Application of Exp-function method to Riccati equation and new exact solutions with three arbitrary functions of Broer-Kaup-Kupershmidt equations. (English) Zbl 1220.37071
Summary: In this Letter, the Exp-function method is used to seek generalized solitonary solutions of Riccati equation. Based on the Riccati equation and its generalized solitonary solutions, new exact solutions with three arbitrary functions of the (2+1)-dimensional Broer-Kaup-Kupershmidt equations are obtained. It is shown that the Exp-function method provides a straightforward and important mathematical tool for nonlinear evolution equations in mathematical physics.
MSC:
| 37L05 | General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations |
| 35Q51 | Soliton equations |
Keywords:
exp-function method; Riccati equation; generalized solitonary solutions; nonlinear evolution equationsReferences:
| [1] | Hirota, R., Phys. Rev. Lett., 27, 1192 (1971) · Zbl 1168.35423 |
| [2] | Miurs, M. R., Backlund Transformation (1978), Springer: Springer Berlin |
| [3] | Weiss, J.; Tabor, M.; Carnevale, G., J. Math. Phys., 24, 522 (1983) · Zbl 0514.35083 |
| [4] | Yan, C. T., Phys. Lett. A, 224, 77 (1996) · Zbl 1037.35504 |
| [5] | Wang, M. L., Phys. Lett. A, 213, 279 (1996) · Zbl 0972.35526 |
| [6] | El-Shahed, M., Int. J. Nonlinear Sci. Numer. Simul., 6, 163 (2005) · Zbl 1401.65150 |
| [7] | He, J. H., Int. J. Nonlinear Sci. Numer. Simul., 6, 207 (2005) · Zbl 1401.65085 |
| [8] | He, J. H., Chaos Solitons Fractals, 26, 695 (2005) · Zbl 1072.35502 |
| [9] | He, J. H., Int. J. Nonlinear Mech., 34, 699 (1999) · Zbl 1342.34005 |
| [10] | He, J. H., Appl. Math. Comput., 114, 115 (2000) |
| [11] | He, J. H., Chaos Solitons Fractals, 19, 847 (2004) |
| [12] | He, J. H., Phys. Lett. A, 335, 182 (2005) |
| [13] | He, J. H., Int. J. Mod. Phys. B, 20, 1141 (2006) · Zbl 1102.34039 |
| [14] | He, J. H., Non-Perturbative Methods for Strongly Nonlinear Problems (2006), dissertation.de-Verlag im Internet GmbH: dissertation.de-Verlag im Internet GmbH Berlin |
| [15] | Abassy, T. A.; El-Tawil, M. A.; Saleh, H. K., Int. J. Nonlinear Sci. Numer. Simul., 5, 327 (2004) · Zbl 1401.65122 |
| [16] | Zayed, E. M.E.; Zedan, H. A.; Gepreel, K. A., Int. J. Nonlinear Sci. Numer. Simul., 5, 221 (2004) · Zbl 1069.35080 |
| [17] | Abdusalam, H. A., Int. J. Nonlinear Sci. Numer. Simul., 6, 99 (2005) · Zbl 1401.35012 |
| [18] | Zhang, S.; Xia, T. C., Commun. Theor. Phys. (Beijing, China), 45, 985 (2006) |
| [19] | Zhang, S.; Xia, T. C., Appl. Math. Comput., 181, 319 (2006) · Zbl 1155.65385 |
| [20] | Zhang, S., Chaos Solitons Fractals, 31, 951 (2007) · Zbl 1139.35392 |
| [21] | Hu, J. Q., Chaos Solitons Fractals, 23, 391 (2005) · Zbl 1069.35065 |
| [22] | Yomba, E., Chaos Solitons Fractals, 27, 187 (2006) · Zbl 1088.35532 |
| [23] | Zhang, S.; Xia, T. C., Phys. Lett. A, 356, 119 (2006) · Zbl 1160.37404 |
| [24] | Zhang, S.; Xia, T. C., Appl. Math. Comput., 182, 1651 (2006) · Zbl 1117.65143 |
| [25] | Liu, S. K.; Fu, Z. T.; Liu, S. D.; Zhao, Q., Phys. Lett. A, 289, 69 (2001) · Zbl 0972.35062 |
| [26] | Fu, Z. T.; Liu, S. K.; Liu, S. D.; Zhao, Q., Phys. Lett. A, 290, 72 (2001) · Zbl 0977.35094 |
| [27] | Parkes, E. J.; Duffy, B. R.; Abbott, P. C., Phys. Lett. A, 295, 280 (2002) · Zbl 1052.35143 |
| [28] | Zhou, Y. B.; Wang, M. L.; Wang, Y. M., Phys. Lett. A, 308, 31 (2003) · Zbl 1008.35061 |
| [29] | Wang, D. S.; Zhang, H. Q., Chaos Solitons Fractals, 25, 601 (2005) · Zbl 1083.35122 |
| [30] | Zhang, S., Phys. Lett. A, 358, 414 (2006) · Zbl 1142.35592 |
| [31] | Zhang, S.; Xia, T. C., Appl. Math. Comput., 183, 1190 (2006) · Zbl 1111.35318 |
| [32] | Zhang, S., Chaos Solitons Fractals, 30, 1213 (2006) · Zbl 1142.35579 |
| [33] | Zhang, S., Chaos Solitons Fractals, 32, 847 (2007) · Zbl 1138.35401 |
