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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)
[3] Parkes, E.J.; Duffy, B.R., Comp. phys. commun., 98, 288, (1996)
[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, ()
[7] Li, Z.B.; Wang, M.L., J. phys. A, 26, 6027, (1993)
[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)
[10] Lou, S.; Huang, G.; Ruan, H., J. phys. A, 24, L584, (1991)
[11] Malfliet, W., Am. J. phys., 60, 650, (1992)
[12] Parkes, E.J., J. phys. A, 27, L497, (1994)
[13] Gudkov, V.V., Comp. math. math. phys., 36, 335, (1996)
[14] Gudkov, V.V., J. math. phys., 38, 4794, (1997)
[15] Wang, M.L.; Li, Z.B., Phys. lett. A, 216, 67, (1996)
[16] Zhang, J.F., Int. J. theor. phys., 35, 1793, (1996)
[17] Fan, E.G.; Zhang, H.Q., Phys. lett. A, 246, 403, (1998)
[18] Kudryashov, N.A., Phys. lett. A, 147, 287, (1990)
[19] Kudryashov, N.A.; Zargaryan, D., J. phys. A, 29, 8067, (1996)
[20] Sivashinsky, G.I., Physica D, 4, 227, (1982)
[21] Kawahara, T., Phys. rev. lett., 51, 381, (1983)
[22] Sachs, R.L., Physica D, 30, 1, (1988)
[23] Satsuma, J., ()
[24] Whilemsson, H., Phys. rev. A, 36, 965, (1990)
[25] Airault, M.; Mckean, H.; Moser, J., Commun. pure appl. math., 30, 95, (1977)
[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)
[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)
[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)
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.