wu, "Exp-function method for nonlinear wave equations," Chaos, Solitons and Fractals, vol. 30, no. 3, pp. 700–708, 2006. 16 S. Zhang, "Exp-function method exactly solving the KdV equation with forcing term," Applied Mathematics and Computation, vol. 197, no. 1, pp. 128–134, 2008. 17 J.-H. He and L.-N. Zhang, "Generalized solitary solution and compacton-like solution of the Jaulent- Miodek equations using the Exp-function method," Physics Letters. A, vol. 372, no. 7, pp. 1044–1047, 2008. 18 S. Zhang, "Application of Exp-function method to a KdV equation with variable coe?cients," Physics Letters. A, vol. 365, no. 5-6, pp. 448–453, 2007. 19 A. H. Salas, "Exact solutions for the general ?fth KdV equation by the exp function method," Applied Mathematics and Computation, vol. 205, no. 1, pp. 291–297, 2008. 10 Mathematical Problems in Engineering 20 A. H. Salas, C. A. G? omez, and J. E. Castillo Herna? ndez, "New abundant solutions for the Burgers equation," Computers & Mathematics with Applications, vol. 58, no. 3, pp. 514–520, 2009. 21 S. Zhang, "Exp-function method: solitary, periodic and rational wave solutions of nonlinear evolution equations," Nonlinear Science Letters A, vol. 1, pp. 143–146, 2010. 22 J.-H. He, G.-C. Wu, and F. Austin, "The variational iteration method which should be followed," Nonlinear Science Letters A, vol. 1.1, pp. 1–30, 2010. 23 Z.-D. Dai, C.-J. Wang, S.-Q. Lin, D.-L. Li, and G. Mu, "The three-wave method for nonlinear evolution equations," Nonlinear Science Letters A, vol. 1, no. 1, pp. 77–82, 2010. 24 H.-M. Fu and Z.-D. Dai, "Double Exp-function method and application," International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 7, pp. 927–933, 2009. 25 A. H. Salas, "Some solutions for a type of generalized Sawada-Kotera equation," Applied Mathematics and Computation, vol. 196, no. 2, pp. 812–817, 2008. 26 A. H. Salas and C. A. G? omez, "Computing exact solutions for some ?fth KdV equations with forcing term," Applied Mathematics and Computation, vol. 204, no. 1, pp. 257–260, 2008. 27 A. H. Salas, C. A. G? omez, and J. G. Escobar, "Exact solutions for the general ?fth order KdV equation by the extended tanh method," Journal of Mathematical Sciences: Advances and Applications, vol. 1, no. 2, pp. 305–310, 2008. 28 A. H. Salas and C. A. G? omez, "A practical approach to solve coupled systems of nonlinear PDE's," Journal of Mathematical Sciences: Advances and Applications, vol. 3, no. 1, pp. 101–107, 2009. 29 A. H. Salas and C. A G? omez, "El software Mathematica en la b? usqueda de soluciones exactas de ecuaciones diferenciales no lineales en derivadas parciales mediante el uso de la ecuaci? on de Riccati," in Memorias del Primer Seminario Internacional de Tecnolog? ?as en Educaci? on Matem? atica, vol. 1, pp. 379– 387, Universidad Pedag? ogica Nacional, Santaf? e de Bogot? a, Colombia, 2005. 30 A. H. Salas, H. J. Castillo, and J. G. Escobar, "About the seventh-order Kaup-Kupershmidt equation and its solutions," September 2008, http://arxiv.org, arXiv:0809.2865. 31 A. H. Salas and J. G. Escobar, "New solutions for the modi?ed generalized Degasperis-Procesi equation," September 2008, http://arxiv.org/abs/0809.2864. 32 A. H. Salas, "Two standard methods for solving the Ito equation," May 2008, http://arxiv.org/abs/0805.3362. 33 A. H. Salas, "Some exact solutions for the Caudrey-Dodd-Gibbon equation," May 2008, http://arxiv.org/abs/0805.3362. 