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The Hirota’s direct method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Ito seventh-order equation. (English) Zbl 1153.65363
Summary: The Sawada-Kotera-Ito seventh-order equation is studied. The tanh-coth method is applied to obtain soliton solution of this equation. The Hirota’s direct method combined with the simplified Hereman’s method are applied to determine the $N$-soliton solutions for this equation. The study highlights the significant features of the employed methods and its capability of handling completely integrable equations.

MSC:
65M70Spectral, collocation and related methods (IVP of PDE)
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SYMMGRP
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References:
[1] Sawada, K.; Kotera, T.: A method for finding N-soliton solutions of the KdV equation and KdV-like equation. Prog. theor. Phys. 51, 355-1367 (1974) · Zbl 1125.35400
[2] Ito, M.: An extension of nonlinear evolution equation of the K-dv (mK-dv) type to higher orders. J. phys. Soc. jpn. 49, 771-778 (1980)
[3] Goktas, U.; Hereman, W.: Symbolic computation of conserved densities for systems of nonlinear evolution equations. J. symb. Comput. 11, 1-31 (1999)
[4] Hirota, R.: The direct method in soliton theory. (2004) · Zbl 1099.35111
[5] Hirota, R.: Exact solutions of the Korteweg-de Vries equation for multiple collisions of solitons. Phys. rev. Lett. 27, No. 18, 1192-1194 (1971) · Zbl 1168.35423
[6] Hirota, R.: Exact solutions of the modified Korteweg-de Vries equation for multiple collisions of solitons. J. phys. Soc. jpn. 33, No. 5, 1456-1458 (1972)
[7] Hirota, R.: Exact solutions of the sine -- Gordon equation for multiple collisions of solitons. J. phys. Soc. jpn. 33, No. 5, 1459-1463 (1972)
[8] Hietarinta, J.: A search for bilinear equations passing Hirota’s three-soliton condition. I. KdV-type bilinear equations. J. math. Phys. 28, No. 8, 1732-1742 (1987) · Zbl 0641.35073
[9] Hietarinta, J.: A search for bilinear equations passing Hirota’s three-soliton condition. II. mkdv-type bilinear equations. J. math. Phys. 28, No. 9, 2094-2101 (1987) · Zbl 0658.35081
[10] Hereman, W.; Zhaung, W.: Symbolic software for soliton theory, acta applicandae mathematicae. Phys. lett. A 76, 95-96 (1980)
[11] Hereman, W.; Nuseir, A.: Symbolic methods to construct exact solutions of nonlinear partial differential equations. Math. comput. Simulat. 43, 13-27 (1997) · Zbl 0866.65063
[12] Weiss, J.: On classes of integrable systems and the Painlevé property. J. math. Phys. 25, No. 1, 13-24 (1984) · Zbl 0565.35094
[13] Goktas, U.; Hereman, W.: Symbolic computation of conserved densities for systems of nonlinear evolution equations. J. symb. Comput. 11, 1-31 (1999)
[14] Dodd, R. K.; Gibbon, J. D.: The prolongation structure of a higher order Korteweg-de Vries equations. Proc. roy. Soc. lond. A 358, 287-300 (1977) · Zbl 0376.35009
[15] Malfliet, W.: The tanh method: a tool for solving certain classes of nonlinear evolution and wave equations. J. comput. Appl. math. 164 -- 165, 529-541 (2004) · Zbl 1038.65102
[16] Malfliet, W.: Solitary wave solutions of nonlinear wave equations. Am. J. Phys. 60, No. 7, 650-654 (1992) · Zbl 1219.35246
[17] Malfliet, W.; Hereman, Willy: The tanh method: I. Exact solutions of nonlinear evolution and wave equations. Phys. scripta 54, 563-568 (1996) · Zbl 0942.35034
[18] Malfliet, W.; Hereman, Willy: The tanh method: II. Perturbation technique for conservative systems. Phys. scripta 54, 569-575 (1996) · Zbl 0942.35035
[19] Wazwaz, A. M.: The tanh method for travelling wave solutions of nonlinear equations. Appl. math. Comput. 154, No. 3, 713-723 (2004) · Zbl 1054.65106
[20] Wazwaz, A. M.: Partial differential equations: methods and applications. (2002) · Zbl 1079.35001
[21] Wazwaz, A. M.: The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations. Appl. math. Comput. 184, No. 2, 1002-1014 (2007) · Zbl 1115.65106
[22] Wazwaz, A. M.: The tanh -- coth method for solitons and kink solutions for nonlinear parabolic equations. Appl. math. Comput. 188, 1467-1475 (2007) · Zbl 1119.65100
[23] Wazwaz, A. M.: New solitary-wave special solutions with compact support for the nonlinear dispersive $K(m,n)$ equations. Chaos solitons fract. 13, No. 2, 321-330 (2002) · Zbl 1028.35131