## Solitary solutions, periodic solutions and compacton-like solutions using the Exp-function method.(English)Zbl 1143.35360

Summary: The Exp-function method is used for seeking solitary solutions, periodic solutions, and compacton-like solutions of nonlinear differential equations. The combined KdV-MKdV equation and the Liouville equation are chosen to illustrate the effectiveness and convenience of the proposed method.

### MSC:

 35Q53 KdV equations (Korteweg-de Vries equations) 35Q51 Soliton equations 35B10 Periodic solutions to PDEs 35C10 Series solutions to PDEs 35-04 Software, source code, etc. for problems pertaining to partial differential equations

### Software:

Matlab; Mathematica
Full Text:

### References:

 [1] He, J.H.; Wu, X.H., Construction of solitary solution and compacton-like solution by variational iteration method, Chaos, solitons and fractals, 29, 108-113, (2006) · Zbl 1147.35338 [2] Odibat, Z.M.; Momani, S., Application of variational iteration method to nonlinear differential equations of fractional order, International journal of nonlinear science and numerical simulation, 7, 27-34, (2006) · Zbl 1401.65087 [3] Bildik, N.; Konuralp, A., The use of variational iteration method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations, International journal of nonlinear science and numerical simulation, 7, 65-70, (2006) · Zbl 1401.35010 [4] Momani, S.; Abuasad, S., Application of he’s variational iteration method to Helmholtz equation, Chaos, solitons and fractals, 27, 1119-1123, (2006) · Zbl 1086.65113 [5] He, J.H., Application of homotopy perturbation method to nonlinear wave equations, Chaos, solitons and fractals, 26, 695-700, (2005) · Zbl 1072.35502 [6] He, J.H., Homotopy perturbation method for bifurcation of nonlinear problems, International journal of nonlinear science and numerical simulation, 6, 207-208, (2005) · Zbl 1401.65085 [7] He, J.H., New interpretation of homotopy perturbation method, International journal of modern physics B, 20, 18, 2561-2568, (2006) [8] El-Shahed, M., Application of he’s homotopy perturbation method to volterra’s integro-differential equation, International journal of nonlinear science and numerical simulation, 6, 163-168, (2005) · Zbl 1401.65150 [9] 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, International journal of nonlinear science and numerical simulation, 7, 7-14, (2006) · Zbl 1187.76622 [10] Cai, X.C.; Wu, W.Y.; Li, M.S., Approximate period solution for a kind of nonlinear oscillator by he’s perturbation method, International journal of nonlinear science and numerical simulation, 7, 109-112, (2006) [11] He, J.H., Variational principles for some nonlinear partial differential equations with variable coefficients, Chaos, solitons and fractals, 19, 847-851, (2004) · Zbl 1135.35303 [12] Liu, H.M., Generalized variational principles for ion acoustic plasma waves by he’s semi-inverse method, Chaos, solitons and fractals, 23, 573-576, (2005) · Zbl 1135.76597 [13] He, J.H.; Wu, X.H., Exp-function method for nonlinear wave equations, Chaos, solitons and fractals, 30, 3, 700-708, (2006) · Zbl 1141.35448 [14] J.H. He, M.A. Abdou, New periodic solutions for nonlinear evolution equations using Exp-function method, Chaos, Solitons and Fractals, in press (doi: 10.1016/j.chaos.2006.05.072) [15] Wazwaz, A.M., The tanh method: solitons and periodic solutions for the dodd – bullough – mikhailov and the tzitzeica – dodd – bullough equations, Chaos, solitons and fractals, 25, 55-63, (2005) · Zbl 1070.35076 [16] Abdusalam, H.A., On an improved complex tanh-function method, International journal of nonlinear science and numerical simulation, 6, 99-106, (2005) · Zbl 1401.35012 [17] El-Sabbagh, M.F.; Ali, A.T., New