广义时变系数Gardner方程的Painlevé分析、李对称和精确解
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摘要
运用Painlevé分析与李对称分析得到该时变系数Gardner方程的可积条件及其在不同条件下的对称,并给出对应的动力学向量场,进而分别基于Painlevé分析和对称约化的思想,将时变系数Gardner方程转化为常系数方程,并结合幂级数法求解约化方程的精确解,得到时变系数Gardner方程的若干精确解。
A generalized Gardner equation with time-dependent coefficients is investigated in this paper, which arise in fluid dynamics, nonlinear lattice and plasma physics. By applying the combination of Painlevé analysis and Lie symmetry analysis method, the integrable conditions, symmetries and corresponding geometric vector fields of the time-dependent coefficient Gardner equation are investigated. Moreover, based on Painlevé analysis and the idea of symmetry reduction, the partial differential equations are reduced to ordinary differential equations. Combined with power series method, exact solutions to the reduced equations and a series of exact solutions to the original equations are obtained.
A generalized Gardner equation with time-dependent coefficients is investigated in this paper, which arise in fluid dynamics, nonlinear lattice and plasma physics. By applying the combination of Painlevé analysis and Lie symmetry analysis method, the integrable conditions, symmetries and corresponding geometric vector fields of the time-dependent coefficient Gardner equation are investigated. Moreover, based on Painlevé analysis and the idea of symmetry reduction, the partial differential equations are reduced to ordinary differential equations. Combined with power series method, exact solutions to the reduced equations and a series of exact solutions to the original equations are obtained.
引文
[1] GARDNER C, GREENE J, KRUSKAL M, et al. Method for solving the Korteweg-de Vries equation[J]. Physical Review Letters, 1967, 19:1095-1097.
[2] ABLOWITZ M, SEGUR H. Soliton and the inverse scattering transform[M]. Philadelphia: SIAM, 1981.
[3] MATVEEV V, SALLE M. Darboux transformations and solitons[M]. Berlin: Springer, 1991.
[4] HIROTA R, SATSUMA J. Avariety of nonlinear network equations generated from the B?cklund transformation for the Toda lattice[J]. Progress of Theoretical Physics Supplement, 1976, 59:64-100.
[5] OLVER P. Applications of Lie groups to differential equations[M]. New York: Springer, 1993.
[6] PUCCI E, SACCOMANDI G. Potential symmetries and solutions by reduction of partial differential equations[J]. Journal of Physics A General Physics, 1993, 26(3):681-690.
[7] QU Changzheng, HUANG Qing. Symmetry reductions and exact solutions of the affine heat equation[J]. Journal of Mathematical Analysis and Applications, 2008, 346(2):521-530.
[8] 李玉, 刘希强. 扩展的KP-Benjamin-Bona-Mahoney方程的对称、约化和精确解[J]. 山东大学学报(理学版), 2017, 52(2):77-84.LI Yu, LIU Xiqiang. Symmetry, reduction and exact solutions of the extended KP-Benjamin-Bona-Mahoney equation[J]. Journal of Shandong University(Natural Science), 2017, 52(2):77-84.
[9] ClARKSON P, KRUSKAL M. New similarity reductions of the Boussinesq equation[J]. Journal of Mathematical Physics, 1989, 30(10):2201-2213.
[10] LI Jibin, LIU Zhengrong. Smooth and non-smooth traveling waves in a nonlinearly dispersive equation[J]. Applied Mathematical Modelling, 2000, 25(1):41-56.
[11] LIU Hanze, LI Jibin. Symmetry reductions, dynamical behavior and exact explicit solutions to the Gordon types of equations[J]. Journal of Computational and Applied Mathematics, 2014, 257:144-156.
[12] WEISS J, TABOR M, CARNEVALE G. The Painlevé property for partial differential equations. II: B?cklund transformation, Lax pairs, and the Schwarzian derivative[J]. Journal of Mathematical Physics, 1983, 24(3):1405-1413.
[13] NEWELL A, TABOR M, ZENG Yanbo. A unified approach to Painlev?e expansions[J]. Physica D: Nonlinear Phenomena, 1987, 29(1):1-68.
[14] CONTE R, MUSETTE M. The Painlevé handbook[M]. Dordrecht: Springer, 2008.
[15] CLARKSON P. Painlev?e analysis and the complete integrability of a generalized variable-coefficient Kadomtsev-Petviashvili equation[J]. Ima Journal of Applied Mathematics, 1990, 44(1):27-53.
[16] LI Juan, XU Tao, MENG Xianghua, et al. Lax pair, B?cklund transformation and N-soliton-like solution for a variable-coefficient Gardner equation from nonlinear lattice plasma physics and ocean dynamics with symbolic computation[J]. Journal of Mathematical Analysis and Applications, 2007, 336(2):1443-1455.
[17] ZHANG Yi, LI Jibin, LYU Yineng. The exact solution and integrable properties to the variable-coefficient modified Korteweg-de Vires equation[J]. Annals of Physics, 2008, 323(12):3059-3064.