34 A. H. Salas S and C. A. G? omez S, "Exact solutions for a third-order KdV equation with variable coe?cients and forcing term," Mathematical Problems in Engineering, vol. 2009, Article ID 737928, 13 pages, 2009. 35 C. A. G? omez S and A. H. Salas, "The Cole-Hopf transformation and improved tanh-coth method applied to new integrable system KdV6 ," Applied Mathematics and Computation, vol. 204, no. 2, pp. 957–962, 2008. Submit your manuscripts at http://www.hindawi.com
下载该文档
文档格式：PDF
更新时间：2014-08-11
下载次数：0
点击次数：1
Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2010, Article ID 498249, 10 pages doi:10.1155/2010/498249 Research Article Exact Solutions for the Generalized BBM Equation with Variable Coef?cients Cesar A. G ? omez1 and Alvaro H. Salas2, 3 1 Department of Mathematics, Universidad Nacional de Colombia, Bogot? a, Colombia 2 Department of Mathematics and Statistics, Universidad Nacional de Colombia, Manizales, Cll 45, Cra 30, Colombia 3 Department of Mathematics, Universidad de Caldas, Manizales, Colombia Correspondence should be addressed to Alvaro H. Salas, asalash2002@yahoo.com Received 23 November 2009; Accepted 21 January 2010 Academic Editor: Jihuan He Copyright q 2010 C. A. G? omez and A. H. Salas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The variational iteration algorithm combined with the exp-function method is suggested to solve the generalized Benjamin-Bona-Mahony equation BBM with variable coe?cients. Periodic and soliton solutions are formally derived in a general form. Some particular cases are considered. 1. Introduction The BBM equation ut uux ux ? μuxxt 0, 1.1 which describes approximately the unidirectional propagation of long waves in certain nonlinear dispersive systems, has been proposed by Benjamin et al. in 1972 1 as a more satisfactory model than the KdV equation 2 ut uux uxxx 0. 1.2 It is easy to see that 1.1 can be derived from the equal width EW-equation 3 : ut uux ? μuxxt 0, 1.3 by means of the change of variable u u 1, that is, by replacing u with u 1. This last equation is considered as an equally valid and accurate model for the same wave phenomena 2 Mathematical Problems in Engineering simulated by 1.1 and 1.2 . On the other hand, some researches analyzed the generalized KdV equation with variable coe?cients ut σ t up ux μ t uxxx 0, 1.4 because this model has important applications in several ?elds of science 4–7 . Motivated by these facts, we will consider here the generalized EW-equation with variable coe?cients ut σ t up ux ? μ t uxxt 0. 1.5 Using the solutions of 1.5 we obtain exact solutions to the generalized BBM equation ut σ t u 1 p ux ? μ t uxxt 0, 1.6 of order p > 0. 2. Exact Solutions to Generalized BBM Equation 2.1. The Variational Iteration Method Consider the following nonlinear equation: Lu x, t Nu x, t g x, t , 2.1 where L and N are linear and nonlinear operators, respectively, and g x, t is an inhomogeneous term. According to the variational iteration method VIM 8–14 , a functional correction to 2.1 is given by un 1 x, t un x, t t 0 θ τ Lun x, τ Nun x, τ ? g x, τ dτ, 2.2 where θ τ is a general Lagrange's multiplier, which can be identi?ed via the variational theory; the subscript n ≥ 0 denotes the nth order approximation and u is a restricted variation which means δu 0. In this method, we ?rst determine the Lagrange multiplier