Exp-function method and its application to nonlinear equations. (English) Zbl 1153.35384

Wu, Xu-Hong (Benn); He, Ji-Huan

Chaos Solitons Fractals 38, No. 3, 903-910 (2008).

Summary: Exp-function method is used to find a unified solution of a nonlinear wave equation. Variant Boussinesq equations are selected to illustrate the effectiveness and simplicity of the method. A generalized solitary solution with free parameters is obtained.

MSC:

35Q53	KdV equations (Korteweg-de Vries equations)
35Q51	Soliton equations

References:

[1]	He, J.H., Variational iteration method – a kind of non-linear analytical technique: some examples, Int J non-linear mech, 34, 4, 699-708, (1999) · Zbl 1342.34005
[2]	He, J.H.; Wu, X.H., Construction of solitary solution and compacton-like solution by variational iteration method, Chaos, solitons & fractals, 29, 1, 108-113, (2006) · Zbl 1147.35338
[3]	He JH. Variational iteration method - Some recent results and new interpretations. J Comput Appl Math [in press].
[4]	He JH, Wu XH. Variational iteration method: new development and applications. Comput Math Appl [accepted].
[5]	Odibat, Z.M.; Momani, S., Application of variational iteration method to nonlinear differential equations of fractional order, Int J nonlinear sci numer simul, 7, 1, 27-34, (2006) · Zbl 1401.65087
[6]	He, J.H., New interpretation of homotopy perturbation method, Int J mod phys B, 20, 18, 2561-2568, (2006)
[7]	He, J.H., Application of homotopy perturbation method to nonlinear wave equations, Chaos, solitons & fractals, 26, 3, 695-700, (2005) · Zbl 1072.35502
[8]	He, J.H., Limit cycle and bifurcation of nonlinear problems, Chaos, solitons & fractals, 26, 3, 827-833, (2005) · Zbl 1093.34520
[9]	He, J.H., Homotopy perturbation method for bifurcation of nonlinear problems, Int J nonlinear sci numer simul, 6, 2, 207-208, (2005) · Zbl 1401.65085
[10]	Rafei, M.; Ganji, D.D., Explicit solutions of Helmholtz equation and fifth-order KdV equation using homotopy perturbation method, Int J nonlinear sci numer simul, 7, 3, 321-328, (2006) · Zbl 1160.35517
[11]	Siddiqui, A.M.; Mahmood, R.; Ghori, Q.K., Thin film flow of a third grade fluid on a moving belt by he’s homotopy perturbation method, Int J nonlinear sci numer simul, 7, 1, 7-14, (2006) · Zbl 1187.76622
[12]	Siddiqui, A.M.; Ahmed, M.; Ghori, Q.K., Couette and Poiseuille flows for non-Newtonian fluids, Int J nonlinear sci numer simul, 7, 1, 15-26, (2006) · Zbl 1401.76018
[13]	Wazwaz, A.M., The tanh method: solitons and periodic solutions for the dodd – bullough – mikhailov and the tzitzeica – dodd – bullough equations, Chaos, solitons & fractals, 25, 55-63, (2005) · Zbl 1070.35076
[14]	Abdusalam, H.A., On an improved complex tanh-function method, Int. J. nonlinear sci. numer. simul., 6, 99-106, (2005) · Zbl 1401.35012
[15]	Bai, C.L.; Zhao, H., Generalized extended tanh-function method and its application, Chaos, solitons & fractals, 27, 1026-1035, (2006) · Zbl 1088.35534
[16]	Abdou, M.A.; Soliman, A.A., Modified extended tanh-function method and its application on nonlinear physical equations, Phys lett A, 353, 6, 487-492, (2006)
[17]	Ibrahim, R.S.; El-Kalaawy, O.H., Extended tanh-function method and reduction of nonlinear Schrödinger-type equations to a quadrature, Chaos, solitons & fractals, 31, 4, 1001-1008, (2007) · Zbl 1139.35396
[18]	El-Wakil, S.A.; Abdou, M.A., New exact travelling wave solutions using modified extended tanh-function method, Chaos, solitons & fractals, 31, 4, 840-852, (2007) · Zbl 1139.35388