| [34] | Zhang, S., Chaos Solitons Fractals, 33, 1375 (2007) · Zbl 1133.91539 |
| [35] | Zhang, S., Appl. Math. Comput., 189, 836 (2006) |
| [36] | Sirendaoreji; Song, J., Phys. Lett. A, 309, 387 (2003) · Zbl 1011.35035 |
| [37] | Zhang, S.; Xia, T. C., J. Phys. A: Math. Theor., 40, 227 (2007) · Zbl 1105.35320 |
| [38] | Zhang, S.; Xia, T. C., Phys. Lett. A, 363, 356 (2007) · Zbl 1197.35008 |
| [39] | Zhang, S., Appl. Math. Comput., 188, 1 (2007) · Zbl 1114.65355 |
| [40] | Zhang, S., Appl. Math. Comput., 190, 510 (2007) · Zbl 1124.35072 |
| [41] | He, J. H.; Wu, X. H., Chaos Solitons Fractals, 30, 700 (2006) · Zbl 1141.35448 |
| [42] | He, J. H.; Abdou, M. A., Chaos Solitons Fractals, 34, 1421 (2007) · Zbl 1152.35441 |
| [43] | X.H. Wu, J.H. He, Exp-function method and its application to nonlinear equations, Chaos Solitons Fractals (2007), doi:10.1016/j.chaos.2007.01.024; X.H. Wu, J.H. He, Exp-function method and its application to nonlinear equations, Chaos Solitons Fractals (2007), doi:10.1016/j.chaos.2007.01.024 · Zbl 1153.35384 |
| [44] | Wu, X. H.; He, J. H., Comput. Math. Appl., 54, 966 (2007) |
| [45] | Zhu, S. D., Int. J. Nonlinear Sci. Numer. Simul., 8, 461 (2007) |
| [46] | Zhu, S. D., Int. J. Nonlinear Sci. Numer. Simul., 8, 465 (2007) |
| [47] | Ebaid, A., Phys. Lett. A, 365, 213 (2007) · Zbl 1203.35213 |
| [48] | Zhang, S., Phys. Lett. A, 365, 448 (2007) · Zbl 1203.35255 |
| [49] | S. Zhang, Application of Exp-function method to high-dimensional nonlinear evolution equation, Chaos Solitons Fractals (2006), doi:10.1016/j.chaos.2006.11.014; S. Zhang, Application of Exp-function method to high-dimensional nonlinear evolution equation, Chaos Solitons Fractals (2006), doi:10.1016/j.chaos.2006.11.014 · Zbl 1142.35593 |
| [50] | Zhang, S., Phys. Lett. A, 371, 65 (2007) · Zbl 1209.65103 |
| [51] | S. Zhang, Exact solutions of a KdV equation with variable coefficients via Exp-function method, Nonlinear Dynam. (2007), doi:10.1007/s11071-007-9251-0; S. Zhang, Exact solutions of a KdV equation with variable coefficients via Exp-function method, Nonlinear Dynam. (2007), doi:10.1007/s11071-007-9251-0 |
| [52] | S. Zhang, Exp-function method exactly solving a KdV equation with forcing term, Appl. Math. Comput. (2007), doi:10.1016/j.amc.2007.07.041; S. Zhang, Exp-function method exactly solving a KdV equation with forcing term, Appl. Math. Comput. (2007), doi:10.1016/j.amc.2007.07.041 · Zbl 1135.65388 |
| [53] | Abdou, M. A.; Solimanm, A. A.; El-Basyony, S. T., Phys. Lett. A, 369, 469 (2007) · Zbl 1209.81091 |
| [54] | M.A. Abdou, Generalized solitonary and periodic solutions for nonlinear partial differential equations by the Exp-function method, Nonlinear Dynam. (2007), doi:10.1007/s11071-007-9250-1; M.A. Abdou, Generalized solitonary and periodic solutions for nonlinear partial differential equations by the Exp-function method, Nonlinear Dynam. (2007), doi:10.1007/s11071-007-9250-1 |
| [55] | El-Wakil, S. A.; Madkour, M. A.; Abdou, M. A., Phys. Lett. A, 369, 62 (2007) · Zbl 1209.81097 |
| [56] | J.H. He, L.N. Zhang, Generalized solitary solution and compacton-like solution of the Jaulent-Miodek equations using the Exp-function method, Phys. Lett. A (2007), doi:10.1016/j.physleta.2007.08.059; J.H. He, L.N. Zhang, Generalized solitary solution and compacton-like solution of the Jaulent-Miodek equations using the Exp-function method, Phys. Lett. A (2007), doi:10.1016/j.physleta.2007.08.059 · Zbl 1217.35152 |
| [57] | Zhu, S. D., Phys. Lett. A, 372, 654 (2008) · Zbl 1217.37064 |
| [58] | Zhu, J. M., Chin. Phys., 14, 1290 (2005) |
| [59] | Cao, L. N.; Wang, D. S.; Zhang, H. Q., Commun. Theor. Phys. (Beijing, China), 47, 270 (2007) |
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