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Layeni, O. P.

A new rational auxiliary equation method and exact solutions of a generalized Zakharov system. (English) Zbl 1180.35445

Appl. Math. Comput. 215, No. 8, 2901-2907 (2009).
Summary: A new rational auxiliary equation method for obtaining exact traveling wave solutions of constant coefficient nonlinear partial differential equations of evolution is proposed. Its effectiveness is evinced by obtaining exact solutions of a generalized Zakharov system, some of which are new. It is shown that the \(G'/G\) and the generalized projective Riccati expansion methods are special cases of the auxiliary equation method. Further, due the solutions obtained, four other new and practicable rational methods are deduced.

Cited in 5 Documents

MSC:

35Q51 Soliton equations
35C07 Traveling wave solutions
35A24 Methods of ordinary differential equations applied to PDEs
35A30 Geometric theory, characteristics, transformations in context of PDEs

Keywords:

rational auxiliary equation method; exact traveling wave solution; generalized Zakharov system
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\textit{O. P. Layeni}, Appl. Math. Comput. 215, No. 8, 2901--2907 (2009; Zbl 1180.35445)
Full Text: DOI

References:

[1] Zeng, X.; Wang, Deng-Shan, A generalized extended rational expansion method and its application to (1+1)-dimensional dispersive long wave equation, Applied Mathematics and Computation, 212, 296-304 (2009) · Zbl 1166.65387
[2] Li, W.; Zhang, H., A new generalized compound Riccati equations rational expansion method to construct many new exact complexiton solutions of nonlinear evolution equations with symbolic computation, Chaos, Solitons & Fractals, 39, 2369-2377 (2009) · Zbl 1197.35238
[3] Kong, C.; Wang, D.; Song, L.; Zhang, H., New exact solutions to MKDV-Burgers equation and (2+1)-dimensional dispersive long wave equation via extended Riccati equation method, Chaos, Solitons & Fractals, 39, 697-706 (2009) · Zbl 1197.35234
[4] Chena, Y.; Li, B., An extended subequation rational expansion method with symbolic computation and solutions of the nonlinear Schrodinger equation model, Nonlinear Analysis: Hybrid Systems, 2, 242-255 (2008) · Zbl 1156.35469
[5] Li, W.; Zhang, H., Generalized multiple Riccati equations rational expansion method with symbolic computation to construct exact complexiton solutions of nonlinear partial differential equations, Applied Mathematics and Computation, 197, 288-296 (2008) · Zbl 1135.65385
[6] Xu, T.; Li, J.; Zhang, H.-Q.; Zhang, Ya-Xing; Yao, Zhen-Zhi; Tian, Bo, New extension of the tanh-function method and application to the WhithamBroerKaup shallow water model with symbolic computation, Physics Letters A, 369, 458-463 (2007)
[7] Wang, B.-D.; Song, Li-Na; Zhang, H.-Q., A new extended elliptic equation rational expansion method and its application to (2+1)-dimensional Burgers equation, Chaos, Solitons & Fractals, 33, 1546-1551 (2007) · Zbl 1132.35343
[8] Zhang, S., A generalized new auxiliary equation method and its application to the (2+1)-dimensional breaking soliton equations, Applied Mathematics and Computation, 190, 510-516 (2007) · Zbl 1124.35072
[9] Song, L.; Zhang, H., New complexiton solutions of the nonlinear evolution equations using a generalized rational expansion method with symbolic computation, Applied Mathematics and Computation, 190, 974-986 (2007) · Zbl 1122.65395
[10] Zheng, Y.; Zhang, Y.; Zhang, H., Generalized Riccati equation rational expansion method and its application, Applied Mathematics and Computation, 189, 490-499 (2007) · Zbl 1123.65107
[11] Wang, D.; Sun, W.; Kong, C.; Zhang, H., New extended rational expansion method and exact solutionsof Boussinesq equation and JimboMiwa equations, Applied Mathematics and Computation, 189, 878-886 (2007) · Zbl 1122.65396
[12] Zheng, Y.; Zhang, Y.; Zhang, H., Generalized Riccati equation rational expansion method and its application, Applied Mathematics and Computation, 189, 490-499 (2007) · Zbl 1123.65107
[13] Wang, D.; Sun, W.; Kong, C.; Zhang, H., New extended rational expansion method and exact solutions of Boussinesq equation and JimboMiwa equations, Applied Mathematics and Computation, 189, 878-886 (2007) · Zbl 1122.65396
