JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2019, Vol. 54 ›› Issue (4): 37-44.doi: 10.6040/j.issn.1671-9352.0.2018.110

Previous Articles     Next Articles

Painlevé analysis, Lie symmetry and exact solutions to the generalized time-dependent coefficients Gardner equation

WANG Qi, LI Lian-zhong*   

  1. School of Science, Jiangnan University, Wuxi 214122, Jiangsu, China
  • Published:2019-04-08

PDF (PC)

434

Abstract: 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.

Key words: Painlevé analysis, Lie symmetry analysis, symmetry reduction, power series solution, exact solution

CLC Number: 

  • O175.2
[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 Bäcklund 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: Bäcklund 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, 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]. 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.
[1] YANG Fei, LIU Xi-qiang. Exact solution of GKP equation with variable coefficients [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(2): 111-120.
[2] LI Yu, LIU Xi-qiang. 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.
[3] LIU Yong, LIU Xi-qiang. Symmetry, reductions and exact solutions of the (2+1)-dimension Caudrey-Dodd-Gibbon equation [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(04): 49-55.
[4] GUO Peng, ZHANG Lei, WANG Xiao-yun, SUN Xiao-wei. [J]. J4, 2012, 47(12): 115-120.
[5] XIN Xiang-peng, LIU Xi-qiang, ZHANG Lin-lin. Symmetry reductions and exact solutions of a (2+1)-dimensional  nonlinear evolution equation [J]. J4, 2010, 45(8): 71-75.
[6] GUO Peng, WAN Gui-xin, WANG Xiao-yun, SUN Xiao-wei. Explicit and exact solution to a nonlinear wave equation in circular-rod waveguide [J]. J4, 2010, 45(11): 122-126.
[7] GUO Xiao-Yi, XU Meng-Yu. An exact solution of unsteady Couette flow of generalized Oldroyd-B fluid [J]. J4, 2009, 44(10): 60-63.
[8] ZHU Hong ,LIU Xiong . White noise functional solutions of a Wick-type stochastic KdV-MKdV equation [J]. J4, 2008, 43(5): 45-49 .
[9] QI Hai-tao,JIN Hui . Exact solutions for the unsteady flow of generalized second [J]. J4, 2006, 41(4): 61-64 .
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
[1] ZHU Lin. Separated monic representations of quivers of type A4and RSS equivalences[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(2): 1 -8 .
[2] . Interval algorithm for mixed integer nonlinear two-level programming problems[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(2): 9 -17 .
[3] YAN Yan, HAO Xiao-hong. Differential privacy partitioning algorithm based on adaptive density grids[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(9): 12 -22 .
[4] LIU Fang-yuan, MENG Xian-jia, TANG Zhan-yong, FANG Ding-yi, GONG Xiao-qing. Android application protection based on smali code obfuscation[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(3): 44 -50 .
[5] LIAO Xiang-wen, ZHANG Ling-ying, WEI Jing-jing, GUI Lin, CHENG Xue-qi, CHEN Guo-long. User influence analysis of social media with temporal characteristics[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(3): 1 -12 .
[6] GU Shen-ming, LU Jin-lu, WU Wei-zhi, ZHUANG Yu-bin. Local optimal granularity selections in generalized multi-scale decision systems[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(8): 1 -8 .
[7] CAO Wei-dong, DAI Tao, YU Jin-biao, WANG Xiao-hong, SHI An-feng. Improvement on the solution of pressure equation based on alternating direction in chemical flooding model[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(10): 88 -94 .
[8] LI Jin-hai, WU Wei-zhi. Granular computing approach for formal concept analysis and its research outlooks[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(7): 1 -12 .
[9] SUN Jian-dong, GU Xiu-sen, LI Yan, XU Wei-ran. Chinese entity relation extraction algorithms based on COAE2016 datasets[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(9): 7 -12 .
[10] YE Xiao-ming, CHEN Xing-shu, YANG Li, WANG Wen-xian, ZHU Yi, SHAO Guo-lin, LIANG Gang. Anomaly detection model of host group based on graph-evolution events[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(9): 1 -11 .
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd