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Extended tanh-function method and its applications to nonlinear equations. (English) Zbl 1167.35331

Summary: An extended tanh-function method is proposed for constructing multiple travelling wave solutions of nonlinear partial differential equations (PDEs) in a unified way. The key idea of this method is to take full advantages of a Riccati equation involving a parameter and use its solutions to replace the tanh function in the tanh-function method. It is quite interesting that the sign of the parameter can be used to exactly judge the numbers and types of these travelling wave solutions. In addition, by introducing appropriate transformations, it is shown that the extended tanh-function method still is applicable to nonlinear PDEs whose balancing numbers may be any nonzero real numbers. Some illustrative equations are investigated by this means and new travelling wave solutions are found.

MSC:

35G20 Nonlinear higher-order PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
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[1] Hereman, W.; Takaoka, M., J. Phys. A, 23, 4805 (1990) · Zbl 0719.35085
[2] Hereman, W., Comp. Phys. Commun., 65, 143 (1991) · Zbl 0900.65349
[3] Parkes, E. J.; Duffy, B. R., Comp. Phys. Commun., 98, 288 (1996) · Zbl 0948.76595
[4] Duffy, B. R.; Parkes, E. J., Phys. Lett. A, 214, 271 (1996) · Zbl 0972.35528
[5] Parkes, E. J.; Duffy, B. R., Phys. Lett. A, 229, 217 (1997) · Zbl 1043.35521
[6] Li, Z. B., Exact solitary wave solutions of nonlinear evolution equations, (Gao, X. S.; Wang, D. M., Mathematics Mechanization and Application (2000), Academic Press)
[7] Li, Z. B.; Wang, M. L., J. Phys. A, 26, 6027 (1993) · Zbl 0809.35111
[8] Tian, B.; Gao, Y. T., Mod. Phys. Lett. A, 10, 2937 (1995)
[9] Gao, Y. T.; Tian, B., Comp. Math. Appl., 33, 115 (1997) · Zbl 0873.76061
[10] Lou, S.; Huang, G.; Ruan, H., J. Phys. A, 24, L584 (1991)
[11] Malfliet, W., Am. J. Phys., 60, 650 (1992) · Zbl 1219.35246
[12] Parkes, E. J., J. Phys. A, 27, L497 (1994) · Zbl 0846.35122
[13] Gudkov, V. V., Comp. Math. Math. Phys., 36, 335 (1996)
[14] Gudkov, V. V., J. Math. Phys., 38, 4794 (1997) · Zbl 0886.35131
[15] Wang, M. L.; Li, Z. B., Phys. Lett. A, 216, 67 (1996)
[16] Zhang, J. F., Int. J. Theor. Phys., 35, 1793 (1996) · Zbl 0862.35110
[17] Fan, E. G.; Zhang, H. Q., Phys. Lett. A, 246, 403 (1998) · Zbl 1125.35308
[18] Kudryashov, N. A., Phys. Lett. A, 147, 287 (1990)
[19] Kudryashov, N. A.; Zargaryan, D., J. Phys. A, 29, 8067 (1996) · Zbl 0901.35090
[20] Sivashinsky, G. I., Physica D, 4, 227 (1982) · Zbl 1194.76054
[21] Kawahara, T., Phys. Rev. Lett., 51, 381 (1983)
[22] Sachs, R. L., Physica D, 30, 1 (1988) · Zbl 0694.35207
[23] Satsuma, J., (Abolowitz, M.; Fuchssteiner, B.; Kruskal, M., Topics in Soliton Theory and Exactly Solvable Nonlinear Equations (1987), World Scientific: World Scientific Singapore) · Zbl 0721.00016
[24] Whilemsson, H., Phys. Rev. A, 36, 965 (1990)
[25] Airault, M.; Mckean, H.; Moser, J., Commun. Pure Appl. Math., 30, 95 (1977) · Zbl 0338.35024
[26] Adler, M.; Moser, J., Commun. Math. Phys., 19, 1 (1978)
[27] Nakamura, A.; Hirota, R., J. Phys. Soc. Jpn., 54, 491 (1985)
[28] Sachs, R. L., Physica D, 30, 1 (1988) · Zbl 0694.35207
[29] Coleman, C. J., J. Aust. Math. Soc. Ser. B, 33, 1 (1992)
[30] Smyth, N. F., J. Aust. Math. Soc. Ser. B, 33, 403 (1992) · Zbl 0758.35042
[31] Clarkson, P. A.; Mansfield, E. L., Physica D, 70, 250 (1993)
[32] Fan, E. G.; Zhang, H. Q., Phys. Lett. A, 245, 389 (1998) · Zbl 0947.35126
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