On the classical and nonclassical symmetries of a generalized Gardner equation

Abstract

In this paper, we consider a generalized Gardner equation from the point of view of classical and nonclassical symmetries in partial differential equations. We perform a complete analysis of the symmetry reductions by using the similarity variables and the similarity solutions which allow us to reduce our equation into an ordinary differential equation. Moreover, we prove that the nonclassical method applied to the equation leads to new symmetries, which cannot be obtained by using the Lie classical method. Finally, we calculate exact travelling wave solutions of the equation by using the simplest equation method.

Solutions and conservation laws of a generalized second extended (3+1)-dimensional Jimbo-Miwa equation

“…Conservation laws have been used to prove global existence theorems and shock wave solutions to hyperbolic systems. They have been applied to problems of stability and have been used in scattering theory and elasticity [17,[26][27][28][29].…”

On a generalized variable‐coefficient Gardner equation with linear damping and dissipative terms

“…Specifically, the Gardner equation has recently been studied by different authors. [1][2][3][4][5][6][7] For instance, in Zhang et al, 7 a large number of solutions of a generalized Gardner equation were constructed. The authors obtained these solutions with the aid of 2 first-order nonlinear ordinary differential equations and a new generalized algebraic method.…”

Symmetry Reductions for a Generalized Fifth Order KdV Equation

“…First of all, Lie point symmetries have been obtained applying the Lie classical method [11,13,20]. Several well-know researchers obtain symmetries because they can be use to obtain systematically exact solutions of the equations.…”