Abstract
This paper is concerned with the variable-coefficient Gardner (vc-Gardner) types of equations, which arise in fluid dynamics, nonlinear lattice and plasma physics. As its special cases, the generalized cylindrical KdV types of equations are considered simultaneously. By using the combination of Painlevé analysis and Lie group classification method, the integrable conditions, Bäcklund transformations and complete group classifications of the vc-Gardner types of equations are obtained. Then, the exact solutions generated from the Painlevé analysis and symmetry reductions are investigated.
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Liu, H., Li, J., Liu, L.: Painlevé analysis, Lie symmetries, and exact solutions for the time-dependent coefficients Gardner equations. Nonlinear Dyn. 59, 497–502 (2010)
Clarkson, P.: Painlevé analysis and the complete integrability of a generalized variable-coefficient Kadomtsev–Petviashvili equation. IMA J. Appl. Math. 44, 27–53 (1990)
Bluman, G., Cheviakov, A., Anco, S.: Applications of symmetry methods to partial differential equations. In: Antman, S., Marsden, J., Sirovich, L. (eds.) Applied Mathematical Sciences, vol. 168. Springer, New York (2010)
Li, J., Xu, T., Meng, X., Zhang, Y., Zhang, H., 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)
Zhang, Y., Li, J., Lv, Y.: The exact solution and integrable properties to the variable-coefficient modified Korteweg–de Vires equation. Ann. Phys. 323, 3059–3064 (2008)
Gardner, C., Greene, J., Kruskal, M., Miura, R.: Method for solving the Korteweg–de Vries equation. Phys. Rev. Lett. 19, 1095–1097 (1967)
Ablowitz, M., Segur, H.: Soliton and the Inverse Scattering Transform. SIAM, Philadelphia (1981)
Matveev, V., Salle, M.: Darboux Transformations and Solitons. Springer, Berlin (1991)
Hirota, R., Satsuma, J.: A variety of nonlinear network equations generated from the Bäcklund transformation for the Toda lattice. Suppl. Prog. Theor. Phys. 59, 64–100 (1976)
Liu, H., Li, J., Chen, F.: Exact periodic wave solutions for the hKdV equation. Nonlinear Anal. TMA 70, 2376–2381 (2009)
Olver, P.: Applications of Lie groups to differential equations. In: Halmos, P., Gehring, F., Moore, C. (eds.) Graduate Texts in Mathematics, vol. 107. Springer, New York (1993)
Pucci, E., Saccomandi, G.: Potential symmetries and solutions by reduction of partial differential equations. J. Phys. A Math. Gen. 26, 681–690 (1993)
Qu, C., Huang, Q.: Symmetry reductions and exact solutions of the affine heat equation. J. Math. Anal. Appl. 346, 521–530 (2008)
Chaolu, T., Pang, J.: An algorithm for the complete symmetry classification of differential equations based on Wu’s method. J. Eng. Math. 66, 181–199 (2010)
Liu, H., Li, J., Liu, L.: Lie group classifications and exact solutions for two variable-coefficient equations. Appl. Math. Comput. 215, 2927–2935 (2009)
Liu, H., Li, J., Liu, L., Wei, Y.: Group classifications, optimal systems and exact solutions to the generalized Thomas equations. J. Math. Anal. Appl. 383, 400–408 (2011)
Liu, H., Li, J., Liu, L.: Complete group classification and exact solutions to the generalized short pulse equation. Stud. Appl. Math. 129, 103–116 (2012)
Liu, H., Geng, Y.: Symmetry reductions and exact solutions to the systems of carbon nanotubes conveying fluid. J. Differ. Equ. 254, 2289–2303 (2013)
Clarkson, P., Kruskal, M.: New similarity reductions of the Boussinesq equation. J. Math. Phys. 30, 2201–2213 (1989)
Li, J., Liu, Z.: Smooth and non-smooth traveling waves in a nonlinearly dispersive equation. Appl. Math. Model. 25, 41–56 (2000)
Liu, H., Li, J.: Symmetry reductions, dynamical behavior and exact explicit solutions to the Gordon types of equations. J. Comput. Appl. Math. 257, 144–156 (2014)
Weiss, J.: The Painlevé property for partial differential equations. II: Bäcklund transformation, Lax pairs, and the Schwarzian derivative. J. Math. Phys. 24, 1405–1413 (1983)
Newell, A., Tabor, M., Zeng, Y.: A unified approach to Painlevé expansions. Phys. D 29, 1–68 (1987)
Conte, R., Musette, M.: The Painlevé Handbook. Springer, Dordrecht (2008)
Liu, H., Li, J.: Lie symmetry analysis and exact solutions for the short pulse equation. Nonlinear Anal. TMA 71, 2126–2133 (2009)
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)
Byrd, P., Friedman, M.: Handbook of Elliptic Integrals for Engineers and Scientists. Springer, Berlin (1971)
Wang, Z., Guo, D.: Introduction to special functions. In: Gao, C. (ed.) The Series of Advanced Physics of Peking University. Peking University Press, Beijing (2000) (in Chinese)
Fokas, A.S.: Symmetries and integrability. Stud. Appl. Math. 77, 253–299 (1987)
Zakharov, V. (ed.): What is Integrability. Springer, Berlin (1991)
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The authors are grateful to the editors and anonymous reviewers for their valuable comments and suggestions.
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This work is supported by the National Natural Science Foundation of China under Grant No. 11171041 and the doctorial foundation of Liaocheng University under Grant No. 31805.
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Liu, H., Li, J. Painlevé analysis, complete Lie group classifications and exact solutions to the time-dependent coefficients Gardner types of equations. Nonlinear Dyn 80, 515–527 (2015). https://doi.org/10.1007/s11071-014-1885-0
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DOI: https://doi.org/10.1007/s11071-014-1885-0
Keywords
- Painlevé analysis
- Integrable condition
- Bäcklund transformation
- Complete group classification
- Symmetry reduction
- Exact solution