θ τ that will be identi?ed optimally via integration by parts. The successive approximation un 1 of the solution u will be readily obtained upon using the determined Lagrangian multiplier and any selective function u0. One of the advantages of the VIM, is the free choice of the initial solution u0 x, t . If we consider a special form to u0 with arbitrary parameters, using the relations un x, t un 1 x, t , ?k ?tk un x, t ?k ?tk un 1 x, t , 2.3 Mathematical Problems in Engineering 3 we can obtain a set of algebraic equations in the unknowns given by the parameters that appear in u0. Solving this system, we have exact solutions to 2.1 . To solve 1.5 , we construct the following functional equation un 1 x, t un x, t t 0 θ τ Lun x, τ Nun x, τ dτ, 2.4 where Lun x, τ un τ x, τ , Nun x, τ σ τ u 1 p ux x, τ ? μ τ uxxτ x, τ . 2.5 Taking in 2.4 variation with respect to the independent variable un, and noticing that δNun 0 we have δun 1 x, t δun x, t δ t 0 θ τ Lun x, τ Nun x, τ dτ δun x, t θ t δun x, t ? t 0 θ τ δun x, τ dτ 0. 2.6 This yields the stationary conditions 1 θ t 0, θ t 0. 2.7 Therefore, θ t ?1. 2.8 Substituting this value into 2.4 we obtain the formula un 1 x, t un x, t ? t 0 Lun x, τ Nun x, τ dτ. 2.9 Using the wave transformation ξ x λt ξ0, 2.10 4 Mathematical Problems in Engineering setting ? ?t u1 ξ ? ?t u0 ξ , 2.11 and performing one integration, 2.9 reduces to λu0 ξ σ t p 1 u p 1 0 ξ ? λμ t u0 ξ 0, 2.12 where for sake of simplicity we set the constant of integration equal to zero. With the change of variable u0 ξ v2/p ξ , 2.13 equation 2.12 converts to λv2 ξ ? 2μ t 2 ? p p2 λ v 2 ? 2μ t p λv ξ v ξ σ t p 1 v ξ 4 0. 2.14 Observe that if v ξ is a solution to 2.14 , then ?v ξ is also a solution to this equation. 2.2. The Exp-Function Method Recently, He and Wu 15 have introduced the Exp-function method to solve nonlinear di?erential equations. In particular, the Exp-function method is an e?ective method for solving nonlinear equations with high nonlinearity. The method has been used in a satisfactory way by other authors to solve a great variety of nonlinear wave equations 15– 21 . The Exp-function method is very simple and straightforward, and can be brie?y revised as follows: Given the nonlinear partial di?erential equation F u, ux, ut, uxx, uxt, utt, . . . 0, 2.15 it is transformed to ordinary di?erential equation F u, u , u , u , uxt, . . . 0, 2.16 by mean of wave transformation ξ x λt ξ0. Solutions to 2.16 can then be found using the expression u ξ d n ?c an exp nξ q n ?p bn exp nξ , 2.17 Mathematical Problems in Engineering 5 where c, d, p, and q are positive integers which are unknown to be determined later, an and bn are unknown constants. After balancing, we substitute 2.17 into 2.16 to obtain an algebraic systems in the variable ζ exp nξ . Solving the algebraic system we can obtain exact solutions to 2.16 and reversing, solutions to 2.15 in the original variables. 3. Solutions to 2.14 by the Exp-Function Method Using the Exp-function method, we suppose that solutions to 2.14 can be expressed in the form v ξ 1 n ?1 an exp nrξ 1 m ?1 bm exp mrξ a?1 exp ?rξ a0 a1 exp rξ b?1 exp ?rξ b0 b1 exp rξ . 3.1 We obtain following solutions to 2.14 : v1 ± 2λk p 1 p 2 2 p 1 p 2 λ exp p/2 μ t ξ ? k2σ t exp ? p/2 μ t ξ , λ λ t , v2 ± 2λk p 1 p 2 σ t k2 exp p/2 μ t ξ ? 2λ p 1 p 2 exp ? p/2 μ t ξ , λ λ t . 