Painlevé analysis, Lie symmetries, and exact solutions for the time-dependent coefficients gardner equations. (English) Zbl 1183.35236

Summary: In this paper, the three variable-coefficient Gardner (vc-Gardner) equations are considered. By using the Painlevé analysis and Lie group analysis method, the Painlevé properties and symmetries for the equations are obtained. Then the exact solutions generated from the symmetries and Painlevé analysis are presented.


35Q51 Soliton equations
76B25 Solitary waves for incompressible inviscid fluids
Full Text: DOI


[1] Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M.: Method for solving the Korteweg-de Vries equation. Phys. Rev. Lett. 19, 1095–1097 (1967) · Zbl 1103.35360
[2] Li, Y.S.: Soliton and integrable systems. In: Advanced Series in Nonlinear Science. Shanghai Scientific and Technological Education Publishing House, Shanghai (1999) (in Chinese)
[3] Hirota, R., Satsuma, J.: A variety of nonlinear network equations generated from the Bäcklund transformation for the Tota lattice. Suppl. Prog. Theor. Phys. 59, 64–100 (1976) · Zbl 1079.35536
[4] Liu, H., Li, J., Chen, F.: Exact periodic wave solutions for the hKdV equation. Nonlinear Anal. 70, 2376–2381 (2009) · Zbl 1162.35312
[5] Olver, P.J.: Applications of Lie groups to differential equations. In: Grauate Texts in Mathematics, vol. 107. Springer, New York (1993) · Zbl 0785.58003
[6] Bluman, G.W., Kumei, S.: Symmetries and Differential Equations. Springer/World Publishing Corp., Berlin (1989) · Zbl 0698.35001
[7] Cantwell, B.J.: Introduction to Symmetry Analysis. Cambridge University Press, Cambridge (2002) · Zbl 1082.34001
[8] Sinkala, W., Leach, P.G.L., O’Hara, J.G.: Invariance properties of a general-pricing equation. J. Differ. Equ. 244, 2820–2835 (2008) · Zbl 1147.91017
[9] Liu, H., Li, J., Zhang, Q.: Lie symmetry analysis and exact explicit solutions for general Burgers’ equation. J. Comput. Appl. Math. 228, 1–9 (2009) · Zbl 1166.35033
[10] Liu, H., Li, J.: Lie symmetry analysis and exact solutions for the short pulse equation. Nonlinear Anal. 71, 2126–2133 (2009) · Zbl 1244.35003
[11] Liu, H., Li, J.: Lie symmetry analysis and exact solutions for the extended mKdV equation. Acta Appl. Math. (2008). doi: 10.1007/s10440-008-9362-8
[12] Liu, H., Li, J.: Lie symmetries, conservation laws and exact solutions for two rod equations. Acta Appl. Math. (2009). doi: 10.1007/s10440-009-9462-0
[13] Clarkson, P., Kruskal, M.: New similarity reductions of the Boussinesq equation. J. Math. Phys. 30(10), 2201–2213 (1989) · Zbl 0698.35137
[14] Clarkson, P.: New similarity reductions for the modified Boussinesq equation. J. Phys. A: Gen. 22, 2355–2367 (1989) · Zbl 0704.35116
[15] Nirmala, N., Vedan, M.J., Baby, B.V.: A variable coefficient Korteweg-de Vires equation: Similarity analysis and exact solution. II. J. Math. Phys. 27(11), 2644–2646 (1986) · Zbl 0632.35062
[16] Li, J., Xu, T., Meng, X.-H., Zhang, Y.-X., Zhang, H.-Q., Tian, B.: Lax pair, Bäcklund transformation and N-soliton-like solution for a variable-coefficient Gardner equation from nonlinear lattice plasma physics and ocean dynamics with symbolic computation. J. Math. Anal. Appl. 336, 1443–1455 (2007) · Zbl 1128.35378
[17] Zhang, Y., Li, J., Lv, Y.-N.: The exact solution and integrable properties to the variable-coefficient modified Korteweg-de Vires equation. Ann. Phys. 323, 3059–3064 (2008) · Zbl 1161.35046
[18] Wiss, J.: The Painlevé property for partial differential equations, II: Bäcklund transformation, Lax pairs, and the Schwarzian derivative. J. Math. Phys. 24, 1405–1413 (1983) · Zbl 0531.35069
[19] Ramani, A., Grammaticos, B.: The Painlevé property and singularity analysis of integrable and non-integrable systems. Phys. Rep. 180, 159–245 (1989)
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.