A generalized \(F\)-expansion method with symbolic computation exactly solving Broer-Kaup equations. (English) Zbl 1122.65095
Summary: A generalized \(F\)-expansion method is applied to construct exact solutions of the \((2 + 1)\)-dimensional Broer-Kaup equations. As a result, many general exact non-travelling wave and coefficient function solutions are obtained including single and combined non-degenerate Jacobi elliptic function solutions, soliton-like solutions and trigonometric function solutions, each of which contains two arbitrary functions. Compared with most existing \(F\)-expansion methods, the proposed method gives new and more general exact solutions. More importantly, with the aid of symbolic computation, the method provides a powerful mathematical tool to solve a large number of nonlinear partial differential equations.
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 35Q51 | Soliton equations |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 68W30 | Symbolic computation and algebraic computation |
Keywords:
generalized \(F\)-expansion method; Jacobi elliptic function solutions; soliton-like solutions; trigonometric function solutions; Broer-Kaup equations; symbolic computationReferences:
| [1] | Ablowitz, M. J.; Clarkson, P. A., Soliton, Nonlinear Evolution Equations and Inverse Scattering (1991), Cambridge University Press: Cambridge University Press New York · Zbl 0762.35001 |
| [2] | Hirota, R., Phys. Rev. Lett., 27, 1192 (1971) · Zbl 1168.35423 |
| [3] | Miurs, M. R., Bachklund Transformation (1978), Springer: Springer Berlin |
| [4] | Weiss, J.; Tabor, M.; Carnevale, G., J. Math. Phys., 24, 522 (1983) · Zbl 0514.35083 |
| [5] | Yan, C., Phys. Lett. A, 224, 77 (1996) · Zbl 1037.35504 |
| [6] | Abassy, T. A.; El-Tawil, M. A.; Saleh, H. K., Int. J. Nonlinear Sci. Numer. Simul., 5, 327 (2004) · Zbl 1401.65122 |
| [7] | Wang, M. L., Phys. Lett. A, 213, 279 (1996) · Zbl 0972.35526 |
| [8] | El-Shahed, M., Int. J. Nonlinear Sci. Numer. Simul., 6, 163 (2005) · Zbl 1401.65150 |
| [9] | He, J. H., Int. J. Nonlinear Sci. Numer. Simul., 6, 207 (2005) · Zbl 1401.65085 |
| [10] | He, J. H., Chaos, Solitons & Fractals, 26, 695 (2005) · Zbl 1072.35502 |
| [11] | He, J. H., Int. J. Nonlinear Mech., 34, 699 (1999) · Zbl 1342.34005 |
| [12] | He, J. H., Appl. Math. Comput., 114, 115 (2000) |
| [13] | He, J. H., Chaos, Solitons & Fractals, 19, 847 (2004) |
| [14] | He, J. H., Phys. Lett. A, 335, 182 (2005) |
| [15] | He, J. H., Int. J. Modern Phys. B, 20, 1141 (2006) |
| [16] | He, J. H., Non-Perturbative Methods for Strongly Nonlinear Problems, Dissertation (2006), de-Verlag im Internet GmbH: de-Verlag im Internet GmbH Berlin |
| [17] | He, J. H.; Wu, X. H., Chaos, Solitons & Fractals, 30, 700 (2006) · Zbl 1141.35448 |
| [18] | Hu, J. Q., Chaos, Solitons & Fractals, 23, 391 (2005) · Zbl 1069.35065 |
| [19] | Chen, Y.; Wang, Q.; Li, B., Commun. Theor. Phys. (Beijing, China), 42, 655 (2004) · Zbl 1167.35461 |