3.2 Some special solutions are obtained if λ λ t ± k2 2 p2 3p 2 σ t . 3.3 This choice gives solutions v3 ± k 2 csch p 2 μ t ξ , λ k2 2 p2 3p 2 σ t , 3.4 v4 k 2 sech p 2 μ t ξ , λ ? k2 2 p2 3p 2 σ t , 3.5 v5 ± k 2 csc p 2 μ t ξ , λ ? k2 2 p2 3p 2 σ t . 3.6 6 Mathematical Problems in Engineering Solution 3.6 follows from 3.4 with the identi?cations μ t → ?μ t and k → ?k √ ?1. v6 ? k 2 sec p 2 μ t ξ , λ ? k2 2 p2 3p 2 σ t . 3.7 Solution 3.7 follows from 3.4 with the identi?cations μ t → ?μ t and k → ?k. 4. Particular Cases 4.1. Case 1: Solutions to 2.14 When p 2 Equation 2.14 takes the form λv2 ξ ? λμ t v ξ v ξ 1 3 σ t v ξ 4 0. 4.1 From 3.2 with p 2: v7 ± 24λk 24λ exp 1/ μ t ξ ? k2σ t exp ? 1/ μ t ξ , v8 ± 24λk k2σ t exp 1/ μ t ξ ? 24λ exp ? 1/ μ t ξ . 4.2 From 3.3 – 3.7 with p 2: v9 ± k 2 csch 1 μ t ξ , λ k2 24 σ t , v10 ± k 2 sech 1 μ t ξ , λ ? k2 24 σ t , v11 ± k 2 csc 1 ?μ t ξ , λ ? k2 24 σ t , v12 ± k 2 sec 1 μ t ξ , λ ? k2 24 σ t . 4.3 Mathematical Problems in Engineering 7 Other exact solutions are: v13 ± 3a2 exp 2 ?2/μ t ξ 2 √ 55a exp ?2/μ t ξ ? 22 k 3a2 exp 2 ?2/μ t ξ 22a exp ?2/μ t ξ 22 , λ ? 1 3 k2 σ t , v14 ± 3a2 ± 2 √ 55a exp ?2/μ t ξ ? 22 exp 2 ?2/μ t ξ k 3a2 22a exp ?2/μ t ξ 22 exp 2 ?2/μ t ξ , λ ? 1 3 k2 σ t , v15 ±k ? ? ?1 ? 44 8 √ 55 3a 11 √ 55 exp ?2/μ t ξ 22 8 √ 55 ? ? ?, λ ? 1 3 k2 σ t , v16 ± k a ± sinh ?2/μ t ξ √ a2 1 ± cosh ?2/μ t ξ , λ ? 1 3 k2 σ t , v17 ± k a ± cosh ?2/μ t ξ √ a2 1 ± sinh ?2/μ t ξ , λ ? 1 3 k2 σ t , v18 ± k cos 2/μ t ξ 1 ± sin 2/μ t ξ , λ k2 3 σ t . 4.4 4.2. Case 2: Solutions to 2.14 When p 4 Equation 2.14 takes the form λv2 ξ μ t λ v 2 ? λ 2 μ t v ξ v ξ 1 5 σ t v ξ 4 0. 4.5 From 3.2 with p 4: v19 ± 60λk 60λ exp 2/ μ t ξ ? k2σ t exp ? 2/ μ t ξ , λ λ t , v20 ± 60λk σ t k2 exp 2/ μ t ξ ? 60λ exp ? 2/ μ t ξ , λ λ t . 4.6 8 Mathematical Problems in Engineering From 3.3 – 3.7 with p 4: v21 ± k 2 csch 2 μ t ξ , λ k2 60 σ t , v22 ± k 2 sech 2 μ t ξ , λ ? k2 60 σ t , v23 ± k 2 csc 2 ?μ t ξ , λ ? k2 60 σ t , v24 ± k 2 sec 2 ?μ t ξ , λ ? k2 60 σ t . 4.7 Other exact solutions are: v25 ± k a exp 2/ ?μ t ξ ? 4 2 a2 exp 4/ ?μ t ξ 16a exp 2/ ?μ t ξ 16 , λ ? 1 5 k2 σ t , v26 ± k 4 exp 2/ ?μ t ξ ? a 2 a2 16a exp 2/ ?μ t ξ 16 exp 4/ ?μ t ξ , λ ? 1 5 k2 σ t , v27 ±2k ? ? ?1 ? 3 2 ± cos 2/ μ t ξ ? ? ?, λ ? 4 5 k2 σ t , v28 ±2k ? ? ?1 ? 3 2 ± sin 2/ μ t ξ ? ? ?, λ ? 4 5 k2 σ t . 4.8 It is clear that using 2.13 we obtain solutions to 1.5 . Finally, observe that if u0 x, t is a solution of 1.5 , then the solutions u x, t to the generalized BBM equation 1.6 are obtained as follows: u x, t u0 x, t ? 1. 4.9 5. Conclusions We have considered the generalized EW-equation with variable coe?cients and the generalized BBM-equation with variable coe?cients. We obtained analytic solutions by using the variational iteration method combined with the exp-function method. With the aid of Mathematica we have derived a lot of di?erent types of solutions for these two models. Combined formal soliton-like solutions as well as kink solutions have been formally derived. Mathematical Problems in Engineering 9 The results obtained show that the technique used here can be considered as a powerful method to analyze other types of nonlinear wave equations. According to 22 , there are alternative iteration alorithms, which might be useful for future work. Furthermore, various modi?cations of the exp-function method have been appeared in open literature, for example, the double exp-function method 23, 24 . Other methods for solving nonlinear di?erential equations may be found in 25–35 . We think that the results presented in this paper are new in the literature. References 1 T. B. Benjamin, J. L. Bona, and J. J. Mahony, "Model equations for long waves in nonlinear dispersive systems," Philosophical Transactions of the Royal Society of London. Series A, vol. 272, no. 1220, pp. 47–78, 1972. 