exact solutions for $$(3 + 1)$$-dimensional kadomtsev – petviashvili equation and generalized $$(2 + 1)$$-dimensional Boussinesq equation, International journal of nonlinear science and numerical simulation, 6, 151-162, (2005) · Zbl 1401.35267 [18] Bai, C.L; Zhao, H., Generalized extended tanh-function method and its application, Chaos, solitons and fractals, 27, 1026-1035, (2006) · Zbl 1088.35534 [19] Abdou, M.A.; Soliman, A.A., Modified extended tanh-function method and its application on nonlinear physical equations, Physics letters A, 353, 6, 487-492, (2006) [20] Ibrahim, R.S.; El-Kalaawy, O.H., Extended tanh-function method and reduction of nonlinear Schrödinger-type equations to a quadrature, Chaos, solitons and fractals, 31, 4, 1001-1008, (2007) · Zbl 1139.35396 [21] El-Wakil, S.A.; Abdou, M.A., New exact travelling wave solutions using modified extended tanh-function method, Chaos, solitons and fractals, 31, 4, 840-852, (2007) · Zbl 1139.35388 [22] Elwakil, S.A.; El-Labany, S.K.; Zahran, M.A.; Sabry, R., Modified extended tanh-function method and its applications to nonlinear equations, Applied mathematics and computation, 161, 2, 403-412, (2005) · Zbl 1062.35082 [23] Lü, Z.; Zhang, H., Applications of a further extended tanh method, Applied mathematics and computation, 159, 2, 401-406, (2004) · Zbl 1062.65111 [24] Pedit, F.; Wu, H., Discretizing constant curvature surfaces via loop group factorizations: the discrete sine- and sinh-Gordon equations, Journal of geometry and physics, 17, 3, 245-260, (1995) · Zbl 0856.58020 [25] Wazwaz, A.M., Exact solutions to the double sinh-Gordon equation by the tanh method and a variable separated ODE method, Computers & mathematics with applications, 50, 10-12, 1685-1696, (2005) · Zbl 1089.35534 [26] Zhao, X.; Wang, L.; Sun, W., The repeated homogeneous balance method and its applications to nonlinear partial differential equations, Chaos, solitons and fractals, 28, 2, 448-453, (2006) · Zbl 1082.35014 [27] Zhaosheng, F., Comment on “on the extended applications of homogeneous balance method”, Applied mathematics and computation, 158, 2, 593-596, (2004) · Zbl 1061.35108 [28] Zhang, J.-F., Homogeneous balance method and chaotic and fractal solutions for the nizhnik – novikov – veselov equation, Physics letters A, 313, 5-6, 401-407, (2003) · Zbl 1040.35105 [29] Zhao, X.; Tang, D., A new note on a homogeneous balance method, Physics letters A, 297, 1-2, 59-67, (2002) · Zbl 0994.35005 [30] Zhang, J.L.; Wang, M.L.; Wang, Y.M., The improved F-expansion method and its applications, Physics letters A, 350, 103-109, (2006) · Zbl 1195.65211 [31] Fan, E.; Zhang, J., Applications of the Jacobi elliptic function method to special-type nonlinear equations, Physics letters A, 305, 6, 383-392, (2002) · Zbl 1005.35063 [32] Hon, Y.C.; Fan, E., Uniformly constructing finite-band solutions for a family of derivative nonlinear Schrödinger equations, Chaos, solitons and fractals, 24, 4, 1087-1096, (2005) · Zbl 1068.35156 [33] Yomba, E., The extended fan’s sub-equation method and its application to kdv – mkdv, BKK and variant Boussinesq equations, Physics letter A, 336, 463-476, (2005) · Zbl 1136.35451 [34] Yomba, E., The modified extended Fan sub-equation method and its application to $$(2 + 1)$$-dimensional dispersive long wave equation, Chaos, solitons and fractals, 26, 785-794, (2005) · Zbl 1080.35096 [35] He, J.H., Non-perturbative methods for strongly nonlinear problems, (2006), Dissertation de-Verlag im Internet GmbH Berlin [36] He, J.H., Some asymptotic methods for strongly nonlinear equations, International journal of modern physics B, 20, 1141-1199, (2006) · Zbl 1102.34039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.