Complete Group Classification and Exact Solutions to the Generalized Short Pulse Equation

Abstract

In this paper, the complete group classification is performed on the generalized short pulse equation, which includes a lot of important nonlinear wave equations as its special cases. In the sense of geometric symmetry, all of the vector fields of the equation are obtained in terms of the arbitrary functions. Then, the symmetry reductions and exact solutions to the equations are investigated. Especially, we develop the analytic power series method for constructing the exact power series solutions to the short pulse types of equations.

Complete Group Classifications and Symmetry Reductions of the Fractional Fifth‐Order KdV Types of Equations

“…This implies that for Equation , there exists a power series solution (33) with the coefficients given by (34). Furthermore, we can show that the convergence of this power series solution (see, e.g., , the details are omitted here). Thus, the solution (33) is the exact analytic solution to Equation .…”

“…In engineering and physical applications, many nonlinear models can be depicted by such equations. In , the FDEs and nonlinear partial differential equations are considered. Recently, we studied some nonlinear partial differential equations by the group classification and dynamical system method, the symmetries, exact solutions, and other properties of the equations are obtained .…”

Symmetry Analysis and Exact Solutions to the Space-Dependent Coefficient PDEs in Finance

“…However, the similarity reductions and exact solutions to such variable-coefficient equations are not considered generally in the aforementioned papers. Recently, we studied some nonlinear PDEs by Lie symmetry analysis and the dynamical system method [8][9][10][11][12][13]; for example, in [8], we considered Lie group classifications and exact solutions to the space-dependent coefficients hanging chain equation and the simplified bond pricing equation. In [9], we investigated the integrable condition and exact solutions to the timedependent coefficient Gardner equations by the Painlevé test and Lie group analysis method.…”

“…In [9], we investigated the integrable condition and exact solutions to the timedependent coefficient Gardner equations by the Painlevé test and Lie group analysis method. In [10][11][12][13], we developed the generalized power series method for dealing with exact solutions to some nonlinear PDEs based on the symmetry analysis method.…”

“…It is known that the Lie symmetry analysis is a systematic and powerful method for dealing with symmetries and exact solutions to partial differential equations (see, e.g., [1][2][3][4][5][6][8][9][10][11][12][13][14][15][16][17][18] and the references therein). Furthermore, we find that the combination of Lie symmetry analysis and the power series method is a feasible approach to investigating exact solutions to nonlinear PDEs [8][9][10][11][12][13]. On the other hand, under the perspective of mathematical physics and Lie symmetry analysis, the space-time dependent coefficients system differs greatly from its time-dependent counterpart, and it is more complicated than the latter.…”