×

Polygons of differential equations for finding exact solutions. (English) Zbl 1133.35084

The authors present a method for finding exact solutions of nonlinear differential equations. Their method is based on application of polygons corresponding to nonlinear differential equations. Main aim is to express solution of the equation studied through solutions of the simplest equations. The method is based on the idea of power geometry. The method can be also applied for finding transformations between solutions of differential equations. To demonstrate the method application exact solutions of several equations are found. These equations are: the KdV-Burgers equation, the generalized Kuramoto-Sivashinsky equation, the fourth-order nonlinear evolution equation, the fifth-order KdV equation, the fifth-order modified KdV equation and the sixth-order nonlinear evolution equation describing turbulent processes. Some new exact solutions of nonlinear evolution equations are given.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35C05 Solutions to PDEs in closed form
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Gardner, C. S.; Greene, J. M.; Kruskal, M. D.; Miura, R. R., Phys Rev Lett, 19, 1095-1097 (1967) · Zbl 1103.35360
[2] Ablowitz, M. J.; Kaup, D. J.; Newell, A. C.; Segur, H., Phys Rev Lett, 31, 125 (1973) · Zbl 1243.35143
[3] Ablowitz, M. J.; Clarcson, P. A., Solitons, nonlinear evolution equations and inverse scattering (1991), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0762.35001
[4] Hirota, R., Phys Rev Lett, 27, 1192-1194 (1971) · Zbl 1168.35423
[5] Kudryashov, N. A., Analytical theory of nonlinear differential equations (2004), Institute of Computer Investigations: Institute of Computer Investigations Moscow-Izhevsk, p. 360 [in Russian]
[6] Polyanin, A. D.; Zaitsev, V. F.; Zhurov, A. I., Methods of solving nonlinear equations of mathematical physics and mekhanics (2005), Fismatlit: Fismatlit Moscow, p. 256 [in Russian]
[7] Weiss, J.; Tabor, M.; Carnevalle, G., J Math Phys, 24, 522 (1983) · Zbl 0514.35083
[8] Conte, R.; Musette, M., J Phys A: Math Gen, 22, 169-177 (1989) · Zbl 0687.35087
[9] Choudhary, S. R., Phys Lett A, 159, 311-317 (1991)
[10] Kudryashov, N. A., J Appl Math Mekh, 52, 361-365 (1988)
[11] Kudryashov, N. A., Rep USSR Acad Sci, 308, 294-298 (1989), [in Russian]
[12] Kudryashov, N. A., Phys Lett A, 155, 269-275 (1991)
[13] Kudryashov, N. A., Phys Lett A, 147, 287-291 (1990)
[14] Kudryashov, N. A., J Appl Math Mekh, 54, 372-376 (1990)
[15] Yan, Z. Y., Chaos, Solitons & Fractals, 21, 1013 (2004) · Zbl 1046.35103
[16] Lou, S. Y.; Huang, G.; Ruan, H., J Phys A: Math Gen, 24, 587-590 (1991) · Zbl 0735.76057
[17] Parkes, E. J.; Duffy, B. R., Comput Phys Commun, 98, 288-300 (1996) · Zbl 0948.76595
[18] Elwakil, S. A.; El-labany, S. K.; Zahran, M. A.; Sabry, R., Phys Lett A, 299, 179-188 (2002) · Zbl 0996.35043
[19] Fan, E. G., Phys Lett A, 277, 212-218 (2000) · Zbl 1167.35331
[20] Fan, E. G., Phys Lett A, 282, 18-22 (2002) · Zbl 0984.37092
[21] Kudryashov, N. A.; Zargaryan, E. D., J Phys A: Math Gen, 29, 8067-8077 (1996) · Zbl 0901.35090
[22] Liu, G. T.; Fan, T. Y., Phys Lett A, 345, 161-166 (2005) · Zbl 1345.35091
[23] Liu, S. K.; Fu, Z. T.; Liu, S. D.; Zhao, Q., Phys Lett A, 289, 69-74 (2001) · Zbl 0972.35062
[24] Fu, Z.; Zhang, L.; Liu, S.; Liu, S., Phys Lett A, 325, 363-369 (2004) · Zbl 1161.35390
[25] Fu, Z.; Liu, S.; Liu, S., Phys Lett A, 326, 364-374 (2004) · Zbl 1138.35393
[26] Yan, C. T., Phys Lett A, 224, 77 (1996) · Zbl 1037.35504
[27] Kudryashov, N. A., Phys Lett A, 324, 99-106 (2005) · Zbl 1222.35054
[28] Kudryashov, N. A., Chaos, Solitons & Fractals, 24, 1217-1231 (2005) · Zbl 1069.35018
[29] Bruno, A. D., Power geometry in algebraic and differential equations (1998), Nauka, Fizmatlit: Nauka, Fizmatlit Moscow, p. 288 [in Russian] · Zbl 0903.34001
[30] Bruno, A. D., Russ Math Surveys, 59, 429-480 (2004) · Zbl 1068.34054
[31] Kudryashov, N. A.; Efimova, O. Yu., Chaos, Solitons & Fractals, 30, 1, 110-124 (2006) · Zbl 1157.34066
[32] Demina MV, Kudryashov NA. Chaos, Solitons & Fractals, in press. doi:10.1016/j.chaos.2005.10.079; Demina MV, Kudryashov NA. Chaos, Solitons & Fractals, in press. doi:10.1016/j.chaos.2005.10.079
[33] Weiss, J., J Math Phys, 24, 1405 (1983) · Zbl 0531.35069
[34] Kuramoto, Y.; Tsuzuki, T., Prog Theor Phys, 55, 356 (1976)
[35] Sivashinsky, G. I., Physica D, 4, 227-235 (1982) · Zbl 1194.76054
[36] Eremenko A. arXiv: nlin.SI/0504053; Eremenko A. arXiv: nlin.SI/0504053
[37] Hone, A. N.W., Physica D, 205, 292-306 (2005) · Zbl 1093.34009
[38] Hone, A. N.W., Physica D, 118, 1-16 (1998) · Zbl 0937.37037
[39] Kudryashov, N. A., Phys Lett A, 224, 6, 353-360 (1997) · Zbl 0962.35504
[40] Kudryashov, N. A.; Soukharev, M. B., Phys Lett A, 237, 206-216 (1998) · Zbl 0941.34077
[41] Kudryashov, N. A., Math Model, 1, 1-6 (1989), [in Russian]
[42] Beresnev, L. A.; Nikolaevskiy, V. N., Physica D, 66, 206-216 (1993)
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.