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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:
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
Software:
SYMMGRP
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[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) · Zbl 1334.35282
[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), Cambridge University Press Cambridge
[5] Hirota, R., Exact solutions of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. rev. lett., 27, 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, 5, 1456-1458, (1972)
[7] Hirota, R., Exact solutions of the sine – gordon equation for multiple collisions of solitons, J. phys. soc. jpn., 33, 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, 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, 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, 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, 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, 3, 713-723, (2004) · Zbl 1054.65106
[20] Wazwaz, A.M., Partial differential equations: methods and applications, (2002), Balkema Publishers The Netherlands · Zbl 0997.35083
[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, 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, 2, 321-330, (2002) · Zbl 1028.35131
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