Abstract
It is generally known that classical point and potential Lie symmetries of differential equations can be different. In a recent paper, we were able to show for a class of nonlinear diffusion equation that the nonclassical potential symmetries possess all nonclassical symmetries of the original equation. We question whether this is true for the power law Harry Dym equation. In this paper, we show that the nonclassical symmetries of the power law Harry Dym equation and an equivalent system still possess special separate symmetries. However, we will show that we can extend the nonclassical method so that all nonclassical symmetries of the original power law Harry Dym equation can be obtained through the equivalent system.
Highlights
Symmetry analysis plays a fundamental role in the construction of exact solutions to nonlinear partial differential equations
A natural question to ask is can we extend the nonclassical method so that the nonclassical symmetries of the equivalent system encompasses the nonclassical symmetries of the single equation? For example, the Harry Dym equation u −2
We were able to show for certain classes of equation, the nonclassical symmetries of an equivalent system captures the nonclassical symmetries of the single equation, in general this is not true
Summary
Symmetry analysis plays a fundamental role in the construction of exact solutions to nonlinear partial differential equations. Arrigo et al [23] asked whether this holds true for nonclassical symmetries; that the nonclassical symmetries of a particular equation and a system equivalent (termed nonclassical potential symmetries) are different. They were able to show that for a large class of nonlinear diffusion equation, the nonclassical potential symmetries contains the nonclassical symmetries of the original equation. They were able to show this for a nonlinear wave equation and highlighted a special case where the opposite was true.
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