An automated \(\tanh\)-function method for finding solitary wave solutions to nonlinear evolution equations. (English) Zbl 0948.76595
Summery: The tanh-function method for finding explicit travelling solitary wave solutions to non-linear evolution equations is described. The method is usually extremely tedious to use by hand. We present a Mathematica package ATFM that deals with the tedious algebra and outputs directly the required solutions. The use of the package is illustrated by applying it to a variety of equations; not only are previously known solutions recovered but in some cases more general forms of solution are obtained.
MSC:
| 76M25 | Other numerical methods (fluid mechanics) (MSC2010) |
| 76B25 | Solitary waves for incompressible inviscid fluids |
References:
| [1] | Drazin, P. G.; Johnson, R. S., Solitons: An Introduction (1989), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0661.35001 |
| [2] | Hereman, W.; Korpel, A.; Banerjee, P. P., Wave Motion, 7, 283 (1985) |
| [3] | Hereman, W.; Banerjee, P. P.; Korpel, A.; Assanto, G.; Van Immerzeele, A.; Meerpoel, A., J. Phys. A: Math. Gen., 19, 607 (1986) · Zbl 0621.35080 |
| [4] | Hereman, W.; Takaoka, M., J. Phys. A: Math. Gen., 23, 4805 (1990) · Zbl 0719.35085 |
| [5] | Coffey, M. W., SIAM J. Appl. Math., 50, 1580 (1990) · Zbl 0712.76025 |
| [6] | Lan, H.; Wang, K., J. Phys. A: Math. Gen., 23, 3923 (1990) · Zbl 0718.35020 |
| [7] | Lou, S.; Huang, G.; Ruan, H., J. Phys. A: Math. Gen., 24, L587 (1991) · Zbl 0735.76057 |
| [8] | Malfliet, W., Am. J. Phys., 60, 650 (1992) · Zbl 1219.35246 |
| [9] | Tian, B.; Gao, Y. T., Mod. Phys. Lett. A, 10, 2937 (1995) |
| [10] | Watanabe, S., J. Phys. Soc. Jpn, 53, 950 (1984) |
| [11] | Kawahara, T., J. Phys. Soc. Jpn, 33, 260 (1972) |
| [12] | Hunter, J. K.; Scheurle, J., Physica D, 32, 253 (1988) · Zbl 0694.35204 |
| [13] | Dai, X.; Dai, J., Phys. Lett. A, 142, 367 (1989) |
| [14] | Yamamoto, Y.; Takizawa, E. I., J. Phys. Soc. Jpn, 50, 1421 (1981) |
| [15] | Kudryashov, N. A., J. Appl. Math. Mech., 54, 372 (1990) |
| [16] | Kudryashov, N. A., Phys. Lett. A, 155, 269 (1991) |
| [17] | Yamamoto, Y.; Takizawa, E. I., J. Phys. Soc. Jpn, 50, 1055 (1981) |
| [18] | Kudryashov, N. A., J. Appl. Math. Mech., 52, 361 (1988) |
| [19] | Kudryashov, N. A., Phys. Lett. A, 147, 287 (1990) |
| [20] | Hooper, A. P.; Grimshaw, R., Phys. Fluids, 28, 37 (1985) · Zbl 0565.76051 |
| [21] | Conte, R.; Musette, M., J. Phys. A: Math. Gen., 22, 169 (1989) · Zbl 0687.35087 |
| [22] | Kuramoto, Y.; Tsuzuki, T., Prog. Theor. Phys., 55, 356 (1976) |
| [23] | Aspe, H.; Depassier, M. C., Phys. Rev. A, 41, 3125 (1990) |
| [24] | Porubov, A. V., J. Phys. A: Math. Gen., 26, L797 (1993) · Zbl 0803.35132 |
| [25] | Caudrey, P. J.; Dodd, R. K.; Gibbon, J. D., (Proc. Roy. Soc. A, 351 (1976)), 407 · Zbl 0346.35024 |
| [26] | Sawada, K.; Kotera, T., Prog. Theor. Phys., 51, 1355 (1974) · Zbl 1125.35400 |
| [27] | Kaup, D. J., Stud. Appl. Math., 62, 189 (1980) · Zbl 0431.35073 |
| [28] | Ito, M., J. Phys. Soc. Jpn, 49, 771 (1980) · Zbl 1334.35282 |
| [29] | Kichenassamy, S.; Olver, P. J., SIAM J. Math. Anal., 23, 1141 (1992) · Zbl 0755.76023 |
| [30] | Wang, X. Y., Phys. Lett. A, 131, 277 (1988) |
| [31] | Duffy, B. R.; Parkes, E. J., Phys. Lett. A, 214, 271 (1996) · Zbl 0972.35528 |
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