2 D. J. Korteweg and G. de Vries, "On the change of long waves advancing in a rectangular canal and a new type of long stationary wave," Philosophical Magazine, vol. 39, pp. 422–443, 1835. 3 P. J. Morrison, J. D. Meiss, and J. R. Cary, "Scattering of regularized-long-wave solitary waves," Physica D, vol. 11, no. 3, pp. 324–336, 1984. 4 R. M. Miura, "Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation," Journal of Mathematical Physics, vol. 9, pp. 1202–1204, 1968. 5 N. Nirmala, M. J. Vedan, and B. V. Baby, "Auto-B¨ acklund transformation, Lax pairs, and Painlev? e property of a variable coe?cient Korteweg-de Vries equation. I," Journal of Mathematical Physics, vol. 27, no. 11, pp. 2640–2643, 1986. 6 Z. Liu and C. Yang, "The application of bifurcation method to a higher-order KdV equation," Journal of Mathematical Analysis and Applications, vol. 275, no. 1, pp. 1–12, 2002. 7 Y. Zhang, S. Lai, J. Yin, and Y. Wu, "The application of the auxiliary equation technique to a generalized mKdV equation with variable coe?cients," Journal of Computational and Applied Mathematics, vol. 223, no. 1, pp. 75–85, 2009. 8 J.-H. He, "Variational iteration method for autonomous ordinary di?erential systems," Applied Mathematics and Computation, vol. 114, no. 2-3, pp. 115–123, 2000. 9 J.-H. He, "Homotopy perturbation method: a new nonlinear analytical technique," Applied Mathematics and Computation, vol. 135, no. 1, pp. 73–79, 2003. 10 J.-H. He, "The homotopy perturbation method nonlinear oscillators with discontinuities," Applied Mathematics and Computation, vol. 151, no. 1, pp. 287–292, 2004. 11 A.-M. Wazwaz, "The variational iteration method for rational solutions for KdV, K 2, 2 , Burgers, and cubic Boussinesq equations," Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 18–23, 2007. 12 E. Yusufoglu and A. Bekir, "The variational iteration method for solitary patterns solutions of gBBM equation," Physics Letters. A, vol. 367, no. 6, pp. 461–464, 2007. 13 J.-M. Zhu, Z.-M. Lu, and Y.-L. Liu, "Doubly periodic wave solutions of Jaulent-Miodek equations using variational iteration method combined with Jacobian-function method," Communications in Theoretical Physics, vol. 49, no. 6, pp. 1403–1406, 2008. 14 C. A. G? omez and A. H. Salas, "The variational iteration method combined with improved generalized tanh-coth method applied to Sawada-Kotera equation," Applied Mathematics and Computation, 2009. In press. 15 J.-H. He and X.-H. Wu, "Exp-function method for nonlinear wave equations," Chaos, Solitons and Fractals, vol. 30, no. 3, pp. 700–708, 2006. 16 S. Zhang, "Exp-function method exactly solving the KdV equation with forcing term," Applied Mathematics and Computation, vol. 197, no. 1, pp. 128–134, 2008. 17 J.-H. He and L.-N. Zhang, "Generalized solitary solution and compacton-like solution of the Jaulent- Miodek equations using the Exp-function method," Physics Letters. A, vol. 372, no. 7, pp. 1044–1047, 2008. 