| [20] | Yomba, E., Chaos, Solitons & Fractals, 27, 187 (2006) · Zbl 1088.35532 |
| [21] | Zhang, S.; Xia, T. C., Phys. Lett. A, 356, 119 (2006) · Zbl 1160.37404 |
| [22] | Chen, Y.; Yan, Z. Y., Appl. Math. Comput., 177, 85 (2006) · Zbl 1094.65104 |
| [23] | Chen, Y.; Wang, Q., Appl. Math. Comput., 177, 396 (2006) · Zbl 1094.65103 |
| [24] | Zhou, Y. B.; Wang, M. L.; Wang, Y. M., Phys. Lett. A, 308, 31 (2003) · Zbl 1008.35061 |
| [25] | Wang, M. L.; Zhou, Y. B., Phys. Lett. A, 318, 84 (2003) · Zbl 1098.81770 |
| [26] | Wang, M. L.; Wang, Y. M.; Zhang, J. L., Chin. Phys., 12, 1341 (2003) |
| [27] | Liu, S. K.; Fu, Z. T.; Liu, S. D.; Zhao, Q., Phys. Lett. A, 289, 69 (2001) · Zbl 0972.35062 |
| [28] | Fu, Z. T.; Liu, S. K.; Liu, S. D.; Zhao, Q., Phys. Lett. A, 290, 72 (2001) · Zbl 0977.35094 |
| [29] | Parkes, E. J.; Duffy, B. R.; Abbott, P. C., Phys. Lett. A, 295, 280 (2002) · Zbl 1052.35143 |
| [30] | Liu, J. B.; Yang, K. Q., Chaos, Solitons & Fractals, 22, 111 (2004) · Zbl 1062.35105 |
| [31] | Wang, M. L.; Li, X. Z., Chaos, Solitons & Fractals, 24, 1257 (2005) · Zbl 1092.37054 |
| [32] | Zhang, S., Chaos, Solitons & Fractals, 30, 1213 (2006) · Zbl 1142.35579 |
| [33] | Zhang, H. Q., Chaos, Solitons & Fractals, 26, 921 (2005) · Zbl 1093.35057 |
| [34] | Wang, D. S.; Zhang, H. Q., Chaos, Solitons & Fractals, 25, 601 (2005) · Zbl 1083.35122 |
| [35] | Ren, Y. J.; Zhang, H. Q., Chaos, Solitons & Fractals, 27, 959 (2006) · Zbl 1088.35536 |
| [36] | Chen, J.; He, H. S.; Yang, K. Q., Commun. Theor. Phys. (Beijing, China), 44, 307 (2005) |
| [37] | Zhang, S.; Xia, T. C., Appl. Math. Comput., 183, 1190 (2006) · Zbl 1111.35318 |
| [38] | Lou, S. Y.; Hu, X. B., J. Phys. A: Math. Gen., 30, L95 (1997) · Zbl 1001.35501 |
| [39] | Lou, S. Y.; Hu, X. B., J. Phys. A: Math. Gen., 27, L207 (1994) · Zbl 0838.35116 |
| [40] | Ruan, H. Y.; Chen, Y. X., Acta Phys. Sinica, 7, 241 (1998) |
| [41] | Yan, Z. Y.; Zhang, H. Q., J. Phys. A: Math. Gen., 34, 1785 (2001) · Zbl 0970.35147 |
| [42] | Zhang, S. L.; Wu, B.; Lou, S. Y., Phys. Lett. A, 300, 40 (2002) · Zbl 0997.76012 |
| [43] | Xia, T. C.; Zhang, H. Q., Chaos, Solitons & Fractals, 16, 167 (2003) · Zbl 1048.35112 |
| [44] | Li, D. S.; Zhang, H. Q., Chaos, Solitons & Fractals, 18, 193 (2003) · Zbl 1055.35099 |
| [45] | Wei, J. Q.; Li, D. S.; Zhang, H. Q., Chaos, Solitons & Fractals, 22, 669 (2004) · Zbl 1062.35143 |
| [46] | Liu, G. T.; Fan, T. Y., Commun. Theor. Phys. (Beijing, China), 42, 488 (2004) · Zbl 1167.35481 |
| [47] | Bai, C. L.; Zhang, H., Chaos, Solitons & Fractals, 23, 777 (2005) · Zbl 1069.35053 |
| [48] | Xie, F. D.; Chen, J.; Lü, Z. S., Commun. Theor. Phys. (Beijing, China), 43, 585 (2005) |
| [49] | Liu, H. C.; Pan, Z. L., Commun. Theor. Phys. (Beijing, China), 44, 15 (2005) |
| [50] | Zhang, S.; Xia, T. C., Commun. Theor. Phys. (Beijing, China), 45, 985 (2006) |
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