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

Abstract

In this paper, the three variable-coefficient Gardner (vc-Gardner) equations are considered. By using the Painlevé analysis and Lie group analysis method, the Painlevé properties and symmetries for the equations are obtained. Then the exact solutions generated from the symmetries and Painlevé analysis are presented.

Travelling Wave Solutions of a Generalized Variable-Coefficient Gardner Equation

“…It is well known due to its many applications in mathematics and physics. Among them, it is noted that symmetry groups can be used to obtain exact solutions of partial differential equations, directly [10,17,20] or by obtaining the similarity variables and the similarity solutions [7,12]; or determine conservation laws [2][3][4]16].…”

Optimal System and Invariant Solutions of a New AKNS Equation with Time-Dependent Coefficients

“…As well as we know, Lie symmetry analysis is a powerful and prolific method for constructing exact solutions for NLPDEs with constant variable [18][19][20]. Recently, the Lie symmetry analysis is extended to find exact solutions of fractional and variable coefficient NLPDEs, such as Time-Fractional Boussinesq-Burgers [21], Gardner equations [22], coupled short pulse equation [23] and so on [24][25][26].…”

Group classifications, optimal systems and exact solutions to the generalized Thomas equations

“…We know that the Lie symmetry analysis is a very powerful method for tackling group classifications and exact solutions to partial differential equations (PDEs) (see e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and the references therein). Furthermore, we find that the combination of Lie symmetry analysis and the power series method is a feasible approach to deal with exact solutions to PDEs [1,[9][10][11][12][13]].…”