×

zbMATH — the first resource for mathematics

Simplest equation method to look for exact solutions of nonlinear differential equations. (English) Zbl 1069.35018
Summary: A new method is presented for the search of exact solutions of nonlinear differential equations. Two basic ideas are in the focus of our approach. One of them is to use the general solutions of the simplest nonlinear differential equations. Another idea is to take into consideration all possible singularities of the studied equation. Applications of our approach to search for exact solutions of nonlinear differential equations is discussed in detail. The method is used to investigate the exact solutions of the Kuramoto-Sivashinsky equation and the equation for the description of nonlinear waves in a convective fluid. New exact solitary and periodic waves of these equations are given.

MSC:
35C05 Solutions to PDEs in closed form
35Q53 KdV equations (Korteweg-de Vries equations)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Gardner, C.S.; Greene, J.M.; Kruskal, M.D.; Miura, R.R., Phys. rev. lett., 19, 1095-1097, (1967)
[2] Hirota, R., Phys. rev. lett., 27, 1192-1194, (1971)
[3] Weiss, J.; Tabor, M.; Carnevalle, G., J. math. phys., 24, 522-526, (1983)
[4] Kudryashov, N.A., J. appl. math. mech., 52, 361-365, (1988)
[5] Conte, R.; Musette, M., J. phys. A.: math. gen., 22, 169-177, (1989)
[6] Choudhury, S.R., Phys. lett. A., 159, 311-317, (1997)
[7] Kudryashov, N.A., Phys lett. A., 155, 269-275, (1991)
[8] Musette, M.; Conte, R., Physica D, 181, 70-79, (2003)
[9] Lou, S.Y.; Huang, G.; Ruan, H., J. phys. A.: math. gen., 24, 587-590, (1991)
[10] Kudryashov, N.A.; Zargaryan, E.D., J. phys. A. math gen., 29, 8067-8077, (1996)
[11] Fan, E.G., Phys lett. A., 227, 212-218, (2000)
[12] Elwakil, S.A.; Ellabany, S.K.; Zahran, M.A., Phys. lett. A., 299, 179-188, (2002)
[13] Liu, S.K.; Fu, Z.T.; Liu, S.D.; Zhao, Q., Phys. lett. A., 289, 69-74, (2001)
[14] Yan, Z.Y., Chaos, solitons & fractals, 15, 3, 575-583, (2003)
[15] Kudryashov, N.A., Phys lett. A., 147, 287-291, (1990)
[16] Porubov, A.V., J. phys. A.: math. gen., 26, L707-L800, (1993)
[17] Landa, P.S., Nonlinear oscillations and waves in dynamical syctems, (1996), Kluwer Academic Publishers, p. 538 · Zbl 0873.34003
[18] Kuramoto, Y.; Tsuzuki, T., Prog. theor. phys., 55, 356, (1976)
[19] Sivashinsky, G.I., Physica D, 4, 227-235, (1982)
[20] Aspe, H.; Depassier, M.C., Phys. rev. A., 41, 3125, (1990)
[21] Garazo, A.; Velarde, M.G., Phys. fluids A, 3, 2295, (1991)
[22] Akhiezer, N.I., Elements of theory of elliptic functions, (1948), OGIZ Moscow, p. 292 (in Russian) · Zbl 0694.33001
[23] Conte, R., The painleve property, one century later, CRM series in mathematical physics, (1999), Springer-Verlag New York, pp. 77-180
[24] Hopf, E., Commun. pure appl. math., 3, 201-230, (1950)
[25] Cole, J.D., Quart. appl. math., 9, 225-236, (1951)
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.