18 S. Zhang, "Application of Exp-function method to a KdV equation with variable coe?cients," Physics Letters. A, vol. 365, no. 5-6, pp. 448–453, 2007. 19 A. H. Salas, "Exact solutions for the general ?fth KdV equation by the exp function method," Applied Mathematics and Computation, vol. 205, no. 1, pp. 291–297, 2008. 10 Mathematical Problems in Engineering 20 A. H. Salas, C. A. G? omez, and J. E. Castillo Herna? ndez, "New abundant solutions for the Burgers equation," Computers & Mathematics with Applications, vol. 58, no. 3, pp. 514–520, 2009. 21 S. Zhang, "Exp-function method: solitary, periodic and rational wave solutions of nonlinear evolution equations," Nonlinear Science Letters A, vol. 1, pp. 143–146, 2010. 22 J.-H. He, G.-C. Wu, and F. Austin, "The variational iteration method which should be followed," Nonlinear Science Letters A, vol. 1.1, pp. 1–30, 2010. 23 Z.-D. Dai, C.-J. Wang, S.-Q. Lin, D.-L. Li, and G. Mu, "The three-wave method for nonlinear evolution equations," Nonlinear Science Letters A, vol. 1, no. 1, pp. 77–82, 2010. 24 H.-M. Fu and Z.-D. Dai, "Double Exp-function method and application," International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 7, pp. 927–933, 2009. 25 A. H. Salas, "Some solutions for a type of generalized Sawada-Kotera equation," Applied Mathematics and Computation, vol. 196, no. 2, pp. 812–817, 2008. 26 A. H. Salas and C. A. G? omez, "Computing exact solutions for some ?fth KdV equations with forcing term," Applied Mathematics and Computation, vol. 204, no. 1, pp. 257–260, 2008. 27 A. H. Salas, C. A. G? omez, and J. G. Escobar, "Exact solutions for the general ?fth order KdV equation by the extended tanh method," Journal of Mathematical Sciences: Advances and Applications, vol. 1, no. 2, pp. 305–310, 2008. 28 A. H. Salas and C. A. G? omez, "A practical approach to solve coupled systems of nonlinear PDE's," Journal of Mathematical Sciences: Advances and Applications, vol. 3, no. 1, pp. 101–107, 2009. 29 A. H. Salas and C. A G? omez, "El software Mathematica en la b? usqueda de soluciones exactas de ecuaciones diferenciales no lineales en derivadas parciales mediante el uso de la ecuaci? on de Riccati," in Memorias del Primer Seminario Internacional de Tecnolog? ?as en Educaci? on Matem? atica, vol. 1, pp. 379– 387, Universidad Pedag? ogica Nacional, Santaf? e de Bogot? a, Colombia, 2005. 30 A. H. Salas, H. J. Castillo, and J. G. Escobar, "About the seventh-order Kaup-Kupershmidt equation and its solutions," September 2008, http://arxiv.org, arXiv:0809.2865. 31 A. H. Salas and J. G. Escobar, "New solutions for the modi?ed generalized Degasperis-Procesi equation," September 2008, http://arxiv.org/abs/0809.2864. 32 A. H. Salas, "Two standard methods for solving the Ito equation," May 2008, http://arxiv.org/abs/0805.3362. 33 A. H. Salas, "Some exact solutions for the Caudrey-Dodd-Gibbon equation," May 2008, http://arxiv.org/abs/0805.3362. 34 A. H. Salas S and C. A. G? omez S, "Exact solutions for a third-order KdV equation with variable coe?cients and forcing term," Mathematical Problems in Engineering, vol. 2009, Article ID 737928, 13 pages, 2009. 35 C. A. G? omez S and A. H. Salas, "The Cole-Hopf transformation and improved tanh-coth method applied to new integrable system KdV6 ," Applied Mathematics and Computation, vol. 204, no. 2, pp. 957–962, 2008. Submit your manuscripts at http://www.hindawi.com