[14] Zhang, X.-L.; Zhang, H.-Q., A new generalized Riccati equation rational expansion method to a class of nonlinear evolution equations with nonlinear terms of any order, Applied Mathematics and Computation, 186, 705-714 (2007) · Zbl 1114.65125
[15] Song, Li-Na; Wang, Q.; Zheng, Y.; Zhang, H.-Q., A new extended Riccati equation rational expansion method and its application, Chaos, Solitons & Fractals, 31, 548-556 (2007) · Zbl 1138.35403
[16] Song, L.; Zhang, H., New exact solutions of the BroerKaup equations in (2+1)-dimensional spaces, Applied Mathematics and Computation, 180, 664-675 (2006) · Zbl 1102.65104
[17] Chen, Y.; Yan, Z., Weierstrass semi-rational expansion method and new doubly periodic solutions of the generalized Hirota-Satsuma coupled KdV system, Applied Mathematics and Computation, 177, 85-91 (2006) · Zbl 1094.65104
[18] Chen, Y.; Wang, Q., A unified rational expansion method to construct a series of explicit exact solutions to nonlinear evolution equations, Applied Mathematics and Computation, 177, 396-409 (2006) · Zbl 1094.65103
[19] Chen, Y.; Wang, Q., A new elliptic equation rational expansion method and its application to the shallow long wave approximate equations, Applied Mathematics and Computation, 173, 1163-1182 (2006) · Zbl 1088.65087
[20] Wang, Q.; Chen, Y.; Zhang, H., A new Riccati equation rational expansion method and its application to (2+1)-dimensional Burgers equation, Chaos, Solitons & Fractals, 25, 1019-1028 (2005) · Zbl 1070.35073
[21] Abdou, M. A., An extended Riccati equation rational expansion method and its applications, International Journal of Nonlinear Science, 7, 1, 57-66 (2009) · Zbl 1161.35477
[22] Javidi, M.; Golbabai, A., Exact and numerical solitary wave solutions of generalized Zakharov equation by the variational iteration method, Chaos, Solitons & Fractals, 36, 309-313 (2008) · Zbl 1350.35182
[23] Li, Ya-zhou; Li, Kai-ming; Lin, C., Exp-function method for solving the generalized-Zakharov equations, Applied Mathematics and Computation, 205, 197-201 (2008) · Zbl 1160.35523
[24] Zhang, H., New exact travelling wave solutions to the generalized Zakharov equations, Report Math Phys, 60, 97-106 (2007) · Zbl 1170.35524
[25] Borhanifar, A.; Kabir, M. M.; Maryam Vahdat, L., New periodic and soliton wave solutions for the generalized Zakharov system and (2+1)-dimensional Nizhnik-Novikov-Veselov system, Chaos, Solitons & Fractals (2009) · Zbl 1198.35216
[26] Yang, X.-L.; Tang, J.-S., Explicit exact solutions for the generalized Zakharov equations with nonlinear terms of any order, Computers and Mathematics with Applications, 57, 1622-1629 (2009) · Zbl 1186.34007
[27] Wang, M.; Li, X., Extended F-expansion and periodic wave solutions for the generalized Zakharov equations, Physical Letters A, 343, 48-54 (2005) · Zbl 1181.35255
[28] Abdou, M. A., A generalized auxiliary equation method and its applications, Nonlinear Dynamics, 52, 1-2, 95-102 (2008) · Zbl 1173.35673
[29] Maugin, G. A., Nonlinear Waves in Elastic Crystals (1999), Oxford University Press: Oxford University Press New York · Zbl 0943.74002
[30] Zakharov, V. E., Collapse of Langmuir waves, Soviet Physics JETP, 35, 908-914 (1972)
[31] Kudryashov, N. A., Simplest equation method to look for exact solutions of nonlinear differential equations, Chaos, Solitons & Fractals, 24, 5, 1217-1231 (2005) · Zbl 1069.35018
[32] Layeni, O. P., Note on exact solutions for a class of nonlinear diffusion equations, Applied Mathematics and Computation, 210, 465-472 (2009) · Zbl 1170.35451
[34] (Scott, A., Encyclopedia of Nonlinear Science (2005), Routledge: Routledge New York) · Zbl 1177.00019
[35] El-Wakil, S. A.; Abdou, M. A.; Hendi, A., New periodic and soliton solutions of nonlinear evolution equations, Applied Mathematics and Computation, 197, 497-506 (2008) · Zbl 1135.65382
[36] Wanga, M.; Lia, X.; Zhang, J., The \((G\)′/\(G)\)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Physics Letters A, 372, 4, 417-423 (2008)
[37] Yan, Z. Y., Generalized method and its application in the higher-order nonlinear Schrodinger equation in nonlinear optical fibres, Chaos, Solitons & Fractals, 16, 759 (2003) · Zbl 1035.78006
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