zbMATH — the first resource for mathematics

Maple packages for computing Hirota’s bilinear equation and multisoliton solutions of nonlinear evolution equations. (English) Zbl 1205.65281
Summary: The Hirota method for generating Hirota’s bilinear equation and constructing soliton solutions of nonlinear evolution equations is discussed and illustrated. Two Maple programs Bilinearization and Multisoliton are presented to automatically calculate Hirota’s bilinear equations for nonlinear evolution equations and to compute their \(N\)-soliton solutions for \(N\) = 1, 2 or 3, respectively. Different kinds of examples are used to demonstrate the effectiveness of the packages.

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
Full Text: DOI
[1] Hirota, R., The direct method in soliton theory, (2004), Cambridge University Press Cambridge
[2] Ito, M., An extension of nonlinear evolution equation of the KdV (mkdv) type to higher orders, J. phys. soc. jpn., 49, 771-778, (1980) · Zbl 1334.35282
[3] Hietarinta, J., A search for bilinear equations passing hirotas three-soliton condition. I. KdV-type bilinear equations, J. math. phys., 28, 8, 1732-1742, (1987) · Zbl 0641.35073
[4] Hietarinta, J., A search for bilinear equations passing hirota’s three-soliton condition. II. mkdv-type bilinear equations, J. math. phys., 28, 9, 2094-2101, (1987) · Zbl 0658.35081
[5] Hietarinta, J., A search for bilinear equations passing hirota’s three-soliton condition. III. sine – gordon-type bilinear equations, J. math. phys., 28, 11, 2586-2592, (1987) · Zbl 0658.35082
[6] Hietarinta, J., A search for bilinear equations passing hirota’s three-soliton condition. IV. complex bilinear equations, J. math. phys., 29, 3, 628-635, (1988) · Zbl 0684.35082
[7] J. Hietarinta, Introduction to the Hirota Bilinear Method, v1. <http://arXiv:solv-int/9708006>, 14 August, 1997. · Zbl 0907.58030
[8] Kazuhiro, N., Hirotas method and the singular manifold expansion, J. phys. soc. jpn., 56, 3052-3054, (1987)
[9] Ablowitz, M.J.; Satsuma, J., Solitons and rational solutions of nonlinear evolution equations, J. math. phys., 19, 10, 2180-2186, (1978) · Zbl 0418.35022
[10] Hirota, R., Soliton solutions of a coupled modified KdV equations, J. phys. soc. jpn., 66, 577-588, (1997) · Zbl 0946.35078
[11] Hereman, W.; Zhuang, W., A MACSYMA program for the Hirota method, (), 842-843
[12] Hereman, W.; Zhuang, W., Symbolic computation of solitons with macsyma, (), 287-296 · Zbl 0765.35048
[13] W. Hereman, W. Zhuang, Symbolic Computation of Solitons via Hirota’s Bilinear Method, Department of Mathematical and Computer Sciences, Colorado School of Mines Golden, Colorado, 1994.
[14] Hereman, W.; Zhuang, W., Symbolic software for soliton theory, Acta applicandae mathematicae, 39, 361-378, (1995) · Zbl 0838.35111
[15] Li, Z.B.; Liu, Y.P., RATH: a Maple package for finding travelling solitary wave solutions to nonlinear evolution equations, Comput. phys. commun., 148, 256-266, (2002) · Zbl 1196.35008
[16] XU, G.Q.; Li, Z.B., Symbolic computation of the Painlevé test for nonlinear partial differential equation using Maple, Comput. phys. commun., 161, 65-75, (2004) · Zbl 1196.35191
[17] Yao, R.X.; Li, Z.B., Conservation laws and new exact solutions for the generalized seventh order KdV equation, Chaos, soliton fractals, 20, 259-266, (2004) · Zbl 1046.35104
[18] Hietarinta, J., Searching for integrable PDE’s by testing hirota’s three-soliton condition, (), 295-300 · Zbl 0925.35148
[19] Wang, W.; Lian, X., A new algorithm for symbolic integral with application, Appl. math. comput., 162, 949-968, (2005) · Zbl 1076.65022
[20] Dye, J.M.; Parker, A., On bidirectional fifth-order nonlinear evolution equations, Lax pairs, and directionally dependent solitary waves, J. math. phys., 42, 6, 2567-2589, (2001) · Zbl 1061.37048
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.