×

Application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations. (English) Zbl 1246.35015

Summary: The application of the Kudryashov method [N. A. Kudryashov, J. Appl. Math. Mech. 54, No. 3, 372–375 (1990; Zbl 0736.76009)] for finding exact solutions of the high order nonlinear evolution equations is considered. Some classes of solitary wave solutions for the families of nonlinear evolution equations of fifth, sixth and seventh order are obtained. The efficiency of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations is demonstrated.

MSC:

35A25 Other special methods applied to PDEs
35C05 Solutions to PDEs in closed form
35G25 Initial value problems for nonlinear higher-order PDEs
35C08 Soliton solutions

Citations:

Zbl 0736.76009

Software:

ATFM
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Kudryashov, N. A., Exact soliton solutions of the generalized evolution equation of wave dynamics, J. Appl. Math. Mech., 52, 361-365 (1988)
[2] Kudryashov, N. A., Exact solutions of the generalized Kuramoto-Sivashinsky equation, Phys. Lett. A, 147, 287-291 (1990)
[3] Kudryashov, N. A., On types of nonlinear nonintegrable equations with exact solutions, Phys. Lett. A, 155, 269-275 (1991)
[4] Kudryashov, N. A., Singular manifold equations and exact solutions for some nonlinear partial differential equations, Phys. Lett. A, 182, 356-362 (1993)
[5] N.A. Kudryashov, Analytical theory of nonlinear differential equations. Moskow - Igevsk. Institute of computer investigations: 2004 [in Russian].; N.A. Kudryashov, Analytical theory of nonlinear differential equations. Moskow - Igevsk. Institute of computer investigations: 2004 [in Russian].
[6] N.A. Kudryashov, On one of methods for finding exact solutions of nonlinear differential equations, 16 Aug 2011. arXiv:1108.3288v1[nlin.SI]; N.A. Kudryashov, On one of methods for finding exact solutions of nonlinear differential equations, 16 Aug 2011. arXiv:1108.3288v1[nlin.SI]
[7] Parkes, E. J., Exact solutions to the two-dimensional Korteweg-de Vries-Burgers equation, J. Phys. A: Math. Gen., 27, L497-L501 (1994), 27 · Zbl 0846.35122
[8] Parkes, E. J.; Duffy, B. R., An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations, Comput. Phys. Commun., 98, 288-300 (1996) · Zbl 0948.76595
[9] Malfliet, W.; Hereman, W., The tanh method: I. Exact solutions of nonlinear evolution and wave equations, Phys. Scr., 54, 563-568 (1996) · Zbl 0942.35034
[10] Fan, E., Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A, 277, 212-218 (2000) · Zbl 1167.35331
[11] Biswas, A., Solitary wave solution for the generalized Kawahara equation, Appl. Math. Lett., 22, 208-210 (2009) · Zbl 1163.35468
[12] Vitanov, N. K.; Dimitrova, Z. I.; Kantz, H., Modified method of simplest equation and its application to nonlinear PDEs, Appl. Math. Comput., 216, 2587-2595 (2010) · Zbl 1195.35272
[13] Kudryashov, N. A.; Louginova, N. B., Extended simplest equation method for nonlinear differential equations, Appl. Math. Comput., 205, 396-402 (2008) · Zbl 1168.34003
[14] Ryabov, P. N., Exact solutions of the Kudryashov-Sinelshchikov equation, Appl. Math. Comput., 217, 3585-3590 (2010) · Zbl 1205.35272
[15] M.M. Kabir, A. Khajeh, E. Abdi Aghdam, A. Yousefi Koma. Modified Kudryashov method for finding exact solitary wave solutions of higher-order nonlinear equations, Math. Meth. Appl. Sci. doi:10.1002/mma.1349; M.M. Kabir, A. Khajeh, E. Abdi Aghdam, A. Yousefi Koma. Modified Kudryashov method for finding exact solitary wave solutions of higher-order nonlinear equations, Math. Meth. Appl. Sci. doi:10.1002/mma.1349 · Zbl 1206.35063
[16] Gepreel, K. A.; Omran, S.; Elagan, S. K., The traveling wave solutions for some nonlinear PDEs in mathematical physics, Appl. Math., 2, 343-347 (2011)
[17] Nikolaevskii, V. N., Dynamics of Viscoelastic Media with Internal Oscillators, (Koh, S. L.; Speciale, C. G., Recent Adv. Eng. Sci. (1989), Springer: Springer Berlin), 210-221
[18] Tribelsky, M. I.; Tsuboi, K., New scenario for transition to turbulence?, Phys. Rev. Lett., 76, 1631-1634 (1996)
[19] Tanaka, D., Chemical turbulence equivalent to Nikolaevskii turbulence, Phys. Rev. E (2004), (015202-1-015202-4)
[20] Kudryashov, N. A.; Migita, A. V., Periodic structures developing with account for dispersion in a turbulence model, Fluid Dyn., 42, 463-471 (2007) · Zbl 1354.76079
[21] Kudryashov, N. A., Nonlinear differential equations with exact solutions expressed via the Weierstrass function, Zeitschrift fur Naturforschung, 59, 443-454 (2004)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.