[18] LIU Hanze, LI Jibin, LIU Lei. Painlevé analysis, Lie symmetries, and exact solutions for the time-dependent coefficients Gardner equations[J]. Nonlinear Dynamics, 2010, 59(3):497-502.
[19] LIU Hanze, LI Jibin. Painlevé analysis, complete Lie group classifications and exact solutions to the time-dependent coefficients Gardner types of equations[J]. Nonlinear Dynamics, 2015, 80(1):515-527.
[20] LIU Hanze, YUE Chao. Lie symmetries, integrable properties and exact solutions to the variable-coefficient nonlinear evolution equations[J]. Nonlinear Dynamics, 2017, 89(3):1989-2000.
[21] ASAMR N H. Partial differential equations with Fourier series and boundary value problem[M]. Beijing: China Machine Press, 2005.
[22] RUDIN W. Principles of mathematical analysis[M]. Beijing: China Machine Press, 2004.
[2] ABLOWITZ M, SEGUR H. Soliton and the inverse scattering transform[M]. Philadelphia: SIAM, 1981.
[3] MATVEEV V, SALLE M. Darboux transformations and solitons[M]. Berlin: Springer, 1991.
[4] HIROTA R, SATSUMA J. Avariety of nonlinear network equations generated from the B?cklund transformation for the Toda lattice[J]. Progress of Theoretical Physics Supplement, 1976, 59:64-100.
[5] OLVER P. Applications of Lie groups to differential equations[M]. New York: Springer, 1993.
[6] PUCCI E, SACCOMANDI G. Potential symmetries and solutions by reduction of partial differential equations[J]. Journal of Physics A General Physics, 1993, 26(3):681-690.
[7] QU Changzheng, HUANG Qing. Symmetry reductions and exact solutions of the affine heat equation[J]. Journal of Mathematical Analysis and Applications, 2008, 346(2):521-530.
[8] 李玉, 刘希强. 扩展的KP-Benjamin-Bona-Mahoney方程的对称、约化和精确解[J]. 山东大学学报(理学版), 2017, 52(2):77-84.LI Yu, LIU Xiqiang. Symmetry, reduction and exact solutions of the extended KP-Benjamin-Bona-Mahoney equation[J]. Journal of Shandong University(Natural Science), 2017, 52(2):77-84.
[9] ClARKSON P, KRUSKAL M. New similarity reductions of the Boussinesq equation[J]. Journal of Mathematical Physics, 1989, 30(10):2201-2213.
[10] LI Jibin, LIU Zhengrong. Smooth and non-smooth traveling waves in a nonlinearly dispersive equation[J]. Applied Mathematical Modelling, 2000, 25(1):41-56.
[11] LIU Hanze, LI Jibin. Symmetry reductions, dynamical behavior and exact explicit solutions to the Gordon types of equations[J]. Journal of Computational and Applied Mathematics, 2014, 257:144-156.
[12] WEISS J, TABOR M, CARNEVALE G. The Painlevé property for partial differential equations. II: B?cklund transformation, Lax pairs, and the Schwarzian derivative[J]. Journal of Mathematical Physics, 1983, 24(3):1405-1413.
[13] NEWELL A, TABOR M, ZENG Yanbo. A unified approach to Painlev?e expansions[J]. Physica D: Nonlinear Phenomena, 1987, 29(1):1-68.
[14] CONTE R, MUSETTE M. The Painlevé handbook[M]. Dordrecht: Springer, 2008.
[15] CLARKSON P. Painlev?e analysis and the complete integrability of a generalized variable-coefficient Kadomtsev-Petviashvili equation[J]. Ima Journal of Applied Mathematics, 1990, 44(1):27-53.
[16] LI Juan, XU Tao, MENG Xianghua, et al. Lax pair, B?cklund transformation and N-soliton-like solution for a variable-coefficient Gardner equation from nonlinear lattice plasma physics and ocean dynamics with symbolic computation[J]. Journal of Mathematical Analysis and Applications, 2007, 336(2):1443-1455.
[17] ZHANG Yi, LI Jibin, LYU Yineng. The exact solution and integrable properties to the variable-coefficient modified Korteweg-de Vires equation[J]. Annals of Physics, 2008, 323(12):3059-3064.
[18] LIU Hanze, LI Jibin, LIU Lei. Painlevé analysis, Lie symmetries, and exact solutions for the time-dependent coefficients Gardner equations[J]. Nonlinear Dynamics, 2010, 59(3):497-502.
[19] LIU Hanze, LI Jibin. Painlevé analysis, complete Lie group classifications and exact solutions to the time-dependent coefficients Gardner types of equations[J]. Nonlinear Dynamics, 2015, 80(1):515-527.
[20] LIU Hanze, YUE Chao. Lie symmetries, integrable properties and exact solutions to the variable-coefficient nonlinear evolution equations[J]. Nonlinear Dynamics, 2017, 89(3):1989-2000.
[21] ASAMR N H. Partial differential equations with Fourier series and boundary value problem[M]. Beijing: China Machine Press, 2005.
[22] RUDIN W. Principles of mathematical analysis[M]. Beijing: China Machine Press, 2004.