SYMMETRY REDUCTIONS AND EXACT SOLUTIONS TO THE SEVENTH-ORDER KDV TYPES OF EQUATION

Year 2015, Volume: 3 Issue: 2, 122 - 130, 01.10.2015

Youwei Zhang

Abstract

In present paper, the seventh-order KdV types of equation is considered by the Lie symmetry analysis. All of the geometric vector elds of the KdV equation are obtained, then the symmetry reductions and exact solutions to the KdV equation are investigated by the dynamical system and the power series method.

Keywords

KdV equation, Lie symmetry analysis, Dynamical system method, Power series method, Exact solution

References

• [1] M. Craddock, K. Lennox, Lie group symmetries as integral transforms of fundamental solutions, J. Differential Equations, 232 (2007), 652-674.
• [2] F. Gungor, C.  Ozemir, Lie symmetries of a generalized Kuznetsov-Zabolotskaya-Khokhlov equation, J. Math. Anal. Appl., 423 (2015), 623-638.
• [3] M. Lakshmanan, P. Kaliappan, Lie transformations, nonlinear evolution equations and Painleve forms, J. Math. Phys., 24 (1983), 795-806.
• [4] H. Liu, J. Li, Lie symmetry analysis and exact solutions for the extended mKdV equation, Acta Appl. Math., 109 (2010), 1107-1119.
• [5] H. Liu, J. Li, F. Chen, Exact periodic wave solutions for the hKdV equation, Nonlinear Anal., 70 (2009), 2376-2381.
• [6] H. Liu, J. Li, L. Liu, Lie symmetry analysis, optimal systems and exact solutions to the fth-order KdV types of equations, J. Math. Anal. Appl., 368 (2010), 551-558.
• [7] H. Liu, J. Li, L. Liu, Conservation law classication and integrability of generalized nonlinear second-order equation, Commun. Theor. Phys. (Beijing), 56 (2011), 987-991.
• [8] H. Liu, Y. Geng, Symmetry reductions and exact solutions to the systems of carbon nanotubes conveying uid, J. Differential Equations, 254 (2013), 2289-2303.
• [9] W. Balser, Multisummability of formal power series solutions of partial differential equations with constant coecients, J. Dierential Equations, 201 (2004), 63-74.
• [10] A.B. Mikhailov, A.B. Shabat, V.V. Sokolov, The symmetry approach to classication of integrable equation, in: What is Integrability?, Springer Series on Nonlinear Dynamics, Berlin, 1991.
• [11] B. Muatjetjeja, C.M. Khalique, Symmetry analysis and conservation laws for a coupled (2+1)- dimensional hyperbolic system, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 1252- 1262.
• [12] P.J. Olver, Applications of Lie Groups to Dierential Equations, Grad. Texts in Math., vol. 107, Springer, New York, 1993.
• [13] P. Razborova, A.H. Kara, A. Biswas, Additional conservation laws for Rosenau-KdV-RLW equation with power law nonlinearity by Lie symmetry, Nonlinear Dynam., 79 (2015), 743- 748.
• [14] W. Sinkala, P. Leach, J. O'Hara, Invariance properties of a general-pricing equation, J. Differential Equations, 244 (2008), 2820-2835.
• [15] P. Winternitz, Lie groups and solutions of nonlinear partial dierential equations, in: Lecture Notes in Physics, CRM-1841, Canada 1993.