Exp-function method for nonlinear wave equations. (English) Zbl 1141.35448

He, Ji-Huan; Wu, Xu-Hong

Chaos Solitons Fractals 30, No. 3, 700-708 (2006).

Summary: A new method, called Exp-function method, is proposed to seek solitary solutions, periodic solutions and compacton-like solutions of nonlinear differential equations. The modified KdV equation and Dodd-Bullough-Mikhailov equation are chosen to illustrate the effectiveness and convenience of the suggested method.

MSC:

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

References:

[1]	Abdusalam, H.A., On an improved complex tanh-function method, Int J nonlinear sci numer simul, 6, 2, 99-106, (2005) · Zbl 1401.35012
[2]	Zayed, E.M.E.; Zedan, H.A.; Gepreel, K.A., Group analysis and modified extended tanh-function to find the invariant solutions and soliton solutions for nonlinear Euler equations, Int J nonlinear sci numer simul, 5, 3, 221-234, (2004) · Zbl 1069.35080
[3]	Bai, C.L.; Zhao, H., Generalized extended tanh-function method and its application, Chaos, solitons & fractals, 27, 4, 1026-1035, (2006) · Zbl 1088.35534
[4]	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, 1, 55-63, (2005) · Zbl 1070.35076
[5]	Li, D.S.; Gao, F.; Zhang, H.Q., Solving the (2+1)-dimensional higher order Broer-Kaup system via a transformation and tanh-function method, Chaos, solitons & fractals, 20, 5, 1021-1025, (2004) · Zbl 1049.35157
[6]	Khuri, S.A., A complex tanh-function method applied to nonlinear equations of schroedinger type, Chaos, solitons & fractals, 20, 5, 1037-1040, (2004) · Zbl 1049.35156
[7]	Yomba, E., The modified extended Fan sub-equation method and its application to the (2+1)-dimensional Broer-Kaup-Kupershmidt equation, Chaos, solitons & fractals, 27, 1, 187-196, (2006) · Zbl 1088.35532
[8]	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, 4, 959-979, (2006) · Zbl 1088.35536
[9]	Wang, D.S.; Zhang, H.Q., Further improved F-expansion method and new exact solutions of konopelchenko-dubrovsky equation, Chaos, solitons & fractals, 25, 3, 601-610, (2005) · Zbl 1083.35122
[10]	Dai, C.Q.; Zhang, J.F., Jacobian elliptic function method for nonlinear differential-difference equations, Chaos, solitons & fractals, 27, 4, 1042-1047, (2006) · Zbl 1091.34538
[11]	Yu, Y.X.; Wang, Q.; Zhang, H.Q., The extended Jacobi elliptic function method to solve a generalized Hirota-Satsuma coupled KdV equations, Chaos, solitons & fractals, 26, 5, 1415-1421, (2005) · Zbl 1106.35087
[12]	Zhao, X.Q.; Zhi, H.Y.; Zhang, H.Q., Improved Jacobi-function method with symbolic computation to construct new double-periodic solutions for the generalized ito system, Chaos, solitons & fractals, 28, 1, 112-126, (2006) · Zbl 1134.35301
[13]	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
[14]	Momani, S.; Abuasad, S., Application of he’s variational iteration method to Helmholtz equation, Chaos, solitons & fractals, 27, 5, 1119-1123, (2006) · Zbl 1086.65113
[15]	Abassy, T.A.; El-Tawil, M.A.; Saleh, H.K., The solution of KdV and mkdv equations using Adomian pade approximation, Int J nonlinear sci numer simul, 5, 4, 327-339, (2004) · Zbl 1401.65122
[16]	El-Sayed, S.M.; Kaya, D.; Zarea, S., The decomposition method applied to solve high-order linear Volterra-Fredholm integro-differential equations, Int J nonlinear sci numer simul, 5, 2, 105-112, (2004) · Zbl 1401.65149
[17]	El-Danaf, T.S.; Ramadan, M.A.; Alaal, F.E.I.A., The use of Adomian decomposition method for solving the regularized long-wave equation, Chaos, solitons & fractals, 26, 3, 747-757, (2005) · Zbl 1073.35010
[18]	Zhu, Y.G.; Chang, Q.S.; Wu, S.C., Construction of exact solitary solutions for Boussinesq-like B(m,n) equations with fully nonlinear dispersion by the decomposition method, Chaos, solitons & fractals, 26, 3, 897-903, (2005) · Zbl 1080.35097
[19]	Liu, H.M., Generalized variational principles for ion acoustic plasma waves by he’s semi-inverse method, Chaos, solitons & fractals, 23, 2, 573-576, (2005) · Zbl 1135.76597
[20]	Liu, H.M., Variational approach to nonlinear electrochemical system, Int J nonlinear sci numer simul, 5, 1, 95-96, (2004)
[21]	Zhang, J.; Yu, J.Y.; Pan, N., Variational principles for nonlinear fiber optics, Chaos, solitons & fractals, 24, 1, 309-311, (2005) · Zbl 1135.78330
[22]	He, J.H., Application of homotopy perturbation method to nonlinear wave equations, Chaos, solitons & fractals, 26, 3, 695-700, (2005) · Zbl 1072.35502
[23]	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
[24]	El-Shahed, M., Application of he’s homotopy perturbation method to volterra’s integro-differential equation, Int J nonlinear sci numer simul, 6, 2, 163-168, (2005) · Zbl 1401.65150
[25]	Rosenau, P.; Hyman, J.M., Compactons: solitons with finite wavelengths, Phys rev lett, 75, 5, 564-567, (1993) · Zbl 0952.35502
[26]	Rosenau, P., Compact and noncompact dispersive patterns, Phys lett A, 275, 3, 193-203, (2000) · Zbl 1115.35365
[27]	Wazwaz, A.M., Two reliable methods for solving variants of the KdV equation with compact and noncompact structures, Chaos, solitons & fractals, 28, 2, 454-462, (2006) · Zbl 1084.35079
[28]	Zhu, Y.; Gao, X., Exact special solitary solutions with compact support for the nonlinear dispersive K(m,n) equations, Chaos, solitons & fractals, 27, 2, 487-493, (2006) · Zbl 1088.35547
[29]	Zhu, Y.; Chang, Q.; Wu, S., Exact solitary-wave solutions with compact support for the modified KdV equation, Chaos, solitons & fractals, 24, 1, 365-369, (2005) · Zbl 1067.35099