[19]	Elwakil, S.A.; El-Labany, S.K.; Zahran, M.A.; Sabry, R., Modified extended tanh-function method and its applications to nonlinear equations, Appl math comput, 161, 2, 403-412, (2005) · Zbl 1062.35082
[20]	Pedit, Franz; Wu, Hongyou, Discretizing constant curvature surfaces via loop group factorizations: the discrete sine- and sinh-Gordon equations, J geomet phys, 17, 3, 245-260, (1995) · Zbl 0856.58020
[21]	Wazwaz, A.M., Exact solutions to the double sinh-Gordon equation by the tanh method and a variable separated ODE method, Comput math appl, 50, 10-12, 1685-1696, (2005) · Zbl 1089.35534
[22]	Zhao, Xiqiang; Wang, Limin; Sun, Weijun, The repeated homogeneous balance method and its applications to nonlinear partial differential equations, Chaos, solitons & fractals, 28, 2, 448-453, (2006) · Zbl 1082.35014
[23]	Feng, Zhaosheng, Comment on “on the extended applications of homogeneous balance method”, Appl math comput, 158, 2, 593-596, (2004) · Zbl 1061.35108
[24]	Zhang, Jie-Fang, Homogeneous balance method and chaotic and fractal solutions for the nizhnik – novikov – veselov equation, Phys lett A, 313, 5-6, 401-407, (2003) · Zbl 1040.35105
[25]	Fan, Engui; Zhang, Jian, Applications of the Jacobi elliptic function method to special-type nonlinear equations, Phys lett A, 305, 6, 383-392, (2002) · Zbl 1005.35063
[26]	Hon, Y.C.; Fan, Engui, Uniformly constructing finite-band solutions for a family of derivative nonlinear Schrödinger equations, Chaos, solitons & fractals, 24, 4, 1087-1096, (2005) · Zbl 1068.35156
[27]	Fan, E.; Hon, Y.C., Applications of extended tanh method to special types of nonlinear equations, Appl math comput, 141, 351-358, (2003) · Zbl 1027.65128
[28]	Wang, M.L.; Li, X.Z., Extended F-expansion method and periodic wave solutions for the generalized Zakharov equations, Phys lett A, 343, 48-54, (2005) · Zbl 1181.35255
[29]	Yomba, E., The extended F-expansion method and its application for solving the nonlinear wave, CKGZ, GDS, DS and GZ equations, Phys lett A, 340, 149-160, (2005) · Zbl 1145.35455
[30]	Ren, Y.J.; Zhang, H.Q., A generalized F-expansion method to find abundant families of Jacobi elliptic function solutions of the (2+1)-dimensional nizhnik – novikov – veselov equation, Chaos, solitons & fractals, 27, 959-979, (2006) · Zbl 1088.35536
[31]	Wang, D.S.; Zhang, H.Q., Further improved F-expansion method and new exact solutions of konopelchenko – dubrovsky equation, Chaos, solitons & fractals, 25, 601-610, (2005) · Zbl 1083.35122
[32]	Yomba, E., The extended fan’s sub-equation method and its application to kdv – mkdv, BKK and variant Boussinesq equations, Phys lett A, 336, 463-476, (2005) · Zbl 1136.35451
[33]	Yomba, E., The modified extended Fan sub-equation method and its application to (2+1)-dimensional dispersive long wave equation, Chaos, solitons & fractals, 26, 785-794, (2005) · Zbl 1080.35096
[34]	He, J.H., Some asymptotic methods for strongly nonlinear equations, Int J mod phys B, 20, 10, 1141-1199, (2006) · Zbl 1102.34039
[35]	He, J.H.; Wu, X.H., Exp-function method for nonlinear wave equations, Chaos, solitons & fractals, 30, 3, 700-708, (2006) · Zbl 1141.35448
[36]	Wu XH, He JH. Solitary solutions, periodic solutions and compacton-like solutions using Exp-function method. Comput Math Appl [accepted].
[37]	He JH, Abdou MA. New periodic solutions for nonlinear evolution equations using Exp-function method, Chaos, Solitons & Fractals [in press, doi:10.1016/j.chaos.2006.05.072.].