Lie Symmetries, Conservation Laws and Exact Solutions for Two Rod Equations

Abstract

In this paper, the Lie symmetry analysis are performed for the two rod equations. The infinite number of conservation laws (CLs) for the two equations are derived from the direct method. Furthermore, the all similarity reductions and exact explicit solutions are provided.

Lie symmetry analysis, optimal systems and exact solutions to the fifth-order KdV types of equations

“…Recently, H. Liu et al considered the periodic wave solutions of a higher-order KdV equation by using the Hirota's direct method [10]. Moreover, by using Lie symmetry analysis and the dynamical system method, we get the symmetries, bifurcations and exact explicit solutions to other nonlinear evolution equations (NLEEs) [11][12][13][14][15][16]. In the present paper, we will consider the fifth-order KdV types of equations:…”

“…(4) We know that Eq. (1) is the general Kawahara equation (see [17,18] and references therein), Eq. (2) is the simplified Kawahara equation, and Eq.…”

Painlevé analysis, Lie symmetries, and exact solutions for the time-dependent coefficients Gardner equations

“…It is known that to find exact solutions of the NLEEs is always one of the central themes in mathematics and physics. In the past few decades, there is noticeable progress in this field, and various methods have been developed, such as the inverse scattering transformation (IST) [1], Darboux and Bäcklund transformations [2], Hirota's bilinear method [2][3][4], Lie symmetry analysis [5][6][7][8][9][10][11][12], CK method [13,14], and so on.…”

Group Classifications, Symmetry Reductions and Exact Solutions to the Nonlinear Elastic Rod Equations

“…In the past few decades, a wealth of methods have been developed to find these exact solutions of the nonlinear PDEs though it is rather difficult. Some of the most important methods are the inverse scattering method [1,2], Darboux and Bäcklund transformations [3,4], Hirota's bilinear method [4][5][6], Lie symmetry analysis [7][8][9][10][11][12][13][14][15][16][17], CK transformation method [18,19], and so on.…”

“…In [17], we considered the symmetry reductions, conservation laws and exact solutions of the two special rod equations. In [21][22][23][24], the existence of solitary wave solutions have been considered for the special cases n = 2 and n = 3.…”