一类组合sinh-cosh-Gordon方程的对称约化、动力学性质与精确解（英文）
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摘要
文章综合运用李对称分析、幂级数解法和动力系统法来求解组合sinh-cosh-Cordon方程的精确解.利用李对称分析得到了组合sinh-cosh-Cordon方程的向量场和相似变换,把难以求解的偏微分问题约化为常微分方程,利用幂级数解法求得了方程的精确解析解.然后用MATLAB画出了约化后方程的相图,最后利用动力系统法分析研究了解的动力学行为,并得到了方程的行波解.
In this paper, we employ Lie symmetry analysis along with the analytic power series method and dynamical method to investigate the exact solutions of the combined sinhcosh-Gordon equation. Through the Lie symmetry analysis, the geometric vector fields and the symmetry reductions of the combined sinh-cosh-Gordon equation are presented, and the difficult problem of partial differential equation is reduced to ordinary differential equation's.We get the exact explicit solutions of the combined sinh-cosh-Gordon equation by the analytic power series method. Especially, the bifurcations of the combined sinh-cosh-Gordon equation are obtained through MATLAB. The dynamical behavior and the exact explicit traveling wave solutions are investigated by the dynamical system method.
In this paper, we employ Lie symmetry analysis along with the analytic power series method and dynamical method to investigate the exact solutions of the combined sinhcosh-Gordon equation. Through the Lie symmetry analysis, the geometric vector fields and the symmetry reductions of the combined sinh-cosh-Gordon equation are presented, and the difficult problem of partial differential equation is reduced to ordinary differential equation's.We get the exact explicit solutions of the combined sinh-cosh-Gordon equation by the analytic power series method. Especially, the bifurcations of the combined sinh-cosh-Gordon equation are obtained through MATLAB. The dynamical behavior and the exact explicit traveling wave solutions are investigated by the dynamical system method.
引文
[1] Cheviakov, A.F., Bluman, G.W. and Anco, S.C., Applications of Symmetry Methods to Partial Differential Equations, New York:Springer-Verlag, 2010.
[2] Geng, Y.X., He, T.L. and Li, J.B., Exact travelling wave solutions for the(n+1)-dimensional double sineand sinh-Gordon equations, Appl. Math. Comput., 2007, 188(2):1513-1534.
[3] Hu, X.R. and Chen, Y., Two-dimensional symmetry reduction of(2+1)-dimensional nonlinear Klein-Gordon equation, Appl. Math. Comput., 2009, 215(3):1141-1145.
[4] Jabbari, A. and Kheiri,H.,The(G'/G)-expansion method for solving the combined and the double combined sinh-cosh-Gordon equation,Acta Univ,Apulensis Math. Inform.,2010,(22):185-194.
[5] Li, J.B., Exact traveling wave solutions and dynamical behavior for the(n+1)-dimensional multiple sineGordon equation, Sci. China Ser. A, 2007, 50(2):153-164.
[6] Liu, H.Z. and Li, J.B., Lie symmetry analysis and exact solutions for the short pulse equation, Nonlinear Anal., 2009, 71(5):2126-2133.
[7] Liu, H.Z. and Li, J.B., Lie symmetry analysis and exact solutions for the extended mKdV equation, Acta Appl. Math., 2010, 109(3):1107-1119.
[8] Milne, A.E., Mansfield, E.L. and Clarkson, P.A., Symmetries and exact solutions of a(2+1)-dimensional sine-Gordon system, Philos, Trans. Roy. Soc. London Ser,A, 1996, 354(1713):1807-1835.
[9] Morgan, A.P., Sommese, A.J. and Wampler, C.W., A power series method for computing singular solutions to nonlinear analytic systems, Numer. Math., 1992, 63(1):191-409.
[10] Wazwaz, A.M, The variable separated ODE and the tanh methods for solving the combined and the double combined sinh-cosh-Gordon equation, Appl. Math. Comput., 2006, 177(2):745-754.
[2] Geng, Y.X., He, T.L. and Li, J.B., Exact travelling wave solutions for the(n+1)-dimensional double sineand sinh-Gordon equations, Appl. Math. Comput., 2007, 188(2):1513-1534.
[3] Hu, X.R. and Chen, Y., Two-dimensional symmetry reduction of(2+1)-dimensional nonlinear Klein-Gordon equation, Appl. Math. Comput., 2009, 215(3):1141-1145.
[4] Jabbari, A. and Kheiri,H.,The(G'/G)-expansion method for solving the combined and the double combined sinh-cosh-Gordon equation,Acta Univ,Apulensis Math. Inform.,2010,(22):185-194.
[5] Li, J.B., Exact traveling wave solutions and dynamical behavior for the(n+1)-dimensional multiple sineGordon equation, Sci. China Ser. A, 2007, 50(2):153-164.
[6] Liu, H.Z. and Li, J.B., Lie symmetry analysis and exact solutions for the short pulse equation, Nonlinear Anal., 2009, 71(5):2126-2133.
[7] Liu, H.Z. and Li, J.B., Lie symmetry analysis and exact solutions for the extended mKdV equation, Acta Appl. Math., 2010, 109(3):1107-1119.
[8] Milne, A.E., Mansfield, E.L. and Clarkson, P.A., Symmetries and exact solutions of a(2+1)-dimensional sine-Gordon system, Philos, Trans. Roy. Soc. London Ser,A, 1996, 354(1713):1807-1835.
[9] Morgan, A.P., Sommese, A.J. and Wampler, C.W., A power series method for computing singular solutions to nonlinear analytic systems, Numer. Math., 1992, 63(1):191-409.
[10] Wazwaz, A.M, The variable separated ODE and the tanh methods for solving the combined and the double combined sinh-cosh-Gordon equation, Appl. Math. Comput., 2006, 177(2):745-754.